Short Fiber Composite Digimat

LS-DYNA Machine Learning-based Multiscale Method for Nonlinear Modeling of

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Final version of this manuscript has been published in Journal of Engineering Mechanics

Ite This Article

Wei, H., Wu, C. T., Hu, W., Su, T. H., Oura H., Nishi, M., Naito T., Chung S., Shen L. (2023). LS-DYNA machine learning-based multiscale method for nonlinear modeling of short-fiber-reinforced composites. Journal of Engineering Mechanics. 149(3): 04023003.

Wei, H., Wu, C. T., Hu, W., Su, T. H., Oura H., Nishi, M., Naito T., Chung S., Shen L. (2023). LS-DYNA machine learning-based multiscale method for nonlinear modeling of short-fiber-reinforced composites. Journal of

Abstract

Short-fiber-reinforced composites (SFRC) are high-performance engineering materials for lightweight structural applications in the automotive and electronics industries. Typically, SFRC structures are manufactured by injection molding, which induces heterogeneous microstructures, and the resulting nonlinear anisotropic behaviors are challenging to predict by conventional micromechanical analyses. In this work, we present a machine learning-based multiscale method by integrating injection molding-induced microstructures, material homogenization, and Deep Material Network (DMN) in the finite element simulation software LS-DYNA for structural analysis of SFRC. DMN is a physics-embedded machine learning model that learns the microscale material morphologies hidden in representative volume elements of composites through offline training. By coupling DMN with finite elements, we have developed a highly accurate and efficient data-driven approach, which predicts nonlinear behaviors of composite materials and structures at a computational speed orders-of-magnitude faster than the high-fidelity direct numerical simulation. To model industrial-scale SFRC products, transfer learning is utilized to generate a unified DMN database, which effectively captures the effects of injection molding-induced fiber orientations and volume fractions on the overall composite properties. Numerical examples are presented to demonstrate the promising performance of this LS-DYNA machine learning-based multiscale method for SFRC modeling.

Keywords: multiscale method, reduced-order modeling, mechanistic machine learning, nonlinear multiscale simulation, deep material network, CAE software, LS-DYNA, short-fiber-reinforced composites, injection-molded thermoplastics, composite structures.

Ntroduction

Short-fiber-reinforced composites (SFRC), such as glass-fiber-reinforced thermoplastics, become increasingly attractive for lightweight structural applications in automotive and electronics industries. Nowadays, mass-production of SFRC parts is achieved by the injection molding process, which inevitably causes inhomogeneous fiber dispersion in the polymer melt, and consequently, location-dependent fiber orientations and volume fractions in the finished products. The heterogeneous microstructural distributions and the distinct properties of each constituent material lead to highly complicated, anisotropic, and nonlinear responses of SFRC (Mortazavian and Fatemi 2015; Hessman et al. 2019), and reliable prediction of the mechanical behaviors of composite products remains challenging. Phenomenological anisotropic constitutive models often require a tedious parameter fitting process to calibrate a large number of model parameters against material data. Measuring these material data from a series of physical experiments is quite time-consuming, and often the fitted model parameters become ineffective for a different microstructure in materials. For instance, conventional constitutive models calibrated for SFRC with low fiber volume fraction is not able to predict SFRC with high fiber volume fraction, even if the fiber orientations remain unchanged. To circumvent such difficulties, multiscale composite material modeling methods have emerged as an effective means to predict macroscopic material properties at the upper length scale from the geometries and properties of the materials at a lower length scale.

Various multiscale methods have been developed for modeling composites. Among these methods, analytical homogenization methods (Eshelby and Peierls 1957; Mori and Tanaka 1973; Nemat-Nasser and Hori 2013; Li and Wang 2008; Huang 2021) are quite popular due to their simplicity and efficiency. For instance, upscaling microscopic material information to obtain homogenized SFRC material properties via analytical micromechanics methods has been studied in (Müller and Böhlke 2016; Tucker III and Liang 1999). Nevertheless, analytical homogenization methods are based on assumptions involving simplified microstructural morphologies and material behaviors, which limits their applicability and accuracy for nonlinear analysis of composites with complicated microstructures. On the other hand, computational homogenization methods become increasing attractive in multiscale design and modeling of composite materials (Fish et al.

2021; Terada et al. 2013). For SFRC, Representative Volume Element (RVE) with realistic fiber distributions can be numerically reconstructed, and multiscale mechanical responses can be predicted through Direct Numerical Simulations (DNS) methods, such as the Finite Element Method (FEM) and Fast Fourier Transformation (FFT) (Naili et al.

