Resin Transfer Molding Rtm Ansys

By means of SMC we construct a benchmark against which we compare performance of a novel regularizing ensemble Kalman algorithm (REnKA) that we propose to approximate the posteriors in a computationally efficient manner under practical scenarios. We investigate the robustness of the proposed REnKA with respect to tuneable parameters and computational cost. We demonstrate advantages of REnKA compared with SMC with a small number of particles.

We further investigate, in both the one-dimensional and two-dimensional settings, practical aspects of REnKA relevant to RTM, which include the effect of pressure sensors configuration and the observational noise level in the uncertainty in the log-permeability quantified via the sequence of Bayesian posteriors.

Key word. Bayesian inverse problems, moving boundary problems, Sequential Monte Carlo method, ensemble Kalman meth- ods, Resin Transfer Molding. 1. Introduction. In this paper we study the Bayesian inverse problem within the moving boundary setting motivated by applications in manufacturing of fiber-reinforced composite materials. Due to their light weight, high strength, as well as their flexibility to fit mechanical requirements and complex designs, such materials are playing a major role in automotive, marine and aerospace industries [4, 3, 26]. The moving boundary problem under consideration arises from Resin Transfer Molding (RTM) process, one of the most commonly used processes for manufacturing composite materials.

Rtm Consists Of The

injection of resin into a cavity mold with the shape of the intended composite part according to design and enclosing a reinforced-fiber preform previously fabricated. The next stage of RTM is curing of the resin-impregnated preform, which may start during or after the resin injection. Once curing has taken place, the solidified part is demolded from the cavity mold. In the present work we are concerned with the resin injection stage of RTM under the reasonable assumption that curing starts after resin has filled the preform. Though the current study is motivated by RTM, the results can be also used for other applications where a moving boundary problem is a suitable model.

We now describe the (from the inverse problem prospective, forward) model (see further details in [3, 46, 37]). Let D∗⊂Rd, d ∈{1, 2}, be an open domain representing a physical domain of a porous medium with the permeability κ(x) and porosity ϕ. The boundary of the domain D∗is ∂D∗= ∂DI ∪∂DN ∪∂DO, where ∂DI is the inlet, ∂DN is the perfectly sealed boundary, and ∂DO is the outlet.

The domain D∗is initially filled with air at a pressure p0. This medium is infused with a fluid (resin) with viscosity µ through an inlet boundary ∂DI at a pressure pI and moves through D∗occupying a time-dependent domain D(t) ⊂D∗, which is bounded by the moving boundary Υ(t) and the appropriate ∗Submitted to the editors DATE.

A. Iglesias, M. Park And M.V. Tretyakov

parts of ∂D. An example of the physical configuration of this problem in 2D is illustrated in Figure 1.1.

No Flow

Figure 1.1. An example of the physical configuration of the moving boundary problem. The forward problem for the pressure of resin p(t, x) consists of the conservation of mass

(1.2)

with the following initial and boundary conditions

(1.9)

Here V (t) is the velocity of the moving boundary Υ(t) in the normal direction, n(x) and n(x, t) are the unit outer normals to the corresponding boundaries. We note that in the considered one and two dimensional cases of this problem, we can view the velocity of the moving boundary as the following

(1.10)

We remark that for definiteness we have assumed that at the initial time the moving boundary Υ(0) coincides with the inlet boundary ∂DI and that the constant pressure condition is imposed at the inlet. It is not difficult to carry over the inverse problem methodology considered in this paper to other geometries and other conditions on the inlet (e.g. constant rate). Further, in two (three) dimensional RTM settings

Bayesian Inversion In Resin Transfer Molding

one usually models permeability via a second (third)-order permeability tensor to take into account anisotropic structure of the media [3, 37] but here for simplicity of the exposition the permeability κ(x) is a scalar function. Again, the developed methodology is easy to generalize to the tensor case.

Let us note that in the one-dimensional case the nonlinear problem (1.1)-(1.9) is analytically simple and admits a closed form solution (see Section 2 and ) but the two and three dimensional cases are much more complicated and analytical solution is in general not available. We remark that in two and three dimensional cases the resin can race around low permeability regions and the front Υ can become discontinuous creating macroscopic voids behind the main front (see further details in [3, 37]) but in this paper we ignore such effects which deserve further study.

It has been extensively recognized [13, 14, 30, 29, 36, 41] that imperfections in a preform that arise during its fabrication and packing in the molding cavity can lead to variability in fiber placement which results in a heterogenous highly-uncertain preform permeability. In turn, these unknown heterogeneities in permeability of the preform give rise to inhomogeneous resin flow patterns which can have profound detrimental effect on the quality of the produced part, reducing its mechanical properties and ultimately leading to scrap.

To limit these undesirable effects arising due to uncertainties, conservative designs are used which lead to heavier, thicker and, consequently, more expensive materials aimed at avoiding performance being compromised. Clearly, the uncertainty quantification of material properties is essential for making RTM more cost-effective. One of the key elements in tackling this problem is to be able to quantify the uncertain permeability.

In this work we assume that D∗, ∂DO, ∂DI, ∂DN, pI, p0, µ and ϕ are known deterministic parameters while the permeability κ(x) is unknown. Our objective is within the Bayesian framework to infer κ(x) or, more precisely, its natural logarithm u(x) = log κ(x) from measurements of pressure p(x, t) at some sensor locations as well as measurements of the front Υ(t), or alternatively, of the time-dependent domain D(t) at a given time t > 0. We put special emphasis on computational efficiency of the inference, which is crucial from the applicable point of view.

1.1. Practical approaches for permeability estimation in fiber-reinforced composites. While the estimation of preform permeability during resin injection in RTM is clearly an inverse problem constrained by a moving boundary PDE such as (1.1)-(1.9), most existing practical approaches pose the estimation of permeability in neither a deterministic nor stochastic inverse problems framework. For example, the very extensive review published in 2010 reveals that most conventional methods for measuring permeability assume that (i) the material permeability tensor is homogenous and (ii) the flow is one-dimensional (including 2D radial flow configurations). Under these assumptions the resin injection in RTM can be described analytically, via expressions derived from Darcy’s law, which enable a direct computation of the permeability in terms of quantities that can be measured before or during resin injection. These conventional methods suffer from two substantial practical limitations. First, they do not account for the heterogenous structure of the preform permeability, and although they provide an estimate of an effective permeability, this does not enable the prediction of the potential formation of voids and dry spots. Second, those conventional methods compute the permeability in an off-line fashion (i.e before RTM) with specific mold designs that satisfy the aforementioned assumptions intrinsic to those methods (e.g. rectangular flat molds). This second limitation is not only detrimental to the operational efficiency of RTM but also neglects the potential changes in permeability that can results from encapsulating the preform in cavities with complex designs.

