Homogenization Composite Ansys

Homogenization of Composite Ferromagnetic Materials

Fran¸Cois Alouges ∗1 And Giovanni Di Fratta†1

1CMAP, ´Ecole Polytechnique, route de Saclay, 91128 Palaiseau Cedex, FRANCE

Abstract

Nowadays, nonhomogeneous and periodic ferromagnetic materials are the subject of a growing interest. Actually such periodic configurations often combine the attributes of the constituent materials, while sometimes, their properties can be strikingly different from the properties of the different constituents.

These periodic configurations can be therefore used to achieve physical and chemical properties difficult to achieve with homogeneous materials. To predict the magnetic behavior of such composite materials is of prime importance for applications.

The main objective of this paper is to perform, by means of Γ-convergence and two-scale convergence, a rigorous derivation of the homogenized Gibbs-Landau free energy functional associated to a composite periodic ferromagnetic material, i.e. a ferromagnetic material in which the heterogeneities are periodically distributed inside the ferromagnetic media.

We thus describe the Γ-limit of the Gibbs-Landau free energy functional, as the period over which the heterogeneities are distributed inside the ferromagnetic body shrinks to zero.

Ntroduction

Composite materials are an important class of natural or engineered heterogeneous media, composed of a mixture of two or more constituents with significantly different physical or chemical properties, firmly bonded together, which remain separate and distinct within the finished structure. Finding a model which considers the composite as a bulk and whose coefficients and terms are computed from suitable averages of those of its constituents and the geometry of the microstructure is the aim of homogenization theory. The study of composites and their homogenization is a subject with a long history, which has attracted the interest and the efforts of some of the most illustrious names in science : In 1824, Poisson, in his first M´emoire sur la th´eorie du magn´etisme , put the basis of the theory of induced magnetism assuming a model in which the body is composed of conducting spheres embedded in a nonconducting material. This paper is the origin of the basic models and ideas that prevailed in the theory of heterogeneous media in almost all domains of continuum mechanics, for almost a century after its appearance . We refer the reader to the papers of Markov and Landauer for more historical details.

Nowadays, nonhomogeneous and periodic ferromagnetic materials are the subject of a growing interest. Actually such periodic configurations often combine the attributes of the constituent materials, while some- times, their properties can be strikingly different from the properties of the different constituents . These periodic configurations can be therefore used to achieve physical and chemical properties difficult to achieve with homogeneous materials. To predict the magnetic behavior of such composite materials is of prime im- portance for applications . From a mathematical point of view, the study of composite materials, and more generally of media which involve microstructures, is the main source of inspiration for the Mathematical Theory of Homogenization which, roughly speaking, is a mathematical procedure which aims at understand- ing heterogeneous materials with highly oscillating heterogeneities (at the microscopic level) via an effective model .

The main objective of this paper is to perform, in the framework of De Giorgi’s notion of Γ-convergence and Allaire’s notion of two-scale convergence (see also the paper by Nguetseng ), a mathe- matical homogenization study of the Gibbs-Landau free energy functional associated to a composite pe- riodic ferromagnetic material, i.e.

a ferromagnetic material in which the heterogeneities are periodically distributed inside the ferromagnetic media. Compared to earlier works related to the subject (see for instance [10, 11, 14, 30]) we consider here the full Gibbs-Landau functional for mixtures of different materials in the three-dimensional space.

Arxiv:1411.1231V1 [Math.Ap] 5 Nov 2014

The Landau-Lifshitz micromagnetic theory of single-crystal ferromagnetic ma-

Terials

According to Landau and Lifshitz micromagnetic theory of ferromagnetic materials (see [6, 8, 18, 23]), the states of a rigid single-crystal ferromagnet, occupying a region Ω⊆R3, and subject to a given external magnetic field ha, are described by a vector field, the magnetization M, verifying the so-called fundamental constraint of micromagnetic theory: A ferromagnetic body is always locally saturated, i.e. there exists a

(1)

The saturation magnetization Ms depends on the specific material and on the temperature T, and vanishes above a temperature (characteristic of each crystal type) known as the Curie point. Since we will assume that the specimen is at a fixed temperature below the Curie point of the material, the value Ms will be regarded as a material dependent function, and therefore as a constant function when working on single- crystal ferromagnets. Due to the constraint (1) in the sequel we express the magnetization M under the form M := Ms(T)m where m : Ω→S2 is a vector field which takes its values on the unit sphere S2 of R3.