2020; Müller et al. 2015). The high-fidelity DNS is especially useful for the design and analysis of composite materials at the RVE level. To model large-scale composite structures, multiscale methods coupling numerical RVE models with structure-level FE Wei, H., Wu, C. T., Hu, W., Su, T. H., Oura H., Nishi, M., Naito T., Chung S., Shen L. (2023). LS-DYNA machine learning-based multiscale method for nonlinear modeling of short-fiber-reinforced composites. Journal of

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models have been developed. While FEM is typically adopted to discretize the macroscale composite structure, different numerical approximation methods have been utilized for the microscale RVE models, and the resulting multiscale methods are referred to as FE2 (Feyel 2003; Feyel and Chaboche 2000; Kouznetsova et al. 2004; Tan et al. 2020) or FE-FFT (Kochmann et al. 2018; Spahn et al. 2014). Although these multiscale methods are particularly advantageous to high-fidelity structural analysis, their applications to industrial scale modeling are limited by the high computational costs (Liu et al., 2020; Xu et al., 2020). For large-scale injection-molded SFRC structures with heterogeneous fiber distributions, high-fidelity FE2 or FE-FFT models will consume extremely high CPU time and memory that are unaffordable especially when nonlinear analyses are desired.

Recent progress in machine learning and data science have brought great opportunities to develop advanced data-driven material modeling and multiscale simulation methods and Humer 2022). To circumvent the limitations of conventional constitutive modeling, the model-free data-driven approach has been developed, which formulates an optimization problem to search for a stress solution directly from the material database characterizing constitutive behaviors subjected to essential physical constraints, such as equilibrium and compatibility conditions (Kirchdoerfer and Ortiz 2016; Ibanez et al. 2018; Eggersmann et al. 2019; He and Chen 2020; He et al. 2020; He et al. 2021a; He et al.

2021b; Xu et al. 2020). This data-driven computing paradigm has been applied for multiscale modeling of fiber-reinforced plastic composites (Huang et al. 2021), biological materials (Mora-Macías et al. 2020; Sanz-Herrera et al. 2021), and granular materials (Karapiperis et al. However, its application to elastoplastic material modeling remains challenging due to difficulties in defining a material database to characterize path-dependent material behaviors. Meanwhile, machine learning techniques have been applied to construct surrogate models of constitutive laws, including Gaussian process modeling (Bostanabad et al. 2018; Chen et al. 2018) and artificial neural networks, such as feedforward neural networks (Ghaboussi et al. 1991; Fritzen et al. 2019; Le et al. 2015; Lu et al. 2019), recurrent neural networks (Ghavamian and Simone 2019; Wang and Sun 2018), and graph/convolutional neural networks (Frankel et al. 2019; Vlassis et al. 2020; Rao and Liu 2020). In addition, neural network-based constitutive models with embedded material physical constraints including material frame invariance (Ling and Jones et al.

2016), symmetric positive definiteness (Xu and Huang et al. 2021), self-consistency (Bonatti and Mohr 2022), and thermodynamics (Vlassis and Sun 2021; Masi and Stefanou et al. 2021; He and Chen 2022) have been developed. Recently, coupling of neural networks with finite element methods and meshfree methods for modeling material damage and strain localization phenomena have also been investigated (Tao et al. 2022; Baek et al. 2022). These studies demonstrate the excellent performance of machine learning methods for modeling complex material physics by exploitation of material data.

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To accelerate multiscale material modeling, several reduced-order modeling methods have been developed through the construction of surrogate models for high-fidelity

On

proper-orthogonal decomposition is developed in (Kaneko et al. 2021; Rocha et al. 2020; Fritzen and Kunc 2018; Goury et al. 2016; Yvonnet and He 2007), self-consistent clustering analysis is developed in (Gao et al. 2020; Liu et al. 2016; Liu et al. 2018; Yu et al. 2019), and a mechanistic machine learning method named Deep Material Network (DMN) is proposed by (Liu et al. 2019a; Liu and Wu 2019). DMN is designed to capture nonlinear microstructural interactions through a binary-tree network structure equipped with physics-based building blocks (Liu et al. 2019a; Liu and Wu 2019; Gajek et al. 2020).

DMN can be trained in an offline stage to learn the microscopic material morphologies and physics hidden in linear composite material data, and afterwards the trained network is able to perform multiscale online prediction of nonlinear constitutive behaviors. DMN has been extended to model woven composites (Wu et al. 2021), porous materials (Nguyen and Noels 2022a; Nguyen and Noels 2022b), cohesive interfacial failure (Liu 2020), and strain localization analysis (Liu 2021). In addition, transfer learning strategies are developed in (Liu et al. 2019b; Liu et al. 2020; Huang et al. 2022) for fast creation of DMN models with minimized training efforts for materials that share similar microstructural morphologies but own different characteristic geometries, such as particle-reinforced composites with different volume fractions. Coupling of DMN to finite elements has also been investigated for multiscale structural simulation (Gajek et al.

