Digimat Rp Reinforced Plastics

Yinshan Chang, Yichao Huang∗, Dang-Zheng Liu, And Xiaolin Zeng

Abstract. The goal of this note is twofold: first, we explain the relation between the isomor- phism theorems in the context of vertex reinforced jump process, discovered in [BHS19, BHS21] and the standard Markovian isomorphism theorems for Markovian jump processes; second, we introduce the vertex reinforced counterpart of the standard Poissonian loop soup developed by Le Jan [LJ10]. To this end, we propose an algorithm that can be viewed as a variant of Wilson’s algorithm with reinforcement. We establish the isomorphism theorems for the erased loops and the random walk from this algorithm, and in particular provide a concrete construction of the reinforced loop soup via a random process with a reinforcement mechanism.

Ntroduction

Kurt Symanzik formulated a framework for Euclidean quantum field theory in [Sym68], establishing a profound connection between Euclidean quantum field theory and the structure of classical statistical mechanics. Especially, he introduced the random path expansion for the Green function [JLM85, Wil77], which was further developed in [BFS82, BFS83, Dyn83, Dyn84, Aiz82].

The goal of this current paper is to consider Symanzik’s loop expansion idea for Euclidean quantum field theory in the context of the so-called H2∣2 supersymmetric hyperbolic sigma model in relation to a class of random processes with reinforcement.

One of the realizations of Symanzik’s loop (or “gas/soup”) expansion for the free field theory is known in the mathematics community under the name of Poissonian loop soup, both in the discrete (random walk loop soup [LTF07], Markovian loop soup [LJ10]) and in the continuum (Brownian loop soup [LW04]). These objects are connected to the discrete Gaussian free field and the continuous Gaussian free field [LJ11, Szn12b]. Loop soup provides an important toolbox for a non-perturbative approach to studying phenomena in quantum field theory. For instance, the Brownian loop soup in dimension two is used as an essential tool to construct the conformal loop ensembles [SW12] and is thus connected to various models of statistical mechanics (especially the interfaces in two-dimensional critical systems). In the case of the Gaussian free fields, the loops in the loop soup are thrown independently as the whole random process can be realized as a Poisson point process on the space of loops. There is a vast literature in the probability community on the Poissonian or Brownian loop soup, in particular on the study of local times [Ray63, Kni63], occupational time fields and isomorphism theorems [Dyn80, Eis05, EKM+00], random interlacements [Szn12a, Lup16] and the list is far beyond our ability to survey here.

It is interesting and important to extend the theory of Symanzik to interacting Euclidean quantum field theory beyond the Gaussian case. It was also Symanzik’s original motivation to consider interacting random paths or random loop ensembles. If there were non-quadratic interaction, e.g. φ4 term or interaction through fermionic variables, the corresponding loop expansion of Symanzik would no longer be Poissonian, and the standard toolbox of Markov processes has to be extended to implement Symanzik’s ideas. Following the loop expansion idea of Symanzik, we introduce and study a non-Poissonian loop soup model with reinforcement, for which the underlying field is a supersymmetric hyperbolic sigma model, first introduced by Zirnbauer [Zir91] inspired by the work of Efetov [Efe99]. The supersymmetric model of Zirnbauer

Yinshan Chang, Yichao Huang∗, Dang-Zheng Liu, And Xiaolin Zeng

is also abbreviated as the H2∣2-model, because the field takes its value in a hyperbolic space which can be parametrized by two bosonic variables x,y and two anti-commuting fermionic variables ξ,η. Roughly speaking, the H2∣2-model is a supersymmetric quantum field theory with the following

J + Y2

j + 2ξjηj. The H2∣2-model has been shown to have a phase transition in its coupling parameter in [DS10, DSZ10] on Z3. Later, it has drawn attention to the probability community because of the unexpected discovery in [ST15] of its relation to the so-called Vertex Reinforced Jump Pro- cess [DV02, Dav90], a non-Markovian jump process which interacts with its own history. The Vertex Reinforced Jump Process, after a suitable random time change, gains an extra property known as partial exchangeability [DF80], and is therefore a mixture of Markov jump processes in some random environment, the law of which is exactly given by some horospherical coordinate of the H2∣2-model. Probabilistic models related to the H2∣2-model are disordered systems (statistical mechanics models with random coupling parameters). See Section 2.8 below for a brief overview of this model, and [DSZ10, DS10, ST15, ACK14, STZ17, MRT19, SZ19] for a far-from-complete list of references.