Some practical methodologies for online (i.e. during resin injection) estimation of heterogenous per- meability have been proposed in [35, 48]. While these approaches seem to address the aforementioned limitations of conventional methods, they also use a direct approach for the estimation of permeability which faces unresolved challenges. As an example, let us consider the recent work of which uses an experimental configuration similar to the one described in Figure 1.1 and which, by using pressure mea-

A. Iglesias, M. Park And M.V. Tretyakov

surements from sensors located within the domain occupied by the preform, computes a finite-difference approximation of the normal flux to the front ∇p(Υ(t), t) · n. In addition, by means of images from CCT cameras, seepage velocity of the resin front is computed in ; this velocity is nothing but V (x, t) defined by (1.5) in the context of the moving boundary problem (1.1)-(1.9). Under the assumption that µ and ϕ are known, the approach proposed in consists of finding

(1.11)

with V (x, t) and ∇p(Υ(t), t) · n computed from measurements as described above. This approach of- fers a practical technique to estimating κ on the moving front and can then potentially infer the whole permeability field during the resin injection in RTM. However, from the mathematical inverse problems perspective, this ad-hoc approach is not recommended as it involves differentiating observations of pres- sure data for the computation of ∇p(Υ(t), t) · n. Indeed, it is well-known that differentiation of data is an ill-posed problem that requires regularization. In addition, rather than an inverse problem, the least-squares formulation in (1.11) is a data fitting exercise that excludes the underlying constraint given by the moving boundary problem and which entails a global effect induced by κ. As a result, the estimate of permeability obtained via (1.11) has no spatial correlation and thus fails to provide an accurate global estimate of the permeability field.

The recent work of demonstrates considerable advantages of using systematic data assimilation approaches to infer permeability during the resin injection of RTM. By means of a standard ensemble Kalman methodology for data assimilation, the approach of uses measurements from visual obser- vations of the front location to produce updates of the preform permeability within the context of a discrete approximation of the moving boundary problem (1.1)-(1.9). While the methodology used in is focused in producing deterministic estimates, the standard Kalman methodology can be potentially used to quantify uncertainty in preform permeability. However, it has been shown that standard Kalman methodologies, such as the one used in , could result in unstable estimates unless further regularisation to the algorithm is applied .

In addition to the lack of an inverse problem framework that can lead to unstable and ultimately inaccurate estimates of the permeability in resin injection of RTM, most existing approaches (i) do not incorporate the uncertainty in the observed variables and (ii) do not quantify uncertainty in the estimates of the permeability of preform. It is indeed clear from our literature review that the estimation of permeability of preform during resin injection deserves substantial attention from an inverse problems perspective capable of quantifying uncertainty inherent to the fabrication and packing of the preform.

the Bayesian approach to inverse problems in order to infer the logarithm of the permeability

U(X) = Log Κ(X), From Observations {Yn}N

n=1 collected at some prescribed measurement/observation times

{Tn}N

n=1 during the resin injection in RTM. At each time tn we observe a vector, yn, that contains noisy measurements of resin pressure from sensors as well as some information of the moving domain (or al- ternatively front location) observed, for example, via CCT cameras or dielectric sensors . In the Bayesian approach, the unknown u(x) is a random function that belongs to a space of inputs X. A prior probability measure µ0(u) = P(u) on u must be specified before the data are collected; this enables us to incorporate prior knowledge which may include design parameters as well as the uncertainty that arises from preform fabrication (i.e. prior to resin injection). In our work we consider Gaussian priors which have been identified as adequate for characterizing the aforementioned uncertainty in log-permeability from the preform fabrication [49, 30, 29] (see also references therein).

At each observation time tn during the infusion of resin in RTM, we then pose the inverse problem in terms of computing, µn(u) = P(u|y1, . . , yn), the (posterior) probability measure of the log-permeability

Bayesian Inversion In Resin Transfer Molding

conditioned on measurements y1 . . , yn. Each posterior µn then provides a rigorous quantification of the uncertainty in the log-permeability field given all available measurements up to the time tn. Knowledge of each of these posteriors during RTM can then be used to compute statistical moments of the log- permeability under µn (e.g. mean, variance) as well as expectations of quantities of interest that may be needed for the optimization of controls (e.g. pressure injection) in RTM.

Although the proposed application of the Bayesian formulation assumes Gaussian priors, the nonlinear structure of the PDE problem, that describes resin injection in RTM, gives rise to a sequence of non-

Gaussian Bayesian Posteriors {Μn}N

n=1 which cannot be characterized in a closed form.

A Sampling

approach is then required to compute approximations of these posteriors.

Among Existing Sampling

methodologies, Sequential Monte Carlo (SMC) samplers [33, 8, 1, 21] are particularly relevant for the formulation of the above described inverse problem as they provide a recursive mechanism to approximate

Starting With J Samples From The Prior U(J)

∼µ0, j = 1, . . , J (i.i.d.), the idea behind SMC is to trans-

J=1

that approximates µn as the new data yn collected at time tn become available. The weights {W (j)

(1.12)

converges to µn as J →∞(δw denotes the Dirac measure concentrated at w). Moreover, if f(u) denotes a quantity of interest of the unknown log-permeability u(x), the weighted particles {W (j)

(1.13)

which converges (see for example ) to the expectation (under µn) of the quantity of interest Eµn(f(u)). The recursive computation of the weighted particles in SMC is suitable for the proposed application in RTM as it allows us to update, potentially in real time, our knowledge of the uncertainty in the log-permeability. However, producing accurate approximations of the Bayesian posteriors {µn}J

N=1 In

the context of the inference of preform log-permeability in RTM represents a substantial computational challenge that arises from the fact that these posterior measures are defined on a (infinite-dimensional) functional space.

Upon discretization, these posteriors could be potentially defined on a very high- dimensional space. Unfortunately, it has been shown [9, 5] that standard Bayesian sampling methodologies such as standard SMC do not scale well with the dimension of the (discretized) unknown; this leads to The recent works [9, 21] developed scalable (dimension independent) sampling algorithms for the

While

these algorithms have a solid theoretical background that ensures their stability and convergence prop- erties, achieving a desirable level of accuracy often comes at extremely high computational cost. More specifically, Bayesian methodologies, that provide approximation of the form (1.12) and that converge asymptotically to the underlying posterior measure µn, often involve solving the forward model thousands or even millions of times. In the context of the inverse problem for RTM, the numerical solution of the moving boundary (forward) problem in 2D or 3D settings is computationally very intensive. Therefore, the sequential approximation of the Bayesian posteriors of preform’s log-permeability must be conducted

A. Iglesias, M. Park And M.V. Tretyakov

with scalable computational efficiency so that it can be realistically used within a near real-time optimiza- tion loop for RTM. In the proposed work we develop a computational inverse framework that possess such computational efficiency with the ultimate aim of the real-time uncertainty quantification of the reinforced preform’s log-permeability.

1.3. Contributions of this work. The contributions of this article are the following: (A) A Bayesian formulation of the inverse problem to infer log-permeability from sequential data collected during resin injection in RTM. Both the 2D forward model described by (1.1)-(1.9) as well as the corresponding 1D version are considered. For the 1D case, we show that application of the infinite-dimensional Bayesian framework of leads to well-posedness of the sequence of Bayesian posteriors.

(B) Application of a state-of-the-art SMC framework for the approximation of the sequence of

Bayesian Posteriors {Μn}J

n=1 that arises from the Bayesian formulation. From this SMC frame- work, we motivate a novel regularizing ensemble Kalman algorithm (REnKA) that aims at ap- proximating this sequence of posteriors in a computationally efficient manner, thus suitable for its implementation in a practical setting of RTM.