2022; Gajek et al. 2021; Liu et al. 2020). Despite the great progress, the usage of mechanistic machine learning techniques for Computer-Aided Engineering (CAE) is still limited within academic research community due to the lack of a general and robust software platform. To bridge this gap, we have developed a unified DMN database for SFRC to cover a full range of injection-molded microstructures, and the trained DMN model is seamlessly integrated in the multiphysics simulation software LS-DYNA for nonlinear multiscale modeling.

The main goal of this paper is to present the LS-DYNA machine learning-based multiscale method for nonlinear modeling of SFRC, and the remainder of this paper is organized as follows. Firstly, an overview of network architecture of DMN is given. Next, we present the details on the integration of DMN into LS-DNA for SFRC modeling, including the offline training of DMN based on an efficient transfer learning scheme for

Based On

injection-molding-induced microstructures, and the nonlinear multiscale prediction of SFRC parts by coupling trained DMN models with finite elements. Lastly, numerical examples are presented to demonstrate the capability of the proposed method, followed by the conclusions.

Overview Of Deep Material Network

Fig. 1. Architecture of a 4-layer Deep Material Network (DMN). Deep Material Network (DMN) proposed in (Liu et al. 2019a; Liu and Wu 2019) is a mechanistic machine learning method for data-driven multiscale material modeling. As illustrated in Fig. 1, a binary-tree network structure is adopted for DMN. Therefore, for a network with 𝑁 layers, there are (2𝑁−1) nodes, where 2𝑁−1 nodes are located at the bottom layer 𝑁. For the 𝑘-th node at layer 𝑖, four network parameters are defined,

𝑘, Where 1 ≤𝑖≤

𝑁 denotes the layer index, and 1 ≤𝑘≤2𝑖−1 denotes the node index within each layer. Different from conventional artificial neural networks, all the network parameters of DMN have clear physical meanings, and thus it is straightforward to apply the rule of

𝑘−1 And 𝝈𝑖+1

2𝑘 are the stresses associated with the two child nodes of the

𝑘−1 And 𝑤𝑖+1

2𝑘 are the nodal weights of these two child nodes, respectively. Accordingly, an averaged material stiffness 𝑪̅𝑖

𝑘 Can Be Expressed As A

function of the nodal weights and material stiffness of its two child nodes:

𝑘−1)𝚨 (2)

where 𝚨 denotes a strain concentration matrix within each DMN building block, which is obtained by enforcing the interfacial equilibrium and kinematic conditions of a two-phase composite material (Liu and Wu, 2019). Derivation of an analytical form for the strain concentration matrix is given in Appendix I of this paper.

𝑘 Are Defined At Each Node Of

the network to capture directional material behaviors due to complicated microstructures. Wei, H., Wu, C. T., Hu, W., Su, T. H., Oura H., Nishi, M., Naito T., Chung S., Shen L. (2023). LS-DYNA machine learning-based multiscale method for nonlinear modeling of short-fiber-reinforced composites. Journal of

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By applying three dimensional rotations to the averaged stiffness and stress, one obtains the following rotated stiffness matrix and stress vector:

𝑘 (4)

where 𝑹 denotes a rotation matrix based on the Euler angles (𝛼𝑖

𝑘). By Applying

the averaging and rotation operations to the physical quantities (stress, strain, stiffness, etc.) in different building blocks, DMN can upscale the microscopic base material behaviors at the bottom layer to predict macroscopic composite behaviors at the top layer.

The Nodal Weight 𝑤𝑖

𝑘 of a parent node is computed by adding up the weights of its two

𝑘 (5)

except for the bottom layer nodes whose weights are activated through the rectified linear

𝑘= 𝑅𝑒(𝑧𝑘) = 𝑚𝑎𝑥(𝑧𝑘, 0) (6)

Therefore, all the nodal weights in DMN can be calculated from the activations 𝑧𝑘. As a result, for a network with 𝑁 layers, its independent network trainable parameters are 2𝑁−1 activations 𝑧𝑘 and 3(2𝑁−1) rotation angles 𝛼𝑖

𝑘, Where 1 ≤𝑖≤

𝑁, 1 ≤𝑘≤2𝑁−1. Once DMN trainable parameters are determined through offline training, the essential microstructural interactions can be learned by the trained network, which can be applied for online prediction of nonlinear composite behaviors under general loading conditions.