Even though the magnitude of the magnetization vector is constant in space, in general it is not the case for its direction, and the observable states can be mathematically characterized as local minimizers of the Gibbs-Landau free energy functional associated to the single-crystal ferromagnetic particle (using the

(2)

The first term, E(m), called exchange energy, penalizes spatial variations of m. The factor aex in the term is a phenomenological positive material constant which summarizes the effect of (usually very) short-range exchange interactions.

The second term, A(m), or the anisotropy energy, models the existence of preferred directions for the magnetization (the so-called easy axes), which usually depend on the crystallographic structure of the mate- rial. The anisotropy energy density ϕan : S2 →R+ is assumed to be a non-negative even and globally lipschitz continuous function, that vanishes only on a finite set of unit vectors (the easy axes).

The third term, W(m), is called the magnetostatic self-energy, and is the energy due to the (dipolar) magnetic field, also known in literature as the stray field, hd[m] generated by m. From the mathematical point of view, assuming Ωto be open, bounded and with a Lipschitz boundary, a given magnetization m ∈

Ω, R3

generates the stray field hd[m] = ∇um where the potential um solves:

(3)

In (3) we have indicated with mχΩthe extension of m to R3 that vanishes outside Ω. Lax-Milgram theorem guarantees that equation (3) possesses a unique solution in the Beppo-Levi space:

(4)

Eventually, the fourth term Z(m), is called the interaction energy (or Zeeman energy), and models the tendency of a specimen to have its magnetization aligned with the external field ha, assumed to be unaffected by variations of m.

The competition of those four terms explain most of the striking pictures of the magnetization that ones can see in most ferromagnetic material , in particular the so-called domain structure, that is large regions of uniform or slowly varying magnetization (the magnetic domains) separated by very thin transition layers (the domain walls).

The Gibbs-Landau energy functional associated to composite ferromagnetic

Materials

Physically speaking, when considering a ferromagnetic body composed of several magnetic materials (i.e. a non single-crystal ferromagnet) a new mathematical model has to be introduced. In fact, as far as the ferromagnet is no more a single crystal, the material depending functions aex, Ms(T) and ϕan are no longer Figure 1: If we assume that the heterogeneities are evenly distributed inside the ferromagnetic media Ω, we can model the material as periodic. As illustrated in the figure, this means that we can think of the material as being built up of small identical cubes Qε, the side length of which we call ε.

constant on the region Ωoccupied by the ferromagnet. Moreover one has to describe the local interactions of two grains with different magnetic properties at their touching interface . From a mathematical point of view, this latter requirement is usually taken into account in two different ways. Either one adds to the model a surface energy term which penalizes jumps of the magnetization direction m at the interface of both grains, or, and we stick on this later on, one simply considers a strong coupling, meaning that the direction of the magnetization does not jump through an interface. We insist on the fact that only the direction is continuous at an interface while the magnitude Ms is obviously discontinuous . Therefore, the natural mathematical setting for the problem turns out to be characterized by the assumption that the magnetization direction m is in the “weak” Sobolev metric space

