Plasticity Composite Ansys

Abstract

Irreversible plastic forming of B19′ martensite of the NiTi shape memory alloy is discussed within the framework of continuum mechanics. It is suggested that the main mechanism arises from coupling between martensite reorientation and coordinated (001)M dislocation slip. A heuristic model is proposed, showing that the (20¯1)M deformation-twin bands, commonly observed in experiments, can be interpreted as a combination of dislocation-mediated kink bands, appearing due to strong plastic anisotropy, and reversible twinning of martensite.

We introduce a term ’kwinking’ for this combination of reversible twinning and irreversible plastic kinking. The model is subsequently formulated using the tools of nonlinear elasticity theory of martensite and crystal plasticity, introducing ’kwink interfaces’ as planar, kinematically compatible interfaces between two differently plastically slipped variants of martensite. It is shown that the (20¯1)M kwink bands may be understood as resultsing from energy minimization, and that their nucleation and growth and their pairing with (100)M twins into specific patterns enables low-energy plastic forming of NiTi martensite.

We Conclude That Kwinking Makes

plastic deformation of B19′ martensite in polycrystalline NiTi possible despite only one slip system being available. Keywords: Shape memory alloy (SMA); Martensitic phase transformation; Plastic forming; Pattern formation; Anisotropic dislocation slip; Crystal plasticity

Ntroduction

While the most important functional properties of the NiTi shape memory alloy originate from the re- versible martensitic transition (Otsuka and Wayman, 1999; Miyazaki et al., 1989; Wei et al., 1998), irre- versible plastic forming of NiTi is the key issue for employing this alloy in technological applications. The relation between plastic forming and the thermo-mechanical performance of the NiTi shape memory alloy is currently an extensively studied topic (Polatidis et al., 2020; Heller et al., 2019; Gao et al., 2017; Šittner et al., 2018); experimental observations indicate that the forming process is immensely complex, possibly involving plastic straining both in the austenite phase and in B19′ martensite. The plastic deformation of the cubic austenite phase by dislocation slip is relatively well described in the literature (see e.g. (Chowd- hury and Sehitoglu, 2016; Ezaz et al., 2013) for a first-principles analysis of all possible slip systems). In contrast, the processes taking place in monoclinic martensite when strained beyond the recoverability lim- its are much less explored yet. As observed first by Nishida et al. (1998) and later by Zhang et al. (2006), such a straining typically leads to formation of (20¯1)M plastic twins, that are never found in thermally induced, self-accommodates martensitic microstructures in B19′. Upon heating, these interfaces are inher- ited in the austenite phase as residual (41¯1)P twins (Ii et al., 2003), where the subscripts M and P denote the lattice indices corresponding to the martensite and the parent phase, respectively. As discussed by Gao et al. (2017); Gao (2019), the (20¯1)M plastic twins are not predictable by the classical mathematical theory arXiv:2305.07125v3 [cond-mat.mtrl-sci] 6 Jul 2023 of martensitic microstructures (Ball and James, 1987; Bhattacharya, 2003; Pitteri and Zanzotto, 2002), as they correspond to straining beyond the given Eriksen-Pitteri neighborhood (EPN,(Bhattacharya, 2003)), i.e., beyond the set of martensitic lattices related to the reference lattice of austenite by a unique lattice correspondence resulting from the transformation path (Zelazny et al., 2011). This agrees well to the fact that the (20¯1)M twins are not ’erased’ by the reverse transition but are, instead, transformed into deformation twins found in the cubic lattice of austenite. To this extent, the importance of the (20¯1)M twins for the plastic forming process is quite well understood.

Nevertheless, the mechanism how the (20¯1)M plastic twins arise in irreversibly strained B19′ has not been sufficiently explained yet, and there are also several features of these twins that call for a more detailed analysis. The most apparent one is that the (20¯1)M twinning provides shearing in the (010)M lattice plane, i.e. in the same plane as several other deformation mechanisms of B19′, namely the M(001)M dislocation slip and the (100)M and (001)M transformation twins. As shown in (Ezaz et al., 2020, 2013), all these mechanisms are energetically favorable over the (20¯1)M twinning, and thus, a question arises under which specific conditions this twinning system is activated, and how does it compare to other mechanism acting on the same plane in terms of achievable strains and their relative energetic expensiveness. Secondly, the (20¯1)M twins are typically observed as a part of particular V-shaped microstructures in combination with (100)M twins ((Zhang et al., 2006; Molnárová et al., 2020), see also Figures 2 and 7). This means that such microstructures must be somehow energetically beneficial, either leading to energy minimization via pattern formation, or enabling a low-energy mechanism of plastic deformation.

In this paper, we propose a micro-mechanical model of plastic straining of NiTi martensite, and for- mulate this model within the framework of continuum mechanics, combining the mathematical theory of martensite and crystal plasticity. The main assumption of the model is that the reversible (100)M and (001)M twinning (i.e. twinning within one EPN) appears simultaneously with highly anisotropic dislocation slip. We show that such a model gives predictions consistent with the experimental observations, leading to energy reduction through formation of (20¯1)M-twin bands and their coupling with (100)M twinning. The most important outcome of the model is the identification of a new plastic straining mechanism that can be described as twinning-assisted formation of symmetric-tilt grain boundaries, or slip-assisted formation of twinned microstructures. This mechanism can arise in a shape memory alloy martensite when the high energy barrier between different EPNs is overcome by dislocation slip; the unified modelling framework allows us to understand it as a result of energy-minimization (or at least energy-reduction). The model suggest that the plastic forming mechanism of B19′ is strikingly similar to the deformation mechanisms of layered media (Wadee et al., 2004; Zhu et al., 2010) or materials with highly anisotropic dislocation slip (Inamura, 2019; Lei and Nakatani, 2015), which explains why also the resulting V-shaped patterns in B19′ clearly resemble the patterns observed in these materials.

The second aim of this paper is to explore the properties of the newly suggested mechanism and to discuss them in the context of the current knowledge of plastic forming of NiTi, both from the theo- retical/modelling point of view and as far as the experimental observations are concerned. Regarding the former, it is worth noting that although the continuum-level description of twinning in martensite and single-crystal plastic slip were well developed in the literature, their combined applications for cap- turing irreversible straining of shape memory alloys have been just sparsely reported so far (Chowdhury and Sehitoglu, 2017), and always with assuming that the dislocation slip appears only in the austenite phase. Motivated by experimental observations by Simon et al. (2010); Delville et al. (2011); Petlon et al. (2012), initiation of plastic slip of the austenite phase at the austenite-martensite interfaces were dis- cussed by Paranjape et al. (2016, 2018) using quite complex multi-scale models, and several others (Mo et al., 2022; Chaugule et al., 2022; Hossain and Baxevanis, 2021). The dominant appearance of the plastic slip in austenite was also assumed for explaining the transformation-plasticity coupling in superelastically strained wires by Šittner et al. (2018); Heller et al. (2018), where this concept was shown to predict cor- rectly the massive plastic slip associated with the reverse transition, as it has been recently confirmed experimentally for thermal cycling (Akamine et al., 2023) as well as mechanical cycling (Sidharth et al., 2023). However, these approaches cannot be applied for the case reported by Zhang et al. (2006) and Molnárová et al. (2020) and discussed in this paper, since the austenite phase is absent here, and plas- tic deformation of martensite must be considered instead. The recent experimental literature concerned with plastic forming of martensite studied on the single-crystal level reveals enormous strain localization into finely microstructured bands (Polatidis et al., 2020) and partial reversibility of the plastic twinning (Chen et al., 2022; Alarcon et al., 2023). These observations do not clarify the mechanism of the (20¯1) twinning, neither do they explain the formation of the V-shaped patterns, but they prove that there are some twinning-plasticity coupling phenomena that go beyond the validity of the simplified models based on partitioning between twinning and slip in individual grains based on their crystallographic orienta- tion and the respective Schmidt factors (see (Richards et al., 2013; Kimiecik et al., 2016; Stebner et al., 2013) for detailed experimental and theoretical treatment of the partitioning). Notice also that while the M(001)M slip has been identified as the possible easiest dislocation slip of the B19′ lattice already by Kudoh et al. (1985) (which has been confirmed by first principles calculations by Ezaz et al. (2011)), no dedicated experimental analysis of plastic anisotropy of NiTi martensite can be found in the available literature so far. By the model developed in this paper, we show that the M(001)M slip is essential for formation of the observed patters, which confirms the considerations of Kudoh et al. (1985) and theoretical results of Ezaz et al. (2011).

On the other hand, there is a quite extensive literature reporting on both experimental research and numerical simulations of NiTi plastic forming on the macro-scale, i.e. polycrystal-scale level. The majority of the works focus on the evolution of the macroscopic response during cyclic loading in pseudelasticity and describe the effective behavior of the material using the approaches developed for transformation/twinning- induced plasticity (TRIP/TWIP) in steels, either applying the approaches based on phenomenological plasticity, e.g. (Lagoudas nad Entchev, 2004; Auricchio et al., 2007; Hartl and Lagoudas, 2009; Zaki et al., 2010; Wang et al., 2017; Petrini and Bertini, 2020; Scalet et al., 2021; Song et al., 2023), or those inspired by the crystal plasticity and using some scale transition technique, e.g. (Manchiraju and Anderson, 2010; Richards et al., 2013; Yu et al., 2015; Xiao et al., 2018). Only a few models of the latter type include plastic processes running in martensite when no transformation occurs, e.g. (Wang et al., 2008; Yu et al., 2014; Dhala et al., 2019; Xu et al., 2021; Ju et al., 2022), and in many of them, the plastic deformation of the martensite phase is assumed to come mainly from the twinning deformation running in 11 twinning systems, i.e. contribution of the (001)M dislocation slip system is neglected (Yu et al., 2014; Dhala et al., 2019; Ju et al., 2022).

Notation, Terminology, And Lattice Parameters

For both the discussion of the experimental observations and the construction of the model, we restrict our considerations to one crystallographic plane, which is the (010)M plane in martensite in the standard notation (used e.g. in (Bhattacharya, 2003)). The reason is that the deformation mechanisms that were experimentally observed as those most active in plastic forming of NiTi B19′ martensite (Zhang et al., 2006; Molnárová et al., 2020; Šittner et al., 2023) have the shearing direction lying in this plane and the invariant plane perpendicular to it. The notation we use for these deformation mechanisms is summarized in Table 1 together with their crystallographic parameters. There is no experimental evidence showing how the material accommodates strains in the direction perpendicular to the given (010)M plane. For the purpose of this work, we assume that there might be a compensation of strains between clusters of neighboring grains, having differently oriented (010)M planes. It is worth noting that the (010)M plane orientations in the martensitic polycrystal are not strictly determined by the texture: the material always has an additional degree of freedom, as different grains can transform (or reorient) into different variants of martensite, which may ease such a compensation. For example, in a strongly (111)P-textured wire (with small equiaxed grains and rotationally symmetric texture (Bian et al., 2022)), which is one of the Table 1: A list of the considered deformation mechanisms acting on a selected (010)M plane in B19′ martensite, and the used notation. The twin components and the shearing magnitudes were taken from (Nishida et al., 1998). In the last row, α denotes the continuum-level slip magnitude, which is arbitrary, non-zero. See also

Α ∈R

aIn figures, for better visibility we often drop the subscript M from the denotations, using e.g.

(100) Twins Instead Of

(100)M twins, etc. cases discussed in this work, there are three variants of martensite that have the M direction oriented perpendicular to the loading axis, and thus, three variants in which the plastic straining in the (010)M plane can directly contribute to the tensile elongation of the wire.

All twinning systems listed in Table 1 are compound twins (in the sense as defined in e.g. (Pitteri and Zanzotto, 2002)), which means that their twinning plane is a lattice plane and the shearing direction is a lattice direction. The shearing magnitudes provided in Table 1 were calculated using the lattice parameters from (Bhattacharya, 2003), which are: a = 2.889 Å, c = 4.622 Å, and β = 96.8◦(Bhattacharya, 2003). We use these parameters for all calculations in this paper, and also for visualizations of the microstructures in figures (with the exception of schematic sketches in Figures 2, 4 and 8); this allows us to discuss the compatibility in the microstructures and the strains carried by the microstructures directly from the visualizations.

From the four deformation mechanisms listed in Table 1, only the (001)M twins and (100)M twins do not affect the corresponding lattice of austenite, i.e., only the strains due to activity of these two mechanisms can be erased by the reverse transition to austenite. This is visualized in Figure 1. The (001)M dislocation slip acts as a (011)P slip in the corresponding austenitic lattice. The (20¯1)M twins correspond to (41¯1)P twins in the parent lattice. In agreement with Gao et al. (2017); Gao (2019), we use the term ’twinning beyond the Eriksen-Pitteri neighborhood (EPN)’ to denote the twinning that affects the corresponding austenite lattice.

The visualization in Figure 1 is created using lattice parameters from (Bhattacharya, 2003) for the shapes of the unit cells, and the positions of the atoms inside of the cells follow the atomic coordinates determined experimentally by Kudoh et al. (1985). It is worth noting that, unlike for the shape of the unit cell, the inner structure in the B19′ cell is not invariant with respect to a 180◦rotation about the M axis. Consequently, for the visualization we need to choose one of two equivalent orientations of the inner structure; these two orientations represent the same crystal structure, but the location of the unit cell in it differs by a translation vector. For the choice done for Figure 1, there is a full mirror symmetry between the visualizations of the adjacent variants for (100)M twining, and a pseudo-mirror symmetry for (001)M twinning, where the shape of the cell is mirror-reflected, while its inner structure is not.

(011)P Slip

Figure 1: The considered deformation mechanisms acting on one (010)M lattice plane. The upper row shows these mechanism in the B19′ martensite lattice, the lower row are the related processes in the parent phase (B2) lattice linked through the lattice correspondence. For the mechanisms within one Eriksen-Pitteri neighborhood (the first two columns), the reference austenite lattice remains unaltered; for the mechanisms straining the lattice outside EPN (the third and the fourth columns), lattice defects and plastic strains are introduced into austenite.

Detailed considerations of whether the atomic positions (and the corresponding motifs) are only mirror- reflected, or also translated play an important role for first-principles calculations of the energy of the twins (Ezaz et al., 2011), but do not affect the continuum-mechanics construction in this paper, as the shape strain (i.e., the quantity that copies from the atomistic level to continuum through the Cauchy-Born hypothesis (Bhattacharya et al., 1999)) in both situations is the same.