(D) Numerical investigation of the accuracy and robustness of the proposed REnKA scheme in the 1D case; this involves constructing, via the SMC sampler of , accurate approximations of the posteriors that we use as Benchmark against which we compare the proposed REnKA. The advantages of REnKA in terms of accuracy vs computational cost are showcased by comparing it with the implementation of a low-resolution SMC whose computational cost is comparable to REnKA’s.

(E) Application of REnKA for further investigations of the Bayesian inverse problem in both 1D and 2D. In particular for the 1D case we conduct a numerical investigation of the added value of assimilating the front location relative to the number of pressure sensors. Since the number of pressure sensors that can be physically deployed for preform permeability monitoring in RTM is usually limited, this investigation aims at providing practitioners with guidelines for the number of sensors that can accurately infer preform permeability alongside with its uncertainty. In addition, for the 1D case we study the effect of the frequency of the observations, as well as the observational noise level on the inference of the log-permeability. We further apply REnKA to the 2D forward model and, analogous to the 1D case, we study the effect that the number of pressure sensors have on the inferred log-permeability.

The rest of the paper is organized as follows. In Section 2 we introduce the Bayesian inverse problem of inferring the permeability of a porous media in a 1D moving boundary problem for resin injection in RTM. In Section 3 we discuss and apply SMC to approximate the Bayesian posteriors that arise from the Bayesian approach. In Section 4 we introduce REnKA and conduct a numerical investigation of its approximation properties relative to its computational cost. In Section 5 we apply REnKA to further investigate relevant practical aspects of the inverse problem in both 1D and 2D; this includes the study of the effect of the number of pressure sensors as well as the noise level on accuracy of the inferred log-permeability and its uncertainty. Some conclusions are presented in Section 6.

2. Bayesian inversion of a one-dimensional RTM model. In this section we apply the Bayesian approach to infer log-permeability in the context of the one-dimensional version of the forward problem defined in (1.1)-(1.9). The corresponding 1D moving boundary problem induces a sequence of forward maps that we define in Section 2.1 and that we aim at inverting with the Bayesian formalism that we introduce in Section 2.2.

This sequence of 1D forward maps admits a closed-form solution that can be numerically approximated at a very low computational cost.

This Will Enable Us In Section 3 To

obtained accurate numerical approximations of the solution to the Bayesian inverse problem; we use this accurate approximations as a benchmark for assessing the approximation properties of the ensemble

Bayesian Inversion In Resin Transfer Molding

Kalman algorithm that we introduce in Section 4 . 2.1. The Forward 1D RTM model. Let us consider a one-dimensional porous media with physical domain D∗≡[0, x∗] ⊂R. As before, we denote by κ(x) (x ∈D∗) and φ > 0 the permeability and porosity of the porous medium, respectively. Resin with viscosity µ is injected at x = 0 at a pressure pI. The pressure at the moving front (outlet) Υ(t) is prescribed and equal to p0. The initial pressure distribution before injection is also set to p0. For convenience of the subsequent analysis, we parameterize the permeability in terms of its natural logarithm u(x) ≡log κ(x). The pressure p(x, t) and the moving front Υ(t) are given by the solution to the following model

(2.5)

The solution to (2.1)-(2.5) can be obtained analytically by the following proposition (see [3, 46, 37]). Proposition 2.1. Given u ∈X ≡C[0, x∗], let us define

Z X

Fu(ξ)dξ. The unique solution Υ(t), p(x, t) of (2.1)-(2.5) is given for t ≥0 by

(2.8)

The quantity of interest arising from the RTM injection model is the so-called filling time: the time it takes the front Υ(t) to reach the right boundary of the domain of interest [0, x∗]. Filling time, denoted by τ ∗, is defined by Υ(τ ∗) = x∗. From (2.7) and the definition in (2.6) it follows that τ ∗is given by

(2.9)

Note that the parameters p0 and pI are prescribed control variables and thus known. In addition, we assume that µ and φ are known constants. As stated earlier, we are interested in the inverse problem of estimating the permeability, or more precisely its natural logarithm u(x) = log κ(x) given time-discrete measurements of the front location as well as the pressure from M sensors located at {xm}M m=1 ⊂[0, x∗].

We Denote By {Tn}N

n=1 the set of N observation times. For fixed (assumed known) parameters pI, p0, φ and µ, the solution to the PDE model (2.1)-(2.5) induces the nth forward map Gn : C[0, x∗] →RM+1

A. Iglesias, M. Park And M.V. Tretyakov

Given u(x) = log κ(x) ∈X, the evaluation of the forward map Gn(u) predicts the location of the front and the pressure at the sensor locations at the time t = tn. Since observation times are prescribed before the experiment, there is no assurance that for a given u, the corresponding filling time satisfies tn ≤τ ∗ for all n = 1, . . , N. In other words, the front could reach the right end of the domain before we observe it at time tn.

In a real experimental setting, the process stops at time τ ∗.

However, In The Inverse

problem of interest here, observation times are selected beforehand, and the search of optimal u’s within the Bayesian calibration of the nth forward map can lead to filling times greater than some observation times. In this case (tn > τ ∗), the definition (2.10) yields Gn(u) = [Υ(τ ∗), {p(xm, τ ∗)}M m=1]T .

The following theorem ensures the continuity of the forward map, which is necessary for justifying the application of the Bayesian framework in Section 2.2.2. Theorem 2.2. The forward map Gn : C[0, x∗] →RM+1 is continuous.

For the proof of this theorem, see Appendix A. In the following subsection we apply the Bayesian framework for inverse problems in order to invert observations of Gn(u).

Remark 2.1. We note that for the present work the porosity ϕ is an assumed known constant; our objective is to infer the log-permeability u(x) = log κ(x). However, the Bayesian methodology that we apply can be extended to the case where the unknown is not only log κ(x) but also ϕ, and can include the case where ϕ = ϕ(x) is a spatial function defined on the physical domain D∗.

2.2. The Bayesian Inverse Problem. Suppose that, at each observation time t = tn, we collect noisy measurements of the front location as well as pressure measurements from sensors. We denoted

N ∈R+ And Yp

n ∈RM, respectively. Our aim is to solve the inverse problem of estimating the log permeability u(x) = log κ(x) given all the data yp

N, Yυ

n up to time t = tn. We assume that the aforementioned observations are related to the unknown u(x), via the forward map

N And Ηp

n are realizations of Gaussian noise with zero mean and covariance ΓΥ

N ∼N(0, Γp

n) (i.i.d.). For simplicity we assume that both measurements of the front location and pressures are uncorrelated in time.

N Are

uncorrelated for all n = 1, . . , N.

(2.14)

Remark 2.2. Due to the nature of the RTM problem, we have that the pressure p(xm, t) at each sensor xm should increase with time as well as the fact that GΥ

N “Unphysical”. In Practice, Observations Need To

be post-processed before using them for the Bayesian inverse problem and unphysical yp

N Should Be

excluded. We leave the question of how to incorporate such a post-processing framework for future study. Here we follow the traditional point of view on data modeled via (2.11)-(2.12) and choose sufficiently

N, Yυ

n being unphysical is very low.

