Impact Analysis Composite Ansys

Abstract

Physics-guided approaches offer a promising path toward accurate and generalisable impact identification in composite structures, especially when experimental data are sparse. This paper presents a hybrid framework for impact localisation and force estimation in composite plates, combining a data-driven implementation of First-Order Shear Deforma- tion Theory (FSDT) with machine learning and uncertainty quantification. The struc- tural configuration and material properties are inferred from dispersion relations, while boundary conditions are identified via modal characteristics to construct a low-fidelity but physically consistent FSDT model. This model enables physics-informed data augmenta- tion for extrapolative localisation using supervised learning. Simultaneously, an adaptive regularisation scheme derived from the same model improves the robustness of impact force reconstruction. The framework also accounts for uncertainty by propagating locali- sation uncertainty through the force estimation process, producing probabilistic outputs.

Validation on composite plate experiments confirms the framework’s accuracy, robustness, and efficiency in reducing dependence on large training datasets. The proposed method offers a scalable and transferable solution for impact monitoring and structural health management in composite aerostructures.

Structural health monitoring; Impact identification; Data-driven FSDT modelling; Physics- augmented machine learning; Physics-adaptive regularisation; Uncertainty quantification

; M.H. Aliabadi

arXiv:2507.13376v1 [physics.data-an] 13 Jul 2025

Ntroduction

Composite aerostructures in service are frequently exposed to various impact threats, such as tool drops, hail, and bird strikes . Among these, low-velocity impacts pose a particular risk due to their ability to induce barely visible impact damage (BVID), which may degrade structural integrity without obvious surface indications. Structural Health Monitoring (SHM) systems, especially those employing passive sensing strategies, are widely used to detect such events. By capturing the dynamic response of the structure through surface-mounted or embedded sensors, these systems aim to infer the impact location, force history, and potential damage. However, accurately identifying impact characteristics from sparse sensor measurements remains a significant challenge, especially for high- energy impacts that lead to BVID .

Impact identification is inherently an inverse problem, wherein unknown impact parameters—location and force—must be inferred from observed structural responses. As summarised in Table 1, existing approaches are broadly classified into model-based and data-driven methods.

Table 1: Impact identification methodologies.

Graph Neural Networks [47, 48]

*q, f(t): impact location and force of the target impact (to be estimated), M: model, L: Sensors’ locations, Q, F(t), S(t): reference impact locations, forces and sensor signals. Model-based techniques rely on physical representations of structural dynamics, including finite element analysis (FEA) , modal analysis , and transfer function formulations .

Dispersion relation-based methods , which capture frequency-dependent wave propagation characteristics, have also been employed to enhance localisation. While these methods offer high physical fidelity, they are often limited by the availability of accurate structural information, includ- ing geometric configurations, material properties, and boundary conditions [49, 50]. Incomplete or inaccurate models degrade identification accuracy, and full-scale simulations typically require costly model validation and updating efforts [18, 51].

In contrast, data-driven approaches construct surrogate models that learn mappings from sensor signals to impact parameters based on reference datasets. These include traditional machine learning models such as multilayer perceptrons (MLPs) , probabilistic frameworks like Gaussian Process Regression (GPR) [35–39, 52], and more recent developments in deep learning, including convolutional (CNN) , recurrent (RNN) [45, 46], and graph neural networks (GNN) [47, 48]. Despite their growing popularity, these methods are hindered by several key limitations: • Training data limitations. Numerical data can be inexpensive but may lack fidelity, whereas experimental data are high-fidelity but costly to acquire. Effective data-driven models require extensive impact testing [41, 44], increasing the cost of deployment.

• Generalisability. Training data cannot feasibly cover all possible impact scenarios.

Onse-

quently, data-driven models struggle to generalise to untrained impact area and impact conditions, where impact force and location may be unknown. • Interpretability. Deep learning models often function as ”black boxes”, making it difficult to extract physically meaningful insights, limiting their acceptance in safety-critical applications.

• Scalability. Data-driven approaches typically require the collection of new data and retraining of the model when applied to different structures, sensor configurations, or structural complexities, reducing their adaptability.

To address the challenges associated with impact identification in composite aerostructures—particularly those related to data scarcity—a hybrid paradigm that integrates physical modelling with data-driven learning is gaining attraction. Rather than relying solely on black-box regression from sensor signals to impact characteristics, this paradigm seeks to approximate the underlying physics of impact dy- namics using interpretable and generalisable models.

For Example, Wave Dispersion Relations Can

significantly improve localisation by characterising the frequency-dependent propagation of guided waves, while force reconstruction can benefit from damping-informed models that capture the energy dissipation inherent in real structures.

This paper presents a physics-augmented framework for impact localisation and force identifica- tion that leverages a data-driven First-Order Shear Deformation Theory (FSDT) plate model. The model is constructed using only observed structural responses, without requiring prior knowledge of geometry, material properties, or boundary conditions. Dispersion curve fitting is employed to estimate effective material properties, while boundary conditions are inferred through modal analy- sis. The resulting FSDT model is physically interpretable and structurally consistent, and it serves as a foundation for two key components: (1) the generation of physics-consistent augmented data to improve the generalisation of impact localisation models, and (2) the formulation of an adaptive regularisation scheme for impact force deconvolution, informed by the system’s modal characteristics.