Summary Of Experimental Observations

The purpose of this subsection is to summarize two most comprehensively documented case studies on B19′ plastic deformation in literature, which are (Zhang et al., 2006) and (Molnárová et al., 2020; Šittner et al., 2023).

We focus on features frequently observed in the experiments, mainly those related to (20¯1)M twinning and pattern formation. Importantly, although these observations were done on very different materials (in terms of chemical composition, grain size, yield strength, etc.), the sequence of the observed microstructures included very similar initial states (mixtures of finely (001)M-twinned domains in self-accommodated, thermally induced martensite), very similar intermediate states (nearly or fully homogeneous fine (001)M lamination when the material was approaching the reversibility limit under tensile loading), and very similar terminal states ((20¯1)M- and (100)M-oriented bands, mostly without the (001)M-twinned substructure, in the plastically formed material). The similarity in the intermediate state is of a particular importance: in this state, the whole grain or a large area of it has one orientation of the (010)M plane with respect to the loading, and thus, the grain can deform through the two-dimensional deformation mechanisms introduced in Figure 1.

The difference can be seen in the reasoning for the appearance of the fine (001)M lamination at the initial and intermediate stages: while Zhang et al. (2006) ascribed this lamination to the Ti-excess and Guinier-Preston zones, Molnárová et al. (2020) have proved that this lamination may originate from the strain-compatibility conditions between neighboring grains.

Additional differences were also in some finer details in how the microstructure evolved between the initial, intermediate and terminal stages, which we ascribe mainly to the different grain size, as discussed below. However, these observations are sufficiently similar to justify our ambition to capture them both by the same microstructural model.

Let us also remark that in the thermally-induced martensite in NiTi polycrystals, Type II twins are typically observed as the most prevalent twinning mode (Matsumoto et al., 1987; Onda et al., 1992), since these twins can help to satisfy compatibility conditions at the austenite-martensite interface during the transition (Hane and Shield, 1999; Bhattacharya, 2003).

N

nanograined polycrystals, however, the bands holding the Type II (or Type I) twin-like relationships are further internally (001)M-compound twinned (Waitz et al., 2008; Molnárová et al., 2020). This internal twinning further reduces the shape strain of the microstructure, and possibly also leads to some elastic softening of the lattice (Wang and Sehitoglu, 2014; Grabec et al., 2021) or results from the rhombohedral R-phase forming during the transition (Zhang and Sehitoglu, 2004). The intermediate state is then reached when the Type II twins are removed by reorientation, while the finer-scale (001)M compound twins persist in the grains (Zhang et al., 2006; Chowdhury and Sehitoglu, 2017; Molnárová et al., 2020; Šittner et al., 2023).

Observations On Melt-Spun Ribbons

Zhang et al. (2006) reported on microstructure evolution in plastically strained melt-spun ribbons (in form of thin films). They observed a typical sequence of microstructures appearing with an increasing load; this sequence, outlined in Figure 2, was later discussed by Chowdhury and Sehitoglu (2017) and many others.

The sequence starts from thermally-induced martensite (Figure 2), consisting of a self-accommodated mixture of variants, often having (001)M compound twins at the finest scale.

When This Mixture Is

subjected to uniaxial tension, it first undergoes martensite reorientation at an approximately constant stress level; this stress level defines the so-called reorientation plateau. At the end of the reorientation plateau, the grains are typically full of fine (001)M compound twins (Figure 2(b)). With further loading the material deforms elastically or by reversible detwinning, until the plastic twins start nucleating in form of V-shaped microstructures. At the nucleation stage (Figure 2(c)), the (20¯1)M and (100)M twin bands include a fine band-like contrast that resembles the herring-bone pattern in thermally induced martensite and provides a continuous connection of the (001)M compound laminate over the newly nucleating bands.

This nucleation of the first V-shaped patterns marks the onset of the plastic deformation and the loss of reversibility of the strains. With a further increase of the load, the (20¯1)M and (100)M twin bands grow, which is accompanied by a simultaneous disappearance of the fine (001)M laminate (Figure 2(d)) surrounding the V-shaped microstructures, as well as of the finer contrast inside of them. Finally, the material reaches the plastic-forming plateau (termed hereafter the yielding plateau), where the plastic deformation proceeds via growth of the (20¯1)M and (100)M twin bands and appearance of secondary twins inside them (Figure 2(e)).

The stages (a-e) in Figure 2 may appear at different stress levels and for different strains, depending on temperature and on the particular microstructure and chemical composition of the alloy. For Ti-rich Ni-Ti thin ribbons (Ni48.9Ti51.5, grain size ∼5 µm) reported by Zhang et al. (2006), the lower plateau appeared at approximately 150 MPa and the stage (b) was reached for axial strain ε ≈4%. The upper plateau (stage (e)) was at approximately 700 MPa and ε ≈12%; the stages (c) and (d) correspond to strains of ε ≈7% and ε ≈9%, respectively.

The coexistence of (20¯1)M and (100)M twins in form of the V-shaped microstructure was reported also for hot-forged and drawn polycrystalline NiTi rods (Ii et al., 2003), for cold-rolled and annealed NiTi sheets (Nishida et al., 1998) or for a single crystal (Ezaz et al., 2020); patterns clearly resembling the (20¯1)M/(100)M wedges were seen also in plastically formed residual bands inside of austenite grains in hot-rolled NiTi sheet with a more than 20 µm average grain size (Polatidis et al., 2020). Several other

(Intermediate State)

Figure 2: A typical tensile stress-strain curve and the corresponding sequence of microstructures appearing in plastically formed large-grain NiTi martensite, as observed by electron microscopy (Zhang et al., 2006): (a) a self-accommodated microstructure originating from the thermally-induced transition; (b) reversibly strained martensite with fine (001)M compound lamination; (c) nucleation of V-shaped microstructures in the laminate with a lamination-like contrast in the newly formed bands; (d) thickening of the new bands and disappearance of the laminate; (e) a final stage with fully formed V-shaped patterns in a single-variant matrix and secondary twinning inside of the bands. For better visibility, the microstructural features (lengthscales ∼10-50 nm) are not drawn in scale with respect to the grain size (∼5 µm). Zhang et al. (2006) also reported on minor appearance of (¯1¯13) twins at the very terminal stages, which we do not consider here; we can speculate that these twins appear because of strain accommodation in direction perpendicular to the considered (010)M plane.

works report on (41¯1)P twin patterns appearing in austenite after a plastically formed B19′ martensite had been transformed to the parent phase due to unloading and/or heating (Nishida et al., 2006; Chen et al., 2019). These patterns are very similar to those of in Figure 2(c), because they are inherited from the martensite phase due to the crystallographic equivalence between the (20¯1)M and (41¯1)P twins, outlined in the third column of Figure 1.

Observations On Cold-Worked (111)A-Textured Wires

A detailed transmission electron microscopy (TEM) analysis of microstructure evolution under plastic forming has been recently reported by Molnárová et al. (2020) for strongly textured NiTi wires (Ni49.5Ti50.5 with a grain size of ∼0.2 µm). The TEM analysis has been complemented with high-resolution TEM (HRTEM) in (Molnárová et al., 2023) (see also the Supplementary material in (Molnárová et al., 2020)), supported by texture evolution observations (Bian et al., 2022), and comprehensively summarized recently in (Šittner et al., 2023).

The pattern formation was observed on the (010)M crystallographic plane in each grain, and, except of for the self-accommodated thermally-induced martensite at the very beginning of the experiment, it utilized only the straining mechanisms acting on this plane and summarized in Figure 1. For the tests performed at 20 ◦C, the reorientation plateau was at approximately 150 MPa again, but the upper plateau was close to 1100 MPa. The axial strains corresponding to (b) and (e) were 7 % and 15 %, respectively.

When the same experiments were performed at 100 ◦C, where the martensite is stress-induced, the plateaus appeared at 400 MPa and 825 MPa, respectively, but exactly the same V-shaped patterns were observed in the material deformed to 15 %. This proves that the mechanism associated with the formation of these patterns is general, and probably is one of the main mechanisms of plastic forming in NiTi. Unlike for the melt-spun ribbons in (Zhang et al., 2006), no nucleation of (20¯1)-oriented bands below the yielding plateau has been observed for the cold-worked wires, that is, the stage (c) in Figure 2 was absent. This can be rationalized by the grain size, as smaller grains pose higher barriers for formation of the microstructure (Waitz et al., 2007, 2008; Kabla et al., 2014), and thus, it appears at much higher driving forces. For this reason, while in the melt-spun ribbons the onset of irreversible strains and appearance V-shaped patterns were spread over the the whole stress-strain curve above the point (c), for the cold-worked wires they became active at the yielding plateau only.

Furthermore, the analysis reported in (Molnárová et al., 2020) revealed some dependence of the pattern formation on the grain orientation with respect to the loading axis. Nearly all grains were finely (001)M- twinned at the end of the reorientation plateau, which was rationalized in (Molnárová et al., 2020) by the compatibility conditions at the grain boundaries for the given (111)P wire texture. This (001)M lamination in all grains with one major variant in each grain has been confirmed also by texture measurements (Bian et al., 2022). Nevertheless, further evolution of the microstructure was different for the grains oriented exactly along the texture, and those specifically inclined from the major orientation. In particular, for p denoting the projection of the loading axis onto the observed (010)M plane, the following dichotomy was

Observed:

1. For the grains where p lied approximately perpendicular to the (001)M-compound twinning planes, the plastic-forming sequence was very similar to the one observed in melt-spun ribbons, with a massive appearance of V-shaped microstructures, and their co-existence with gradually disappearing (001)M laminates. For this direction of the vector p, the variants forming the (001)M-laminate were energetically equivalent with respect to the loading; for this reason, some (001)M-twinned regions outside the (20¯1)M and (100)M twin bands persisted in the microstructure even at the yielding plateau. At the same time, some deformation bands observed in the grains were not twin bands, but bands of rotated lattice (so-called kink bands, Figure 7 in (Molnárová et al., 2020)) as usual in plastically formed materials with highly anisotropic plastic slip (Hess and Barrett, 1949; Inamura, 2019). The presence of kink bands confirmed the massive activity of the (001)M dislocation slip in these grains.

2. For the grains where p lied approximately along the [10¯1]M direction of the major variant in the (001)M-compound laminate, only few V-shaped microstructures were observed, while the dominant objects were the (20¯1)M twin bands. In contrast, no isolated (100)M twin bands were observed.

In these grains, one of the variants forming the (001)M laminate is strongly energetically preferred by the loading; this laminate was completely absent in microstructures representing the material strained beyond the start of the yielding plateau. In other words, the main mechanism of the plastic forming along the yielding plateau in these grains was the growth of (20¯1)M plastic twin bands in an otherwise fully detwinned oriented martensite.

For the former case (p perpendicular to (001)M), the microstructures were becoming more and more complex with a further progress of plastic forming (Molnárová et al., 2020, 2023; Šittner et al., 2023), until the grain became finally fully filled with various deformation bands. The planar interfaces between the bands were identified as either providing the (100) or (20¯1) twin relations between the neighboring lattices, or being just tilt interfaces between two differently rotated lattices belonging to the same variant of martensite.

Another important observation of Molnárová et al. (2020, 2023); Šittner et al. (2023) were the ori- entation relationships between the lattices forming the (20¯1)M twins, and the resulting geometry of the (20¯1)M-twin bands. For compound-type twins lying along a crystallographic plane and representing a shear along a lattice direction, one would expect a quite strictly defined orientation relationship and small deviations from the exact twinning plane only. Instead, a quite strong scatter from the theoretically pre- dicted orientations was reported (see (Molnárová et al., 2023), we discuss this in more detail in Section 4), also there was a considerable disorder seen in the crystal structure surrounding the (20¯1)M-twin interfaces seen in the HRTEM images (Ii et al., 2003; Molnárová et al., 2023). The misorientations and local dis- orders are then also inherited in austenite during the reverse transition. As mentioned in (Molnárová et al., 2020), the V-shaped patterns formed by {41¯1} twins in austenite also deviate from being attached to exact lattice planes and there is some scatter in the tilt angles between the lattices; The diffuse character of {41¯1} twins is seen in (Ii et al., 2003). All these observations indicate that the character of the (20¯1)M twins in martensite might be somehow different from other compound twins commonly observed in shape memory alloys, revealing, again, the need of a more detailed analysis of their origin and their contribution to plastic forming of NiTi.

Otivation For The Kwinking Model

The theoretical model capturing the above summarized experimental observations is in detail formulated in Section 3. Here, we explain the motivation for main features of the model, using particular examples from the experiment. This subsection gives a heuristic illustration of how essential is the role of the dislocation slip and how the slip can be coupled with the twinning; in Section 3, we discuss how this importance of the slip and the coupling can be incorporated in a unified, general model.

Nucleation and growth of the deformation bands in a (001)M-compound laminate First we focus on the nucleation stage of the V-shaped patterns in melt-spun ribbons (Figure 2(c)), where the (20¯1)M and (100)M twin bands are formed inside of a (001)M-twinned lattice. As clearly documented by Zhang et al. (2006), both (20¯1)M and (100)M bands cross the fine (001)M-compound twins existing at the end of the reorientation plateau, i.e., the newly formed bands are also internally twinned and there is a continuous, one-to-one connection between the twins inside the bands and the (001)M-compound twins in the matrix (as seen in several places in Figure 3(a)). From their crystallographic orientations, and also because there is no other reversible twinning system available in the given plane, it is clear that the fine twins inside the bands are, again, (001)M-compound twins. Hence, the first requirement we put on the model (in the heuristic approach) is that the model must be able to capture the compatible connections of (001)M-compound twins over the (20¯1)M and (100)M twin bands that nucleate at the very first stages of plastic forming.