Bayesian Inversion In Resin Transfer Molding

We adopt the Bayesian framework for inverse problems where the unknown u(x) = log κ(x) is a random field and our objective is to characterize the sequence of distributions of u conditioned on the observations which we express as u|y1, . . , yn. In other words, at each observation time t = tn we aim at computing the Bayesian posterior µn(u) = P(u|y1, . , yn).

From This Distribution We Can Obtain

point estimates of the unknown that can be used in real time to, for example, modify controls (e.g.. pI). More importantly, as we stated in the Introduction, the aforementioned distribution enables us to quantify uncertainty not only of the unknown but also of quantities of interest that may be relevant to an optimization of resin injection in RTM.

Even though for the illustrative purposes the model presented in this section is discretized on a rela- tively low dimensional space (e.g. 60 cells), our aim is to introduce a general computational framework independent of the size of the discretized domain. We therefore consider an infinite-dimensional formu- lation of the Bayesian inverse problem for which the unknown u belongs to a functional space X. The discretization of the Bayesian inverse problem will be conducted at the last stage of the computational algorithm, when the posteriors are sampled/approximated. Thus, we are aiming at robust mesh-invariant 2.2.1. The Prior. For the Bayesian approach that we adopt in this work, we require to specify a prior distribution µ0(u) = P(u) of the unknown, before the data are collected. This distribution comprises all our prior knowledge of the unknown and may include, for example, the regularity of the space of admissible solutions to the inverse problem. For the present work we consider Gaussian priors which have been used to characterize the uncertainty in the (log) permeability that arises from the preform fabrication [49, 30, 29] (see also references therein). In particular, here we consider stationary Gaussian priors µ0 = N(u, C) with covariance operator C that arises from the Wittle-Matern correlation function

,

where Γ is the gamma function, l is the characteristic length scale, σ2

Is An Amplitude Scale And Kν Is

the modified Bessel function of the second kind of order ν. The parameter ν controls the regularity of the samples. It can be shown [11, 42] that, for any ν > 0, if u ∼µ0, then u ∈C[0, x∗] almost-surely, i.e.

µ0([0, x∗]) = 1. This requirement, together with the continuity of the forward map ensures the well-posedness of the Bayesian inverse problems as we discuss in the next subsection. In the context of composite preform’s permeability, it is natural to choose the mean u according to the log-permeability intended by the design of the composite part .

For computational purposes we use the prior to parametrize the unknown u in terms of its Karhunen-

(2.16)

with coefficients uk and where λk and vk are the eigenvectors and eigenfunctions of C, respectively. A random draw from the prior u ∼N(u, C) can then be obtained from (2.16) with drawing uk ∼N(0, 1) i.i.d.

2.2.2. The Posterior. From (2.13) and our Gaussian assumptions on the observational noise, it follows that for a fixed u ∈X, we have yn = Gn(u) + ηn ∼N(Gn(u), Γn). Therefore, the likelihood of

A. Iglesias, M. Park And M.V. Tretyakov

At a given time t = tn, the Bayesian posterior µn(u) = P(u|y1, y2, . . , yn) is defined by the following infinite-dimensional version of Bayes’s rule. Theorem 2.3 (Bayes Theorem ).

Et {Gs}N

s=1 be the sequence of forward maps defined by (2.10)

And Let {Ls(U; Ys)}N

s=1 be the corresponding likelihood functions (2.17). Let µ0 = N(u, C) be the prior distribution with correlation function (2.15). Then, for each n ∈{1, . . , N}, the conditional distribution of u|y1, · · · , yn, denoted by µn, exists. Moreover, µn ≪µ0 with the Radon-Nikodym derivative

(2.19)

Proof: The proof follows from the application of Theorem 6.31 in and the continuity of the forward

Maps (Theorem 2.2) On A Full Μ0-Measure Set X. □

Note that from our assumption of independence of η1, . . , ηn, the right hand side of (2.18) is the likelihood of y1, . , yn|u. Remark 2.3. Due to the assumption of independence between front location and pressure measure-

(2.21)

This enables us to define two particular cases of the inverse problem. The first case corresponds to the

Assimilation Of Only Pressure Measurements Yp

n, while in the second case only front location measurements

Yυ

n are assimilated. Similar arguments to those that led to Theorem 2.3 can be applied (with lβ

N)

instead of ls(u, yn)) to define the Bayesian posteriors µp

N Associated To These Two Bayesian Inverse

added value of assimilating observations of the front location with respect to assimilating only pressure measurements. 3. Approximating the posteriors via Sequential Monte Carlo method. In the previous section we have established the well-posedness of the Bayesian inverse problem associated to inferring the log- permeability in the one-dimensional moving boundary problem (2.1)-(2.5). The solution of this inverse problem is the sequence of posterior measures {µn}N

N=1 Defined By Theorem 2.3. As We Discussed In

Section 1, these posteriors cannot be expressed analytically and so a sampling approach is then required to compute the corresponding approximations. Note that the sampling of each posterior µn (n = 1, . . , N) can be performed independently by, for example, Markov chain Monte Carlo (MCMC) methods. However, we reiterate that, for the present application SMC samplers are rather convenient as they exploit the sequential nature of the considered inverse problem by enabling a recursive approximation of the posterior measures as new data (in time) become available. Such recursive approximations of the posterior could enable practitioners to update their probabilistic knowledge of preform’s log-permeability which is, in turn, essential to develop real-time optimal control strategies for RTM under the presence of uncertainty.

Recognizing that the inverse problem under consideration involves inferring a function potentially discretized on a very fine grid, it is vital to consider the application of SMC samplers such as the one

Bayesian Inversion In Resin Transfer Molding

introduced in , carefully designed for approximating measures defined on a high-dimensional space. In this section we review and apply this scheme for the approximation of the Bayesian posteriors {µn}N

N=1

that we defined in the previous section. The aims of this section are to (i) provide a deeper quantitative understanding of the accuracy of the fully-Bayesian methodology of with respect to its computational cost under practical computational conditions; (ii) provide a motivation for the proposed REnKA that we propose from this SMC sampler in Section 4; and (iii) define accurate approximations of {µn}N

N=1

which we use as a benchmark for testing our REnKA scheme. In Section 3.1 we briefly discuss the essence of the standard SMC that we then use in Sections 3.2-3.3 to review methodological aspects of the adaptive-tempering SMC sampler for high-dimensional inverse problems of . We then apply this SMC in Section 3.4 for the solution of the Bayesian inverse problem in the 1D case defined in the previous section. In Section 3.5 we assess practical limitations of the SMC.

3.1. Standard SMC for Bayesian inference. As we discussed in the Introduction, starting with the prior µ0, the objective of SMC is to recursively compute an approximation of the sequence of Bayesian

Posteriors {Μn}N

n=1 in terms of weighted particles. More specifically, assume that at the observation time

= 1/J, J = 1, . . . , J),

which provides the following particle approximation of µn−1(u) = P(u|y1, . . , yn−1):

(3.1)

The objective now is to construct a particle approximation of µn(u) = P(u|y1, . . , yn), which includes the new data yn collected at time tn. In a standard SMC framework [32, 8, 10], this particle approximation is constructed by means of an importance sampling step with proposal distribution µn−1. To illustrate

(3.3)

which can be obtained directly from Theorem 2.3. An approximation of (3.2) can be obtained by

(3.5)

From (3.4) we see that the importance (normalized) weights W (j)

N−1 Define

the following empirical (particle) approximation of µn:

A. Iglesias, M. Park And M.V. Tretyakov

However, the accuracy of such empirical approximation relies on µn−1 being sufficiently close to µn; when this is not the case, after a few iterations (observation times) the algorithm may produce only a few particles with nonzero weights. This is a well-known issue of weight degeneracy that often arises from the application of empirical (importance sampling) approximations within the context SMC samplers .