In [BHS19, BHS21], several isomorphism theorems have been discovered as the Vertex Rein- forced Jump Process counterparts of the standard Markovian forms of respectively BFS-Dynkin isomorphism, generalized second Ray-Knight theorem, and Eisenbaum’s isomorphism. In these three isomorphism theorems, the counterpart of the Gaussian free field in the above-mentioned theorems becomes the H2∣2 supersymmetric hyperbolic sigma model. This is an important step towards implementing Symanzik’s idea in a supersymmetric quantum field theory with non-trivial interaction, thus with interacting random paths. We feel that it is useful to also investigate Symanzik’s loop expansion theory of the H2∣2-model as a non-perturbative approach to under- standing certain integrable aspects of this supersymmetric hyperbolic sigma model. We propose a candidate for the counterpart of the standard Poissonian loop soup in the case of H2∣2-model, which we call the reinforced loop soup. While the reinforced loop soup can no longer be realized as a Poisson point process, one of our main contributions is the mathematical construction of this object, using a variant of Wilson’s algorithm [Wil96, PW96] with a natural reinforcement mechanism. It turns out that many features of the Poissonian loop soup still hold when the Gaussian free field is replaced by the H2∣2 supersymmetric hyperbolic sigma model. In particular, we establish a Dynkin-type isomorphism theorem, relating the occupation time field of reinforced loop soup to natural observables of the H2∣2 field. To show all these isomorphism theorems, we introduce a key tool called the supersymmetric Bayes formula. This formula allows one to translate quite systematically any standard Markovian isomorphism theorem into its Vertex Reinforced Jump Process counterpart. We now give a summary of our approach and main results.

1.1. Main results. Our contribution in this paper is resumed as follows: (1) A reinforced Wilson’s algorithm. We conceive a variant of Wilson’s algorithm with a reinforcement mechanism, which we use to concretely construct the reinforced loop soup.

In short, we replace the Markovian exploration in the standard Wilson’s algorithm with the Vertex Reinforced Jump Process, and in such a way obtain a reinforced spanning tree and a reinforced random loop collection. The reinforced random loop collection from this algorithm gives a trajectory explanation to the Dynkin-type reinforced loop soup isomorphism below.

(2) Connection of the Dynkin-type isomorphism theorems for the H2∣2-model to the standard

Reinforced Loop Soup Via Wilson’S Algorithm

H2∣2-model discovered in [BHS19, BHS21] can be obtained as annealed versions of the corresponding standard Markovian Dynkin-type isomorphism theorems, by integrating them in the random environment generated by the Vertex Reinforced Jump Process. This idea is based on the connection between the H2∣2-model and the Vertex Reinforced Jump Process discovered by Sabot and Tarrès [ST15], and our main tool is a supersymmetric Bayes formula.

(3) Dynkin-type isomorphism theorem for the reinforced loop soup. Using the reinforced Wilson’s algorithm and our method of proof for the H2∣2 isomorphism theorems for the Vertex Reinforced Jump Process, we propose and prove a H2∣2 isomorphism theorem for the reinforced loop soup. Indeed, we show that the occupation time field of the loops obtained from the reinforced Wilson’s algorithm is the correct H2∣2 counterpart in Le Jan’s isomorphism theorem for the Poissonian loop soup. This gives a natural construction for a non-Poissonian loop soup model and establishes an integrability result thereof.

(4) Complete reconstruction of the reinforced loop soup with the reinforced Wilson’s algorithm. We use the reinforced Wilson’s algorithm to give a complete reconstruction of reinforced loop soups with arbitrary parameter α > 0, based on the idea of a Poisson-Dirichlet decomposition of random loops due to Le Jan. We also extend the H2∣2 isomorphism theorem for the reinforced loop soup to higher-dimensional H2k∣2k versions with positive integer k.

We hope that the definition and the study of the reinforced loop soup can lead to a better understanding of the H2∣2-model, and many interesting questions remain to be addressed. To list a few, our method should give precise information on the reinforced random interlacements in the H2∣2-model. Another more challenging direction would be to investigate the continuum limit of the reinforced loop soup, towards a definition of the reinforced Brownian loop soup or reinforced loop ensembles for the H2∣2-model.