As shown by the mathematical theory of martensite, the so-called crossing-twins microstructure is energetically admissible only if the twinning components of the involved twins satisfy a quite strict set of algebraic conditions (Bhattacharya, 2003; Seiner et al., 2014). The set is treated in Section 4.1 by an explicit calculation. However, to show that this set is not satisfied (not even in a approximate sense) for a combination of (001)M and (100)M compound twins from one EPN, and also if the band undergoes exact (20¯1)M twinning outside the given EPN, it is fully sufficient to use visualizations, as done in Figures 3(b) and (c). Figure 3(b) shows a matrix (denoted as Variant 1) which includes two bands of Variant 2 that are expected to cross each other. One band holds a (001)M twin relation with the matrix and the second one the (100)M twin relation. It is seen that the area where the bands intersect cannot hold twin relations to the lattices inside both bands and a geometric misfit (i.e., an incompatibility) appears. A similar situation appears for a (20¯1)M twin band, as shown in Figure 3(c). Here, the major variant of the band is some Variant 2′, which belongs to another EPN and holds a (20¯1)M twin relation to the matrix; the (001)M twin relation in this EPN transforms Variant 2′ into Variant 1′. Again, the Variant 1′ in the area where the bands intersect is not compatibly connected to the (001)M band in the matrix, and there is an incompatibility.

For close-to-compatible microstructures, where the misfit angle is ≲1◦(Balandraud and Zanzotto, 2007; Heczko et al., 2013), the compatibility can be achieved by elastic strains. Here, however, the misfit angles are ≈27◦for the situation sketched in Figure 3(b) and ≈10◦for the situation sketched in Figure 3(c). Hence, plastic slip is required to compensate the misfit. The B19′ martensite has only one slip system available, which is the (001)M slip; however, this one slip system is enough for the Variant 1 inside of the area where the bands intersect to reach such strain that the incompatibility is significantly reduced (although not fully removed, as discussed below). This situation is sketched in Figure 3(d): the new microstructure consists of (001)M and (100)M twins, plus there are dislocation cores appearing between Variant 1 and Variant 2 that result from the plastic slip.

The newly formed planar interface between Variant 2 and plastically slipped Variant 1 requires some additional comments. It is a twin interface with an additional rotation resulting from the presence of the dislocation cores. Hence, this interface combines the character of tilt interfaces that typically encapsulate kink bands in layered media or media with highly anisotropic dislocation slip (such as long period stacking order magnesium alloys) with the twin interfaces that encapsulate twin bands, as usual in shape memory alloys. We will later show that such combined interfaces and such combined kink-twin bands may play a crucial role in the discussed plastically formed patterns in NiTi. Because of the lack of any established terminology, we call these interfaces kwink interfaces (or simply kwinks, which comes from combining the words twin and kink), and we call the resulting bands the kwink bands.

The situation in Figure 3(d) is idealized, assuming that while the material in the intersection area becomes plastically sheared, all other regions remain intact and are just translated to stick to each other in the resulting microstructure. This leads to just a partial relaxation of the incompatibility; the material still needs to be elastically strained to reach the compatibility conditions over the kwink interfaces. The reason is that the extrapolations of the (001)M slip planes in the intersection area do not exactly meet the (001)M planes in the rest of the (001)M-twin band. However, the microstructure now has additional degrees of freedom, through which the full compatibility can be reached: the (100)M twin band in the lower part of the domain can move to the left or to the right by martensite reorientation, and the kwink interfaces can tilt accordingly, ensuring the continuity of the band. As shown in Section 4.1, the full compatibility is achieved if the lattices forming a kwink hold a twin-like relation which is quite close to a (20¯1)M-twin relation, and also the orientation of the resulting kwink band is quite close to the (20¯1)M plane. This is approximately drawn in Figure 3(e), where the kwinking planes are exactly (20¯1)M with respect to the lattice of Variant 2. In other words, the slip inside the (001)M can enable a fully compatible, i.e., energetically cheap, penetration of this band by a (100)M twin band, and then, as a result, new interfaces running along planes close to (20¯1)m form, possessing mirror symmetry between the neighboring lattices.

Hence, we observe that the (20¯1)M-twin relation can be (approximately) achieved without any (20¯1)M- twinning mechanism overcoming the barrier between different EPNs, but, instead, by combining two very simple straining mechanisms, which are the (100)M compound twinning and the (001)M slip. The reason is that the coordinated (001)M slip, in fact, acts as a mechanism that translates the given material point between different EPNs.

For the case in Figure 3(f), a similar compensation of the incompatibility is possible by activation of the plastic slip in Variant 1′. Again, a kwink interface forms between Variant 1′ and Variant 2, and the orientation relationship between these two variants over the kwink interface is quite close to the (100)M compound twin relation. Nevertheless, in the light of the observation done for Figures 3(b) and 3(d), we can alternatively assume that the discussed morphology may arise without the (20¯1)M twinning, i.e. by combining reversible twinning with plastic slip. Notice that the nucleation of the (100)M bands and the (20¯1)M bands appears at the same stress level, which means that if the construction in Figures 3(b) and 3(d) is correct, the slip is already massively active at this level, and there is no reason why it could not be creating the (20¯1)M-oriented bands themselves. Then, as sketched in Figure 3(g), there is a kwink band with an approximately (20¯1)M orientation, that can compatibly cross with a (001)M twin

Plane

Figure 3: (a) an experimental micrograph of (100)M and (20¯1)M twin bands crossing with the initial (001)M twin laminate, micrograph modified after Zhang et al. (2006) with permission from Elsevier; (b,c) visualization of the incompatibility resulting from the crossing; (d,e,f,g) visualizations of the compensation of the incompatibility through a (001)M dislocation slip. The structure in (d) has been created from (b) by dislocation slip in the intersection area only, assuming that all other regions and their boundaries remain unaltered; in (e), additional energy relaxation is enabled through allowing martensite reoreintation in these other regions. In (b-f), lattice parameters of B19′ introduced in Section 2.1 were used for the visualizations; the kwinking plane orientations in (d-g) are just illustrative and approximate, their exact parameters are calculated in section 4.1.

band. Again, the microstructure is composed just of Variant 1 and Variant 2, plus there is a (001)M plastic slip, and thus, the situations outlined in Figures 3(d) and (g) are utilizing the same mechanism to enable a compatible crossing of a (001)M twin band with a (100)M twin band and a (20¯1)M kwink band, respectively. As a result, the whole microstructure observed in Figure 3 can be constructed by combining the slip-enabled compatible crossings from Figures 3(d) and (g); in other words, introducing the kwinking enables explaining this microstructure in its whole richness just by combining three simple deformation mechanism: (100)M compound twinning, (001)M compound twinning, and (001)M dislocation slip.

Because of formulating the model at the continuum length-scale (with large strains and large rotations considered, as usual for the mathematical theory of martensitic microstrutures (Ball and James, 1987, 1992; Bhattacharya, 2003)), we do not discuss here the mechanism the penetration of (100)M twin bands by (001)M in the topological sense, that is, in terms of twinning dislocations and twinning disconnections that interact at the interfaces. That would definitely be needed for any more quantitative modelling, considering the energy barriers for the penetration as well as the driving forces on the individual line defects. The penetration models at this level are well developed in literature, with the main ideas owing to Müllner and Pirouz (1997); Müllner (2006). In these topological models, the incompatibility visualized in Figure 3(b,c) would be represented by a pair of disclination dipoles in an otherwise connected material, and the dislocation walls appearing at the kwink interface would be understood as arrays of defects, the stress fields of which compensate the dipoles (cf. (Müllner and Romanov, 2000; Müllner and King, 2010; Müllner, 2013)). Such a description would be, in many respects, fully equivalent to our all-continuum approach, possibly elucidating more details on how the line defects responsible for the growth of the (001)M and (100)M twins (i.e., the twinning dislocations) couple with the slip dislocations into new line defects substantiating the nucleation and growth of the kwink interfaces. At the same time, with a topological model we would loose the advantage of representing all active deformation mechanisms by simple two- dimensional deformation gradients, as done in Section 3. Nevertheless, the concept (and terminology) of disclinations and their compensation is applicable also at the continuum level, and we will utilize it Sections 2.3.4 and 4.3, when discussing the formation of the V-shaped patterns.

The kwinks, as introduced above, allow for compatible crossing between the (001)M and (100)M twins. This enables us to interpret the transition from the (001)M laminate into the pattern consisting of (100)M twins and (20¯1)M kwinks, as described in Section 2.2. This interpretation is outlined in Figure 4. We consider first a (001)M compound twin laminate, consisting of two variants, Variant 1 being the major one. The laminate is inside a material loaded in tension beyond the reorientation plateau. For simplicity, we assume that the loading direction is perpendicular to the (001)M planes; this means that neither the (001)M compound twins, nor the (001)M plastic slip can relax the external loads, as these mechanisms have both zero Schmid factors. In (111)M-textured NiTi wires, the (001)M compound laminate at the end of the reorientation plateau is stabilized due to compatibility conditions between grains (Molnárová et al., 2020), and the loading direction points rather along the M lattice vector. That situation is too complex to be captured by our model, but even in such a case, there is a large component of the loading that is perpendicular to the (001)M planes, and the (001)M compound twins and the (001)M plastic slip cannot fully relax the external loads, so the sequence outlined in Figure 4 may still give a quite good approximation of what happens in individual grains. Notice that this sequence explains only how the microstructure in Figure 4(a) transforms into the microstructure in Figure 4(d) under the external load, not how the V-shaped patterns accommodate tensile strains in the loading direction, or how does the material tackle stress singularities appearing at the tips of the patterns (i.e. in the regions where the (100)M twins and (20¯1)M kwinks meet). This will be discussed separately, first heuristically in subsection 2.3.4, and then by an explicit calculation in section 4.

For the material in Figure 4(a) loaded in the vertical direction, the only available mechanism within the given EPN that has a non-zero Schmid factor is the (100)M twinning. And thus, the material needs to utilize this mechanism to relax the stress. However, the (100)M twin bands need to intersect with the (001)M compound laminate, which is not possible due to compatibility reasons, as discussed above. If

Compatible Crossing

Figure 4: A simplified evolution of the microstructure from a (001)M laminate (a) into the V-shaped patterns (d), as suggested for the demonstration of the main features of the developed model. The compatible crossings in (b) are enabled by the (001)M slip inside of the bands; after the detwinning (c), the microstructure is composed just of (001)M twin bands and (20¯1)M kwink bands, the latter being decorated by dislocation cores.

For simplicity and better visual clarity, this figure is using a schematic geometry, not utilizing real shape strains or real orientation relations in the lattice. we admit that there is a (001)M slip active in the microstructure, and that it enables formation of kwink interfaces, there are two types of deformation bands that utilize the (100)M twinning: one holds the (100)M-twin relation with the major variant (and becomes a kwink band with respect to the minor variant), and the second one holds the (100)M-twin relation with the minor variant (and becomes a kwink band with respect to the major variant). Because they are created by the same mechanism, we can assume that they nucleate simultaneously in the microstructure (Figure 4(b)); in fact, as these two bands provide shearing along two different planes, inclined symmetrically with respect to the loading direction, their combination is needed to provide tensile strain along this direction. Bands of both types cross compatibly with the (001)M laminates, which means they do not cost any extra elastic energy, and also their faces are partially composed of (100)M twinning planes, so they are cheaper in terms of the surface energy than any kink or kwink bands with other orientations (see Section 3.3). To nucleate the bands, the external stress must be large enough to trigger a massive, coordinated plastic slip that overcomes the strong plastic anisotropy of the B19′ lattice; that is why the yielding starts at much higher stress levels compared with the reorientation plateau.

Once the bands are nucleated, the (001)M planes inside of them are tilted with respect to the loading axis, which leads to non-zero Schmid factors for the (001)M compound twinning, and creates a driving force for (001)M-detwinning, as well as for the (001)M dislocation slip. The detwinning inside the bands must be accompanied by detwinning outside bands, because of the compatible crossing between the laminates. Simultaneously, the bands grow (Figure 4(c)), as both the detwinning inside the bands and thickening of the bands themselves help to accommodate the tensile strains. Finally, the (001)M compound laminate completely disappears (Figure 4(d)), and further plastic straining would lead to formation of additional V-shaped microstructures, their growth and collisions, and secondary twinning inside of the bands. None of these mechanisms is affected by the compatible crossing between the deformation bands and the (001)M twins anymore; the main morphology of the pattern is, nevertheless, inherited from the nucleation stage, where this feature is essential.

General Comments On The Concept Of Kwinks

The advantage of considering the (20¯1)M interfaces as kwinks instead of twins for the modelling purposes is quite obvious (and it will be more detailed in Section 3): while the (100)M twinning can be captured by the mathematical theory of martensitic microstructures, and the unidirectional plastic slip by conventional

B19' Matrix

Figure 5: Atomistic-scale visualization of the difference between a (20¯1)M twin and a (20¯1)M kwink. Both deformation mechanisms result in the same shape strains and the same twin-relationship between the neighboring lattices. However, while for the twin this relationship is achieved through a complex twinning path (shown schematically in the middle, see (Gao et al., 2017; Ezaz et al., 2020) for more details), in the kwink the relationship results from an array of dislocation cores arranged along an originally (100)M twin interface.

crystal plasticity tools, which are both well-established modelling approaches, the twinning outside the EPN would lead to several open theoretical questions, such as the loss of the reference configuration (cf. (Baggio et al., 2019)). Also, the structure and height of the energy barrier between different EPNs is not sufficiently understood yet. Moreover, there also are several indications that the (20¯1)M interfaces indeed arise rather by kwinking than by irreversible twinning. At the atomistic scale, the difference between these two approaches is drawn in Figure 5. For the plastic (20¯1)M twinning, the unit cells need to undergo extensive shuffling of the atoms in addition to the shear deformation, where the individual shuffling atoms are expected to move over distances comparable to the lattice spacing (see also (Christian and Mahajan, 1995) for a similar twinning path for the (41¯1) deformation twinning in B2 austenite); the distances the atoms need to move are by an order of magnitude larger than the shuffling amplitudes commonly accompanying plastic twinning e.g. in hexagonal materials. As shown by Ezaz et al. (2020), such a large- amplitude shuffling might be energetically very demanding, which makes the (20¯1)M twinning energetically disadvantageous compared with the reversible (100)M and (001)M twinning. In contrast, the (001)M plastic slip is geometrically simple and energetically quite cheap, and so is the (100)M twinning, so their combination as of two coupled processes might be energetically favorable over the (20¯1)M twinning at the atomistic level.