, I.E. On The Metric

subspace of H1(Ω, R3) constituted by the functions constrained to take values on the unit sphere of R3 and endowed with the L2(Ω) metric. It is in this framework that we will conduct our work from now on. We start by recalling the basic idea of the mathematical theory of homogenization. Let Ω⊂R3 be the region occupied by the composite material. If we assume that the heterogeneities are regularly distributed, we can model the material as periodic. As illustrated in Fig.1, this means that we can think of the material as being built up of small identical cubes, the side length of which being called ε. Let Q = [0, 1]3 be the unit cube of R3. We let for y ∈Q, aex(y), Ms(y), ϕan(y, m) be the periodic repetitions of the functions that describe how the exchange constant aex, the saturation magnetization Ms and the anisotropy density energy ϕan(y, m) vary over the representative cell Q (see Fig. 1). Substituting x/ε for y, we obtain the ≪two-scale≫functions aε(x) := aex(x/ε), Mε(x) := Ms(x/ε) and ϕε(x, m) := ϕan(x/ε, m) that oscillate periodically with period ε as the variable x runs through Ω, describing the oscillations of the material dependent parameters of the composite. At every scale ε, the energy associated to the ε-heterogeneous ferromagnet, will be given by the following generalized Gibbs-Landau energy functional

(5)

The asymptotic Γ-convergence analysis of the family of functionals (Gε

Statement Of The Main Result

The main purpose of this paper is to analyze, by the means of both Γ-convergence and two-scale convergence techniques, the asymptotic behavior, as ε →0, of the family of Gibbs-Landau free energy functionals (Gε

)Ε∈R+

expressed by (5). Let us make the statement more precise. We consider the unit sphere S2 of R3 and, for every s ∈S2, the tangent space of S2 at a point s will be

S2

. The class of admissible maps we are interested in is defined as

,

where we have denoted by τ the Lebesgue measure on R3. We consider H1(Ω, S2) as a metric space endowed with the metric structure induced by the classical L2(Ω, R3) metric. For every positive real number t > 0, we set Qt := [0, t]3 and Q := Q1 = [0, 1]3. We recall that a function u : R3 →R is said to be Q-periodic if u(·) = u(· + ei) for every ei in the canonical basis (e1, e2, e3) of R3.

For the energy densities appearing in the family (Gε

)Ε∈R+ We Assume The Following Hypotheses:

[H1] The exchange parameter aex is supposed to be a Q-periodic measurable function belonging to L∞(Q) which is bounded from below and above by two positive constants cex > 0, Cex > 0, i.e. 0 < cex ⩽ aex(y) ⩽Cex for τ-a.e.

y ∈Q. In the setting of classical Calculus of Variations, this hypothesis guarantees that the exchange energy density, which has the form g(y, ξ) := aex(y)|ξ|2, ξ ∈R3×3, is a Carath´eodory integrand satisfying the following quadratic growth condition for τ-a.e. y ∈Q

∀Ξ ∈R3×3

cex|ξ|2 ⩽g(y, ξ) ⩽Cex(1 + |ξ|2).

(6)

Then we set aε(x) := aex(x/ε). [H2] The anisotropy density energy ϕan : R3 × S2 →R+ is supposed to be a Q-periodic measurable function belonging to L∞(Q) with respect to the first variable, and globally lipschitz with respect to the second one (uniformly with respect to the first variable), i.e. ∃κL > 0 such that

(7)

We then set ϕε(x, s) := ϕan(x/ε, s). The hypotheses assumed on ϕan are sufficiently general to treat the most common classes of crystal anisotropy energy densities arising in applications. As a sake of example, for uniaxial anisotropy, the energy density reads as

(8)

the spatially dependent unit vector u(·) being the easy axis of the crystal. For cubic type anisotropy,

(9)

the mutually orthogonal unit vectors ui(·) being the three easy-axes of the cubic crystal. Note that the anisotropy depends on the material both in strength (κ(y)) and in direction (ui(y)). [H3] The saturation magnetization Ms is supposed to be a Q-periodic measurable function belonging to L∞(Q), and we set Mε(·) = Ms(·/ε).