Offline Training Of Dmn For Sfrc

Rewriting the independent network trainable parameters in a vector form as 𝒛∈ℝ2𝑁−1, 𝜶∈ℝ2𝑁−1 , 𝛃∈ℝ2𝑁−1 , 𝛄∈ℝ2𝑁−1 , we can express the overall material stiffness

𝑪1

1 of a two-phase composite at the top node of a 𝑁-layer DMN as:

𝑪1

1 = 𝒇(𝑪̂𝑓, 𝑪̂𝑚, 𝒛, 𝜶, 𝛃, 𝛄) (7) in which 𝑪̂𝑓 and 𝑪̂𝑚 represent stiffness matrices of the fiber phase and the matrix phase of short-fiber-reinforced composites. During the offline training process, they are

(8)

To determine the network trainable parameters, an optimization problem is formulated based on the mean square error (MSE), and the cost function is given by (Liu and Wu

(9)

where ‖⋯‖ denotes the Frobenius norm, 𝜆 is a positive hyper-parameter associated with the regularization term, which is set to be 0.001 in the present study to ensure the well-posedness of the optimization problem, 𝑗 denotes the index of the material sample in the training dataset, and 𝑁𝑠 is the total number of material samples. Hence, 𝑪̂𝑗

𝑪̂𝑗

𝑐 represent the fiber stiffness, matrix stiffness, and composite stiffness of the 𝑗th material sample, all of which are considered as linear elastic material properties. To minimize the cost function, the mini-batch gradient descent algorithm is employed, where gradients of the cost function with respect to the trainable parameters 𝛻𝐽 are derived by the backpropagation algorithm, as analytical functions are available in DMN building blocks.

Offline training data for DMN, i.e., linear elastic macroscopic stiffness tensors of the composites and microscopic stiffness tensors of the material constituents, can be gathered from both experimental measurements and numerical predictions. In the present study, high-fidelity computational homogenization of SFRC is employed to generate training data due to the lack of experimental data. To this end, RVE models are reconstructed Wei, H., Wu, C. T., Hu, W., Su, T. H., Oura H., Nishi, M., Naito T., Chung S., Shen L. (2023). LS-DYNA machine learning-based multiscale method for nonlinear modeling of short-fiber-reinforced composites. Journal of

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based on SFRC microstructures. In practice, SFRC products may contain heterogeneous microstructures due to numerous combinations of fiber orientations and fiber volume fractions, depending on the injection-molding layout and material design. Therefore, it is infeasible to reconstruct a new SFRC RVE for each individual microstructural geometry.

To reduce the cost of DMN training, we introduce a transfer learning scheme (Liu et al. 2019b; Liu et al. 2020; Huang et al. 2022) to quickly generate DMNs for new SFRC microstructures by transferring the knowledge of a few pre-trained networks. To consider the effects of different fiber orientation states, we need to reconstruct 3 SFRC RVE geometries with the same fiber volume fraction, including RVEs with random 3D, random 2D, and unidirectional (UD) fiber orientation states. These three special orientation states are chosen for the offline training because a linear combination of their corresponding second-order orientation tensors (Advani and Tucker III, 1987) is sufficient for parametrization of all other possible fiber orientation states, as explained in Appendix II of this paper. Furthermore, an additional RVE geometry with a UD fiber orientation state and a high fiber volume fraction is reconstructed to capture the fiber volume fraction effect on the composite response. In total, we have reconstructed 4 SFRC RVE geometries with a fiber aspect ratio around 20 for the offline training of DMN, as shown in Fig. 2.

Fig. 2. Short-fiber-reinforced composite microstructures for transfer-learning-based offline training of DMN models, where FVF denotes the fiber volume fraction. For each SFRC microstructural geometry, we define 500 material samples containing different microscopic stiffness tensors for the fiber phase and matrix phase, and computational homogenization is employed to obtain the corresponding macroscopic stiffness tensors for the composites. Material samples are assigned with linearly elastic microscopic stiffness tensors with sufficient phase contrast and material anisotropy (Liu and Wu 2019) for the material network to learn the topological representation of SFRC. In the present study, each microstructure is discretized by 10-node tetrahedron finite elements in LS-DYNA and the *RVE_ANALYSIS_FEM keyword is used to Wei, H., Wu, C. T., Hu, W., Su, T. H., Oura H., Nishi, M., Naito T., Chung S., Shen L. (2023). LS-DYNA machine learning-based multiscale method for nonlinear modeling of short-fiber-reinforced composites. Journal of

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automatically impose the periodic displacement boundary conditions for homogenization (Wei and Lyu et al. 2022). Since 4 microstructural geometries are considered for offline training, 2000 linear elastic finite element models are generated in total. To calculate the macroscopic composite stiffness tensor, 6 orthogonal loading conditions are imposed to each finite element model, respectively. As a result, 12000 linear elastic finite element simulations are performed, for which the total CPU time is approximately 670 hours with 32 processors used for each simulation.