Additionally, the proposed framework incorporates uncertainty quantification by propagating lo- calisation errors into the force reconstruction process, enabling probabilistic estimations with as- sociated confidence bounds.

This capability enhances the reliability of predictions in real-world applications, particularly under conditions of limited sensor coverage or unseen impact scenarios. Experimental validation is conducted on a composite plate subjected to controlled low-velocity impact events. Results demonstrate that the proposed methodology significantly improves robustness, scalability, and interpretability compared to conventional data-driven approaches, while also reducing the need for exhaustive training datasets. The framework is well-suited for deployment in operational composite structures where prior model information may be incomplete or unavailable.

The key contributions of this study are as follows: • Data-driven FSDT modelling: A method for constructing a FSDT plate model from ob- served data, with material and boundary conditions inferred via dispersion and modal analyses, respectively.

• Physics-augmented machine learning: Use of the identified physical model to generate augmented training data, enabling impact localization models to generalise beyond the training domain.

• Adaptive regularisation: A novel, physics-informed regularisation approach for impact force deconvolution, that adaptively applies penalisation to stabilise inverse force estimation. • Uncertainty quantification: Integration of probabilistic modelling to propagate localisation uncertainty into force estimates, yielding confidence-aware predictions suitable for safety-critical applications.

The remainder of this paper is organised as follows: Section 2 describes the proposed data-driven FSDT modelling framework, including physics-augmented impact localisation and physics-adaptive regularisation for force reconstruction. Experimental validation using low-velocity impact testing is detailed in Section 3. The corresponding results for FSDT model construction, impact localisation, force reconstruction and uncertainty quantification are presented in Section 4.

Finally, The Main

conclusions and potential directions for future research are discussed in Section 5. Physics-guided impact identification via sparse data-driven

Fsdt Modelling

In practical applications, plate-like carbon fiber-reinforced polymer (CFRP) laminated composite structures—such as flat plates, stiffened panels, and sandwich configurations—are extensively used in aerospace and related industries. These structures are therefore the primary focus of this chapter.

The geometric dimensions of such components, specifically the length a and width b, are typically determined during the design phase to satisfy manufacturing constraints and ensure effective sensor deployment. Sensor locations, denoted by L, are generally fixed based on these design considerations.

However, in more complex configurations such as stiffened plates and sandwich structures, the plate thickness d exhibits spatial non-uniformity. For instance, stiffened regions may consist of in- creased local thickness ds, compared to the nominal plate thickness dp. Similarly, sandwich structures often integrate aluminium cores with composite laminate facesheets, introducing additional hetero- geneity. In such cases, the effective material density ρ is approximated as a volume-weighted average of the constituent materials to account for spatial variation in structural properties.

Given these conditions, the impact identification task within the proposed data-driven framework

(1)

where the plate dimensions (a, b), sensor locations L, reference impact data (Q, F(t), S(t)), and measured target responses s(t) are treated as observed inputs. In contrast, uncertain parameters such as the plate thickness d ∈[dlb, dub] and effective density ρ ∈[ρlb, ρub] are treated as bounded equivalents to accommodate material and geometric variability. The goal is to infer the unknown structural parameters Θ and the target impact characteristics (q, f(t)) from the observed data.

To enhance the generalisability of impact identification across various types of plate-like CFRP structures, a physics-based model is integrated into the framework. The FSDT is adopted to approxi- mate the behaviour of the target structure as an equivalent flat plate. Despite geometric complexities, such as stiffeners or sandwich cores, FSDT provides a tractable yet sufficiently accurate representa- tion of wave propagation and structural dynamics for these classes of composite systems . This model forms the foundation for coupling physics-based insights with data-driven techniques, thereby enabling robust physics-augmented impact localisation and physics-informed force reconstruction, as illustrated in Fig. 1. For completeness, the equations of motion governing symmetrically laminated FSDT plates are provided in Appendix A, with detailed derivations available in standard references [55, 56].

Physics-Regularised Impact Force Deconvolution

Figure 1: Physics-augmented impact identification and uncertainty quantification based on sparse data-driven FSDT modelling. The construction of the FSDT-based physics model is conducted in two sequential phases: 1. Material property identification, achieved by fitting the dispersion curve of the A0 mode; 2. Boundary condition calibration, performed by matching modal characteristics such as nat- ural frequencies and transmissibility.

These features are extracted from the reference dataset (Q, F(t), S(t)), which provides time difference of arrival (TDOA) data for dispersion analysis and modal information for boundary condition tuning. Once constructed, the FSDT model enables physics-informed impact identification through two

Integrated Pathways:

1. Physics-guided impact localisation: Dispersion analysis →Group velocity profile (GVP) →Synthetic data generation →Data-augmented impact localisation. 2. Physics-guided impact force estimation: Modal analysis →Transfer function estimation →Frequency-dependent adaptive regularisation →Force deconvolution.

For impact localisation, the estimated GVP is used to synthesise propagation-aware datasets, enabling a data-augmented multi-fidelity learning framework that enhances generalisability across the structural domain—even in regions without direct reference impacts.