Theorem 1.1 Let (Gε

L)ε∈R+ be a family of Gibbs-Landau free energy functionals satisfying the hypotheses

(10)

The four terms that appear in (10) have the following expressions: The homogenized exchange energy is

(11)

where Ahom is the ≪classical≫homogenized tensor Ahom given by the average Ahom := ⟨aex(y)(I + ∇ϕ(y))T (I + ∇ϕ(y))⟩Q, ϕ := (ϕ1, ϕ2, ϕ3),

(12)

where for every j ∈N3 the component ϕj is the unique (up to a constant) solution of the following scalar unit

(14)

while the homogenized magnetostatic self-energy is given by

(15)

where, for every x ∈Ω, the scalar function vm : Ω× Q →R, is the unique solution of the cell problem:

#(Q).

Finally, the homogenized interaction energy is given by

(17)

The paper is organized as follows: In Section 2 we give a brief survey of the main mathematical concepts and results used throughout the paper. The equicoercivity of the family (Gε

)Ε∈R+ Is Established In Section 3;

the Γ-limit of the exchange energy family of functionals (Eε)ε∈R+ is computed in Section 4; in Section 5 it is shown that the family of magnetostatic self-energies (Wε)ε∈R+ continuously converges to Whom, while in Sec- tion 6 it is established the continuous convergence of the family of anisotropy energies (Aε)ε∈R+ to Ahom and the continuous convergence of the family of interaction energies (Zε)ε∈R+ to the functional Zhom. Eventually, the proof of mht (Theorem 1.1) is completed in Section 7, and some well-known results in homogenization theory, though somewhat difficult to find in the literature are given in the appendix.

Athematical Preliminaries

The purpose of this section is to fix some notations and to give a survey of the concepts and results that are used throughout this work. All results are stated without proof as they can be readily found in the references given below.

Γ-Convergence Of A Family Of Functionals

We start by recalling De Giorgi’s notion of Γ-convergence and some of its basic properties (see [12, 9]). Throughout this part we indicate with (X, d) a metric space and, for every m ∈X, with Cd(m) the subset of all sequences of elements of X which converge to m.

Definition 2.1 (Γ-convergence of a family of functionals) Let (Fn)n∈N be a sequence of functionals defined on X with values on R. The functional F : X →R is said to be the Γ-lim of (Fn)n∈N with respect to the

(19)

In this case we write F = Γ- limn→∞Fn. If (Fε)ε∈R+ is a family of functionals, we say that F : X →R is the Γ-lim of (Fε)ε∈R+ as ε →0, if for every εn ↓0 one has F = Γ- limn→∞Fεn. In this case we write F = Γ- limε→0 Fε.

The condition (19) is sometimes referred to in literature as the existence of a recovery sequence. One of the most important properties of Γ-convergence, and the reason why this kind of variational con- vergence is so important in the asymptotic analysis of variational problems, is that under appropriate com- pactness hypotheses it implies the convergence of (almost) minimizers of a family of equicoercive functionals to the minimum of the Γ-limit functional. More precisely, the following result holds: Theorem 2.2 (Fundamental Theorem of Γ-convergence) If (Fε)ε∈R+ is a family of equicoercive functionals Γ-converging on X to the functional F. Then F is coercive and lower semicontinuous (therefore there exists a minimizer for F on X) and we have the convergence of minima values

(20)

Moreover, given εn ↓0 and (mn)n∈N a converging sequence such that

(21)

its limit is a minimizer for F on X. If (21) holds, the sequence (mn)n∈N is said to be a sequence of almost- minimizers for F. Let us recall now that given two families of functional (Fε)ε∈R+ and (Gε)ε∈R+ Γ-converging respectively to F and G, it is in general not the case (see ) that Γ-limε→0(Fε +Gε) = F +G . A sufficient condition for that property to hold is that at least one of the two families of functionals satisfies a stronger type of convergence: Definition 2.3 We say that a family of functionals (Gε)ε∈R+ is continuously convergent in X to a functional

Lim

(m,ε)→(m0,0) Gε(m) = G(m0).

