Delamination Analysis Ansys

Abstract

Isogeometric cohesive elements are presented for modeling two and three dimensional delaminated composite structures. We exploit the knot insertion algorithm offered by NURBS (Non Uniform Rational B-splines) to generate cohesive ele- ments along delamination planes in an automatic fashion. A complete computational framework is presented including pre-processing, processing and post-processing. They are explained in details and implemented in MIGFEM–an open source Matlab Isogemetric Analysis code developed by the authors. The composite laminates are modeled using both NURBS solid and shell elements. Several two and three dimensional examples ranging from standard delamination tests (the mixed mode bending test), the L-shaped specimen with a fillet, three dimensional (3D) double cantilever beam and a 3D singly curved thick-walled laminate are provided. To the authors’ knowledge, it is the first time that NURBS-based isogeometric analysis for two/three dimensional delamination modeling is presented. For all examples considered, the proposed framework outperforms conventional Lagrange finite elements.

isogeometric analysis (IGA), B-spline, NURBS, finite elements (FEM), CAD, delamination, composite,

Ntroduction

Isogeometric analysis (IGA) was proposed by Hughes and his co-workers in 2005 to reduce the gap between Computer Aided Design (CAD) and Finite Element Analysis (FEA). The idea is to use CAD technology such B-splines, NURBS (Non Uniform Rational B-splines), T-splines etc. as basis functions in a finite element (FE) framework. Since this seminal paper, a monograph has been published entirely on the subject and applications have been found in several fields including structural mechanics, solid mechanics, fluid mechanics and contact mechanics. It should be emphasized that the idea of using CAD technologies in finite elements is not new. For example in , B-splines were used as shape functions in FEM and subdivision surfaces were adopted to model shells .

Due to the ultra smoothness provided by NURBS basis, IGA has been successfully applied to many engineering problems ranging from contact mechanics, see e.g., [5, 6, 7, 8, 9], optimisation problems [10, 11, 12, 13], structural mechanics [14, 15, 16, 17, 18, 19, 20], structural vibration [21, 22, 23, 24], to fluids mechanics [25, 26, 27], fluid-structure interaction problems [28, 29]. In addition, due to the ease of constructing high order continuous basis functions, IGA

Arxiv:1305.2738V1 [Math.Na] 13 May 2013

has been used with great success in solving PDEs that incorporate fourth order (or higher) derivatives of the field variable such as the Hill-Cahnard equation , explicit gradient damage models and gradient elasticity . The high order NURBS basis have also found potential application in the Kohn-Sham equation for electronic structure modeling of semiconducting materials . We refer to for an overview of IGA and its implementation aspects.

In the context of fracture mechanics, IGA has been applied to fracture using the partition of unity method (PUM) to capture two dimensional strong discontinuities and crack tip singularities efficiently [35, 36]. In an explicit isogeometric enrichment technique is proposed for modeling material interfaces and cracks exactly. Note that this method is contratry to PUM-based enrichment methods which define the cracks implicitly. A phase field model for dynamic fracture has been presented in where adaptive refinement with T-splines provides an effective method for simulating fracture in three dimensions. There are, however, only a few works on cohesive fracture in an IGA framework . The method hinges on the ability to specify the continuity of NURBS/T-splines through a process known as knot insertion. Highly accurate stress fields in cracked specimens were obtained with coarse meshes.

Delamination or interfacial cracking between composite layers is unarguably one of the predominant modes of failure in laminated composite. This failure mode has therefore been widely investigated both experimentally and numerically. Delamination analyses have been traditionally performed using standard low order Lagrange finite elements, see e.g., [40, 41, 42, 43] and references therein. The two most popular computational methods for the analysis of delamination are the Virtual Crack Closure Technique (VCCT) [44, 45] and interface elements with a cohesive law (also known as decohesion elements) [40, 41, 42]. The latter is adopted in this contribution for it can deal with initiation and propagation of delamination in a unified theory. The Element Free Galerkin, which is a meshfree method, with the smooth moving least square basis was also adopted for delamination analysis . In order to alleviate the computational expense of cohesive elements, formulations with enrichment of the FE basis was proposed in [47, 48]. The extended finite element method (XFEM) have been adopted for delamination studies e.g., [50, 51, 52] which makes the pre- processing simple for the delaminations can be arbitrarily located with respect to the FE mesh. The interaction between the delamination plane and the mesh is resolved during the solving step by using enrichment functions. More recently, in high order B-splines cohesive FEM with C0 continuity across element boundary were utilized to efficiently model delamination of two dimensional (2D) composite specimens. In the referred paper, it was shown that by using high order B-spline (order of up to 4) basis functions, relatively coarse meshes can be used and 2D delamination benchmark tests such as the MMB were solved within 10 seconds on a laptop.