On the other hand, as discussed in (Gao et al., 2017; Gao, 2019), the barrier between the EPNs might be not as high as predicted in (Ezaz et al., 2020), because the (20¯1)M-twinning path goes over the orthorhombic structure B33, which is frequently identified as the ground state structure of NiTi by ab- initio calculations (Wagner and Windl, 2008; Huang et al., 2003). Further atomistic-scale considerations also reveal that the two discussed mechanisms, (20¯1)M twinning and kwinking, are essentially very similar at this scale, and might be even seen as just two different descriptions of the same phenomenon: for example, the (yellow) parent cell for the B19′ twin unit cell sketched in the middle part of Figure 5 can be turned into the twin unit cell (up to the shuffle) not only by shearing along the (20¯1)M plane, but equivalently by slipping along the (001)M plane. And, again, such a slipping passes through a structure which is geometrically very close to B33.

At the same time, the kwinking mechanism at the atomistic scale poses several intriguing questions, such as how do the dislocation cores enter and leave the kwinks, or what barriers the kwinks need to overcome during their motion. Resolving these questions would require a detailed atomistic-topological analysis of the kwinks, as done for example for Type II twins in B19′ NiTi by Mohammed and Sehitoglu (2020a,b, 2021).

For these reasons, we restrict further distinguishing between kwinking and twinning only to the con- tinuum scale, where we can understand kwinking as a result of simultaneously appearing martensite reorientation and a massive coordinated (001)M plastic slip, and twinning as a separate mechanism, a pure shear strain localized to the twinning plane and oriented along it that can exist independently of the plastic slip in the surrounding material. In this sense, when formulating the model in Section 3, we define kwink interfaces as interfaces between different variants of martensite that, at the same time, represent a jump of plastic slip amplitude, and we stick to this definition throughout the rest of the paper.

At the continuum lengthscale, the heuristic insight into the nature of the observed microstructures comes from the pattern formation. The V-shaped patterns (Figure 3(a)) clearly resemble the patterns forming in layered media, in particular those observed in a compression-loaded pile of papers by Wadee et al. (2004), but also the typical plastic kinks that appear in magnesium alloys (Inamura, 2019; Lei and Nakatani, 2015). On the other hand, such patterns are not typically observed in other shape memory alloys or in NiTi not strained beyond the recoverability limit. In these materials, regular 1st or higher order lamination is typically observed, arising from energy minimization (Bhattacharya, 2003) and from the nucleation and growth mechanism of the martensite phase (Schwabe et al., 2021). This suggests that the underlying mechanism is similar, i.e., the observed patterns in NiTi martensite arise due to the plastic slip as well. We will discuss the activity of the plastic slip during the formation of the V-shaped patterns in more details in Section 4.2.

The experimental observations that the orientation relationship over the (20¯1)M-oriented interfaces is not exactly a (20¯1)M-twin relation (as discussed in Section 2.1, see also Section 4.2 and (Molnárová et al., 2023) for quantitative data), may be understood as another supporting argument for the existence of the kwinks. In Section 3.3, we will suggest that the surface energy of a kwink may be reduced by increasing locally the number of coherent sites at the atomistic scale, which creates a driving force trying to align segments of the kwinks exactly along the (20¯1)M planes. Nevertheless, the compatibility conditions, such as those arising from the compatible crossing with the (001)M twins, are probably the leading energy terms here, and thus, at larger than atomic lengthscales the misorientation is observed. Some misorientations may arise also from the fact that the creation of the kwink bands is related to the dislocation slip, which is an energy dissipating process, and the dissipation may prevent the system from reaching the exact energy-minimizing configuration. Again, such an effect cannot be expected for classical twinning.

Let us point out that the misorientation for the (20¯1)M twins are then also inherited by the (41¯1)P twins in the parent phase after the reverse transformation; a large scatter of the observed orientations from those predicted theoretically was reported in (Molnárová et al., 2020). This agrees well with the fact that the (41¯1)P-twinning mechanism in the B2 structure is also quite energetically demanding and including a large-amplidute shuffle (Christian and Laughlin, 1988); hence, as an alternative mechanism we suggest that the (41¯1)P twins are, in fact, not twins but symmetric-tilt grain boundaries that are obtained from kwinks formed in martensite.

Plastic Forming Of Oriented Martensite

The second requirement we put on the model is that it must be able to capture the inelastic straining of the B19′ lattice in case of fully detwinned oriented martensite, loaded along the [10¯1]M direction, which is the direction of the largest transformation strain in the (010)M plane. This situation appears for example at the yielding plateu in cold-worked NiTi wires reported in (Molnárová et al., 2020) in grains with the vector p pointing along [10¯1]M. This loading direction corresponds to the P direction in austenite, which means the dominant axial direction in the textured wire. From a more general point of view, this is the most fundamental requirement to be put on any model of plastic forming of NiTi martensite: it should be able to identify what mechanisms are activated once all reversible twinning systems are exhausted.

We consider a single variant of martensite subjected to tensile loading along [10¯1]M (Figure 6(a)). The (001)M and (100)M twins have negative Schmid factors, i.e., they cannot contribute to any further elongation of the lattice along the loading direction. Hence, the only available deformation mechanisms

(D)

Figure 6: Visualization of plastic forming of a grain consisting only of fully oriented, detwinned martensite: (a) a single-variant grain loaded along the direction of maximum transformation strain in the given (010)M plane, the dash-dot lines show the traces of invariant planes for the slip and twinning/kwinking, p is the projection of the wire axis onto the given (010)M plane; (b) the change of the shape of the grain induced by the (001)M- dislocation slip, the red lines denote the planes at which the slip occurred; (c) the change of the shape of the grain induced by the (20¯1)M plastic twinning (or kwinking); (d) comparison of the resulting shapes of the grain from (b) and (c) with the shape representing area-preserving elongation along the vector p.

with positive Schmid factors are the (001)M dislocation slip (possibly leading to formation of kink bands) and the (20¯1)M plastic twinning (or kwinking, as introduced above). As outlined in Figures 6(b) and (c), respectively, both these mechanisms provide elongation along the [10¯1]M direction, but also a shear deformation in a perpendicular direction; the perpendicular shear deformation is quite pronounced, since the angle between the invariant planes and the loading direction is larger than π/4 in both cases. According to the Taylor’s hypothesis (Bhattacharya and Kohn, 1996; Ono et al., 1989; Shu and Bhattacharya, 1998), the shape change of individual grains in the polycrystal must comply with the macroscopic straining of the wire. This requirement follows from the fact that the grains are anchored in the microstructure, i.e., they cannot freely rotate with respect to each other. In other words, the Taylor’s hypothesis requires the grain to achieve pure elongation along the loading direction without the perpendicular shear strains, which may happen only if the (001)M dislocation slip and the (20¯1)M twining/kwinking appears simultaneously, such that the perpendicular strains compensate (Figure 6(d)). Hence, the (20¯1)M twinning/kwinking in this case is triggered if and only if the lattice, at the same time, plastically slips along the (001)M planes. Nearly exact compensation of the shears is possible, since the invariant planes for both mechanisms make quite similar angles with the loading axis (Figure 6(a)).

At the yielding plateau (Figure 2(e)), the (20¯1)M bands continuously grow, which means that also the whole lattice continuously slips such that the shears in the perpendicular direction are compensated. This is an additional motivation for not considering the (20¯1)M plastic twinning as an independent mechanism from the dislocations slip. The concept of kwinking assumes, instead, that the (20¯1)M bands are direct consequences of the slip, and thus, the simultaneous (20¯1)M plastic twinning and plastic slip are, in fact, one straining mechanism, with the (001)M dislocations moving everywhere in the material: homogeneously in the matrix and inside the bands, the motion of the kwink (or kink) interfaces between the matrix and the bands itself can be also seen as a coordinated motion of slip dislocations. The kwink interfaces are composed of dislocation cores, and they are, therefore, probably prone to absorbing or releasing more dislocations, enabling the plastic slip to take place on both sides of the interface. In contrast, the twin interfaces typically act as obstacles against the dislocation slip. Hence, again, we can conclude that the desired feature of the model is more easily achieved when the (20¯1)M are considered to be kwink interfaces instead of twin interfaces.

In Section 4.2, we will show by an explicit calculation that there exist kwink bands that enable an exact compensation of the perpendicular shears from the (001)M slip, and that these bands are, again, oriented close to the (20¯1)M planes. In some sense, this observation is quite similar to the observation

Boundary

Figure 7: (a) TEM image of a representative grain in plastically formed NiTi; (b) notation and the spatial arrangement of the three lattices (M, A, and B) forming the V-shaped pattern; (c) orientation relationships within the pattern; (d) orientation relationships for the secondary (20¯1)M twinning (lattice C); (e) mirror symmetry of the (001)M slip planes with respect to the A-B interface, with τ representing the tilt angle. The shapes of the unit cells as well as the orientations of the assumed twin interfaces in (b) correspond to the lattice parameters from (Bhattacharya, 2003), the orientation of the A-B interface with respect to the twin interfaces and lattice orientations for (b) and (e) is taken from the micrograph in (a). The loading direction (or, more precisely, its projection onto the given (010)M plane) is approximately perpendicular to the (001)M planes in lattice M.

done in the previous subsection, where the approximately (20¯1)M-oriented kwink bands were shown to be able to cross compatibly with the (001)M twin bands; here we assume that they can also co-exist with the (001)M slip that carries shear strains of the same orientation as the (001)M twinning.

V-shaped patterns and symmetric-tilt grain boundaries The last heuristic argument we use for justification of the proposed model is that this model is able to fully capture the pattern formation observed in the experiments in the given (010)M martensite plane, and to explain it in terms of energy reduction. The TEM observations by Molnárová et al. (2020) prove that there are no other patterns created at the yielding plateau than the bands and V-shaped microstructures in the (010)M plane. And thus, if the model can capture them, it can fully capture the whole plastic forming process.

We focus on grains that are loaded in direction approximately perpendicular to (001)M, because of the richer patterns appearing in them. For the argument, we use the microstructure in one selected grain from (Molnárová et al., 2020), containing prototypical V-shaped patterns and secondary (20¯1)M twinning. The grain is shown in Figure 7(a) (see Figure 10 in (Molnárová et al., 2020) for the details of the indexing of individual lattices forming the pattern). The grain is composed of the matrix lattice (M) and lattices (A) and (B) holding (100)M and (20¯1)M twin relationships to the matrix lattice, respectively. The both twin relationships are of the compound character, which means that the lattice A is mirror-symmetric to M over the (100)M plane, and so is the lattice B with respect to M over the (20¯1)M plane. The arrangement of these lattices in a simple V-shaped microstructure is visualized in Figure 7(b). In addition, there is the lattice C, which holds a (20¯1)M twin relationship with the lattice B, that is, C is a result of the secondary (20¯1)M twinning, reported already in (Zhang et al., 2006).

Let us now analyze the characters of the individual interfaces between the lattices M, A, B, and C, and discuss whether they can arise from the simple mechanisms we build our model from, which are the martensite reorientation through reversible (100)M and (001)M twinning and the (001)M dislocation slip. For the primary twinning, this analysis is visualized in Figure 7(c). The A-M twinning is directly twin relation) can be decomposed into two steps: one is a simple rotation in the (010)M plane, the second is the (100)M twinning. Importantly, the rotation angle is the same as the rotation relating the lattices A and B, which is a necessary condition for the whole microstructure being geometrically admissible. For the secondary twinning (Figure 7(d)), similar decompositions can be done for relationships between the lattice C and all other lattices. The C-A relationship is the same as B-M, which has been discussed above. The C-M relationship is a simple rotation, which can be understood as merging the C-A twinning and A-M twinning, and finally the C-B relationship can be decomposed into the C-A relationship and a rotation, again identical to the rotation between A and B. In summary, each lattice from the observed pattern can be translated into any other lattice from the pattern by a sequence of two mechanisms: (100)M twinning and a specific rotation, equivalent the rotation between lattices A and B.

To confirm that the formation of the observed pattern can be indeed explained by the proposed model, we need to show that this specific rotation can be a result of the (001)M dislocation slip. As it can be concluded both from crystallographic considerations (Starkey, 1968) and from continuum-mechanics compatibility conditions (Inamura, 2019), the kink interface between two mutually rotated lattices resulting from unidirectional dislocation slip must always be a mirror plane between orientations of the slip planes on both sides of the kink interface. This follows from a simple requirement that the interface must be equally distorted and rotated in both lattices. The interfaces satisfying this condition are the so-called symmetric-tilt grain boundaries (STGB, see e.g. (Han et al., 2016)), that can be characterized by a single scalar parameter, the tilt angle τ. For the needs of the model formulated in Section 3, we define this angle as the angle between the slip planes in the neighboring lattices, where for a STGB both slip planes make an angle of τ/2 with the interface. Notice that for low-symmetry lattices, such as the monoclinic lattice of B19′, the mirror-symmetry for STGBs holds only for the slip plane orientations, not for the lattices themselves. This distinguishes here the STGBs from the twins.

In our case, this condition means that the A-B interface should be a mirror-symmetry plane between (001)M planes in lattices A and B. Figure 7(e) shows that, within the precision of ±2◦with which the orientation of the interface can be assessed from the micrograph, there is a perfect mirror symmetry between the (001)M planes, with τ ≈146◦. And thus, indeed, the A-B rotation for the given orientation of the A-B can arise as a result of the (001)M dislocation slip. We will further support this conclusion by an explicit calculation in Section 4.3. For the needs of the current heuristic discussion, nevertheless, we can conclude that the A-B interfaces can be captured by the proposed model, and so can also all other interfaces observed in the pattern, because (as it can be easily checked) all of them satisfy the requirement of the mirror symmetry between the slip planes. In other words, all these interfaces are admissible as the outputs of the model.

It is also worth noting that this may not be true for all planar interfaces found in martensitic mi- crostructures. Inamura et al. (2017) observed interfaces created by head collisions of mutually incompat- ible martensitic plates in a Ti-Nb-Al alloy formed within the phase transition from austenite. Although being visually similar to the V-shaped patterns in NiTi, these microstructures were of a very different origin and behavior. While the V-shaped patterns gradually grow during the plastic forming, exploit- ing the twin/kink/kwink character of all interfaces, the martensite plates in (Inamura et al., 2017) stop growing along the junction plane immediately as they collide, which results in fine triangular microstruc- tures. The reason is that the collision-created interfaces violate the compatibility conditions, and are, thus, energetically expensive.