We Then Have (See For A Proof):

Proposition 2.4 Let F = Γ- limε→0 Fε. Suppose that the family of functionals (Gε)ε∈R+ continuously con- verges to G, and that Gε and G are everywhere finite on X. Then G = Γ- limε→0 Gε and

Γ- Lim

ε→0(Fε + Gε) = F + G. In particular if Z : X →R is a continuous functional then Γ- limε→0(Fε + Z) = F + Z and Z is called a continuous perturbation of the Γ-limit.

Two-Scale Convergence

The aim of this section is to present in a schematic way the main properties of two-scale convergence, a notion that is first due to Nguetseng , developed as a methodology by Allaire and further investigated by many others (see and references therein for instance).

We Denote By C∞

# (Q) the set of infinitely differentiable real functions over R3 that are Q-periodic and

#(Q) Has The Same Trace On

the opposite faces of Q. A generalized version of the Riemann-Lebesgue Lemma holds for the weak limit of rapidly oscillating functions. For a proof we refer the reader to .

Proposition 2.5 Let Ω⊂R3 be any open set. Let 1 ⩽p < ∞and t > 0 be a positive real number. Let u ∈Lp(Qt) be a Qt-periodic function. Set uε(x) := u(x/ε) τ-a.e. on Ω. Then, if p < ∞, as ε →0

U Dτ

weakly∗in L∞(Ω). Definition 2.6 Let Ωbe an open set Ω⊂R3, and let (εk)k∈N be a fixed sequence of positive real numbers (when it is clear from the context we will omit the subscript k) converging to 0. The sequence of functions (uε) ∈L2(Ω) is said to two-scale converge to a limit u ∈L2(Ω× Q), if for any function ϕ ∈D[Ω; C∞

(22)

In this case we write uε ↠u. We say that (uε) in L2(Ω) strongly two-scale converges to a limit u ∈

∥U∥Ω×Q = Lim

ε→0 ∥uε∥Ω. The importance of this new notion of convergence relies on the following compactness results. Proposition 2.7 For each bounded sequence (uε) ∈L2(Ω), there exists an u ∈L2(Ω× Q) such that, up to a subsequence, uε ↠u.

Moreover, for bounded sequences in H1(Ω) we have the following result: Proposition 2.8 Let (uε) be a sequence in H1(Ω) that converges weakly to a limit u ∈H1(Ω). Then uε ↠u

#(Q)/R] Such That, Up To A Subsequence:

∇uε ↠∇u + ∇yv. Next we recall that if the sequence (uε) is bounded in L2(Ω), it is possible to enlarge the class of test functions used in the definition of two-scale convergence.

Proposition 2.9 Let (uε) be a bounded sequence in L2(Ω) which two-scale converges to u ∈L2(Ω× Q).

U(X, Y)Φ(X, Y) Dy Dx

for every ϕ ∈L2[Ω; C#(Q)]. Finally we recall a simple criteria that permits to ≪bypass≫the problem concerning the convergence of the product of two L2(Ω)-weakly convergence sequences (cfr.[3, 20]).

Proposition 2.10 Let (uε) and (vε) be sequences in L2(Ω) that respectively two-scale converge to u and v in L2(Ω× Q). If at least one of them strongly two-scale converges, then uεvε ↠uv.

In particular, if (uεvε) is bounded in L2(Ω), from the previous proposition, we have

U(X, Y)V(X, Y)Φ(X, Y) Dy Dx

for every ϕ ∈L2[Ω; C#(Q)]. TheequicoercivityofthecompositeGibbs-Landaufreeenergyfunc-

Tionals

This section is devoted to the proof of the equicoercivity of the family of Gibbs-Landau free energy func-

Tionals (Gε

L)ε∈R+ expressed by (5). Equicoercivity has an important role in homogenization theory. In fact, the metric space in which to work, must be able to guarantee the equicoercivity of the family of functionals under consideration, i.e. the validity of the Fundamental Theorem of Γ-convergence.