Weight degeneracy is routinely measured in terms of the Effective Sample Size (ESS) statistic :

(3.7)

which takes a value between 1 and J; ESS = J when all weights are equal and ESS = 1 when the distribution is concentrated at one single particle. A common approach to alleviate weight degeneracy is, for example, to specify a threshold for the ESS below which resampling (often multinomially) according

N }J

j=1 is performed. Resampling discards particles with low weights by replacing them with several copies of particles with higher weights. The approximation of a sequence of measures via the combination of the importance sampling step followed with resampling leads to the Sequential Importance Resampling (SIR) scheme .

It is important to note that the aforementioned resampling step in SIR can clearly lead to the lack of diversity in the population of resampled particles. This is, in turn, detrimental to the approximation of the sequence of posteriors. The general aim of the standard SMC approach is to diversify these particles by a mutation step with involves replacing them with samples from a Markov kernel Kn with invariant distribution µn.

In the following subsection we provide a discussion of the aforementioned mutation in the context of the SMC sampler for high-dimensional inverse problem . We refer the reader to [33, 32, 8, 10] for a thorough treatment of more standard SMC samplers.

pling step described above is more pronounced when the two consecutive measures µn−1 and µn differ substantially from each other. This has been particularly associated with complex (e.g. multimodal) mea- sures defined in high-dimensional spaces. When the change from µn−1 to µn is abrupt, the importance sampling step can result in a sharp failure, whereby the approximation of µn is concentrated on a single particle . Recent work for high-dimensional inference problems has suggested [21, 1] that further sta- bilization of the importance weights is needed by defining a smooth transition between µn−1 and µn. For the present work, we consider the annealing approach of [34, 33], where qn intermediate artificial measures

{Μn,R}Qn

r=0 are defined such that µn,0 = µn−1 and µn,qn = µn. These measures can be bridged by introduc- ing a set of qn tempering parameters denoted by {φn,r}qn r=1 that satisfy 0 = φn,0 < φn,1 < · · · < φn,qn = 1 and defining each µn,r as the probability measure with density proportional to ln(u, yn)φn,r with respect

(3.9)

Note that when qn = 1, φn,1 −φn,0 = 1 and so expression (3.9) reduces to (3.3). We now follow the SMC algorithm for high-dimensional inverse problems as described in . 3.2.1. Selection Step. The first stage of the SMC approach of is a selection step which consists of careful selection of the tempering parameters which define the intermediate measures {µn,r}qn

Bayesian Inversion In Resin Transfer Molding

are in turn approximated by the application of the SIR scheme described above. Let us then assume that at an observation time tn and iteration level r−1, the tempering parameter φn,r−1 has been specified, and

That A Set Of Particles U(J)

n,r−1 provides the following approximation (with equal weights) of the intermediate

Δu(J)

n,r−1(u) ≈µn,r−1(u). From (3.9) we can see that the new tempering parameter φn,r must be selected to ensure that φn,r−φn,r−1 is sufficiently small, so that the subsequent measure µn,r is close to µn,r−1 thus preventing a sharp failure of the empirical approximation of µn,r (3.6). In particular, once the next tempering parameter φn,r is specified, we note from expression (3.9) that the importance weights for the approximation of µn,r are

(3.11)

Recognizing that the ESS in (3.7) quantifies weight degeneracy in SIR, the approach of (see also ) proposes to define on-the-fly the next tempering parameter φn,r by imposing a fixed, user-defined value Jthres on the ESS. More specifically, φn,r is defined by the solution to the following equation:

(3.12)

which may, in turn, be solved by a simple bisection algorithm on the interval (φn,r−1, 1]. An approximation of µn,r is then given by the weighted particle set {u(j)

J=1. If At The R −1 Level, We Find That

ESSn,r(1) > Jthresh, it implies that no further tempering is required and thus one can simply define φn,r = 1. We note that the number of tempering steps qn is random. While the tempering approach described above is aimed at preventing ESS from falling below a spec- ified threshold Jthres and thus avoiding a sharp failure of the empirical approximation of µn,r, resampling is still required to discard particles with very low weights. Let us then denote by ˆu(j)

N,R (J = 1, . . . , J) The

particles, with equal weights, that result from resampling with replacement of the set of particles u(j)

According To The Weights W (J)

n,r. 3.2.2. Mutation Phase. As stated in the preceding subsection, at the core of the SMC methodology is a mutation phase that adds diversity to the population of the resampled particles ˆu(j)

N,R. In The Context

of the tempering approach described above, this mutation is conducted by means of sampling from a Markov kernel Kn,r with invariant distribution µn,r. Similar to the approach of , here we consider mutations given by running Nµ steps of an MCMC algorithm with µn,r as its target distribution. More specifically, we consider the preconditioned Crank-Nicolson (pcn)-MCMC method from with target distribution µn,r and reference measure µ0. Formally, these two measures are related by

(3.13)

The pcn-MCMC method for sapling µn,r is summarised in Algorithm B.1 (see Appendix B). Under rea- sonable assumptions this algorithm produces a µn,r-invariant Markov kernel . The resulting particles

N,R ∼Kn,R(ˆU(J)

n,r, ·)) then provide the following particle approximation of µn,r:

(3.14)

where the convergence is proven in a suitable metric for measures .

Note That At The End Of The

iteration r = qn, the corresponding particle approximation µJ

N,Qn Provides The

desired approximation of the posterior that arises from the Bayesian inverse problem of interest. This SMC sampler is summarized in Algorithm B.2 (see Appendix B). Remark 3.1. For simplicity, here we use the resampling step at every iteration of the SMC sampler.

However, whenever ESSn,r(1) > Jthresh (and so φn,r = 1) the resampling step can be skipped; this involves using the corresponding weighted particle approximation and modifying the formula for the incremental weights as discussed in [21, Section 4.3].

3.3. A note on tempering. Let us define the following inverse of the increment in tempering param-

(3.15)

and note that 0 ≤φn,r ≤1 implies αn,r ≥1. In addition, expression (3.9) can be written as

(3.16)

where we have used the definition of the likelihood in (2.17). Informally, we can then interpret each iteration of the SMC sampler (at a given observation time tn) as the solution of a Bayesian inverse problem that consists of finding µn,r given the prior µn,r−1 and the data:

(3.17)

From (3.15) and the fact that 0 ≤φn,r ≤1, it follows that αn,r ≥1. Therefore, (3.17) is nothing but the original problem (2.13) albeit with a noise ˜ηn,r that has an inflated covariance αn,rΓn. We also note that αn,r plays the role of a regularization parameter in the sense that it controls the transition between µn,r−1 and µn,r. The larger the αn,r the smoother this transition. Alternatively, we can see that αn,r can be interpreted as a “temperature” in the tempering scheme which, in turn, flattens out the likelihood function at the observation time tn. Clearly, more tempering will be required whenever ||(Γn)−1/2(yn −Gn(u))||2 is large; this can for example happen if the observational data are accurate (i.e small Γn) and/or many observations are available.