After the finite element simulations, the homogenized composite stiffness and the corresponding microscopic fiber and matrix stiffness data are collected, and then the material data for each RVE microstructural geometry are separated into two datasets, of which 400 data points are defined as the training dataset, and the remaining 100 data points are defined as the testing dataset. The training dataset is utilized with a gradient-based optimization to calculate the network parameters of DMN models during the offline training stage, whereas the testing dataset is used to assess the generalization performance of a trained model.

Fig. 3. Workflow for the 4-stage offline training of DMN for SFRC. Transfer learning-based offline training of DMN for SFRC consists of the following four stages, as illustrated in Fig. In stage 1, the RVE microstructure with 8% fibers uniformed oriented in 3D is considered, and DMN is trained with randomly initialized trainable parameters. After this stage, we obtain a trained DMN model, and we transfer it to initialize the networks for the random 2D and the UD RVEs with 8% fibers in stage 2 and stage 3, respectively. Finally, in stage 4 we transfer the network trained for the UD RVE with a low fiber volume fraction at 8% to the UD RVE with a high fiber volume fraction at 35%. For each stage, we use 20000 epochs to train a DMN model, where one Wei, H., Wu, C. T., Hu, W., Su, T. H., Oura H., Nishi, M., Naito T., Chung S., Shen L. (2023). LS-DYNA machine learning-based multiscale method for nonlinear modeling of short-fiber-reinforced composites. Journal of

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epoch refers to one round of evaluation on all the training samples. During the optimization process, the 400 training samples for each SFRC microstructure are divided randomly into 10 mini-batches, so there are 10 training steps in each epoch. In addition, the bold driver method is employed to adapt the learning rate (i.e., the multiplier on the step size) by comparing the training error to its previous value after each epoch. Parallel computing with 10 processors is adopted for the offline training, and it takes around 200 hours to finish all the 4 training stages.

Histories of the average training and testing errors for the offline training process are plotted in Fig. 4, where the average errors are defined in (Liu and Wu 2019). DMN for the first RVE with the random 3D microstructure begins with a large training error since the trainable parameters are randomly initialized without any prior knowledge about the microstructure. For the other three RVEs, the training starts from a much lower error thanks to the knowledge transferred from the pre-trained network, which demonstrates the enhanced training efficiency of the employed transfer learning scheme.

Fig. 4. Histories of the average training and testing errors for transfer-learning-based training of DMN models for SFRC. Wei, H., Wu, C. T., Hu, W., Su, T. H., Oura H., Nishi, M., Naito T., Chung S., Shen L. (2023). LS-DYNA machine learning-based multiscale method for nonlinear modeling of short-fiber-reinforced composites. Journal of

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Table 1. Training results of DMN for SFRC microstructures

%

Table 1 shows the training results for DMN with 8 layers, where the accuracy is measured by the scaled mean absolute error. As can be seen from the table, training errors of all the DMN models are less than or equal to 0.33%. In addition, we can observe that the training error decreases from the random 3D fiber orientation to random 2D fiber orientation, and it further decreases for the UD fiber orientation state. Increasing the fiber volume fraction, however, induces a higher training error. The levels of testing errors on unseen data points are quite close to the training error levels, suggesting that there is no overfitting issue. The strong generalization performance of DMN is attributed to the essential physics embedded in the two-layer building block, which enhances the extrapolation capability to unknown material and loading spaces. To further examine the ability of trained DMN for capturing material anisotropic effects, DNS and DMN are employed to predict the homogenized linear elastic material stiffness for UD and random 2D SFRC microstructures, respectively, for which we adopt a set of fiber and matrix properties unseen in the training process. Using the method described by (Nordmann et al.

2018), direction-dependent Young’s modulus calculated from the anisotropic stiffness can be visualized as a 3D surface, as shown in Fig. 5. The good agreement between DNS and

The

microstructure-induced directional dependency of SFRC properties. Wei, H., Wu, C. T., Hu, W., Su, T. H., Oura H., Nishi, M., Naito T., Chung S., Shen L. (2023). LS-DYNA machine learning-based multiscale method for nonlinear modeling of short-fiber-reinforced composites. Journal of

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Fig. 5. 3D representation of Young’s modulus for anisotropic SFRC microstructures predicted by DNS and DMN, where the radius (vector measured from the origin to the surface) in any direction is proportional to the magnitude of the Young’s modulus in that direction, and the magnitude of the Young’s modulus is also conveyed by a color mapping applied to the surface.