Simultaneously, the FSDT model serves as a forward solver for transfer function prediction, facili- tating uncertainty quantification in the radial basis function (RBF) interpolation of transfer functions . These quantified interpolation errors inform a frequency-adaptive regularisation strategy, which significantly improves the accuracy of impact force reconstruction when compared to conventional ℓ2 regularisation [57, 58] with a fixed penalty parameter.

FSDT modelling phase 1: elastic property identification Maximum likelihood estimation of elastic properties The dispersion relations of a structure are inherently governed by its elastic properties while remaining independent of its boundary conditions. This property enables a hierarchical approach to identifying both structural elastic properties and boundary conditions using sparse reference impact data. Given that low-velocity impacts predominantly generate flexural waves [59, 60], this study focuses specifically on the analysis of out-of-plane flexural wave propagation. For symmetrically laminated composites, the dispersion relation of flexural wave group velocity is formulated as a function of the structural stiffness parameters D, As and inertia terms I1, I3, as detailed derived in Appendix B:

(2)

where vg(ω, θ) represents the group velocity of out-of-plane flexural waves, dependent on wave fre- quency ω and wave direction θ due to wave dispersion and structural anistropy. The bending stiffness matrix D, share stiffness matrix As and the inertia terms I1, I3 are detailed in [55, 61–63].

The group velocity profile (GVP) vg is governed by the plate-level material properties Θp, which can be further derived from the lamina-level properties Θl. For most-widely used transversely isotropic lamina, these lamina-level properties include the elastic constants of individual plies E1, E2, G12, ν12, ν23, the lamina thickness t and the stacking sequence Ψ: Θl = {ρ, Ψ, t, E1, E2, G12, ν12, ν23} →Θp = {D, As, I1, I3}.

(3)

Here, Ψ is a vector of 2Nl elements that define the laminate stacking configuration, while the total plate thickness is given by 2Nltply. The relationship between reference impact data and wave dispersion is established through time difference of arrival (TDOA) measurements as follows:

(4)

where ∆t1j(ω) represents the extracted TDOA between the j-th sensor and the i-th sensor, and vg(θj|ω, Θl) denote the wave propagation group velocity at frequency ω from the impact point to the j-th sensor. The uncertainties in TDOA, denoted e(ω), are random variables influenced by wave frequency ω, typically modeled as Gaussian with zero mean and variance σ2(ω) .

Since plate-level properties Θp primarily capture out-of-plane wave behaviour and do not account for in-plane dynamics, a more comprehensive identification of lamina-level properties Θl is necessary. Given the sensor location Lj = (xj, yj), independent reference impact locations Qi = (xi, yi) and TDOA estimates ∆ti(ω) from sensor measurements, the material properties at the lamina level can be estimated using a multi-frequency maximum log-likelihood approach :

(5)

where ζ denotes the probability density function of a standard Gaussian distribution, and ˆσ2(ωm) is the estimated variance of TDOA at frequency ωm, obtained by maximising the likelihood function. The likelihood function Lm at each frequency ωm is derived based on the wave propagation formulation in Eq. (4).

This multi-frequency identification approach leverages the full dispersion curve, reducing depen- dency on extensive reference datasets and large sensor arrays. Consequently, it enhances the feasibil- ity of sparse data-driven applications in SHM by improving the accuracy and robustness of material property estimation while minimising experimental overhead.

Esign Space Of Elastic Properties Of Cfrp Lamina

In the maximum likelihood estimation of the elastic properties of CFRP lamina based on dispersion relations, the definition of the probability spaces for these properties is crucial. CFRP composite lamina are anisotropic, meaning their mechanical properties vary depending on the direction relative to the fibre orientation. The longitudinal modulus, E1 (along the fibre direction), is significantly higher than the transverse modulus, E2 (perpendicular to the fibres).

For Unidirectional Cfrp

lamina, E1 typically ranges from 120 to 250 GPa , encompassing low-modulus lamina such as T800H fibre/epoxy , high-modulus lamina such as GY-70/epoxy , and high-tenacity lamina such as T300/T400/T700S fibres/epoxy and AS4/AS6 fibres/epoxy .

The ratio E2/E1, which depends on both the fibre and matrix properties, typically ranges from 0.05 to 0.1 for CFRP . Another key ratio, G12/E1, representing the in-plane shear modulus G12, generally falls between 0.02 and 0.05 due to fibre-matrix interactions. These ranges align with values observed in widely used unidirectional lamina . Collectively, these bounds define the probability space for the elastic properties of CFRP lamina, summarised in Table 2.

Table 2: Probability space of the elastic properties of CFRP lamina and stacking sequence

Dlb

*Note: UB: upper bound, LB: lower bound E1: longitudinal modulus along the fibre direction, E2: transverse modulus perpendicular to the fibre direction, G12: in-plane shear modulus, ν12: in-plane Poisson’s ratio, ρ: material density, d: plate equivalent thickness, Nl: half of lamina number, tply: thickness of lamina.