In this manuscript, prompted by our previous encouraging results reported on and the work in , we present an isogeometric framework for two and three dimensional (2D/3D) delamination analysis of laminated composites. Both the geometry and the displacement field are approximated using NURBS, therefore curved geometries are exactly represented. We use knot insertion algorithm of NURBS to duplicate control points along the delamination path.

Meshes of zero-thickness interface elements can be straightforwardly generated. The proposed ideas are implemented in our open source Matlab IGA code, MIGFEM4, described in . Several examples are provided including the mixed mode bending test, a L-shaped curved composite specimen test [54, 55], 3D double cantilever beam and a 3D singly curved thick-walled laminate. Moreover, isogeometric shell elements are used for the first time, at least to the authors’ knowledge, to model delamination. Our findings are (i) the proposed IGA-based framework reduces significantly the time being spent on the pre-processing step to prepare FE models for delamination analyses and (ii) from the analysis perspective, the ultra smooth high order NURBS basis functions are able to produce highly accurate stress fields which is very important in fracture modeling. The consequence is that relatively coarse meshes (compared to meshes of lower order elements) can be adopted and thus the computational expense is reduced.

The remainder of the paper is organized as follows. Section 2 briefly presents NURBS curves, surfaces and solids. Section 3 is devoted to a discussion on knot insertion and automatic generation of cohesive interface elements followed 4available for download at https://sourceforge.net/projects/cmcodes/ by finite element formulations for solids with cohesive cracks given in Section 4. Numerical examples are given in Section 5. Finally, Section 6 ends the paper with some concluding remarks.

Nurbs Curves, Surfaces And Solids

In this section, NURBS are briefly reviewed. We refer to the standard textbook for details. A knot vector is a sequence in ascending order of parameter values, written Ξ = {ξ1, ξ2, . . , ξn+p+1} where ξi is the ith knot, n is the number of basis functions and p is the order of the B-spline basis. Open knots are used in this manuscript.

Given a knot vector Ξ, the B-spline basis functions are defined recursively starting with the zeroth order basis

(2)

This is referred to as the Cox-de Boor recursion formula. Figure 1 illustrates some quadratic B-splines functions defined on an open non-uniform knot vector. Note that the basis functions are interpolatory at the ends of the interval thanks to the use of open knot vectors and also at ξ = 4, the location of a repeated knot where only C0-continuity is attained. Elsewhere, the functions are C1-continuous. The ability to control continuity by means of knot insertion is particularly useful for modeling discontinuities such as cracks or material interfaces as will be presented in this paper. In general, in order to have a C−1 continuity at a knot, its multiplicity must be p + 1.

−1

Figure 1: Quadratic (p = 2) B-spline basis functions for an open non-uniform knot vector Ξ = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5}. Note the flexibility in the construction of basis functions with varying degrees of regularity.

(3)

where Ni,p(ξ) denotes the ith B-spline basis function of order p and wi are a set of n positive weights. Selecting appropriate values for the wi permits the description of many different types of curves including polynomials and circular arcs. For the special case in which wi = c, i = 1, 2, . . , n the NURBS basis reduces to the B-spline basis. Note that for simple geometries, the weights can be defined analytically see e.g., . For complex geometries, they are obtained from CAD packages such as Rhino .

Given two knot vectors (one for each direction) Ξ = {ξ1, ξ2, . . , ξn+p+1} and H = {η1, η2, . , ηm+q+1} and a control net Bi,j ∈Rd, a tensor-product NURBS surface is defined as

(7)

Derivatives of the B-splines and NURBS basis functions can be find elsewhere e.g., [1, 2].

Knot Insertion

It should be emphasized that knot insertion does not change the B-spline curves or surfaces geometrically but a direct influence on the continuity of the approximation where knots are repeated. Let us consider a knot vector defined by Ξ = {ξ1, ξ2, . . , ξn+p+1} with the corresponding control net denoted by B. A new extended knot vector given by ¯Ξ = {¯ξ1 = ξ1, ¯ξ2, . , ¯ξn+m+p+1 = ξn+p+1} is formed where m knots are added. The n + m new control points ¯Bi are

(9)

Considering a quadratic B-spline curve with knot vector Ξ = {0, 0, 0, 0.5, 1, 1, 1} and control points as shown in Fig. 2 (left). On the right of the same figure, two new knots ξ = 0.25 and ξ = 0.75 were added. Consequently, two new control points were formed. Although the curve is not changed geometrically and parametrically, the basis functions are now richer and may be more suitable for the purpose of analysis.