1.2. Structure of the paper. This paper is organized as follows. In Section 2 we gather the necessary backgrounds for this paper, including the standard forms of Wilson’s algorithm, the definition of the standard Poissonian loop soup, the supersymmetric fields and the Vertex Reinforced Jump Process. In Section 3 we introduce a variant of Wilson’s algorithm with reinforcement mechanism. In Section 4 we present a useful lemma (called supersymmetric Bayes formula) relating supersymmetric free field expectations with the H2∣2 expectations. In Section 5, we give alternative proofs of the three BFS-Dynkin type isomorphisms for the Vertex Reinforced Jump Process. In Section 6 we use the reinforced Wilson’s algorithm to construct the reinforced loop soup occuptional time field and establish the corresponding H2∣2 isomorphism theorem thereof. In Section 7 we reconstruct the reinforced loop soup process with arbitrary parameter α > 0 using the reinforced Wilson’s algorithm, and extend the isomorphism theorem for the reinforced loop soup to higher-dimensional supersymmetric hyperbolic sigma models.

Acknowledgement. Y.C. is supported by National Natural Science Foundation of China (grant 2023YFA1010103) and by NSFC-12301164, and thanks the Institut de Recherche Mathématique Avancée (IRMA) of the Université de Strasbourg for their kind hospitality. D-Z.L. is supported by the National Natural Science Foundation of China #12371157 and #12090012. X.Z. acknowledges the Agence Nationale de la Recherche for their financial support via ANR grant RAW ANR-20- CE40-0012-01 and IRMIA++.

Backgrounds

We collect some elementary backgrounds for the main objects studied in this paper.

Yinshan Chang, Yichao Huang∗, Dang-Zheng Liu, And Xiaolin Zeng

2.1. Graph Laplacian. Consider a finite graph of vertex set V = {i,j,k,...} and non-oriented edges E(V ) = {(ij)}. To each edge (ij) ∈E is associated a non-negative weight Wij = Wji ≥0. When there is no edge between two vertices i,j ∈V , it is equivalent in this paper to add the edge (ij) to the edge set E(V ) and declare that Wij = Wji = 0 (and implicitly Wii = 0). To each configuration of weights on the graph is associated a square symmetric Laplacian matrix ∆W of

∑K∈V ∖{I} Wik

i = j . We sometimes consider a distinguished vertex δ called the root vertex of the graph. This can either be done by assigning a vertex to be the root, or equivalently, we can augment the graph (V,E(V ),W) in the following way: the vertex set is augmented tõ V = V ∪{δ}, the edge set is augmented to E(̃V ) = E(V )∪{(iδ),i ∈V } and we impose that the augmented edge weights satisfy the condition that at least one of the {Wiδ}i∈V is positive (so that δ is not disconnected from the original graph). The augmented graph will be denoted bỹ V ,E(̃V ), and̃ W for respectively its vertex set, edge set, and edge weight set. All corresponding definitions on the augmented graph

Will Be Decorated With An Extrã

. 2.2. Uniform spanning tree. Consider a finite weighted connected graph V . Here, the con- nectness of V means that one can move from any vertex i ∈V to another vertex j ∈V using only positively weighted edges. A spanning tree T is a tree of V (that is, a connected subgraph of V without loops) whose set of vertices is exactly V . The weight W(T) of the spanning tree T is the product of all the weights of its edges, i.e.

(Ij)∈E(T )

Wij. When V is finite, the number of spanning trees on V is large (more precisely ∣V ∣∣V ∣−2 by Cayley’s formula), but it remains finite. A random spanning tree T is called uniform if a sample T is chosen with probability proportional to W(T), the weight of T. In the literature, T constructed above is often called weighted uniform spanning tree (when no weight is specified, the standard uniform spanning tree is the special case of constant weight W ≡1): our graphs will always be weighted and we drop the term “weighted” in the sequel. Notice that it makes no difference whether V is rooted or not in the above definition, and in sections dealing with Wilson’s algorithms, we write a rooted graph as V instead of̃ V for simplicity and to be conform with existing conventions.