The amount of tempering is controlled by the number of parameters obtained via (3.12). The greater the number of tempering parameters, the larger the αn,r’s which in turn indicates that more regularization is needed to ensure a stable transition between those measures. This has also, in turn, an associated increase in iterations and thus in computational cost.

3.3.1. Computational aspects of SMC. The main computational cost of the SMC sampler previ- ously discussed is attributed to the mutation step for which Nµ steps of the pcn-MCMC algorithm are performed. At each observation time tn and iteration r, the SMC sampler then requires J Nµ evaluations

Bayesian Inversion In Resin Transfer Molding

of the nth forward map Gn. Therefore, the computational cost of computing µn is qngnJ Nµ, where gn denotes the computational cost of evaluating Gn which, in turn, corresponds to solving the moving boundary problem from time t = 0 up to time tn. The total computational cost of computing the full

(3.18)

which is expressed in terms of gN, the cost of evaluating GN (i.e. solving the forward model from time zero up to the final observation time). The work of has suggested that accurate approximations of the posterior via SMC samplers require, for example, values of Nµ = 20 and J = 104.

F We Assume For A Moment That Only One

observation time N = 1 is considered and that only one tempering step q1 = 1 is required to compute µ1, the computational cost in this case would be approximately 105 times the cost of solving the forward model from time t = 0 up to time t1. Such cost would be clearly computationally prohibitive for practical applications, where the aforementioned forward simulation may take several minutes of CPU time. In particular, for the 2D or 3D version of the RTM process, the high computational cost of the SMC sampler becomes impractical. While reducing the values of J and Nµ may result in a more affordable computational cost, this is substantially detrimental to the level of accuracy of the SMC sampler as we show via numerical experiments in Section 3.5.

3.4. Numerical examples with SMC. In this subsection we report the results from the numerical application of the SMC sampler discussed in the previous subsection. The objective is to approximate the sequence of Bayesian posteriors that arise from the 1D moving boundary problem defined in Section 2 for the experimental set-up described in Section 3.4.1. In Section 3.4.2 we discuss the numerical results obtained via the SMC sampler with a very high number of particles which results in accurate approx- imations of the Bayesian posteriors. These approximations are then used in Section 3.5 to assess the practical limitations of the scheme under certain choices of tunable parameters and number of particles.

These limitations motivate the approximate methods that we propose in Section 4. 3.4.1. Experimental set-up. We consider a dimensionless version of the one-dimensional model (2.1)- (2.5) which together with its numerical approximation is described in Appendix C. The dimensionless values for the control variables are p0 = 1 and pI = 2. We use a Gaussian prior distribution µ0 = N(u, C) with the covariance operator C that arises from the covariance function defined in (2.15). We numerically solve (off-line) the eigenvalue problem associated to the matrix that results from discretizing C; the corresponding eigenvector/eigenvalues are then stored for subsequent use in the parameterization of the log-permeability in the SMC sampler. The KL expansion (2.16) becomes a truncated sum with a number of elements equal to the the total number of eigenvalues of this matrix; these are, in turn, equal to the number of cells used for the discretization of the domain D∗= [0, 1]. No further truncation to this KL expansion is carried out. A few samples from the prior are displayed in Figure 3.1 (right). Pointwise percentiles (0.02, 0.25, 0.5, 0.75 and 0.98) of the prior are displayed in Figure 3.2 (top-left). Tuneable parameters of the prior for the present experiments are σ2

= 0.5, Ν = 1.5, L = 0.05 And U(X) = 0.0 For All

x ∈D∗. In order to generate synthetic data, we define the “true/reference” log permeability field u† whose graph (red curve) is displayed in Figure 3.2 (top-left); this function is a random draw from the prior described above. We use u = u† in the numerical implementation of (2.1)-(2.5) in order to compute the true pressure field p†(x, t) as well as the true front location Υ†(t). The plot of p†(x, t) is shown in Figure 3.1 (left) together with the space-time configuration of M = 9 pressure sensors and N = 5

Observation Times. The Graphs Of {P(X, Tn)}5

n=1 are shown in Figure 3.1 (middle). The true locations of

-0.5

Figure 3.1. Left: True pressure field p†(x, t) and space-time measurement configuration with M = 9 sensors and N = 5 observation times. Middle: True pressure at observation times {p†(x, tn)}5

Synthetic Data Yp

n (dots). Right: Samples from the prior.

The Front {Υ†(Tn)}5

n=1 are 0.21 ,0.40, 0.58, 0.73 and 0.87. Synthetic data are then generated by means of

N Are Gaussian Noise (See Subsection

2.2) with standard deviations equal to 1.5% of the size of the noise-free observations. Synthetic pressure

N }5

n=1 are 0.21, 0.39, 0.59, 0.74, 0.86. In order to avoid inverse crimes, synthetic data are generated by using a finer discretization (with 120 cells) than the one used to approximate the posteriors (with 60 cells).

3.4.2. Application of SMC. In this subsection we report the application of the SMC sampler of (see Algorithm B.2 in Appendix B) which, as described in the preceding section, provides a particle approximation of each posterior that converges to the exact posterior measure µn as the number of particles J goes to infinity.

In order to achieve a high-level of accuracy we use J = 105 number of particles which is substantially larger compared to the number of particles (e.g. 103 to 104) often used in existing applications of SMC for high-dimensional inverse problems [8, 21]. In addition, we consider the selection of tunable parameters Nµ = 20 and Jthresh = J/3 similar to the ones suggested in . For each observation time tn, we store the ensemble of particles {u(j)

J=1 That Approximates The Corresponding

posterior µn. From this ensemble, we compute the 0.02, 0.25, 0.5, 0.75, 0.98 posterior percentiles displayed in Figure 3.2 (top-middle to bottom-right), where we also include the graph of the true log-permeability (red curve).

The vertical line in these figures indicate the true location of the front Υ†(tn) at each observation time tn. We can clearly appreciate that the uncertainty band defined by these percentiles is substantially reduced as more observations (in time) are assimilated. In fact, the main reduction of the uncertainty is observed in the region of the moving domain D†(tn) = [0, Υ†(tn)] at the corresponding observation time tn.

It is then clear that at each observation time tn, measurements collected from pressure sensors with xm ∈D∗\D†(tn) are not very informative of the log-permeability field. This comes as no surprise when we recognize that the pressure field given by (2.8) depends on the permeability field only in the region of the moving domain D(t). In other words, the values of the permeability in the region defined by D∗\ D(tn) have no effect on p(x, tn); hence the nth likelihood function is independent of u in this region. We can indeed observe from Figure 3.2 that the percentiles of the log-permeability in this region (see domain to the right of the vertical lines) is similar to those from the prior. However, due to the regularity of the log-permeability enforced in the prior µ0, there is a smooth transition in the uncertainty band at the interface defined by the front location Υ(tn).