Figure 2: Knot insertion on a quadratic B-spline curve. The curve is not changed geometrically. Control points are denoted by filled green circles. Points corresponding to the knot values are denoted by red circles. These points divide the curve into segments or elements from an analysis standpoint.

Let us now consider a quadratic B-spline defined using Ξ = [0, 0, 0, 1, 1, 1]. The three basis functions for this curve are given in Fig. (3a). Now suppose that we need to have a discontinuity at ξ = 0.5. This can be achieved by inserting a new knot ¯ξ = 0.5 three (= p + 1) times. The new knot vector is then given by Ξ′ = [0, 0, 0, 0.5, 0.5, 0.5, 1, 1, 1] and the new basis functions are shown in Fig. (3b). Let us build a B-spline curve with the control net defined by B as shown in Eq. (10). The new control net that is defined by B′ is also given in Eq.

= B′

4. The B-spline curve corresponds to the original and new basis is the same

And Given In Fig. (3C). Imagine Now That Point B′

4 slightly moves vertically, the resulting B-spline curve with a strong discontinuity at x = 0.5 is plotted in Fig. (3d). This technique of inserting knot values p + 1 times was used in to model the decohesion of material interfaces. The application of this method in two/three dimensions resemble the usage of zero-thickness interface elements by doubling nodes in the FE framework.

We demonstrate the technique to generate a discontinuity into a NURBS surface by a simple example. The studied surface is a square of 10 × 10 and suppose that one needs a horizontal discontinuity line in the middle of the square

The Original Position

Figure 3: p + 1 times knot insertion for a quadratic B-spline curve to introduce a C−1 discontinuity at ξ = 0.5. as shown in Fig. (4a). The coarsest mesh consists of one single bi-linear NURBS element with Ξ = H = {0, 0, 1, 1} and p = q = 1. To insert the desired discontinuity, the following steps are performed: (1) perform order elevation to p = q = 2; (2) perform knot insertion for H , the new knot is H = {0, 0, 0, 0.5, 0.5, 0.5, 1, 1, 1} (Fig. (4b)); and (3) perform knot insertion to refine the mesh if needed. In Fig. (4c,d) the duplicated control points were moved upward to show the effect of discontinuity. In order to use these duplicated nodes in a FE context, one can put springs connecting each pair of nodes or employ zero-thickness interface elements. In this manuscript the latter is used. With a small amount of effort, the connectivity matrix for the interface elements can be constructed using a simple Matlab code as given in Listing 1. It is obvious that, due to the simplification made in line 2 of Listing 1, this code snippet applies only for a horizontal/vertical discontinuity line. However, it is straightforward to extend this template code to general cases by changing line 2. Such refinements are certainly problem dependent and hence not provided here. We refer to Fig. (5) for one example of a curved composite panel made of two plies.

Listing 1: Matlab code to build the element connectivity for 1D interface elements

C

Figure 4: Example of introducing a horizontal discontinuity in a NURBS surface.

N

Figure 5: L-shaped composite sample of two plies with a fillet modeled with a bi-quadratic NURBS: red circles denote duplicated nodes. For this case, it suffices to find the index of node S–the first node on the discontinuity curve. By virtue of the tensor-product nature of NURBS, the indices of other discontinuity nodes can then be found with ease.

= Lowernodes ( S C T R ) ;

iElements ( i , p+2:end) = upperNodes ( s c t r ) ;

End

Listing 2: Matlab code to build the element connectivity for 2D interface elements

= Lowernodes ( Ielements ( E , : ) ) ;

iElements ( e , ( p+1)∗(q+1)+1:end) = upperNodes ( iElementS ( e , : ) ) ;