2.3. The standard Markovian Wilson’s algorithm. Given a finite connected graph V with root vertex δ, the celebrated Wilson’s algorithm [Wil96, PW96] generates a random (uniform) spanning tree T of V using a procedure known as the loop erasure. For later purposes, we follows [LP16, Chapter 4].

2.3.1. Wilson’s algorithm and loop erasure. The most well-known form of the standard Markovian Wilson’s algorithm uses the idea of loop erasure (or cycle erasure). Let P ∶i0 →i1 →i2 →... be a path in V , with (ik)k=0,1,... vertices of V appearing in the order in which they are visited. The loop erased path LE(P) is the path obtained by chronologically erasing loops that appear in P.

It is clear from the construction that LE(P) is self-avoiding, that is, it does not contain any loop. The standard Markovian Wilson’s algorithm with loop erasure procedure can be described as follows. First, order the vertices arbitrarily, say V = (i0,i1,...,i∣V ∣−1), and declare δ = i0 to be the root of V . Let T(0) = {δ} be the initial tree, and inductively grow the tree T(i) in the following way. If T(i) spans V , then we stop. Otherwise, let x be the vertex with the smallest subscript 1The terminology “standard Markovian” is used in opposition to “reinforced” or “supersymmetric”.

Reinforced Loop Soup Via Wilson’S Algorithm

not in T(i), and consider an independent Markov random walk starting from x with edge weights (We)e∈E(V ) until the first time it hits T(i). Perform the loop erasure operation on the path of this Markov random walk, and add the self-avoiding branch obtained by loop erasure to the tree T(i). This constructs the next T(i + 1), which is a tree by self-avoidedness. The final tree T is spanning and random, and the observation of Wilson is the following (see [LP16, Theorem 4.1]): Lemma A (Wilson’s algorithm with loop erasure). The random spanning tree T constructed via the loop erasure algorithm is the uniform spanning tree of V .

In particular, T is independent of the choice of the root and of the way we numbered the vertices. The next construction shows that at each step when T(i) is constructed, we can reorder the remaining vertices, and the resulting T will still be the uniform spanning tree. This last observation (see the paragraph under [LP16, Theorem 4.1]) seems to be less well known and is sometimes omitted in some textbooks, but will be crucial in our construction of the reinforced version of Wilson’s algorithm in Section 3.

2.3.2. Wilson’s algorithm and cycle popping. We recall an alternative description of the standard Markovian Wilson’s algorithm. The key idea is usually called cycle popping. Consider a finite connected graph V with root vertex δ. Suppose that a stack of cards {Si

K}K≥1

is situated at each vertex i ∈V except at the root vertex δ, with the card Si

On Top Of Si

3 etc. On each card, an arrow is given, which tells a walker where to go for the next step. The card is “popped” (or removed) and the next card revealed when the walker uses this card to go to the next vertex. If the arrow on the card Si

K Is Sampled Independently According To

the transition probability at the vertex i to its neighbors using the weights (Wij)j∈V , i.e.

,

then the law of the walker is that of a standard Markov chain with edge weights (Wij)i,j∈V and root vertex δ. If further the walker stays at the vertex i with the correct exponential transition time before jumping (that is, an independent exponential variable with parameter ∑k∈V Wik), then we obtain the standard continuous Markov jump process on V with edge weights (Wij)i,j∈V and killed at the root vertex δ.

Now we introduce the idea of cycle popping. Consider a configuration of stacks as above, where we can observe one visible card at the top of the stack at each vertex except at the root vertex δ. Connecting each vertex to its neighbor indicated on the visible card, we obtain an oriented graph of V rooted at δ. If this oriented graph has no cycle, then it is a (oriented) spanning tree with root δ. If this oriented graph has a cycle, then we “pop” this cycle by removing all the visible cards in this cycle, revealing the next cards in the respective stacks. We obtain a new oriented graph, and we continue this cycle popping procedure until it stops. If this algorithm stops, we unorient the oriented spanning tree, and obtain a random spanning tree of V rooted at δ.2 The standard Markovian Wilson’s algorithm with cycle popping is the following statement (see [LP16,

Emma 4.2]):

Lemma B (Wilson’s algorithm with cycle popping). The random spanning tree T constructed via the cycle popping algorithm is the uniform spanning tree of V . In particular, T is independent of the order in which the cycles are popped. It is important to note that the core argument [LP16, Lemma 4.2] is deterministic: given any sample of stacks, the order in which the cycles are popped is irrelevant for the final spanning tree. Therefore, the previous loop erasure form of the standard Markovian Wilson’s algorithm can be regarded as a 2If we want to run Wilson’s algorithm on an oriented graph, choose the outgoing weights in the definition of the jump rates above, and do not unorient the final spanning tree.