The in-plane Poisson’s ratio, ν12, represents the ratio of transverse to longitudinal strain under uniaxial loading along the fibre direction. While carbon fibres themselves exhibit a low Poisson’s ratio (νfibre ≈0.2 −0.3, table 4.4-4 in ), their influence on ν12 is minimal, as the transverse response is primarily governed by the matrix. The epoxy matrix, which deforms more in the transverse direction, typically has a Poisson’s ratio of νmatrix ≈0.3 −0.4 (table 3.2-6 in ). For aerospace-grade CFRP (fibre volume fraction 60%-70%, table 4.4-5 in ) and commercial-grade CFRP (50%-60%), ν12 generally falls between 0.25 and 0.35.

The density of CFRP lamina or laminate is determined by the volume-weighted average of fibre and matrix densities. Due to the higher fraction of fibres, they predominantly influence the composite density. Standard carbon fibres and high-modulus carbon fibres have densities of 1750-1900 Kg/m3 and 1900-2000 Kg/m3, respectively (table 3.3-1 in ), while commonly used epoxy resins have lower densities, typically between 1100 and 1300 Kg/m3 (table 3.2-7 in ). Consequently, for fibre volume fractions of 50%-70%, the overall density of CFRP lamina typically ranges from 1500 to 1700 Kg/m3.

The equivalent plate thickness is bounded based on the plate type. As illustrated in Fig. 2, for a flat plate of thickness d, the equivalent thickness is constrained within [d−ε, d+ε], where ε represents measurement uncertainty. For a sandwich plate of total thickness d and face sheet thickness dfs, the equivalent thickness falls within [2dfs, d]. For a stringer-stiffened plate, the equivalent thickness is naturally bounded between the thickness of the unstiffened region and that of the stringer toes.

(C)

Figure 2: Three type of laminated composite plates with bounded thickness: (a) flat plate, (b) sandwich plate, (c) stringer-stiffened plate. The stacking sequence of the lamina is a key factor in defining the structural anisotropy and mechanical performance of composite laminates. In this study, four of the most commonly used ply orientations in practical applications are considered: 0◦, 45◦, −45◦, and 90◦. This selection simplifies the design space of the stacking sequence vector Ψ while maintaining practical relevance. Variations in fibre orientation across individual plies influence both the in-plane and out-of-plane stiffness of the laminate, thereby altering its dispersion relations. The defined parameter spaces for material properties and lamina layup provide a structured basis for the data-driven identification of elastic properties.

The material property identification process, as formulated in Eq. (5), is a mixed-integer optimi- sation problem, where the four candidate ply angles are represented as discrete integer values ranging from 1 to 4. Given that the fitness function, defined as the log-likelihood, can be evaluated at rel- atively low computational cost—primarily involving calculations of location-to-sensor distances and wave propagation velocities—a mixed-integer genetic algorithm is employed to efficiently solve this optimisation problem and determine the optimal material properties.

FSDT modelling phase 2: boundary condition identification The identification of the elastic properties of laminated composites facilitates the subsequent deter- mination of boundary conditions based on modal characteristics.

Forward modal analysis by Rayleigh-Ritz variational method with general bound-

Ary Conditions

Based on the constitutive equations of FSDT plate, the strain energy U, kinetic energy T, and potential energy V of external forces for a symmetric plate subjected to a transverse impact load f(t)

(6)

where Ωrepresents the structural spatial domain in the x-y plane, ϵ0, κ and γ are vectors of midplane strains, plate curvatures, and shear strains, respectively. The boundary conditions also store energy during impact events, depending on their nature.

N Real-World Structures, Boundary Conditions

are rarely purely free, simply supported, or fully clamped. To model general boundary conditions, artificial springs are introduced to simulate the shear force Vs and bending moment Mb acting on the boundary Γ. This is expressed in terms of the spring stiffness as follows:

(7)

where kt and kr denote the translational and rotational stiffness, respectively. Here, d = [u0, v0, w0, φx, φy]T represents the displacements resisted by the springs, and −→

N Is The Tangential Direction Along The

boundary. Consequently, the additional strain energy stored in the elastic boundary is given by:

(8)

Using the Rayleigh-Ritz variational approximation approach, the displacements are assumed to take

The Form:

d = [u0, v0, w0, φx, φy]T = [U0BT , V0BT , W0BT , ΦxBT , ΦyBT ]T = D ⊗BT ,

(9)

where the basis function bj(x) is chosen as Legendre polynomials due to their favourable convergence properties , and D represents the coefficients of these basis functions. The operator ⊗denotes the block Kronecker product. Substituting Eq. (9) into Eq. (6) and Eq. (8), and applying Hamilton’s principle, the equations of motion are obtained as:

(10)

where M and F denote the generalised mass matrix and force vector, respectively. Kp, Kb are the stiffness matrixes for the plate structure, boundary conditions, respectively. The expressions of these matrices are detailed in Appendix C. Solving the eigenvalue problem by letting F = 0 in Eq. (10) leads to the estimation of structural mode shapes and natural frequencies.

Boundary condition identification by matching modal characteristics What modal information can be extracted from the reference impact data Q, F(t), S(t)? By analysing the structural response by mode superposition method, a key outcome is the identification of signif- icant lower-mode natural frequencies. In addition to natural frequencies, transmissibility serves as another modal property that can be inferred from reference impact data. Two types of transmissibility are considered: sensor transmissibility and excitation transmissibility.