Finally, we need to suggest how the formation of the V-shaped patterns can lead to low-energy ac- commodation of the tensile strains applied to the grain. Here we utilize the knowledge of the deformation mechanisms in mechanical systems with unidirectional slip (Lei and Nakatani, 2015; Zhu et al., 2010; Inamura, 2019; Wadee et al., 2004). As shown above, all interfaces present in the microstructure are either twin interfaces or kink interfaces or their combinations, and, as such, they can be expected to appear in energy-minimizing patterns. This does not, however, guarantee that the whole pattern can be also an energy minimizer or even its close approximation. The interfaces meet in triple junctions, where certain ge- ometric conditions between the shearing directions (Bhattacharya, 1991, 2003; Balandraud and Zanzotto, 2007) need to be satisfied. If they are not, a stress singularity (a so-called disclination (Inamura, 2019; Müllner and Pirouz, 1997; Müllner and Romanov, 2000; Müllner and King, 2010)) arises at the junction.

This would lead to a tendency to merge pairs of triple junctions together such that the disclinations get compensated (Inamura, 2019), which has not been ever reported from the experiments on NiTi martensite (cf. (Zhang et al., 2006; Molnárová et al., 2020)).

We will show in Section 4.3 that the conditions for the shearing vectors cannot be satisfied, regardless of the exact orientation of the involved twin and kwink (or kink) bands. Hence, there must be an additional mechanism, not visible from the micrographs, that compensates the stress singularity at the disclinations and enables, thus, the V-shaped patterns to form without any excessive increase in elastic energy in the grain. One possibility is that the disclination is compensated by the (001)M plastic slip. A simple mechanism how this can be achieved is schematically sketched in Figure 8 (notice that for better visibility of the strains in this sketch, the used geometry is not derived from real lattice parameters, unlike in most of the visualizations in this paper). In this figure, we consider first the matrix lattice as the reference (undeformed) configuration, and we suggest a pattern composed of seven regions, neighboring to each other over planar interfaces (Figure 8(a)). There are no triple junctions in this construction, but quadruple junctions, for which the compatibility conditions are easier to achieve1 (Bhattacharya, 2003; Seiner et al., 2014); this is reached by adding an extra band (regions (6) and (7)) to the V-shaped geometry.

In the deformed configuration (Figure 8(b)), the regions (1), (2) and (3) remain in the matrix lattice, while the other gain homogeneous deformation gradients such that the continuity of the domain as a whole remains preserved. All involved deformation gradients can be polar-decomposed into a shape strain in the shown plane (the shape strain here is pure shear and the amplitude of this shear is used for the colors in Figure 8(b)) and a rotation in this plane. The bands (4) and (5) become the twin band and the kwink band, respectively, with opposite signs of the shear deformation. The plastically slipped band (6) is oriented parallel to the slip planes, and that is why there are no visible interfaces between this band and the regions (1) and (3) in the micrograph. The role of the band (6) is mainly in bringing extra plastic strain into the triangular region (7), such that the disclination is fully compensated. As a result, the dislocation-induced shear strain in this triangle is of a higher amplitude than in the rest of the kwink band (5), but because the boundary between (7) and (5) is parallel to the slip plane, it is invisible in the micrograph. The dislocation cores that move though the band (6) might originate either from a grain boundary, or possibly from another V-shaped microstructure, oriented upside-down with respect to the discussed one (see (Inamura, 2019) for a similar situation for kink ridge patterns). These dislocation cores enter the region (7) and stop at its boundary with the twin band (4), where contribute to the rotation between the adjacent lattices of the same variant of martensite.

To confirm that the construction in Figure 8(a) enables creation of the V-shaped microstructures appearing in the experiments, we need to show that the interfaces (6)-to-(7) and (3)-to-(5), although not parallel in the reference configuration (Figure 8(a)), become always perfectly parallel to each other in the deformed configuration (Figure 8(b). This is done in Section 4.3, where also the explicit amount of dislocation slip in the band (6) needed to compensate the disclination is calculated for B19′ NiTi martensite.

From Figure 8(b), it is also seen how the V-shaped pattern helps to accommodate tensile strains along the loading direction. The region (2) is translated upwards, along the twin boundary interface, which is enabled by the horizontal translation of the region (3). Such a mechanism, involving twinning, kwinking 1A somehow simplified reasoning is that a triple junction can exist without a disclination only if the meeting regions are all sheared in the direction parallel to the junction line only. While this is possible in some wedge-shaped martensitic microstructures (Bhattacharya, 1991), it cannot be achieved here, because we assume that all involved mechanisms act only on the (010)M plane. For the quadruple junctions, the condition is softer, as the four meeting deformation gradients can pairwise compensate.

-150%

Figure 8: A schematic sketch of the pattern formation mechanism: (a) reference configuration and (b) deformed configuration of a representative volume accommodating tensile loads in the vertical direction by formation of a V-shaped microstructure. In (a), the numbers (1)-(7) denote regions gaining different homogeneous deformation gradients and/or translations in the deformed configuration. The color code in (b) represents the magnitude of shear along the planes that are horizontal in (a), that is, the slip planes. The red lines represent two such planes in the deformed configuration; for a real material, these planes would be (001)M planes along which the slip in the M direction occurs. It is seen that these planes are mirror-symmetric with respect to each interface they intersect. While in real V-shaped patterns in B19′ martensite the shear strain magnitude in the (100)M twin bands is εtwin = 0.2385 (see Table 1), for this sketch a more than six-times higher amplitude εtwin = 1.5 is used for better visibility of the elongation of the domain along the loading direction.

and plastic slip, results in elongating the considered domain, although the loading is applied perpendicular to the slip plane. This explains why the V-shaped patterns are seen mainly in grains with the (001)M plane oriented perpendicular to the loading direction in nanocrystalline textured NiTi wires (see Section 2.2.2).

In grains oriented such that the loading is applied along the [10¯1] direction, the combination of dislocation slip and kwinking is sufficient for accommodating the strains, as described in the previous Section 2.3.2.. Consequently, only the kwink bands appear.

Summary Of The Heuristic Argument

We have above discussed three particular experimentally observed processes occurring in plastically strained NiTi, and in all cases we realized that these processes require the (001)M dislocation slip to take place. Once the slip is active, the lattice can locally easily translate between different EPNs, and so distinguishing between ’twinning inside one EPN’ and ’twinning beyond the EPN’ becomes not important. As a substi- tute for the concept of ’twinning outside the EPN’ we proposed the new concept of kwink bands, which are twin bands with additional rotations coming from the dislocation cores present at the interfaces, and additional shear coming from the dislocation slip in the adjacent variants of martensite. Approximately (20¯1)M-oriented kwinks appear to be of high importance: they can cross the (001)M-compound twins without additional elastic energy, which enables their easy nucleation beyond the end of the reorientation plateau, and they can also compensate the perpendicular shears coming from the (001)M dislocation slip in oriented martensite, which enables them to act as a part of the mechanism of plastic forming at the yielding plateau (we support both these claims by direct calculations in Section 4). Finally, the (20¯1)M- oriented kwinks can compose into V-shaped patterns with (100)M twins, provided that there is additional dislocation slip available, enabling compensation of the disclinations at the triple points. These heuristic results serve as a basis for the model developed in the following sections, where we show the appearance of specifically oriented kwinks and their co-existence with other twins and with massive dislocation slip can be understood as a result of energy minimization.

Odel Formulation

For the continuum-level description of the plastic forming mechanisms, we use the framework and formal- ism developed for martensitic microstructures within the theory of non-linear elasticity (Ball and James, 1987, 1992; Bhattacharya, 2003). In this theory, the individual variants of martensite are represented by deformation gradients constructed from the lattice correspondence and lattice parameters via the Cauchy- Born hypothesis. The same formalism can be also applied for plastic slip, as utilized in crystal plasticity (Roters et al., 2010; Gurtin, 2000; Acharya and Bassani, 2000). As discussed in detail in Section 2, the plastically formed B19′ NiTi martensite undergoes mainly (100)M, (001)M and (20¯1)M twinning and the (001)M dislocation slip (see Table 1). If we restrict the construction of the model just to these mecha- nisms, the problem becomes essentially two-dimensional, as all these mechanism represent straining just in one (010)M plane (one {110}P plane in the parent phase), shared by all involved variants. This significantly simplifies the description.

Ariants Of Martensite And Deformation Mechanisms

We take one variant of martensite as the reference configuration (Variant 1 in Figure 9), let the lattice in the (010)M plane be generated by vectors (e1 and e2). We refer further to this variant as to the matrix (M) and assume that this variant is dominant in the given grain. The deformation gradient representing this variant is the identity (I), i.e. we fix the rotation of this variant. The matrix sets the coordinate system xi in which we describe the evolution of the microstructure, using two-dimensional deformation gradients acting on the x1x2 plane, see Figure 9.

The second variant sharing the same (010)M plane with the matrix is the Variant 2 (on the right in Figure 9), that can be achieved from Variant 1 by shearing. Two different shearing paths between Variant 1 and Variant 2 can be considered. The first is through the deformation gradient F and keeps the lattice in the given EPN. This path corresponds to the classical compound twinning, as explained below. The second path is through the deformation gradient F′ which goes beyond the given EPN. This path (suggested by Gao et al. (2017); Gao (2019)) can represent either the irreversible twinning, or the (001)M dislocation slip of Variant 2 with a specific slip amplitude. To highlight that this path results in the same variant of martensite but corresponding to another EPN, we use the denotation Variant 2′ in Figure 9.

In the given coordinate system, the deformation gradients are

(1)

where γ = 0.2385 and γ′ = −0.3910 for the lattice parameters of monoclinic B19′ martensite introduced in Section 2.1. The lattice of Variant 2 (or Variant 2′) is generated by vectors Fe1 and Fe2 (or F′e1 and F′e2). Notice, however, that for Variant 2′ the deformation gradient F′ describes only the change of the shape of the unit cell, not the rearrangement of the atoms inside it. In other words, the reorientation of martensite outside the given EPN requires also additional shuffling, as discussed in more details in Section 2.3.

In addition to the deformation gradients representing the twinning, we also consider a deformation gradient representing the slip. Here we adopt the assumption of Kudoh et al. (1985) that the easiest slip in the B19′ martensite lattice is the (001)M slip. According to the first-principles calculations

X1

Figure 9: Notation of the variants, the used coordinate system (xi axes in the shaded bar), and the definition of the deformation gradients F and F′. The shapes of the unit cells correspond to the lattice parameters of B19′ martensite from (Bhattacharya, 2003).

(Ezaz et al., 2020), the energy barriers for activation of this slip are comparable to those for all considered twins, and by an order of magnitude lower than those for the hypothetical (100)M or [¯10¯2](20¯1)M dislocation slips. In the given coordinate system, the two-dimensional deformation gradient representing

(2)

where α is the slip amplitude representing the average amount of slip in the given volume. Notice that there are specific slip amplitudes α = γ and α = γ′ for which P(γ) = F and P(γ′) = F′, i.e. the same effective deformation of the lattice can be achieved by slip of Variant 1, and by reorientation of Variant 1 into Variant 2 or Variant 2′.

Plastic slip can be further combined with martensite reorientation, resulting in the deformation gradient

(3)

and equivalently for F′. Note that the commutability follows from the specific form of P(α), F, and F′.

Ompatibility Conditions And Coherent Interfaces

When a microstructure is formed, different regions of the material undergo different deformations. To achieve the kinematic compatibility (i.e., displacement field continuity) between these regions, additional rotations may be needed. For this reason, we assume henceforth the deformation gradients of form RF, RF′ and RP(α) as those of which the analyzed patterns are composed, where R ∈SO(2) is the orthogonal rotation matrix. In the given two-dimensional setting, the rotation can be represented by one angle θ,

Authors:

Peder EZ Larson 1, 2,* , Jenna ML Bernard1, James A Bankson 3, Nikolaj Bøgh 4, Robert A Bok1, Albert P. Chen 5, Charles H Cunningham 6,7, Jeremy Gordon1, Jan-Bernd Hövener 8, Christoffer Laustsen 4, Dirk Mayer 9,10, Mary A McLean11 12, Franz Schilling13, James Slater1, Jean-Luc Vanderheyden5, 14, Cornelius von Morze 15, Daniel B Vigneron1, 2, Duan Xu1, 2, and the HP 13C

94143, Usa.

Denmark. 5 GE Healthcare, Menlo Park, California, USA. 6 Physical Sciences, Sunnybrook Research Institute, Toronto, Ontario, Canada.

8 Section Biomedical Imaging, Molecular Imaging North Competence Center (MOIN CC), Medicine, Baltimore, MD, USA. Cambridge, United Kingdom.

14Jlvmi Consulting Llc, Dousman, Wi, Usa

#See Acknowledgements for a list of all HP 13C MRI Consensus Group Members This work was supported by the ISMRM Hyperpolarized Media MR Study Group, the ISMRM Hyperpolarization Methods & Equipment Study Group, and the Hyperpolarized MRI Technology Resource Center (NIH/NIBIB grant P41EB013598).

Abstract

MRI with hyperpolarized (HP) 13C agents, also known as HP 13C MRI, can measure processes such as localized metabolism that is altered in numerous cancers, liver, heart, kidney diseases, and more. It has been translated into human studies during the past 10 years, with recent rapid growth in studies largely based on increasing availability of hyperpolarized agent preparation methods suitable for use in humans. This paper aims to capture the current successful practices for HP MRI human studies with [1-13C]pyruvate - by far the most commonly used agent, which sits at a key metabolic junction in glycolysis. The paper is divided into four major topic areas: (1) HP 13C-pyruvate preparation, (2) MRI system setup and calibrations, (3) data acquisition and image reconstruction, and (4) data analysis and quantification. In each area, we identified the key components for a successful study, summarized both published studies and current practices, and discuss evidence gaps, strengths, and limitations. This paper is the output of the “HP 13C MRI Consensus Group” as well as the ISMRM Hyperpolarized Media MR and Hyperpolarized Methods & Equipment study groups. It further aims to provide a comprehensive reference for future consensus building as the field continues to advance human studies with this metabolic imaging modality.

Keywords: Hyperpolarized MRI, metabolic imaging, carbon-13, pyruvate, dissolution dynamic

Introduction

MRI with hyperpolarized 13C agents, also known as hyperpolarized (HP) 13C MRI, has shown great potential as a novel imaging modality, particularly for its ability to probe metabolic processes in real time. The first human studies with HP [1-13C]pyruvate were performed in 2011 in prostate cancer patients (1).