Proposition 3.1 The Family (Gε

L)ε∈R+ of Gibbs-Landau energy functionals is equicoercive on the metric

. Proof 3.2 According to the hypotheses [H1], [H2] and [H3], there exist positive constants cex, Cex, Cs, Can ∈

S|Ω|, Cs|Ω|1/2∥Ha∥Ω, Can|Ω|

. Now observe that for every constant in space magnetization u and for every ε > 0 on has Gε

(24)

To finish, we simply observe that, due to Rellich–Kondrachov theorem, K is a compact subset of

The Γ-Limit Of Exchange Energy Functionals Eε

The fundamental constraint of micromagnetic theory, i.e. the fact that the domain of definition of the family Eε is a manifold value Sobolev space, plays a fundamental role in the homogenization process. In fact, although the unconstrained problem has been fully investigated (see [7, 21, 25]), it is not possible to get full information about the manifold constrained Γ-limit by just looking at the unconstrained one. This is due to the well-known fact (see ) that if (Fε) is a family of functionals defined in the metric space X and Γ-converging to F, and G = Γ- lim[Gε := Fε|Y ] where (Gε) := (Fε|Y ) represents the family of functionals obtained as the restriction of (Fε) to some (metric) subspace Y of X, then F|Y ⩽G. Thus the identification of the manifold constrained Γ-limit requires more effort.

The Tangential Homogenization Theorem

In what follows we make use of the following theorem due to Babadjian and Millot (see ) in which the dependence of the Γ-limit from the tangent bundle of the manifold is taken into account via the so-called tangentially homogenized energy density. We state the tangential homogenization theorem (thm) in a bit less general form which is adequate for our purposes.

Proposition 4.1 (thm) Let M be a connected smooth submanifold of R3 without boundary and g : R3 × R3×3 →R+ be a Carath´eodory function such that 1. For every ξ ∈R3×3 the function g(·, ξ) is Q-periodic, i.e. such that if (e1, e2, e3) denotes the canonical

Α|Ξ|2 ⩽G(Y, Ξ) ⩽Β(1 + |Ξ|2)

for a.e. y ∈R3 and all ξ ∈R3×3.

(25)

defined in the metric space (H1(Ω, M), dL2(Ω,M)) Γ-converges to the functional

(26)

where for every s ∈M and ξ ∈[Ts(M)]3, Qt := [0, t]3,

(27)

is the tangentially homogenized energy density. We refer the reader to for a more general version and the proof. The role of tangent bundle.

Let us emphasize why the tangent bundle [T (M)]3 := ⊔s∈M[Ts(M)]3 plays a role. In order to understand this, it is convenient to develop a minimizer mε of Eε under the so-called multiscale expansion

(28)

where m0, m1 are respectively a minimizer of the Γ-limit of Eε and the null average first order corrector. Clearly, due to the constraint mε(x) ∈M for a.e. x ∈Ω, we get

(29)

where we have denoted by n the local normal field defined around mε(x) ∈M. By passing to the two-scale limit in both terms of (29), we formally reach the equality 0 ≡n[m0] · (∇m0 + ∇ym1) ≡n[m0] · ∇ym1, which shows that n[m0(x)] · m1(x, y) does not depend on y. Then, passing to the average over Q we get m1(x, y) ∈Tm0(x)(M). The rigorous formulation of the previous idea is the object of the next Proposition: Proposition 4.2 Let M be a connected smooth submanifold of R3, and let (mε) be a sequence in H1(Ω, M)

And

mε ↠m. Moreover there exists a null average function v ∈L2[Ω; H1

And

v(x, y) ∈Tm0(x)(M) for a.e. (x, y) ∈Ω× Q. Proof 4.3 In view of Proposition 2.8, we only need to prove that v(x, y) ∈Tm0(x)(M) for a.e. (x, y) ∈Ω×Q. To this end, let us denote by n(m) the normal vector at m ∈M and observe that it is sufficient to prove that the scalar function n(m(x)) · v(x, y) does not depend on the y variable, i.e. that in the language of

(30)

Indeed, as far as n(m(x)) · v(x, y) is independent from the y variable, since by assumption ⟨v(x, ·)⟩Q = 0 for a.e. x ∈Ω, one has n(m(x)) · v(x, y) = n(m(x)) · ⟨v(x, ·)⟩Q = 0 and therefore v(x, y) ∈Tm(x)(M) for a.e. (x, y) ∈Ω× Q.