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LS-DYNA Nonlinear Multiscale Online Prediction for Injection-Molded SFRC Integration of DMN models for SFRC with the engineering simulation software LS-DYNA is implemented for multiscale structural analysis of SFRC. In dynamic finite element analysis, the spatial domain 𝑉 of the global SFRC structure is discretized into a collection of subdomains 𝑉𝑒, where 𝑒= 1, ⋯, 𝑁𝑒, 𝑁𝑒 denotes the total number of finite elements, and the global displacement field 𝒖(𝑿, 𝑡) ∈ℝ3 is approximated by

(10)

where 𝐼 denotes the global nodal index, 𝑁𝑛 denotes the total number of nodes in the finite element mesh, 𝑿∈ℝ3 and 𝑡 denote the spatial position and time, respectively,

N𝐼

𝑢(𝑿) and 𝐔𝐼(𝑡) ∈ℝ3 denote the shape function and the displacement vector associated with node 𝐼, respectively. The nodal displacements, velocities, and accelerations can be obtained by solving the global semi-discrete momentum equation:

(11)

where 𝐔̈ ∈ℝ3𝑁𝑛 is the global nodal acceleration vector, which is the 2nd-order material time derivative of the global nodal displacement vector 𝐔∈ℝ3𝑁𝑛, 𝐌∈ℝ(3𝑁𝑛)×(3𝑁𝑛), 𝐅ext ∈ℝ3𝑁𝑛, and 𝐅int ∈ℝ3𝑁𝑛 are the lumped mass matrix, the global external nodal force vector, and the global internal nodal force vector, respectively. 𝐅int is obtained by

(12)

where 𝓑𝐼 denotes the shape function gradient matrix associated with node 𝐼. Evaluation of the internal nodal force in Eq.(12) is based on numerical integration. To this end, the macroscopic stress 𝛔 is calculated at every quadrature point 𝛏𝑞 of the finite element model, where the subscript 𝑞 denotes the quadrature point index. In the present LS-DYNA multiscale method, the macroscopic stress 𝛔 is predicted by DMN coupled with finite elements, where each quadrature point 𝛏𝑞 has an associated network corresponding to the local fiber orientation and volume fraction. To model injection-molded SFRC parts, where nonuniform fiber distributions are induced by different molding conditions (e.g., part geometry, injection gate position, filling time, and mold temperature), it is desirable to create the online DMN models in an efficient manner, instead of performing offline training for each individual microstructure. For this reason, Wei, H., Wu, C. T., Hu, W., Su, T. H., Oura H., Nishi, M., Naito T., Chung S., Shen L. (2023). LS-DYNA machine learning-based multiscale method for nonlinear modeling of short-fiber-reinforced composites. Journal of

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the transfer learning method proposed in (Liu et al. 2019b; Liu et al. 2020; Huang et al. 2022) is adopted for creating DMN models in LS-DYNA during online computation. Under the transfer learning framework, the base topological structures of all the four DMN models obtained from offline training are analogous, which enables a continuous migration between different networks through direct interpolation of their trainable parameters. Let us define a data point (𝑿∗, 𝒀∗), where the superscript (∗) denotes an intermediate state, 𝒀∗ denotes the unknown DMN trainable parameters:

(13)

and 𝑿∗ denotes the geometric descriptors of the intermediate SFRC microstructure:

∗ Are Two Largest

eigenvalues of the second-order fiber orientation tensor (Advani and Tucker III, 1987), which describes the orientation state of short fibers. Note that the three eigenvalues 𝑎11, 𝑎22, and 𝑎33 of any fiber orientation tensor 𝒂 satisfy 𝑎11 ≥𝑎22 ≥ 𝑎33 and 𝑎11 + 𝑎22 + 𝑎33 = 1, as described in Appendix II. The values of fiber orientation tensor and volume fraction can be either measured from experiments or predicted through injection molding simulation of the melt flow process (Wang et al. 2018). Similarly, we can define the known trainable parameters of pre-trained DMN models as 𝒀1, 𝒀2, …, 𝒀𝑁, and the geometric descriptors of microstructures used in the offline training as 𝑿1, 𝑿2, …, 𝑿𝑁.