Since then, there have been over 60 papers published with imaging results of human subjects from 13 different sites, with applications including prostate cancer, brain tumors, breast cancer, kidney cancer, pancreatic cancer, metastatic disease, liver disease, ischemic heart disease, diabetes and cardiomyopathies. The vast majority of these studies used [1-13C]pyruvate (1–63), where [2-13C]pyruvate (64) and 13C-urea (56) have been demonstrated too.

As clinical HP 13C MRI advances, there is a growing need to build consensus for best practices, which are critical for comparing data across sites, performing multi-site trials,deploying methods to new sites, partnering with vendors, and potentially for obtaining broader regulatory approvals.

In March 2022, we initiated an effort to build consensus within the HP 13C MRI community with this opportunity in mind, and it was greeted with strong enthusiasm. The “HP 13C MRI Consensus Group”, containing over 55 members from 27 sites, identified the area of greatest need and opportunity for consensus building to be HP [1-13C]pyruvate human

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Pyruvate is the most mature and widely used HP agent and has the most significant translational evidence emphasizing the potential clinical impact.

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Clinical trials, particularly multi-site trials, have the strongest need for consensus methods to ensure that data can be combined across sites. This work is a Position Paper for which the goal is to describe current successful practices and study methods for HP [1-13C]pyruvate human studies along with justification to support those practices. This is divided into four major topic areas: (1) HP 13C-pyruvate preparation, (2) MRI system setup and calibrations, (3) data acquisition and image reconstruction, and (4) data analysis and quantification (Fig. 1). The current successful practices and study methods include a literature review of published peer-reviewed journal papers showing human HP [1-13C]pyruvate study data, up to September 2022 (1–63), as well as new unpublished information from surveys of HP 13C study sites. Based on this information, we also highlight the evidence gaps, strengths, and limitations of current practices which are summarized at the end of each section.

Figure 1: Illustration of the HP 13C MRI human study process, including the 4 major areas covered in this paper: Hyperpolarized 13C-pyruvate preparation, MRI system setup and calibration, Acquisition and Reconstruction, and Data Analysis and Quantification.

Figure 2: Anatomical targets of HP [1-13C]pyruvate MRI human studies published up to September 2022.

Hyperpolarized 13C-Pyruvate Preparation

This section covers the processes for creating the HP agent, 13C pyruvate, and will include many aspects and considerations that are needed to safely and effectively prepare doses for metabolic imaging studies in human subjects. These include material, personnel, equipment and facility, fluid path preparation, quality control, and release.

It is helpful to understand that the specifications of a dose of 13C pyruvate suitable for in vivo MR HP metabolic imaging were shaped in part by early preclinical studies performed by GE HealthCare summarized in Ref. (65). In short, the safety of the two novel drug components, 13C pyruvate and the electron paramagnetic agent (EPA) AH111501, were demonstrated in those studies. The more precise formulation of the dose suitable for human use was then determined from clinical studies (66) that included two Phase 1 clinical trials in young and elderly healthy volunteers without hyperpolarization of the 13C nuclei and another Phase 1/2a dose escalation and imaging feasibility study with HP 13C pyruvate in 31 prostate cancer patients at the With the exception of the first HP 13C imaging clinical trial, which utilized a prototype device in a cleanroom (1), all HP 13C studies performed in humans to date have utilized the SPINlab polarizer (manufactured by GE HealthCare). Consequently all doses of the HP 13C pyruvate delivered by SPINlab have been produced using the “SPINlab Pharmacy Kit” that serves as the container-closure system for the various drug components (13C pyruvic acid and EPA mixture, dissolution medium, and neutralization and dilution medium) during sample polarization, dissolution and quality control (QC) processes. Thus many aspects of the HP sample preparation considerations discussed below are related to the SPINlab instrument and the consumables designed to be used with it (67).

General Considerations

While more than 860 patients or healthy subjects having been injected with HP 13C pyruvate as of January 2022 without reports of any serious adverse events (68), HP 13C pyruvate injection remains an investigational MR contrast agent and can only be administered by those with Investigational New Drug (IND) exemption from the Food and Drug Administration (FDA) in the USA, a Clinical Trial Application (CTA) in Canada, approval from National Research Ethics Committee Services in the UK, or approval from the relevant local regulatory body. Thus, methods and processes involved to produce a dose should have patient safety as the first priority. Since utilizing dissolution dynamic nuclear polarization (dissolution-DNP) for human use is still a relatively new development, there are no existing published regulatory guidelines specifically for this method.

There are two major production styles that determine how various sites approach the agent preparation. In the US, the most common approach is to rely on a sterilizing filter (“Terminal Sterilization”) to ensure sterility of the final product, akin to PET tracer production, where a starting molecule with a radioisotope is processed using various other ingredients to make the final, desired and injectable contrast agent within a necessarily short amount of time (69). For these sites, sterilization of the components and accessories upstream of this filter are not required, although many of them were manufactured and tested following Good Manufacturing Practice (GMP) or Good Laboratory Practice (GLP) requirements. The filling process is usually performed under an ISO 5 laminar flow hood, but a clean room or an isolator is not required.

This approach is typically accompanied by testing the integrity of the sterilizing filter prior to release of the dose for injection. Typically, post release endotoxin and sterility tests are performed using an aliquot reserved from each released dose.

In the UK and EU, the most common approach is to more-closely follow sterile pharmaceutical compounding guidelines (70), where all components and ingredients are required to be sterile or manufactured under GMP guidelines and are assembled and filled within a clean room environment or an isolator system (“Sterile Preparation”). Typically a batch of Pharmacy Kits for HP 13C pyruvate injection are prepared together. The sterility of the final dose is also ensured by batch validation testing, in addition to the sterility of the ingredients and the sterile compounding process. The endotoxin and sterility testing are performed for the process validation but are not performed for each injected dose.

Some institutions fill and assemble the Pharmacy Kit required for a specific study on the same day or the day prior to polarization, dissolution, and patient administration, but others have also demonstrated the feasibility of preparing a batch of kits, keeping them in a -20ºC freezer and using them over a period of a few months.

Beyond the obvious requirements that the process and the facility has to ultimately produce a dose that is safe to inject into a human, regulatory authorities will also focus on the question “Are you in control of your processes?”. To be in control of your process requires an in-depth and broad understanding of all processes involved in pre, post, and during the production process.

Personnel

It is typical and may be required to have licensed personnel involved in the production process depending on local regulations.Typically a pharmacist, radiopharmacist or other similarly qualified person (QP), in charge of the facility where the Pharmacy Kit filling and preparation is taking place, is responsible for the overall process and the release of the injectable dose.

Qualified cleanroom technicians are often involved in the Pharmacy Kit filling under the supervision of the pharmacist or QP. As is required for pharmaceutical compounding or PET tracer production, training requirements and training records for all personnel need to be maintained and available for audit by the FDA or equivalent.

Equipment And Facility

The facility and all equipment need to have standard operating procedures (SOPs) that describe how equipment is used, maintained, and calibrated to comply with relevant legislation. Currently, almost all the filling of the Pharmacy Kit takes place within a compounding laminar flow hood or isolator (typically ISO 5). At some sites, the filling is conducted within a cleanroom, while at others, it is conducted in a dedicated non-cleanroom space, reflecting differences in cleanroom approach and specifications between regulators worldwide (71). Some equipment or facilities, such as the compounding hood or cleanroom, may require external certified laboratories for testing.

Material Handling

Material handling guidelines (69,70) require SOPs detailing a system to track all of the materials involved in the HP production process for a particular patient dose, similar to current good manufacturing practice (cGMP) requirements for material handling for drug compounding. This includes acceptance standards, storage conditions, amount used in the patient dose for each ingredient and materials used in the assembly of the fluid path and Pharmacy Kit. Currently some users choose to open and inspect and sometimes modify the Pharmacy Kits upon arrival, but some users keep them in the sealed packaging until they are required for dose preparation.

Pharmacy Kit Filling And Assembling

As required by an IND or its equivalent, the preparation of the doses of HP 13C agent are detailed in the Chemistry, Manufacturing, and Control (CMC) section of an applicable regulatory submission; an example of this has been made available (72). It describes the processes of filling the Pharmacy Kit with the different components that make up the final drug product, and of assembling the final kit for either storage or immediate use in the polarizer. Special attention should be given to the laser welding process in order to satisfy installation qualification (IQ) and operational qualification (OQ). Typically, the final developed process is validated by process qualification (PQ) runs, during which 3 or more Pharmacy Kits are filled and used and the final HP 13C products are tested for endotoxin and sterility and to confirm that they meet the dose specifications for injections (usually including pyruvate concentration, residual EPA concentration, pH, liquid state polarization level and dose temperature). The data from 3 consecutive PQ runs are submitted as part of the IND submission (or its equivalent), and are often also reviewed by the Institutional Review Board (IRB) where the studies are conducted.

Quality Control And Dose Release

The quality control (QC) and dose release can be separated into two aspects: one is the QC and release of the filled Pharmacy Kit, and second is the QC and release of the HP 13C agent for injection, after polarization and dissolution. For institutions filling a batch of kits and storing them to use over a period of time, typically the batch can be released based on initial validation, environmental monitoring data from the day of kit production, and if filters are used during preparation of any of the components, filter integrity testing. But in some cases one or more kits are used for validation before the batch of kits are released for future use. For institutions that fill only the kits required for specific studies shortly before the experiment, the filled kits often do not go through separate release tests before they are used.

The quality control of the HP 13C pyruvate solution post dissolution is primarily performed to ensure that the agent meets the dose specifications (Table 1) before it is administered to the subject. These specifications target both safety (pH, residual EPA, temperature) and efficacy (pyruvate concentration, polarization, volume). Typically, the pyruvate concentration, residual EPA concentration, pH, dose temperature, dose volume, and liquid state polarization are measured by the QC accessory associated with the SPINlab polarizer. Some users perform a secondary measurement for one of the parameters, such as pH, using a different instrument or pH paper. For sites that do not go through a separate release testing process for batch filled kits, the integrity of the sterilization assurance filter, a part of the Pharmacy Kit, is typically tested as a part of the dose release. It is also common for these users to preserve an aliquot of the final HP 13C pyruvate solution for post-release endotoxin and sterility testing. This testing cannot be completed fast enough to test an individual dose prior to injection, but this is why other processes such as PQ runs and validation testing are done to minimize the chance a subject could be injected with a contaminated dose.

The Final Dose Release And Injection

should be done under the supervision of a licensed professional, based on local regulations.

Some Key Challenges

Many of the challenges associated with HP 13C pyruvate preparation can be attributed to the conditions required for the dissolution-DNP method of high magnetic field (~3-7 T) and very low temperature (~1 K) during polarization, with pressurized and superheated water necessary for the rapid dissolution event. These extreme conditions are quite challenging for the design of the container-closure and fluid path system. In particular, the cryogenic temperature in the polarizer requires special attention to any moisture or ambient (moist) air introduced into that portion of the fluid path, which can form an ice block at ~1 K. This ice can lead to flow restriction during the dissolution event and reduce the strength of the laser welded bond between the cryovial and its cap. This can ultimately produce failures in the dissolution step, including variations in final pyruvate concentration and pH that may fail to meet QC release criteria as well as fluid path ruptures that provide no available dose and result in polarizer down-time.

The polarization of the HP 13C pyruvate sample decays quickly over the span of a few minutes after dissolution, and thus the process of dissolution, QC for release, and injection should be completed as fast as possible to preserve the high polarization level achieved. Any delays in the preparation process, such as transportation time or equipment malfunction, can significantly reduce the final polarization and result in lower quality imaging data.

Current Practices

A summary of data collected from all sites performing clinical trials with HP 13C-pyruvate is shown in Fig. 3 and Table 1, including the specification of the final dose and how the quality control and release of the final dose are performed. There is a split in the Production Style, described in the General Considerations section above, with 8/13 sites using Sterile Preparation versus 5/13 using Terminal Sterilization. While many of the dose specifications show notable differences in acceptable ranges, all of these variations listed in tables have been successfully and safely been used to perform HP 13C pyruvate studies in humans. Their differences depend on the institutions’ preferences, resources and their particular regulatory situation. There is high similarity in pyruvate ranges, temperature ranges, EPA limits, and volume limits. There is modest variability in pH ranges and large variability in the endotoxin test limit. There is a 3-fold difference in acceptable polarization levels, which are measured to ensure a futile dose is not injected since the polarization is directly proportional to SNR. This reflects the decision by several sites to believe that useful data can be still be obtained with suboptimal polarizations.

Figure 3: Hyperpolarized agent preparation methods reported by sites currently performing HP

In House

Table 1: HP 13C-pyruvate preparation parameters, methods, and dose specifications used for quality control testing and release as well as validation. These were obtained from a survey of all sites performing clinical trials with HP [1-13C]pyruvate. The parameters used for product release are noted in bold text, otherwise these parameters are measured for batch validation or other QC measurements. The endotoxin and sterility testing are performed during process validation of the batch and/or post-injection, and largely depends on the agent production approach.

Summary

The overall safety record of HP 13C-pyruvate has been very strong, and the SPINlab hyperpolarizer has proven to provide high polarizations at human sized doses while meeting numerous QC and release criteria. A weakness remains the failure modes of the SPINlab Phamacy Kits (e.g. ice blocks, path ruptures), which are placed under extreme requirements particularly during dissolution. The preparation process still requires a high degree of expertise.

Therefore, there is a significant need to improve the reliability, robustness, and ease of operation for generating HP 13C-pyruvate doses for human studies. Furthermore, there is a divide between manufacturing and sterile compounding style preparation as well as other site-specific practices, resulting in variations in SOPs and justification required to relevant regulatory bodies. There have also been no comparisons between these approaches. It is also unclear what release criteria and QC parameters are truly required to ensure patient safety.

However, all of the reported methods are acceptable and approved by the appropriate regulatory authorities, and have led to the rapid expansion of successful human studies in recent years.

Mri System Setup And Calibrations

This section covers the MRI system setup, including the imaging system, RF coils, phantoms, and prescan calibration methods.