To prove (30) we note that since mε →m in L2(Ω), one also has n(mε) →n(m) in L2(Ω). Therefore the family n(mε) strongly two-scale converges to n (m) and moreover 0 = (n(mε)·∇, mε)Ω→(n(m)·∇, m)Ω.

[M(X) · V(X, Y)] Divy Φ(X, Y) Dx,

i.e. the desired relation (30). The role of tangentially homogenized energy density Although the following considerations are not completely rigorous, they give a very good explanation of the idea behind the expression of the tangentially homogenized energy density (27); they are mainly based on the notion of two-scale convergence while relies on Γ-convergence.

(31)

and let us denote by EM its Γ-limit in the metric space (H1(Ω, M), dL2(Ω,M)). Since H1(Ω, M) is a metric

Subspace Of H1 Ω, R3

, from the properties of Γ-convergence under restriction to subspaces (see ), we know that for every m0 ∈H1(Ω, M) and for every (mε) ∈CdL2(Ω,M)(m0)

(32)

Now, let M be a convex smooth manifold, i.e. a smooth manifold lying on some subset of the boundary of a convex bounded domain ΘM. Next, define the tangent cone of M at s ∈M by the position

,

and denote by ΠM the nearest point projection on ΘM. Since ΠM is a (Lipschitz) non-expansive map, one has ΠM[u] ∈H1(Ω, M) for every u ∈H1(Ω, Θc

(33)

Let us now suppose that m0 is sufficiently smooth so that for every test function m1 ∈D[Ω; C∞

# (Q, Tm0(M))],

the family mε(x) := m0(x) + εm1(x, x/ε) belongs to H1(Ω, Θc

Πm[Mε] →M0

in (H1(Ω, M), dL2(Ω,M)). Therefore, taking into account the estimates (33), (32) and the fact that ∇ym1 is an admissible test function (see , Remark 1.11), we get (passing to the two-scale limit):

Since M1 ∈D[Ω; C∞

# (Q, Tm0(M))] is an arbitrary test function, passing to the infimum we finish with the following upper and lower bound for the manifold constrained homogenized functional:

Since The Functional I[Ξ, ·] : C∞

# (Q, Ts(M)) →R+ is continuous with respect to the H1

# (Q, Ts(M)) Is Dense In H1

#(Q, Ts(M)), the infimum in (34) can be taken over H1 #(Q, Ts(M)). Remark 4.4 Quite remarkably, as we prove below, when M := S2 the infimum appearing in the right-hand side of (34) does not depend on the s-variable and coincides with the infimum in the left-hand side of (34).

Thus for M := S2 also the lower bound to EM is sharp and ER3 ≡ES2.

The Tangentially Homogenized Exchange Energy Ehom

Let us go back to the application of the tangentially homogenization result in our setting.

We Consider

the family of exchange energy functionals, all defined in H1(Ω, S2), given by (Eε)ε∈R+. Since [H1] holds, Proposition 4.1 ensures that the family (Eε)ε∈R+ Γ-converges in the metric space

, i.e. with respect to the topology induces on H1(Ω, S2) by the strong L2(Ω, R3) topology, to the functional

Equivalently, Since It[Ξ, ·] : H1

0(Qt, Ts(S2)) →R+ is a continuous functional, and the subspace W 1,∞

Is Dense In H1

0(Qt, Ts(S2)), we have for every s ∈S2 and every ξ ∈[Ts(S2)]3

(39)

This latter formulation is a more convenient one.

Ndeed, It Can Be Easily Verified, By Lax-Milgram

theorem, that for every t ∈N, every s ∈S2 and every ξ ∈[Ts(S2)]3, there exists a unique solution

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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