Sensor transmissibility is defined as the ratio of the spectral amplitudes of responses recorded by two sensors at a given natural frequency when the structure is excited by an impact.

When

piezoelectric (PZT) sensors monitor impacts and record strain responses, the sensor transmissibility between sensor j and sensor n across all reference impacts i = 1, ..., Ni is given by:

(11)

where ˆs(ω; qi, Lj) and ˆh(ω; qi, Lj) denote the spectral response and transfer function at sensor location

J When Excited At Qi, Respectively. Here, Φu

m denotes m-th mode shape in term of u-displacement, and

M

∂x (Lj) is its spatial derivative with respect to the x, evaluated at sensor location Lj. Essentially, sensor transmissibility reflects the ratio of mode shape gradients. However, sensor characteristics, including frequency response variations due to degradation or poor attachment, may introduce errors in the transmissibility estimate.

To address this limitation, excitation transmissibility is defined based on transfer functions and depends on the availability of reference impact data rather than sensor characteristics. Considering two independent impacts at locations qi and qk, excitation transmissibility is defined as the ratio of the spectral transfer functions ˆh(ω; qi, L) and ˆh(ω; qk, L) at natural frequency ωm, evaluated across

(12)

For both PZT sensors and accelerometers, excitation transmissibility directly corresponds to the ratio of mode shapes. This definition provides a more robust means of estimating modal properties, particularly in cases where sensor performance may be affected by degradation or attachment quality.

By leveraging both natural frequencies and transmissibility, boundary conditions can be identified by minimising the discrepancies between the extracted modal properties and those predicted using the Rayleigh-Ritz method. This optimisation problem is formulated as:

(13)

where Θk represents the stiffness of artificial springs applied at the boundaries. Lnf and Let are the mean sqaure error loss of natural frequencies and excitation transmissibility, respectively. Nsm denotes the number of significant identified modes from reference impact data. By optimising the artificial spring stiffness, the mode shapes and natural frequencies of the FSDT plate can be determined, facilitating the process of impact force identification.

Physics-guided impact localisation and uncertainty quantification The GVPs vg(θ; ω) are identified by optimising the structural material properties in the FSDT mod- elling, allowing the generation of extended data and facilitating impact localisation.

Multi-fidelity Gaussian process regression for impact localisation The integration of physics-generated data enhances the generalisability of the model. By leveraging the identified GVPs vg(θ; ω), from the FSDT model, low-fidelity synthetic TDOA data ∆tF SDT (ωm) can be generated across the entire structure.

Since These Gvps Are Inferred From Experimental

data, the simulated TDOAs maintain an intermediate level of accuracy, capturing the primary wave propagation characteristics while exhibiting some deviations from high-fidelity experimental data. To improve the reliability of impact localisation, a multi-fidelity GPR framework is employed, which integrates both simulated and experimental data. This approach enables accurate localisation across the structure by modelling the high-fidelity output as a scaled low-fidelity prediction with an

(14)

where yLF (x) represents the low-fidelity model trained on the synthetic TDOA data ∆tF SDT (ωm). The scaling factor α adjusts for systematic discrepancies between low- and high-fidelity data, while the correction term δ(x) is modelled as a GP trained on sparse high-fidelity TDOA data from experimental measurements. This multi-fidelity learning framework effectively compensates for the limitations of the low-fidelity model, enabling robust and accurate impact localisation across the structure.

To effectively fuse multi-frequency TDOA data within a single GPR model, a tailored multi- frequency kernel is designed as a summation of uni-frequency radial basis function (RBF) kernel:

(15)

The RBF kernel krbf contains a length-scale parameter: lrbf. A key consideration is whether these length scales should be independently modelled for each frequency, particularly since TDOAs at higher frequencies tend to have lower magnitudes, significantly affecting the kernel values for a fixed lrbf.

However, increasing the number of hyperparameters in the GPR model elevates model complexity and raises the risk of overfitting. The simulated impact locations are typically derived from structured experimental designs, such as uniform sampling over a 20 mm by 20 mm grid. Given the intermediate dataset size, reducing model complexity is crucial to achieving high generalisability. Therefore, it is preferable to model the multi-frequency composite kernel using a shared length scales lrbf. To facilitate this, normalisation preprocessing of the TDOA data is required to ensure comparability across different frequencies. This normalisation is achieved using the dispersion relations vg(ω; θ): ∆tnor(ωm) = ∆t(ωm)vg(ωm; θ = 0).

(16)

By combining physics-augmented data generation with multi-frequency kernel design, the proposed GPR model is expected to achieve accurate and robust impact localisation with enhanced generalis- ability while also providing uncertainty quantification.

Physics-guided impact force deconvolution and uncertainty quantifi-

Cation

The force reconstruction integrates the constructed FSDT model and high-fidelity (HF) reference im- pact data. This integration enables estimation of the transfer function ˆh(ω; q, L) at a target location q, facilitating physics-guided impact force deconvolution via adaptive frequency-domain regularisation.