Imaging System

The main prerequisite for a given MRI scanner to be capable of supporting studies with HP 13C is its “broadband” capability to transmit and receive radiofrequency (RF) signal at the frequency of 13C, which is around 4 times lower than 1H. This does not come as a default on clinical MR devices. The transmit power of the broadband amplifier should also be sufficient to support the intended flip angle and RF pulse shape with the employed transmission RF coil(s) for 13C. Most studies to date use relatively low flip angles (< 90 degrees) for HP 13C in order to preserve polarization for time-resolved imaging. The capability to receive 13C signal on multiple channels is also desirable to increase SNR, as discussed further in the “RF coils” section.

The choice of magnetic field strength is primarily dependent on the metabolites’ frequency separation due to chemical shift dispersion and 1H imaging. High field strengths do not enhance hyperpolarized 13C signal as they do for 1H because the signal strength in a HP experiment relies on manipulating the population of quantum energy states outside of the MRI scanner.

However, the injected HP 13C-pyruvate and its metabolic products have greater frequency separation at higher fields, and it may thus be easier to separate and quantify these resonances at higher fields. This comes at the cost of a reduction in the achievable T2* and often reduced T1. As the initial polarization is independent of the imaging field strength it has been proposed that the increased T2* at 1.5T can potentially be exploited to increase SNR by adapting the acquisition bandwidth or reduce off-resonance imaging effects in cases when the decay of the transverse magnetization is dominated by T2* (73). In practice, 3T has been used in all published human 13C-pyruvate studies surveyed (Supporting Table S1), and comprises the majority of scanners currently in use for human studies (Table 3). A field strength of 3T is well-suited for 1H MRI anatomical reference and correlative imaging.

Stronger and more rapidly slewing magnetic field gradients support more rapid spatial encoding, particularly for metabolite-specific single-shot imaging using echo-planar imaging (EPI) or spiral imaging (See “Acquisition and Reconstruction”). Although the spatial resolution acquired for HP 13C imaging is typically much coarser than for 1H MRI, the factor of ~4 in gyromagnetic ratio leads to the same reduction factor in performance of the gradient system, so 13C experiments are potentially more limited by gradient hardware performance. To date, all human studies have used the commercially-available integrated gradient systems provided in clinical MRI scanners.

Optimization of scanner design has understandably focused on minimization of artifacts in 1H MRI, where devices such as room lights, the gradient amplifiers, and the motors driving the patient bed are checked to ensure that they do not produce RF interference at the 1H frequency, but artifacts may arise at other frequencies. Eddy current compensation is also not always appropriately adjusted for nuclei at other frequencies (74). In order to optimize for 13C, many sites have performed checks on phantoms for RF interference, gradient artifacts, and eddy currents (74), including the use of post-hoc gradient impulse response function characterisation and correction, and some vendors have fixed these issues as well.

Rf Coils

For HP 13C imaging studies in humans, RF coils for both 1H and 13C nuclei are needed, with 1H MRI providing an anatomical reference for registration and optional additional multiparametric MRI readouts. At the Larmor frequency of 13C nuclei, the relative contributions from coil noise compared to sample noise increase compared to 1H (73,75), although sample noise still is likely the dominant contributor for human-sized coils at 32.1MHz - the resonance frequency of 13C nuclei at 3T.

The key requirement for human 13C-pyruvate RF coils are that the coil geometry and sensitive volume must cover the volume of interest in the subject. Table 2 and Figure 4 shows coil configurations that have been used and optimized for applications in different anatomic regions.

Volume resonators are most commonly used for transmit, as they surround the subject to

Provide B1 Transmit Across The Fov (B1

+). While 1H relies on a large birdcage (“body”) coil built into the scanner, 13C transmit coils must be placed inside the bore. This takes up valuable space within the magnet, and also has led to the use of designs with relatively inhomogeneous

B1

+. Many human studies have used Helmholz pair resonators for transmit, including the “clamshell coil”, which has a notably inhomogeneous B1

+ Profile But Has Been Used Because Of

relatively easy integration into the scanner bore. B1

+ Variation Results In Variations In The Flip

angles that control the use of the hyperpolarized magnetization and creates errors in common HP metrics (9,76). The exception are head coils, where birdcage designs with highly

Homogeneous B1

+ can be placed around the head while easily fitting inside the bore. As with 1H MRI, higher SNR can typically be achieved by smaller receive coil elements, such as surface coils or phased arrays, and the majority of 13C receive coils used have layouts similar to 1H phased arrays.

RF coil quality control is important to ensure proper functioning of the coils to provide consistent imaging quality, especially with limited natural abundance 13C signal in vivo. It typically involves 1) a physical integrity check of the coil cables and connectors and 2) phantom SNR tests to check the coil’s performance and to monitor it over time (see Phantoms below). An useful reference for RF coil quality control is outlined in the MRI accreditation program of the American College of Radiology (77) and can be adapted for 13C coils.

Notably, configurations for brain and prostate studies used dual-tuned 1H/13C coil designs, which greatly simplify workflow and registration of 1H and 13C images, as no switching of coils is needed.

(1)

Table 2: RF coil configurations reported for human HP [1-13C]pyruvate studies.

Tx = Transmit

coil, RX = receive coil. The commonly used “clamshell” TX coil is a Helmholz pair design. For 1H RF configurations, all used the Body coil for TX unless otherwise noted, and “repositioned” indicates the 13C coil was removed for 1H imaging. One representative reference is listed for each configuration. The RF coil configurations reported in the reviewed papers are shown in Supporting Table S1.

Figure 4: Examples of RF coil configurations used for human HP [1-13C]pyruvate brain studies. (A,B) 13C Clamshell TX (Helmholz pair) and 2× 4-channel paddle RX arrays. (C) 13C Birdcage volume TX and 32-channel RX array (RX array slides into TX coil). (D) 13C Birdcage volume TX and 24-channel RX array, combined with a 1H 8-channel RX array. Image reproduced with permission from Ref (16).

Phantoms

Since hyperpolarized magnetization is non-renewable, phantoms containing 13C nuclei are important to: 1) test the multi-nuclear capabilities of the imaging system, including all parts of the signal excitation and receive chain; 2) perform calibration measurements before a scan with hyperpolarized nuclei; and 3) perform necessary pre-scan adjustments (see “Prescan Calibration” section). The phantoms currently in use are listed in Table 3. Their composition must provide sufficient 13C signal, with additional considerations of conductivity, stability, chemical shift(s) present, potential for dynamic imaging, and cost. The phantom geometries are typically either compact, in order to be used alongside the subject during a HP scan, or large enough to mimic the inner volume of a RF coil for system testing.

One popular compact design contains enriched 13C-urea at high concentration, typically 8 M, which provides a single resonance, placed inside a small container ~1 mL. The most common recipe mixes 13C-urea in a 90% water/10% glycerol solution, with glycerol used to increase the urea solubility and doping with a Gd-based contrast agent to shorten T1 which increases the potential SNR per unit time. For example, when Dotarem is added at a 3:1000 volume ratio the 13C-urea T1 is around 500 ms and T2 is around 100 ms. However, when testing pulse sequences influenced by T1 and T2, doping should be used carefully. This phantom is suitable for frequency calibration, transmit gain calibration, sequence testing, and as a fiducial marker when placed next to a patient. However, enriched 13C-urea has a relatively high cost compared to natural abundance compounds.

For larger volumes (>100 ml), the phantoms most often used contain undiluted ethylene glycol, glycerol, or dimethyl silicone. These compounds have sufficiently high carbon concentrations to provide sufficient 13C signal even with the 1.1% natural abundance of 13C. These larger phantoms matching the inner volume of an RF coil are useful for coil testing, including transmit

+) And Receive (B1

-) coil profile mapping, as well as to mimic acquisitions using in vivo FOV requirements. In this case, size and conductivity should match the expected subject size in order to mimic coil loading and get a realistic estimation of B1+. Large-volume natural abundance urea phantoms have also been used by some sites, but suffer from higher conductivity compared to biological tissues. Typically, it is easier to increase the conductivity and hence coil loading of the non-conductive phantom by adding NaCl to match physiological loading (16,78).

Dynamic phantoms that aim to mimic metabolite kinetics have also been developed (79–81), and have the potential to more closely mimic the HP experiment, but so far these are not widely used.

Prescan Calibration

Prior to performing an MRI acquisition, the so-called prescan procedure is used to set the shim parameters to maximize B0 homogeneity over the field of view (FOV) or a specific region of interest (ROI), the scanner center frequency (CF), the RF transmit gain, and the receiver gain.

While this calibration procedure is usually automated for 1H, the lack of sufficient natural abundance 13C signal prevents use of automated methods. (Although natural abundance 13C lipid signal has been detected, there are so far no reports on using this signal for prescan.) Table 3 shows current practices across sites.

Maximizing B0 homogeneity is independent of the nucleus and is therefore performed prior to 13C imaging using the 1H water signal and existing shimming tools, such as by a standard automated process (“Auto Shimming”) or using high order shimming routines. Similarly, the 13C CF can be calculated from the 1H CF using a predetermined scaling factor that depends on the target chemical shift (82). Another common approach used is to have a small, high-concentration 13C phantom, e.g. 8M 13C-urea, integrated in the RF coil or placed next to the scan subject (1). The reference frequency can also be based on real-time measurements after the HP injection but prior to imaging (83). Both the CF and B0 shimming are critical when using spectrally-selective RF pulses, as inmetabolite-specific imaging methods, where the desired excitation bandwidths are typically very narrow and frequency offsets can lead to a failure mode that is only apparent after injection.

The calibration of the RF transmit power is typically performed on a small, high-concentration 13C phantom placed near the region of interest during the scan or on a large 13C phantom of similar size and coil loading as the subject, prior to the subject scan. Reference power is often done by sweeping the power in a pulse-acquire sequence (53,62), or the Bloch-Siegert method (52,84). When using a small phantom, the location of the phantom, B1

+ Inhomogeneity As Well

as any shielding effects, e.g., when the phantom is integrated into a coil (1), may degrade the accuracy. Other methods include real-time Bloch-Siegert method measurements after the HP injection (83), and using the stronger natural abundance 23Na signal that is close enough to the 13C resonance frequency to be detected by 13C coils (82).

The receiver gain is predetermined, either systematically based on independent phantom measurements and assuming the dose and polarization of the HP compound is known prior to injection, or based on past HP imaging studies.

Power [Kw]

Phantom(s) - during study Phantom(s) - before study 13C Frequency

8

13C-bicarbonate doped with dimethyl silicone, various

Power [Kw]

Phantom(s) - during study Phantom(s) - before study 13C Frequency

Maximum Values

Table 3: Summary of the imaging systems, phantoms, and prescan procedures used at sites currently performing HP 13C-pyruvate human studies. These were obtained from a survey of all sites performing clinical trials with HP [1-13C]pyruvate. *Previously performed studies with a Siemens 3T Tim Trio. The imaging systems, phantoms, and prescan procedures reported in the reviewed papers are shown in Supporting Table S1.

Summary

Commercially available 3T MRI systems are by far the most commonly used for human HP 13C-pyruvate studies, although a systematic investigation of the impact of B0 has only recently been investigated (73). The multi-nuclear RF transmit and receive chain has proven sufficient for current acquisition strategies, although many sites have observed artifacts due to RF interference, gradient interference, and residual eddy currents when operating at the 13C frequency. A variety of 13C RF coils, tailored for numerous anatomical targets, have been successfully demonstrated, with the main limitation that most transmit coils take up a lot of additional space inside the bore and provide relatively inhomogeneous B1

+ Profiles. The

phantoms used have converged into generally 2 categories - small phantoms containing 13C-enriched compounds that can be used during the study and human-sized phantoms containing compounds with high carbon concentrations but without 13C enrichment that are used to test and calibrate the coils. There are no standardized compositions or geometry, and dynamic phantoms that recapitulate in vivo kinetics would be desirable but are still an emerging area. Prescan calibration procedures were not well defined in most publications, so we surveyed individual sites to determine current practices. Calibration procedures for the B0 field (13C CF and shimming) for most sites take advantage of 1H signal and methods, while methods

For Calibration Of B1

+ is more variable across sites, likely a reflection of remaining challenges in how to perform this calibration. Standardization of both phantoms and calibration procedures would synergistically improve the robustness and reproducibility of HP 13C studies.

Acquisition And Reconstruction

Data acquisition strategies in human HP [1-13C]pyruvate MRI studies must account for multiple chemical shifts, efficiently utilize the non-renewable HP magnetization, and acquire data quickly relative to metabolism and relaxation decay processes. These studies require spectral encoding to separate metabolites, necessitating pulse sequences that efficiently encode up to 5D data (3 spatial + 1 spectral + 1 temporal dimension). RF pulses must efficiently sample without immediately saturating the non-renewable HP magnetization, and sequences must acquire data quickly and be robust to both experimental and physiologic variation (e.g. B1

+ Inhomogeneity,

variation in perfusion) to ensure reproducibility and minimize scan-to-scan variability. This section covers current successful practices for data acquisition in human [1-13C]pyruvate studies, and accompanying 1H imaging, from different anatomic regions, including scan parameters and image reconstruction.

Acquisition And Reconstruction Methods

The acquisition methods used in human [1-13C]pyruvate studies can be classified into 3 categories: 1) MR spectroscopy or MR spectroscopic imaging (“MRS/I”), 2) chemical shift encoding methods, and 3) metabolite-specific imaging (Fig. 5).

Mrs/I Methods Specifically

resolve a spectrum that can be analyzed to extract expected as well as unexpected resonances, making this approach very robust. It was used in many initial studies (1).

Chemical Shift

encoding methods, most commonly the Iterative Decomposition of water and fat with Echo Asymmetry and Least-squares estimation (IDEAL) method, use imaging sequences acquired with multiple TEs and rely on a model-based separation of expected chemical shifts (85).

Metabolite-specific imaging methods use specialized RF pulses that are spatially and spectrally selective to excite individual metabolites which are then typically imaged with fast k-space trajectories such as echo planar imaging (EPI) or spirals (86).

Their Application To Different

organ systems is described below. The image reconstruction methods used in human [1-13C]pyruvate studies have typically been conventional methods (e.g. FFT, non-uniform FFT, or equivalent). The incorporation of accelerated imaging and advanced reconstruction methods including parallel imaging (4,57,87) and compressed sensing (7) has also been applied in human studies for improved spatial resolution, temporal resolution and coverage, but have the potential for additional artifacts as well as SNR losses due to ill-conditioning of the reconstruction (e.g. g-factor).