End

The technique introduced so far can be straightforwardly extended to three dimensions, see Listing 2 and Fig. (6) for an example. The discontinuity surface lies in the X −Y plane. Line 7 of this Listing builds the element connectivity array for a 2D NURBS mesh, we refer to for a detailed description of these Matlab functions. These pre-processing techniques are implemented in our open source Matlab IGA code named MIGFEM, desribed in , which is available at https://sourceforge.net/projects/cmcodes/. In order to support IGA codes which are based on the B´ezier extraction [58, 59], see also Section 4.5, MIGFEM computes the 1D, 2D and 3D B´ezier extractors. In summary the pre-processing code writes to a file with (1) coordinates of control points (including duplicated ones), (2) connectivity of continuum elements, (3) connectivity of interface elements, (4) 2D/3D B´ezier extractors for continuum elements and (5) 1D/2D extractors for interface elements. It should be emphasized that inserting interface elements into a Lagrange FE mesh is a time-consuming task even with commercial FE packages. Due to that fact, a free mesh generator for cohesive modeling was developed by the first author and presented in .

Dis

Figure 6: A 3D bar with a discontinuity surface in the middle: modeled by a tri-quadratic NURBS solid. Remark 3.1. In the proposed framework, interface elements are inserted a priori, therefore delaminations only grow along predefined paths.

For laminates built up by plies of unidirectional fiber reinforced composites, the fracture toughness of the plies is much greater than the fracture toughness of the ply interfaces. Therefore, delaminations only grow along the ply interfaces which are known a priori. And that justifies our assumption.

Sogeometric Analysis

According to the IGA the field variable (which is, in this paper, the displacement field) is approximated by the same B-spline/NURBS basis functions used to exactly represent the geometry. Therefore, in an IGA context, one writes for

(11B)

where xI are the nodal coordinates, uiI is the i (i = 1, 2, 3) component of the displacement at node/control point I and NI denotes the shape functions which are the B-spline/NURBS basis functions described in Section 2.

Phy

Figure 7: Definition of domains used for integration in isogeometric analysis. Elements are defined in the parametric space as non-zero knot spans, [ξi, ξi+1] × [ηj, ηj+1] and elements in the physical space are images of their parametric counterparts.

Elements are defined as non-zero knot spans, see Fig. (7), which are elements in the parameter space (denoted by ˆΩe). Their images in the physical space obtained via the mapping, see Eq. (11), are called elements in the physical space (denoted by Ωe) that resemble the familiar Lagrange elements. From our experiences, it is beneficial to work with elements in the parameter space. Numerical integration is also performed on a parent domain as in Lagrange FEs.

Fe Discrete Equations

The semi-discrete equation for a solid with cohesive cracks is given by

(12)

where a is the acceleration vector, M denotes the consistent mass matrix, f ext is the external force vector , the internal force vector is denoted as f int and the cohesive force vector f coh. The elemental mass matrix, external and internal force vectors are computed from contributions of continuum elements and given by

(15)

where ρ is the density, Ωe is the element domain, Γe t is the element boundary that overlaps with the Neumann boundary, b and ¯t are the body forces and traction vector, respectively. The shape function matrix and the strain-displacement matrix are denoted by N and B; σ is the Cauchy stress vector.

The cohesive force vector is computed by assembling the contribution of all interface elements. It is given by for an

(16)

in which tc denotes the cohesive traction, Nint represents the shape function matrix of interface elements. The subscripts +/- denote the upper and lower faces of the interface element. The displacement of the upper and lower faces of an interface element, let say the first element in Fig. (8)-left read

(17)

with NI (I = 1, 2, 3, 4) are the quadratic NURBS shape functions. Figure 8 also explains the difference between Cp−1 and C0 high order elements–for the same number of elements, Cp−1 meshes have less nodes. We refer to for more information on this issue. The latter was used in with B-spline basis for 2D delamination analysis.

Having defined the displacement of the upper and lower faces of the interface, it is able to compute the displacement

(19)

The displacement jump will be inserted into a cohesive law (or traction-separation law) to compute the corresponding traction tc. We refer to [40, 41, 42] and references therein for other aspects of interface cohesive elements.

The

implementation for three dimensional problems i.e., 2D interface elements is straightforward, for example in Eq. (17),

N2

Figure 8: Illustration of Cp−1 NURBS interface elements (left) and C0 NURBS interface elements (right): For Cp−1 ele- ments, the connectivity of the first element is [1, 2, 3, 5, 6, 7] while the connectivity of the second element is [2, 3, 4, 6, 7, 8]. instead of using univariate NURBS basis one uses bivariate basis NI(ξ, η).