Regularised force deconvolution in frequency domain Given the estimated transfer functions ˆh(ω; q, L) at the target location q, the impact force in the frequency domain can be reconstructed using the classical least-squares deconvolution:

(17)

where |·| represents the absolute operator, ˆs(ω; q, Lj) is the j-th sensor response in the frequency domain, and ˆh∗(ω; q, Lj) is the complex conjugate of the transfer function. However, this inversion problem is typically ill-posed that the possible presence of small values in ˆh(ω; q, L) can amplify noise and destabilise the solution. To mitigate this, regularisation techniques are employed to stabilise the inverse problem. A widely used stabilisation method is frequency-domain Tikhonov regularisation—also known as Wiener deconvolution [14, 57, 58]—which reformulates the inverse problem as a penalised least-squares optimisation:

(18)

where ∥·∥represent the Euclidean norm operator, λreg is the regularisation parameter that imposes a global smoothing effect. This penalisation yields a closed-form solution:

(19)

The regularisation parameter λreg controls the trade-off between fidelity to the measured signal and smoothness of the reconstructed force. Its selection is critical and can be automated using data-driven approaches such as the L-curve criterion [72, 73] or Generalised Cross Validation (GCV) [74, 75]. GCV, in particular, provides a principled and automated approach for selecting λreg without requiring ground truth information. The GCV functional for frequency-domain deconvolution across

(20)

with the optimal λreg obtained by minimising the above expression. While ℓ2 regularisation is effective in stabilising the inversion, it often leads to underestimation of the true force magnitude , especially at low frequencies where penalisation is overly aggres- sive. To mitigate this, frequency-dependent regularisation schemes are adopted, which allow adaptive penalisation across the spectrum and improve reconstruction fidelity.

In addition to frequency-domain ℓ2-based methods, several time-domain techniques exploit the temporal sparsity of impact forces, such as sparse regularisation (ℓ1-based) [77, 78] and wavelet-based regularisation [79, 80]. These methods offer better temporal resolution and robustness, especially when only a few sensor signals are available. Nonetheless, both ℓ1 and ℓ2 regularisers tend to under- estimate force magnitudes , and their accuracy is constrained when frequency response variability is significant.

Fidelity of transfer function estimation from FSDT and reference impact data The data-driven FSDT model is calibrated via sequential estimation of material properties and bound- ary conditions. However, it cannot perfectly replicate sensor responses to actual impacts due to several

Limitations:

• Model simplifications: The FSDT plate model neglects transverse normal strain and assumes a constant transverse shear strain through the plate thickness.

Additionally, The Boundary

conditions are modelled using uniform spring stiffness along each edge. These simplifications limit its ability to fully capture the real-world structural dynamics. • Sparse reference data: The accuracy of material property and boundary condition identification depends on the availability of reference data, which may be insufficiently dense to ensure precise reconstruction.

• Sensor frequency responses function variability: The sensor’s inherent frequency responses func- tion is affected by factors such as degradation and attachment quality, introducing further discrepancies.

Therefore, while the FSDT model provides a low-fidelity (LF) approximation of the transfer function, reference impacts offer high-fidelity (HF) information. Impact force reconstruction must prioritise HF data, with LF models used to provide physical constraints or regularisation guidance.

Strategies for multi-fidelity transfer function estimation To estimate the transfer function ˆh(ω; q, L) at a new location q, the following strategies combine LF

Physics And Hf Reference Data:

1. Physical prior approach: Utilise the transfer function estimated by the FSDT model as an LF physical prior, while the HF reference impact data are employed to model the residual between the physical prior and the reference measurements.

2. Modal kernel design: Incorporate physical modal characteristics into GPR or RBF interpolation . For instance, integrating modal similarity into the GPR kernel ensures that interpolation is influenced not only by spatial proximity but also by modal resemblance, thereby improving accuracy.

3. RBF interpolation with adaptive regularisation: Perform interpolation/extrapolation of the transfer function separately using both HF reference transfer functions and LF FSDT transfer functions. An adaptive regularisation factor is introduced, defined based on the discrepancy between the interpolated FSDT results and theoretical FSDT predictions.

The choice among these depends on the FSDT model’s fidelity and reference impact coverage: • Strategy 1 is preferable for accurate FSDT models with sparse reference data. • Strategy 2 improves interpolation accuracy but requires careful kernel design and a reliable modal basis.

• Strategy 3 offers a flexible compromise when FSDT accuracy is moderate and HF data are sparse. In this work, Strategy 3 is adopted due to the intermediate-to-low accuracy of the data-driven FSDT model. While the LF model cannot be used directly as a prior mean or to calibrate HF estimates, it can still provide valuable information about the spatial complexity of local dynamics and the frequency- dependent quality of interpolation. Therefore, it is utilised to guide adaptive regularisation in force deconvolution.