The number of intermediate tempering distributions that SMC adaptively computed to approximate

(3.19)

We use these numbers in (3.18) to compute the total computational cost of approximating the sequence

{Μn}5

n=1. The values of gn (i.e. cost of evaluating each Gn) are estimated by the average CPU time from 1000 simulations computed with different log-permeabilities sampled from the prior. We obtain that total cost is approximately 1.5 × 107 times the cost of evaluation the 5th forward map G5 (i.e. at the final observation time). Clearly, this computational cost is prohibitive for the two and three dimensional problems where, as stated earlier, evaluating the forward map can take several minutes of CPU time. For the present one-dimensional case we are able to afford this cost due to the relatively low cost associated with solving the 1D moving boundary problem.

P50

Figure 3.2. Top-left: Percentiles of the prior log-permeability µ0.

Posteriors {Μn}5

n=1 obtained via SMC with large number of samples J = 105. Solid red line corresponds to the graph of the true log-permeability u†. Vertical dotted line indicates the location ot the true front Υ†(tn). 3.5. Reducing the cost of SMC by adjusting tunable parameters. Given the high computational cost of computing accurate approximations of the posteriors with SMC, it is reasonable to ask whether its computational cost can be reduced by adjusting the tunable parameters in (3.18). By reducing either the number of particles J and/or the number of MCMC steps Nµ, we can achieve a substantial decrease in the computational cost. The selection of Jthresh also determines the computational cost as it, in turns, defines the number of tempering steps for each posterior. However, it is essential to understand the effect of decreasing these tunable parameters on accuracy of the SMC sampler. In this subsection we aim at understanding this effect by comparing the application of the SMC sampler with smaller number of particles J and different choices of the tunable parameters Nµ and Jthresh. This requires creating a Benchmark against which we can compare performance of SMC. The Benchmark is obtained by the highly-resolved characterization of the posteriors that we computed in the preceding section by using the SMC sampler with large number of particles (J = 105). In Appendix B we provide further discussions of the performance and diagnostics of the SMC sampler applied to approximate these posterior measures.

A. Iglesias, M. Park And M.V. Tretyakov

These diagnostics offer evidence that the SMC sampler has been successfully applied, thereby providing accurate characterization of the posterior that we may use as a Benchmark to compare against the posteriors computed via algorithms with lower resolution/accuracy. The numerical investigation below is aimed at assessing SMC with different selections of ensemble size J as well as the tunable parameters Nµ and Jthresh.

For the reasons stated above, through the rest of the this and the following sections, we refer to the aforementioned highly-resolved SMC particle approximations (with J = 105) as the “exact” sequence of

Posteriors {Μn}5

n=1 that we use for subsequent comparisons purposes. Moreover, for these comparisons we assume that the sample mean and variance of these SMC samples are exact approximations of the mean Eµn and variance Vµn of the posterior µn. In other words, we assume

(3.21)

Let us now consider the application of the SMC sampler for the following choices of small number of particles: J = 50, 100, 200, 400, 800, 1600, 3200, 6400. We also consider three choices of the tunable parameter Jthresh (Jthresh = J/3, J/2, 2J/3) and two choices of Nµ (Nµ = 5, 20). In Figure 3.3 we show percentiles of the log-permeability posteriors µn (for n = 1, 3 and 5) obtained using the aforementioned SMC sampler for some of those choices of the number particles J, and with the same selection of tunable parameters Nµ = 20, Jthresh = J/3 that we used for the highly-resolved SMC with large particles; percentiles from the latter are included in the right column of Figure 3.3 for comparison purposes. We can see that as the ensemble of particle increases, the approximation of SMC improves when compared to the one provided by the highly-resolved SMC. Note that very small number of particles results in very poor approximations of these percentiles.

In order to quantify the level of approximation obtained with SMC with the aforementioned selections of parameters, we compute the L2(D∗)-relative errors of the mean and variance with respect to the posterior measure approximated with the highly-resolved SMC computed as described in the preceding

(3.22)

where Eµn and Vµn are the µn-posterior mean and variance characterized via SMC with large J from (3.20)-(3.21). In the previous expressions un,J and σ2

N }J

j=1 computed via SMC for the choices of small J stated above and with the aforementioned selections of tunable parameters. In addition, we consider the estimator of the true log-permeability defined by the ensemble mean un,J and thus we monitor the corresponding

N And Εn

J are random variables that depend on the initial ensemble of particles that we generate from the prior µ0. We thus report these quantities (for each n = 1, 3, 5) averaged over 15 experiments corresponding to different selections of the initial ensemble of particles. In Figure 3.4 we

N (Middle) And Εj

n (bottom) for (from left to right) n = 1, 3, 5 as a function of the aforementioned selections of ensemble size J. For brevity we omit the results for n = 2, 4 as they display similar behaviour. The total computational cost of computing the full sequence of posteriors (i.e. CSMC from (3.18)) is shown in Figure 3.5 (left). We reiterate that this cost is expressed in terms of the number of evaluations of the 5th forward map G5.

While the numerical analysis of the convergence of the SMC sampler is beyond the scope of this work, the results presented in this section are aimed at understanding the level of accuracy of SMC with relatively small number of particles and for a selection of tunable parameters which may enable the use of this method in more practical scenarios. From these results it is clear that the selections of Jthresh have no substantial effect on the accuracy of the scheme in terms of approximating the mean and variance of each posterior. Similarly, the computational cost with respect to our selections of Jthresh does not seem to vary significantly. It is evident that the main effect in terms of accuracy is the choice of MCMC steps (i.e. parameter Nµ). Indeed, note that the error obtained with Nµ = 5 is considerably larger than the one with Nµ = 20 although the computational cost of the former is one quarter of the computational cost of the latter. We conclude that even though decreasing Nµ can offer computational affordability, it is detrimental to the approximation properties of the scheme. This comes as no surprise as it is well known that the mutation step that involves running MCMC is crucial for the accuracy of any SMC methodology.

The behavior of the SMC sampler with respect to the number of particles J is as expected. On the one hand, an increase in J corresponds to a decrease in the error with respect to the mean and variance. On the other hand, the computational cost, CSMC, increases with J. Note that there is a clear linear relationship between these two variables which is, in turn, obvious from (3.18) provided that qn is invariant with respect to J. Indeed, for the cases considered here, the number of intermediate tempering distributions (not reported) computed at each observation time, is invariant with respect to our choices of J. This is somewhat an expected outcome since our choice of Jtresh in (3.12) is always a fraction of J.

It is also worth mentioning that the effect of J is less noticeable when we look at the error with respect to the truth. At each observation time, we notice that the ϵn seems to converge to a nonzero value as J increases. Note that convergence to the truth is not ensured due to the limited number of measurements inverted and the potential lack of identifiability of the log-permeability.

The results reported in this subsection suggest that achieving a reduction in the computational cost by reducing Nµ has a severe detrimental effect in the accuracy of the SMC sampler with small number of particles. In addition, Jthresh does not seem to have a substantial effect in either the accuracy or computational cost. Clearly, we are only then limited to the number of particles J to control the computational cost of the sampler without severely compromising accuracy of the approximate posteriors.