The Majority Of

published studies do not use accelerated imaging indicating the resolution and coverage achievable without acceleration is currently adequate for successful data collection. Performing coil combination, even with fully sampled data has also been shown to have specific challenges for HP human images: using naive sum-of-squares methods suffer from high noise amplification in the relatively low SNR regime of HP [1-13C]pyruvate (compared to 1H), motivating several HP 13C-specific methods that include data-driven coil sensitivity estimation which have shown obvious improvements over sum-of-squares (11).

More recently denoising techniques have been applied as post-processing of human HP data(41,42,44). The techniques applied are based on spatial-temporal singular value decomposition for unsupervised estimation of signal and noise components. They have shown improvements in apparent SNR in the brain and liver, while care must be taken to choose parameters such as the rank threshold to avoid oversmoothing and overfitting to the estimated signal components.

Prostate Studies

Prostate cancer was the first human application of HP [1-13C]pyruvate (1), and data was acquired with MRS/I methods: 1D dynamic MRS, single-slice 2D dynamic echo-planar spectroscopic imaging (EPSI), and single time point 3D EPSI. Advances in imaging strategies led to the development and application of new acquisition schemes, including undersampled 3D EPSI with compressed-sensing (7), model-based chemical shift encoding methods that use a priori information (47,59), and metabolite-specific EPI (10), all of which can provide volumetric whole-organ coverage and dynamic acquisitions.

The pyruvate bolus arrival in the prostate can vary by ± 10 s between patients, necessitating dynamic imaging to reliably and consistently capture the pyruvate bolus (18). For this reason, all currently ongoing studies acquire dynamic data. While MRS/I, chemical shift encoding, and metabolite-specific imaging can all achieve dynamic imaging, chemical shift encoding and metabolite-specific imaging provide greater dynamic and volumetric coverage (85). For scan prescriptions, the FOV is designed to provide full prostate coverage and typically to match the orientation of the anatomic imaging used for registration. Flip angles used in current studies are constant through time, as quantification with a variable-through-time flip scheme is highly sensitive to bolus timing (8) and errors in the RF transmit (B1 +) field (76).

Heart Studies

Data acquisition methods for 13C imaging in the heart must be designed to meet the demands of significant cardiac motion and blood flow. To cope with the periodic cardiac motion, most human heart studies to date used gating to the diastolic window, the longest cardiac cycle interval, which has reduced motion (2,22,28,30,35,36,38,45,52). The duration of the diastolic window limits the available data sampling time, making cardiac acquisitions the most time-constrained of the HP 13C MRI applications. The most common acquisition approach is metabolite-specific imaging with spiral k-space trajectories (2). Their single-shot imaging capability makes these methods particularly robust to motion effects. Furthermore, spiral k-space trajectories provide rapid k-space coverage and relatively benign flow and motion artifacts. The majority of studies have used 2D multi-slice acquisitions, but 3D encoding has also been used successfully (35).

Brain Studies

For HP 13C MRI of the human brain, the majority of studies have also used 2D (slice selective) acquisitions (10–12,14,16,28,33,40,41,44,51,53,60), with a trend toward volumetric coverage using 2D multi-slice metabolite-specific imaging. 3D metabolite-specific imaging of the whole brain, with phase encoding of the slice direction (34,57), has been shown to provide similar SNR efficiency (88) compared with multislice imaging. A number of studies have employed MRS/I (5,6,29,31–33,50,55) resulting in a spectrum from each voxel, which has the advantage of not requiring a priori information about which peaks to encode. This was important in early brain studies when it was not known which peaks would be detectable. Chemical shift encoding, using a set of images with different echo times and an iterative reconstruction of the individual resonances (i.e. the IDEAL approach (85)), has also been used (12,49,54), with the drawback that coverage in the slice direction was limited due to the time required to acquire multiple echo time images.

Abdomen And Breast Studies

The fundamental approaches to data acquisition and reconstruction in the abdomen and breast are largely similar to the aforementioned applications, but demand attention to particular challenges associated with these anatomic regions, especially relating to respiratory motion.

Although it has been shown that a basic 2D MRSI approach based on phase encoding and FID readout can be successfully applied for HP 13C imaging in breast (15) and kidney (13), major advantages in terms of spatiotemporal resolution and coverage have been realized using tailored approaches based on metabolite-specific imaging (43,62) and chemical shift encoding (43), which have facilitated multi-slice or 3D dynamic acquisitions over large FOVs in the abdomen (4,37,46).

The significant respiratory motion encountered in these regions can directly blur 13C images, and has further favored these rapid acquisition strategies. Motion also degrades B0 homogeneity, which can shift frequency-selective excitation profiles and introduce artifacts into rapid imaging readouts. This makes accurate determination of the acquisition center frequency and shimming essential in these regions which often cover large FOVs. (See “Prescan Calibration” section for more information). In some studies, breath-holding was used to minimize motion effects and enforce frame-to-frame data consistency (42). A pragmatic and reasonably effective approach for dealing with respiratory motion during 13C data acquisition is an initial breath-hold (as long as can be tolerated), followed by free-breathing (46,62).

1H Imaging

Collection of 1H imaging data is essential both for prescribing the 13C acquisition and for interpretation of the resulting 13C data. Multi-planar 1H scouts are acquired prior to 13C acquisition to enable graphical prescription of the 13C imaging region. All human HP 13C-pyruvate imaging studies acquire conventional MRI scans (e.g. T1- and T2-weighted volumes) for anatomic reference, aiming to cover at least the full 13C FOV. Acquiring these anatomic scans as close as possible to the time of 13C imaging (immediately before or after) minimizes potential misregistration between the data sets. Depending on the application, other advanced 1H sequences are also acquired (e.g. diffusion-weighted imaging for cancer imaging).

When contrast-enhanced data is acquired, it is done after 13C imaging, as paramagnetic contrast agents will accelerate 13C relaxation.

Reported Study Parameters

Figures 5 and 6, and Supporting Table S2 shows the reported acquisition study parameters for human HP [1-13C]pyruvate studies published as of September 2022. Figure 5 shows a mixture of MRS/I, metabolite-specific imaging, and chemical shift encoding methods have been successfully used, where spectroscopy-based methods have become less prevalent in recent studies. Figure 6 shows the acquisition timing, including the important start time and interval/temporal resolution, is quite variable across studies.

Figure 5: Acquisition methods used in published HP [1-13C]pyruvate human studies published up to September 2022, classified into: MR spectroscopy and spectroscopy imaging (MRS/I); chemical shift encoding methods, such as IDEAL, that use multiple TEs and model-based reconstructions; and metabolite-specific imaging methods that use spectrally-selective excitation to image a single resonance at a time.

Figure 6: Temporal acquisition characteristics reported in HP [1-13C]pyruvate human studies published up to September 2022. (a) Reported referencing of acquisition start times.

(B)

Acquisition start times reported when using dynamic imaging and when timing was reported relative to the end of the injection. (c) Temporal resolutions. “Not Applicable” indicates dynamic imaging was not used.

Summary

Three general categories of acquisition strategies have been used successfully for human HP 13C-pyruvate studies: MRS/I, model-based chemical shift encoding (e.g. IDEAL) methods, and metabolite-specific imaging methods. These have enabled successful studies in the prostate, heart, brain, abdomen, and breast. Recent studies increasingly have used the imaging-based strategies of metabolite-specific imaging and chemical shift encoding which are the fastest methods, although a heads-to–head comparison between techniques has not been performed.

Metabolite-specific imaging is quite popular because of its speed and compatibility with single-shot imaging, but is sensitive to B0 field variations and thus requires careful calibrations. Nearly all studies surveyed acquired data dynamically, allowing measurement of the bolus and metabolite kinetics. The exact timings and associated flip angles vary quite widely across reported studies, with no consensus yet as to how to choose these parameters. Image reconstruction is typically done directly using Fourier Transform methods, and accelerated imaging strategies are uncommon.

Data Analysis And Quantification

This section covers the analysis of data from human HP [1-13C]pyruvate studies, including modeling and metrics, visualization, as well as considerations for how to store data and metadata. Depending on study design, the analysis may need to give quantitative or semi-quantitative output reflecting a biological process or may just reflect a contrast between different regions of interest for quantitative evaluation.

Metrics

Figure 7: HP [1-13C]pyruvate raw data (A) have typically been quantified using four categories of metrics depending on the acquisition. Data acquired as a single time point are often quantified using normalized metabolite images or metabolite ratios (B). Dynamic data can be quantified using normalized metabolite images or metabolite ratios (B), or with metabolite timings such as time-to-peak (TTP) or pharmacokinetic (PK) models (C). The latter two require the data to be time-resolved. [1-13C]alanine and 13C-bicarbonate are analyzed similarly to [1-13C]lactate but omitted here for display.

Metabolite images are commonly used as summary metrics for HP MRI data, often including some form of normalization as well as summed over time as an area under the time curve (AUC) (17). These are analogous to the visual evaluation that is most used for routine clinical work (89,90). In these metabolite images, we expect that the [1-13C]pyruvate AUC signal is predominantly weighted towards perfusion and uptake, while [1-13C]lactate, [1-13C]alanine and 13C-bicarbonate AUCs represent metabolic conversion. The strength of this approach lies in its simplicity and relatively few underlying assumptions. Limitations to the use of single-metabolite images or AUCs include sensitivity to inhomogeneous coil profiles (57,87,91), the acquisition strategy and acquisition parameters, pyruvate polarization and concentration level, and signal relaxation rates (92). Further, the reader must be careful to interpret all the images in conjunction to better understand the underlying biology; for example, increased [1-13C]lactate in the presence of decreased [1-13C]pyruvate delivery can have a very different meaning compared to increased [1-13C]lactate with increased [1-13C]pyruvate delivery.

In an attempt to address variations in coil sensitivity, polarization level, and pyruvate delivery, AUC images are often computed by normalizing to a specified parameter, such as the maximum pyruvate or average lactate signals, or presented as a ratio such as lactate/pyruvate or divided by “total Carbon” - the sum total of HP 13C signal observed across all metabolites. The AUC ratios between metabolites and pyruvate are proportional to the corresponding forward kinetic rates (81,93), but are not directly comparable to rate constants when magnetization loss rates (e.g. relaxation and losses due to signal excitation) differ between studies. Similarly, the ratios between the produced metabolites (e.g. bicarbonate/lactate) can reflect the balance between downstream metabolic pathways (12,55). Care must be taken to consider how AUC images are calculated and normalized before comparing values between studies.

To further quantify the interpretation, pharmacokinetic (PK) modeling approaches were developed to compute the apparent kinetics of pyruvate-to-metabolite exchange (92,94–99). These yield semi-quantitative to quantitative apparent rate constants, given in s-1. Some models require a vascular input function, while others avoid this requirement (95). PK models can explicitly account for acquisition-specific details such as excitation angle and repetition time, and thus may reduce the effects of these details on quantification. An input-less model, provided in the Hyperpolarized-MRI-Toolbox (https://github.com/LarsonLab/hyperpolarized-mri-toolbox) (100) and thus frequently employed for human data, has been shown to fit well and robustly to prostate and brain data (8,20). PK models are quantitative in nature, arguably provide more relevant biological information (8,20), and appear to be reproducible across sites (51). However, rate constants derived from PK models are still apparent rates, and likely do not reflect a single biological characteristic.

Some additional considerations include whether complex or magnitude data is used, as the noise behaviors will impact the analysis differently. Additionally, cut-off thresholds or other criteria may be used to identify and avoid voxels with insufficient SNR before analysis to improve robustness (20,41).

Regardless of the analysis approach, the underlying biology is not always clearly represented by the data; instead, the metrics may be influenced by perfusion, barrier permeability, intercellular shuttles, enzyme activities, co-substrate concentrations, or combinations thereof, depending on the organ and disease of interest (19,43,94,101–103). This may be addressed by incorporating complementary information. As an example, HP 13C pyruvate data is influenced by perfusion, and thus addition of perfusion MRI could be important for interpretation (98,104,105).

All the methods outlined above have been explored in clinical studies, described in Supporting Table 3 and summarized in Figure 8. As of September 2022, approximately 52% of studies involving human subjects report rate constants derived from a PK model with a few different models reported. A nearly equal fraction (51%) of the studies report AUC ratio values.

Approximately 66% of these studies report metabolite-specific images or AUC values. About 40% report SNR values; this metric is particularly frequent in manuscripts that describe technical developments for clinical HP MRI. Approximately 16% of these studies summarize model-free metrics, and 10% report measurements from a single timepoint. Most studies report a combination of quantities.

Figure 8: Reported metrics used for analysis in HP [1-13C]pyruvate human studies published up to September 2022.

Visualization

A wide variety of approaches have been used for visualizing data from human HP 13C-MRI studies. The challenges and practical considerations are: 1) choosing the appropriate metrics to display, 2) how to encode the parameters (e.g. the colormap), and 3) choosing how to provide anatomical context and other multi-parametric data. The choice of visualization also depends on the goal which could be for diagnostic interpretation, but also quality control, reproducibility among readers and publication.

Metrics

The choice of HP 13C metrics is described in detail above. At this stage in HP 13C development where there is no standardized metric, often a combination of metabolite images and ratios or PK model parameters are shown.

Parameter Encoding

The mapping function chosen should provide an adequate, often quantitative, impression of the parameter mapped. There is a consensus in the visualization field that perceptually uniform maps are best suited to visualize continuous parameters, like the greyscale typically used by radiologists as well as other monochrome (black to blue) and color ranges (fire-type, rainbow-type) (106,107). Multi-color heatmaps have been the most frequently employed method for HP 13C data, while greyscale has infrequently been used but it ensures there is no coloring-based bias as well as facilitating later reuse (Fig. 9a). Among the color schemes employed in the clinical HP 13C literature, fire-type scheme seems to be the most common [similar to “Plasma” or “Inferno” in matplotlib.org]. Next most commonly employed is the rainbow-type scheme [similar to “Rainbow” in matplotlib.org].

Anatomical Context

HP MRI faces the challenge that it does not necessarily depict the anatomical features, similar to PET, and thus requires an anatomical reference. Most often, a grayscale anatomical image is overlaid with a HP colormap (Fig. 9c,d). This approach is very intuitive, but can skew perception as the grey-scale anatomical reference may affect the brightness of the HP data (e.g. signal in the skull). This bias does not occur when showing adjacent maps (Fig. 9a, b). Here, anatomical outlines may help to provide reference (Fig. 9b).

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