Transfer function estimation and force deconvolution with adaptive regularisation Let the HF and LF transfer function estimates at the reference impact locations Q be denoted by:

• ˆH(Ω; Q, L): From Experimental Data;

• ˆHF SDT (ω; Q, L): from the FSDT model. Using RBF interpolation, the estimated transfer functions at a new location q are denoted as: • ˆhRBF (ω; q, L): interpolated from HF reference data,

• ˆHrbf

F SDT (ω; q, L): interpolated from LF FSDT data. The explicit RBF formulations for these interpolants are provided in Appendix D.2, following the methodology in . Briefly, each interpolant combines a weighted sum of radial basis functions and a low-order polynomial to ensure well-posedness and smoothness:

(21)

where ψ(·) represents the chosen radial basis function, and pj(·) denotes a polynomial term included to ensure the well-posedness of the RBF interpolation. The theoretical transfer function predicted by the FSDT model at location q is denoted as ˆhF SDT (ω; q, L). The discrepancy between the RBF-interpolated transfer function and the theoretical FSDT prediction is quantified by their difference:

(22)

where cumsum denotes the cumulative sum operator applied across the frequency range. This adap- tive, frequency-dependent regularisation term, λreg(ω), reflects the degree of mismatch between the physics-based (FSDT) model and the data-driven RBF interpolation. By incorporating this discrep- ancy into the regularisation term of the frequency-domain deconvolution formulation (see Eq. (19)), the solution is penalised more heavily in frequency bands where the interpolation error is high, thereby improving the robustness and accuracy of impact force reconstruction.

Experimental Validation By Impact Testing

Experimental impact tests were conducted to capture the structural response and impact force his- tory of a sensorised fibre-reinforced composite plate subjected to hammer impacts, as illustrated in Fig. 3(a). The composite plate, fabricated from M21/T800 prepreg material, measured 290 × 200 × 4 mm and had a quasi-isotropic layup of [0/ + 45/ −45/90]2s. During impact testing, this composite plate was clamped along its two longer edges, leaving the two shorter edges free. Impact excitation was generated using a PCB Piezotronics 086C03 hammer, which has a weight of 160 g. The layout of the composite plate and the hammer impact locations are illustrated in Fig. 3(b), where the impact points are marked as black solid squares. These impacts were applied at the vertices of a 20 × 20 mm grid. During each hammer strike, four PZT sensors recorded the structural responses, while the ICP quartz force sensor integrated within the hammer measured the impact force history. For reference, detailed reference elastic properties of M21/T800 lamina are listed in [53, 65]. As this study focuses on a purely data-driven approach, these reference material properties are used solely for the accuracy validation of the estimated properties.

(B)

Figure 3: Impact testing using handheld hammer.

Aterial Property Identification

As demonstrated in the authors’ previous study , the combination of four reference impacts and a sparse sensor network consisting of four PZT sensors enables the identification of GVPs with high accuracy. Given this effectiveness, the same configuration is expected to yield reliable estimates of the material properties. Considering the dominance of low-frequency modes in impact dynamics, material property identification is focused on the dispersion characteristics within the frequency range of 1–10 kHz. By matching the estimated dispersion relations with experimental measurements obtained from the four reference impacts, the material properties of the FSDT plate are identified through the maximisation of the log-likelihood function defined in .

As illustrated in Fig. 4(a), the log-likelihood function is maximised based on four reference impacts numbered 12, 14, 22, 24 shown in Fig. 3(b), across three frequency ranges, where Nf denotes the number of frequency points considered, spanning from 1 kHz to Nf kHz in 1 kHz increments. With 100 generations in the GA, all three cases (Nf = 2, Nf = 5, and Nf = 10) converge to a stable maximum.

(C)

Figure 4: Material properties identification with four reference impacts. The comparison between the GVPs derived from the optimised material properties and those corresponding to the reference properties is presented in Fig. 4(b), where the reference GVPs are depicted as solid magenta lines and the optimised GVPs as dashed lines. Among the three cases, the Nf = 10 optimisation yields the most accurate GVPs, as the inclusion of a wider frequency range allows the estimated GVPs to align more closely with those obtained from the reference material properties.

A noticeable trend is observed: as the frequency increases, the reference GVPs tend to be higher than the optimised ones. This discrepancy can be attributed to two main factors: (1) the FSDT model inherently overestimates GVPs at higher frequencies compared to experimental measurements, and (2) at lower frequencies, where the TDOA values are larger, the optimisation process prioritises the low-frequency components to maximise the likelihood function. The comparison of dispersion relations at θ = 0◦between the optimised and reference models is illustrated in Fig. 4(c).

For

Nf = 2 and Nf = 5, significant deviations from the reference dispersion relations occur at higher frequencies, whereas for Nf = 10, the estimated dispersion relations remain more consistent across the entire frequency range. This improvement can be attributed to the integration of a wider frequency spectrum in the optimisation, leading to enhanced accuracy in the high-frequency range.

Despite the observed discrepancies between the reference and optimised GVPs, particularly at higher frequencies, the optimised GVPs remain highly accurate for impact localisation, especially in the low-frequency range. This is because they are directly optimised based on reference experimental impact data, ensuring reliability in practical applications.

Table 3 summarises the identified plate-level material properties for different frequency ranges. The reference values (REF) represent the known material properties of the composite plate, while the optimised values correspond to different Nf cases. The density (ρ) and plate thickness (d) remain relatively consistent across the cases, exhibiting only slight variations due to the optimisation process.