Accordingly, the regression function for the new data point (𝑿∗, 𝒀∗) can be expressed as 𝒀∗(𝑿∗) = 𝒓(𝑿∗|(𝑿1, 𝒀1), (𝑿2, 𝒀2), ⋯, (𝑿𝑁, 𝒀𝑁)) (15) To determine the unknown trainable parameters (i.e., [𝒛∗, 𝜶∗, 𝛃∗, 𝛄∗]) for a linear regression model with three independent geometric descriptors (i.e., [𝑣𝑓

∗]), We

need four linearly independent data points (𝑿𝑖, 𝒀𝑖), which correspond to the four RVE geometries created in the offline training stage. Therefore, N=4 is chosen in Eq. (15). Wei, H., Wu, C. T., Hu, W., Su, T. H., Oura H., Nishi, M., Naito T., Chung S., Shen L. (2023). LS-DYNA machine learning-based multiscale method for nonlinear modeling of short-fiber-reinforced composites. Journal of

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Fig. 6. Illustration of the DMN-based nonlinear multiscale simulation framework for short-fiber-reinforced composite (SFRC) structures, where microstructural data from Moldex3D are mapped by LS-PrePost to LS-DYNA finite elements coupled with DMN.

Since the online network creation is guided by the microstructures at quadrature points, it is essential to gather the injection-molded microstructure information. In practice, microstructural distribution in SFRC products can be obtained through injection molding simulation (Wang et al. 2018) using the software Moldex3D, and the predicted fiber orientation and volume fraction data can be mapped from the molding simulation mesh to the LS-DYNA structural simulation mesh using the pre-processing software LS-PrePost.

After mapping, the DMN online prediction module will create a new DMN model at each quadrature point specific to the local microstructure, and then the network will be dynamically coupled to the finite elements in LS-DYNA for nonlinear multiscale online prediction. An illustration of the overall multiscale simulation framework (Wei et al. 2021) is depicted in Fig. 6. Note that this online DMN creation process does not involve RVE reconstruction or DNS. In addition, the new DMN models are created only once at the beginning of the online prediction stage, so the associated computational cost is negligible in the overall multiscale simulation.

After the creation of microstructure-based DMN models, LS-DYNA multiscale structural simulations will be carried out, where finite element modeling for the global structures and DMN prediction of the local composite materials are tightly coupled. At each time step, finite element equations are solved to calculate the nodal accelerations, velocities, and displacements at the global structural level. In the present work, an explicit time integration algorithm has been adopted, which has been proven to be highly efficient and Wei, H., Wu, C. T., Hu, W., Su, T. H., Oura H., Nishi, M., Naito T., Chung S., Shen L. (2023). LS-DYNA machine learning-based multiscale method for nonlinear modeling of short-fiber-reinforced composites. Journal of

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robust for nonlinear dynamic problems involving contact-impact and large deformations (Belytschko et al. 2014). Afterwards, the macroscopic strain at each quadrature point is evaluated and transferred to DMN. With the macroscopic strain increment, backward de-homogenization and forward homogenization of material information are performed within DMN to predict the multiscale material response. The incremental stress-strain relationship associated with DMN’s 𝑘th node at layer 𝑖 takes the following form:

𝑘 Denotes The Strain Increment Of Dmn’S

𝑘th node at layer 𝑖. In multiscale structural analysis, macroscopic rate-of-deformation increments computed by the finite element method are assigned to the top layer node of DMN at the corresponding quadrature point. Strain increments of nodes at other layers are calculated through backward de-homogenization. d𝝈̅𝑖

𝑘 Denotes A Correction To The

incremental stress, which should vanish if material nonlinearities of composites are omitted. In nonlinear composite modeling, however, d𝝈̅𝑖

𝑘 Is Not Necessarily Equal To Zero

and is calculated through forward propagation from a lower layer of the network:

𝑘−1 And 𝑤𝑖+1

2𝑘 are the corresponding nodal weights, and the vector 𝝌 depends on the material stiffness matrices and stress corrections of the two child nodes, for which an analytical expression can be found in (Liu and Wu, 2019). d𝝈𝑖+1

D𝝈𝑖+1

2𝑘 are the rotated stress corrections of child nodes, which are obtained by applying a rotation operation to the averaged stress correction:

𝑗, 𝛾𝑖

𝑗) denotes the rotation matrix based on the Euler angles (𝛼𝑖

𝑗)

of the network. At the bottom layer of DMN, material stiffness matrices 𝑪𝑁

𝑘, And The Correction D𝝈𝑁

𝑘 are evaluated using microscopic constitutive laws for the fiber phase and the matrix phase. While linear elastic constitutive laws are adopted during the offline stage to learn the essential physics, elastoplastic microscopic constitutive laws can be adopted in the online structural analysis stage to capture nonlinear composite material behaviors. For SFRC, a linear elastic law is usually sufficient for modeling the fiber phase, whereas an elastoplastic law with isotropic hardening can be adopted for modeling the nonlinear matrix phase. After the microscopic material law evaluation, stress and state variables (e.g., equivalent plastic strain/EPS) are Wei, H., Wu, C. T., Hu, W., Su, T. H., Oura H., Nishi, M., Naito T., Chung S., Shen L. (2023). LS-DYNA machine learning-based multiscale method for nonlinear modeling of short-fiber-reinforced composites. Journal of