4. Approximating the posteriors via a regularizing ensemble Kalman algorithm. In the previous section we have demonstrated, by means of numerical examples, that an accurate approximation of the Bayesian posteriors via the state-of-the art SMC samplers results in a very high computational cost; hence it is unfeasible for practical applications such as the 2D resin injection in RTM introduced in Section 1.

In this section we propose a regularizing ensemble Kalman algorithm (REnKA) that aims at providing an accurate approximation of the sequence of Bayesian posteriors at a much lower computational cost. In Section 4.1 we introduce REnKA as a Gaussian approximation from the SMC sampler of discussed in the preceding section. The proposed REnKA in the context of existing ensemble Kalman methods is discussed in Section 4.2. A numerical investigation of the convergence properties of REnKA is reported in Section 4.3.

For the subsequent development of the proposed scheme, we extend the domain of definition of the sequence of forward maps Gn introduced in (2.10). More specifically, we assume Gn : X →RM+1 where X is a Hilbert space such that X = C[0, x∗] ,→X (compactly). We denote by < ·, · >X and < ·, · > the inner products in X and RM+1, respectively. In addition, we define Z ≡X × RM+1 with inner product

A. Iglesias, M. Park And M.V. Tretyakov

Figure 3.3. Percentiles of the posteriors µn’s (n = 1, 3, 5) obtained via SMC with (from left to right) J

=

50, 400, 1600, 105. Solid red line corresponds to the graph of the true log-permeability u†.

Ertical Dotted Line Indicates

the location ot the true front Υ†(tn). denoted by < ·, · >Z. 4.1. Motivation for REnKA. Motivated by the SMC tempering approach described in the previous section, we now propose an ensemble Kalman algorithm whose aim is to approximate {µn,r}qn

Sequence Of Gaussian Measures {Νn,R}Qn

r=1 which are, in turn, characterised by a set of particles with equal weights. Suppose that, at time t = tn we have an ensemble {u(j)

J=1 Of J Samples From A Gaussian

measure νn,r−1 that approximates µn,r−1, and a prescribed tempering parameter φn,r−1. We may then solve (3.12) for the new φn,r and define the regularization parameter αn,r in (3.15). We now wish to make a transition from νn,r−1 to a Gaussian measure νn,r that approximates µn,r. To this end, let us define

(4.1)

and note that, in terms of this variable, we may rewrite (3.17) as

(4.2)

where H = (0, I) and I is the identity operator. One can see that by reformulating the inverse problem in terms of the augmented variable, the resulting forward map (i.e. H) acting on this variable is linear. From (4.1) we define the following augmented particles

Nµ = 5

Figure 3.4. SMC approximations. Top and middle: Relative errors of mean (top) and variance (middle) of the posteriors µn, (from left to right) n = 1, 3, 5, obtained with SMC with different choices of small (log) ensemble size log(J) and tunable (SMC) parameters Jthresh and Nµ. Bottom: Relative errors with respect to the truth u† of the ensemble mean.

Log(6400)

Figure 3.5. Total computational cost in terms of G5-forward model evaluations. Left: Total computational cost obtained via SMC with different choices of Nµ and Jthresh. Right: Comparison of total computational cost obtained via REnKA with different choices of Jthresh = J/3 against the cost of SMC with different selection of tunable parameters Nµ and Jthresh.

The Gaussian measure ˆνn,r−1(z) is used to approximate the measure, denoted by ˆµn,r−1(z), that arises from pushing forward µn,r−1(u) under (4.1). By using this Gaussian approximation of ˆµn,r−1(z), we then provide a Bayesian formulation of the inverse problem given by (4.2). More specifically, we wish to compute ˆνn,r(z) ≡P(z|yn) given ˆνn,r−1(z) and the data from (4.2). A formal application of Bayes

(4.7)

Moreover, from (4.4) and the linearity of the forward map H (on the augmented variable), it follows by

Where

Kn,r ≡Cn,r−1HT (HCn,r−1HT + αn,rΓ)−1.

(4.9)

Let us then note that Cn,r−1 in (4.6) can be written as

U(J)

n,r−1. Informally, we use the block structure of (4.10) and define νn,r, the approximation of µn,r, as the marginal

(4.15)

Although the measure (4.14) is fully characterised by its mean and covariance, for the subsequent tem- pering step we need a particle approximation of νn,r(u). We can obtain those particles by updating the

(4.17)

Indeed, under the standard assumption that the noise η(j)

Gn(U(J)

n,r−1), it can be shown by the standard arguments in Kalman-based methods (see for example )

(4.18)

Expression (4.16) and selection of the regularisation parameter αn,r based on the adaptive tempering approach discussed in Section 3.2 constitutes the proposed scheme summarized in Algorithm 4.1. Remark 4.1. Note that the key assumption for the proposed scheme is the Gaussian approximation of ˆµn,r−1(z) provided by (4.8). It is clear that the measure ˆµn,r−1(z) is, as a rule, non-Gaussian and the aforementioned assumption will result in a methodology that will, in general, not converge to the posteriors µn as the ensemble size J →∞. Nevertheless, we will show via numerical examples that this approximation provides reasonably accurate estimates using only a small number of particles.

It is not difficult to see that the main computational cost of REnKA, in terms of the cost of evaluating the forward model at the final observation time, is given by

(4.20)

where, as before, gn denotes the computational cost of evaluating the Gn-forward map.

As We Will

demonstrate via numerical experiments, for the moving boundary problem of Section 2.1, REnKA offers a computationally affordable and thus practical approach to approximate the solution to the Bayesian inverse problem that arises from RTM.

A. Iglesias, M. Park And M.V. Tretyakov

Algorithm 4.1 Regularizing ensemble Kalman algorithm (REnKA)

,0}J

j=1 ∼µ0 be the initial ensemble of J particles. Define the tunable parameter Jthresh.

Ompute The Nth Likelihood (2.17) Ln(U(J)

n,r−1, yn) for j = 1, . . , J.

Else

compute φn,r such that ESSn,r(φ) ≈Jthresh using a bisection algorithm on (φn,r−1, 1].

N,R−1, Cww

n,r−1 defined by expressions (4.11)-(4.12).

End For

It is important to mention that, at a given observation time tn and iteration level r, the value

J=1 Ln(U(J)

n,r−1, yn)1−φn,r−1 may be zero to machine precision. In this case, the tempering parame- ter φn,r is not be computable, via a bisection scheme on (φn,r−1, 1], as stated in Algorithm 4.1. This computational issue is more likely to arise at the early iterations of the scheme for which the value of φn,r−1 is not sufficiently close to one.

This can be overcome, for example, by simply adapting the bisection algorithm in order to first compute a φ∗such that PJ

N,R−1, Yn)Φ∗−Φn,R−1 > 0. If

minφ∈(φn,r−1,φ∗] ESSn,r(φ) > Jthresh we then set φn,r = φ∗; otherwise, we find φn,r by solving (3.12) via a bisection algorithm on (φn,r−1, φ∗]. For the numerical experiments reported in the present work, zero

J=1 Ln(U(J)

n,r−1, yn)1−φn,r−1 were only encountered where a large number of measurements were inverted in the 2D setting of Section 5.2. Kalman methods for inverse/calibration problems have been widely used in the last decades . More recently, using iterative Kalman methods with a regularization parameter (e.g. αn,r in (4.19)) have been

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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