Ohesive Laws

In this work, we adopt the damage-based bilinear cohesive law developed in [61, 62]. This is a cohesive law in which the fracture toughness is a phenomenological function, rather than a material constant, of mode mixity as formulated by Benzeggagh and Kenane . Herein we briefly recall the cohesive law of which implementation details can be found in . Denoting d as the damage variable (0 ≤d ≤1), the cohesive law reads in the local coordinate system attached

(20)

where K is the dummy stiffness. The damage variable d is a function of the equivalent displacement jump, the onset

[[U]]0

eq and the propagation equivalent displacement jump [[u]]f

Eq, Mode I

and II fracture toughness GIc, GIIc, the mode mixity and η which is a curve fitting value for fracture toughness tests performed by Benzeggagh and Kenane .

Numerical Integration

In this manuscript, full Gaussian integration schemes are used. Precisely, for 2D solid elements of order p × q, a (p + 1) × (q + 1) Gauss quadrature rule is adopted and for cohesive elements of order p, a (p + 1) Gauss scheme is utilized. A similar rule was used for 3D solid elements and 2D cohesive elements.

Mplementation Aspects

There are at least two approaches to incorporating IGA into existing FE codes–with and without using the B´ezier extraction. The former, which relies on the B´ezier decomposition technique, was developed in [58, 59] and provides data structures (the so-called B´ezier extractor sparse matrices) that facilitate the implementation of IGA in existing FE codes. Precisely, the shape functions of IGA elements are the Bernstein polynomials (defined on the standard parent element) multiplied by the extractors. We refer to for a discussion on both techniques.

For curved geometries, the post-processing of IGA is more involved than Lagrange FEs due to two reasons (1) some control points locate outside the physical domain (hence the computed displacements at control points are not nodal values) and (2) existing post-processing techniques cannot be applied directly to NURBS meshes. Interested reader can refer to for a discussion on some post-processing techniques for IGA. For completeness we discuss briefly is constructed. The nodes of this mesh are the intersections of the ξ and η knot lines in the physical space. We then extrapolate the quantities at Gauss points to these nodes and perform nodal averaging if necessary. Figure 9 summarizes the idea.

Gp

Figure 9: Post-processing in Isogeometric Analysis.

Examples

Since we are introducing a computational framework for delamination analyses rather than a detailed study of the delamination behaviour of composite materials, intralaminar damage (matrix cracking and fiber damage) is not taken into account leading to an orthotropic elastic behaviour assumption for the plies. Note that matrix cracking can however be efficiently modeled using extended finite elements as shown in [65, 64] and can be incorporated in our framework without major difficulties. Besides, inertia effects are also skipped. In order to trace equilibrium curves we use either a displacement control (for problems without snapbacks) and the energy-based arc-length control [66, 67].

Interested reader can refer to [53, 65] for the computer implementation aspects of this arc-length solver. A full Newton- Raphson method was used to solve the discrete equilibrium equations. Unless otherwise stated, a geometrically linear formulation is adopted.

We use a C++ code for computations since Matlab is not suitable for this purpose. Whenever possible, validation against theoretical solutions are provided.

Four Numerical Examples Are Provided Including

• Mixed mode bending test (MMB), 2D simple geometry, implementation verification test; • L-shaped specimen, single and multiple delamination, NURBS curved geometry; • 3D double cantilever beam, to verify the implementation; • Singly curved thick-walled laminate, 3D curved geometry.

And in an extra example, we present NURBS parametrization for other commonly used composite structures–glare panel with a circular initial delamination, open hole laminate and doubly curved composite panel.

Ixed Mode Bending Test (Mmb)

Figure 10 shows the mixed mode bending test of which the geometry data are L = 100 mm, h = 3 mm; the beam thickness B is equal to 10 mm; the initial crack length is a0 = 20 mm. The plies are modeled with isotropic material to make a fair comparison with analytical solutions which are valid for isotropic materials only. The properties for the isotropic material are E = 150 GPa and ν = 0.25. The properties for the cohesive elements are GIc = 0.352

= 80 Mpa, Τ 0

3 = 60 MPa. The interface stiffness is K = 106 N/mm3 and η = 1.56. In order to prevent interpenetration of the two arms, in addition to cohesive elements, frictionless contact elements are placed along the initial crack. The loads applied are P1 = 2Pc/L and P2 = P(2c + L)/L, where L is the beam length, c is the lever arm length, and P is the applied load. From these relationships, it is clear that the applied loads P1 and P2 are proportional i.e., P2/P1 = (2c + L)/L. We choose c = 43.72 mm so that the mixed-mode ratio GI/GII is unity.