Page 18 Of 41

stored at the bottom layer, while the stiffness matrices and stress corrections are propagated to an upper layer of the network. Due to material nonlinearities, forward homogenization and backward de-homogenization of stresses and strains are iterated in the network. To check convergence for the network iteration, an L2 norm of the difference in two successive strains is computed at the bottom layer:

≤𝜖𝑡𝑜𝑙 (19)

where the superscripts (𝑖𝑡𝑒) and (𝑖𝑡𝑒+ 1) denote iteration counts, and 𝜖𝑡𝑜𝑙 is a convergence tolerance. Once convergence is achieved, the microscopic stress and state variables (e.g., equivalent plastic strain) of each bottom node are updated, and the stress

Increment ∆𝝈1

1 of the DMN’s top layer node is employed to update the macroscopic stress 𝛔 at the finite element’s quadrature point 𝛏𝑞:

(𝛏𝑞) (20)

where the superscript 𝑡𝑛 and 𝑡𝑛+1 denote two different time instants during the time integration of the momentum equation. Upon the completion of the DMN-based multiscale stress computation, finite elements in LS-DYNA will gather the macroscopic stress from different quadrature points to evaluate the internal force vector 𝐅𝐼

Int By

Eq.(12) for nonlinear finite element analysis. After the internal force computation, the resulting finite element equations for composite structures can be solved for the next time step. A flowchart for the DMN-based internal force calculation in LS-DYNA is given in Box 1. It is noteworthy to mention that, in addition to applying DMN in the nonlinear finite element modeling, it is also feasible to couple DMN with meshfree methods (Wang et al. 2009; Wu et al. 2020; Huang et al. 2020; Pasetto et al. 2021) for accelerated multiscale analysis of structures undergoing extreme deformations.

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Box 1. Flowchart for DMN-based internal force calculation in FEA a.

Oop Over Finite Elements 𝑒= 1, ⋯, 𝑁𝑒

i. Gather element nodal displacements and velocities ii. Loop over quadrature points 𝛏𝑞∈ℝ3 with quadrature weights 𝜛𝑞(𝛏𝑞) 1. Initialize Deep Material Network (DMN) parameters if time t𝑛= 0 1.1 Import fiber orientation 𝒂(𝛏𝑞) ∈ℝ3×3, volume fraction 𝑣𝑓(𝛏𝑞) 1.2 Regression-based transfer learning to get new network parameters 𝒛, 𝜶, 𝛃, 𝛄, 𝒘 based on SFRC microstructure at point 𝛏𝑞 Retrieve DMN parameters 𝜶, 𝛃, 𝛄, 𝒘 stored at point 𝛏𝑞 if time t𝑛> 0 2. Compute macroscopic rate-of-deformation increment ∆𝐃(𝛏𝑞)

Ompute Cauchy Stress Increment ∆𝛔(𝛏𝑞) By Dmn

3.1 Evaluate microscopic constitutive equations to get stress ∆𝝈̅𝑁

Stiffness 𝑪̅𝑁

𝑘, and material state variables of bottom-layer nodes

𝑘

3.5 Check network convergence. If not converged, go to 3.1 4. Update macroscopic Cauchy stress 𝛔(𝛏𝑞) ←𝛔(𝛏𝑞) + ∆𝝈1

Page 20 Of 41

Applications for Nonlinear Modeling of Short-Fiber-Reinforced Composites In this section, two numerical examples are presented to demonstrate the effectiveness and performance of the present DMN-based multiscale method. In the first example, we verify the accuracy and efficiency of the method by comparing with direct numerical simulations of RVE, where both the microstructural geometries and nonlinear microscopic material laws are unseen in the DMN offline training. In the second example, nonlinear multiscale analysis is performed for a short-fiber-reinforced thermoplastic part by integrating injection molding-induced fiber orientations and volume fractions, which demonstrates the capability of the present method for industrial applications where capturing the microstructural effects is essential.

Verification Against Direct Numerical Simulation of SFRC RVE Fig. 7. Reconstructed SFRC microstructures for direct numerical simulation. Table 2. SFRC microstructures analyzed in the online prediction

%

In this example, we present nonlinear online prediction results of DMN at a single macroscopic material point level for different SFRC microstructures. The two analyzed

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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