However, the bending stiffness components (D11, D22, and D66) show more pronounced differences, particularly for D11, which exceeds the reference value. These variations highlight the inherent complexity of the optimisation problem, which features multiple local minima that yield dispersion relations and GVPs closely matching experimental data.

Given that experimental measurements may not perfectly align with the theoretical reference prop- erties due to factors such as manufacturing variations and measurement noise, achieving an exact match with the reference material properties is nearly impossible. Instead, the optimisation process converges to a local minimum that ensures highly accurate GVPs, particularly in the low-frequency range, where wave propagation characteristics are most critical for impact localisation.

Table 3: Identified plate-level material properties.

Boundary Condition Identification

The modal characteristics of a structure can be estimated from the responses of PZT sensors and transfer functions when the impact forces are known.

Mpact Forces, Such As Half-Sine Impulses,

exhibit their highest spectral amplitude in the low-frequency range. This amplifies the contribution of low-frequency components in the PZT sensor signals, making it easier to identify low-frequency modes from these signals compared to transfer functions.

N Contrast, Transfer Functions Tend To

exhibit an energy shift towards higher frequencies, facilitating the identification of higher-frequency modes. To ensure that each reference impact contributes equally to the modal identification process, the spectra and transfer functions of the reference impacts are normalised before averaging. Fig. 5(a) illustrates mode identification using PZT sensor signals and transfer functions obtained from the sparse reference impact data. A comparison of the spectral distributions of both the PZT sensor response and the transfer function (TF) reveals that lower-frequency components (below 2000 Hz) dominate the PZT sensor responses.

The relatively low spectral amplitude at higher frequencies makes it challenging to accurately identify high-frequency modes from PZT sensor signals alone. In contrast, the transfer function exhibits a spectral energy shift towards higher frequencies, enabling the identification of high-frequency modes while struggling to capture the lowest mode due to a small peak.

(B)

Figure 5: Mode identification from PZT sensor signals and transfer functions. The PZT sensor response and the transfer function provide complementary modal information. As shown in Fig. 5(b), combining both methods allows for the identification of modes across a broader frequency range, capturing both low-frequency and high-frequency modes more effectively.

This

combined approach enhances the accuracy of mode identification, which is crucial for excitation transmissibility estimation and subsequent boundary condition identification. As shown in Fig. 3, the composite plate was clamped along its longer edges (though not perfectly) while the shorter edges remained free. To model these boundary conditions in a simplified manner, artificial translational and rotational springs were introduced at the plate edges. The design variables consist of four spring stiffness values, one for each edge, with the assumption that the translational and rotational stiffnesses are identical for a given edge.

Although the two shorter edges of the experimental plate are nominally free, their stiffness values were still included in the model, constrained within the range [0, 0.5].

Eanwhile, The Stiffness

values of the two clamped edges were bounded within [0, 1]. To ensure numerical stability and avoid excessively large stiffness values in optimisation, the stiffness parameters were normalised using the

(23)

where a maximum real stiffness value of 1e10 corresponds to a perfectly clamped edge. Beyond this value, further increases in stiffness do not affect the simulated natural frequencies in the FSDT model. Fig. 6 presents the Pareto front of the minimisation process, balancing the natural frequency loss Lnf and excitation transmissibility loss Let. The Pareto front exhibits an L-curve, where significant variations in one loss function occur along the two distinct edges of the ’L’, while the other loss remains nearly unchanged. This characteristic suggests that the optimal solution should be selected at the transition point of the L-curve, where a balanced trade-off between the two loss functions is achieved.

Figure 6: Boundary condition identification. At the transition point, marked by the red circle in Fig. 6, the identified normalised stiffness values for the two longitudinal edges (clamped but not perfectly rigid) were 0.687 and 0.688. These correspond to real stiffness values more than 100 times greater than those of the two shorter edges, which were nominally free. This indicates that the longitudinal edges serve as the effective boundary conditions, while the shorter edges have a negligible influence. The identified boundary condition configuration aligns well with the experimental setup, thereby validating the effectiveness of the proposed boundary condition identification method.

To further validate the identified boundary conditions, Table 4 presents the natural frequencies of the first 10 modes, extracted from the reference impact data and compared with those identified using the optimised FSDT model. The results indicate an average discrepancy of approximately 80 Hz between the experimentally extracted and model-predicted natural frequencies. This frequency deviation suggests that while the identified boundary conditions provide a reasonable approxima- tion, the transfer function estimated from the constructed FSDT model may exhibit slight peak shifts compared to the experimentally obtained transfer function from reference impact data. These discrepancies may arise from unmodelled factors such as localised boundary flexibility, material in- homogeneities, or additional damping effects, which could be further refined in future studies.

Table 4: Identified natural frequencies.

Estimated ¯Ωm (Hz)

Impact localisation based on physics-augmented multi-fidelity GPR Based on sparse experimental reference impact data and the identified GVPs (approximated physics), a GPR model was constructed and trained to locate a total of 35 experimental impacts on the composite plate.

The approximated physics enables the generation of low-fidelity (LF) physics- augmented data, which provides additional training information for the GPR model. For the target structure, these physics-augmented data points were generated using uniform sampling on a 20 × 20 mm grid across the structure, ensuring comprehensive spatial coverage.

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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