The external force vector is therefore f ext = λ[1, −2.1436]T (a unit force was assigned to P1) in which the variable load scale λ is solved together with the nodal displacements using the energy based arc-length method [66, 67, 53].

P1

Figure 10: Mixed Mode Bending (MMB): geometry and loading.

Geometry And Mesh

For those who are not familiar to B-splines/NURBS, we present how to build the beam geometry using B-splines. It is obvious that the beam can be exactly represented by a bilinear B-spline surface with 4 control points locating at order that suits the analysis purpose, see line 10 of the same Listing. The delamination path locates in the midline of the beam i.e., η = 0.5 and note that q = 2, in order to introduce a discontinuity one simply has to insert 0.5 three (= q + 1) times into knot vector H (knot vector which is perpendicular to the delamination plane). For point load P2 one needs a control point at the location of the force which corresponding to insert 0.5 three times (equals p = 3) into knots Ξ. Line 13 does exactly that. In order to differentiate cohesive elements and contact elements (remind that contact elements are put along the initial crack to prevent interpenetration), a knot 1 −a0/L is added to Ξ p times (see line 14). The final step is to perform a h-refinement to refine the mesh and extract element connectivity data for the interface elements using the code given in Listing 1.

Listing 3: Matlab code to build the beam using B-splines

Vknot = ;

s o l i d = nrbmak( controlPts , { uKnot vKnot } ) ; % build

S O L I D

= nrbkntins ( s o l i d ,{[1 −a0/L 1−a0/L 1−a0/L ]

Analyses With Varying Basis Orders

We use meshes with two elements along the thickness direction and the basis order along this direction is fixed to 2 (quadratic basis). The notation 2 × 128 B2 × 3 indicates a mesh of 2 × 128 elements of orders 2 × 3. The order of basis functions along the length direction, p, varies from two to five. Firstly we perform a mesh convergence test for quartic-quadratic elements and the result is given in Fig. (11a). Mesh 2 × 64 is simply too coarse to accurately capture the cohesive zone and mesh 2 × 128 is sufficient to get a reasonable result. Next, the mesh density is fixed at 2 × 128 and p is varied from 2 to 5, the result is plotted in Fig. (11b). We refer to for a throughout study on the excellent performance of high order B-splines elements compared to low order Lagrange finite elements for delamination analyses.

-Shaped Composite Panel With A Fillet

For the second example, we analyze the L-shaped composite specimen which was studied in [55, 54] using Lagrange finite elements. The geometry and loading configuration is given in Fig. (12). Contrary to the previous example, in this example NURBS surfaces are used to exactly represent the curved geometry (to be precise circular arcs). The structure is built up by 15 plies of a unidirectional fiber reinforced carbon/epoxy material. The plies are oriented in alternating 0◦and 90◦orientation, where the angle is measured from the xy plane. The inner ply and the outer ply are oriented in the 0◦direction. Material constants are given in Table 1 which are taken from [55, 54]. A plane strain condition

(B)

Figure 11: Mixed Mode Bending (MMB): (a) mesh convergence test and (b) varying basis order in the length direction on meshes of 2 × 128 elements. is assumed. For this problem, unless otherwise stated, we use bi-quadratic NURBS elements for the continuum and quadratic NURBS interface elements for the delamination.

Pa

Table 1: L-shaped specimen: material properties.

Geometry And Mesh

The L-shaped geometry can be exactly represented by a quadratic-linear NURBS surface as shown in Fig. (13) that consists of 7 × 2 control points. The Matlab code used to build the NURBS is given in Listing 4. It is easy to vary the number of plies (see line 4 of the same Listing). Listing 5 gives code to perform p-refinement (to a bi-quadratic NURBS surface) and knot insertion at ply interfaces (two times) to create C0 lines so that the strain field is discontinuous across the ply interfaces. Next, knot insertion is performed again to generate discontinuity lines at the desired ply interfaces.

Two cases are illustrated in the code–interface elements locate along the interface between ply 5 and 6 (line 10) and interface elements at every ply interfaces (line 12-16). Listing 4: Matlab code to build NURBS geometry of the L-shaped specimen.

Layout

Figure 12: L-shaped specimen: boundary and geometry data. There are 15 plies (0◦and 90◦). The ply orientation is measured with respect to the x −y plane. Figure 13: L-shaped specimen: quadratic-linear NURBS geometry with control points (filled circles) and control polygon.

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

●

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

●

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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