Yinshan Chang, Yichao Huang∗, Dang-Zheng Liu, And Xiaolin Zeng

convenient choice of popping cycles in a deterministic order, but any algorithm that explores the graph properly and pops every possible cycle is equivalent to the standard Markovian Wilson’s algorithm. In Section 3.1 we will introduce an appropriate form of Wilson’s algorithm with a reinforcement mechanism, suitable to the purpose of this article.

2.4. Poissonian loop soup. We now recall the standard Markovian construction of Poissonian loop soup, following the language of [LJ24] and [Szn12b]. Consider a finite graph V with edge weights (Wij)i,j∈V . A based loop with base point i is a trajectory starting and ending at the same vertex i ∈V , of the form

Ðð→

with 0 < t1 < t2 < ⋅⋅⋅< tk < t and ik vertices of V , where the quantities on the arrows denote the time spent at the previous vertex before the jump to the next vertex. The last arrow means that we stay at i for a time equal to t −tk without jumping and the trajectory ends. For example, i

Tð→

is the trivial loop that makes no jump. Given a Markov process with generator L(i,j), we can define the bridge measure of the above

T

(dl) = L(i,i1)L(i1,i2)...L(ik−1,i)e−t1L(i,i)−(t2−t1)L(i1,i1)−⋅⋅⋅−(t−tk)L(i,i)dt1dt2 ...dtk. The based loop measure (associated to the generator L) is an infinite measure on the space of

T

(dl)dt. We can work on the equivalence class of based loops by forgetting the base point (via rerooting). The image of the loop measure µ under this equivalence relation is denoted µ○, and is called the (unbased) loop measure. For this paper, all computations are the same using either measure.

The Poissonian ensemble of loops Lα for α > 0 is defined as the Poisson point process in the space of loops with intensity measure αµ○. This means that, for any functional F on the space of loops vanishing on loops with arbitrarily small length,

= Exp(Α∫(Eif (L) −1)Μ○(Dl))

where the integral on the right-hand side is over the space of all admissible unbased loops. Since the measures µ○and µ coincide on loop functionals, equivalently one can use the intensity measure αµ and the space of based loops in the above display.

The occupation time field of a (based or unbased) loop l iŝ

∀I ∈V,

and the occupation time field of a collection of random loops L iŝ L = ∑l∈L̂ l. In particular, the occupation field of the Poissonian loop soup L1 with α = 1 will be denoted̂ L1. 2.5. Loop soup and Wilson’s algorithm. In the standard Markovian Wilson’s algorithm, one can look at the resulting uniform spanning tree as well as the erased loops during the algorithm.

There are deep symmetries involving these two random processes similar to the boson-fermion correspondence. Here we only collect some useful properties and refer to [LJ24, Chapter 8] for a detailed exposition.

Consider the standard Markovian Wilson’s algorithms reviewed in Section 2.3. The result that will be most relevant for us is the following identification [LJ24, Corollary 8.1]:

Reinforced Loop Soup Via Wilson’S Algorithm

Lemma C (Occupation field of the random loops in the standard Markovian Wilson’s algorithm). Consider a symmetric Markov jump process with generator̃ A on the augmented graph̃ V with killing at the root vertex δ. The occupation time field defined by the random set of (based) erased loops during the standard Markovian Wilson’s algorithm oñ V with root δ is independent of the random spanning tree and of the ordering of the vertices, and has the same distribution as the occupation time field̂ L1 of the Poissonian loop soup on V with generator̃ A∣V of parameter α = 1.

The standard Markovian Wilson’s algorithm can be refined to give a complete reconstruction of the Poissonian loop soup L1 by applying a Poisson-Dirichlet decomposition of loops, see [LJ24, Section 8.3] for more information on this reconstruction. See also Section 7.1 for the construction of general Poissonian loop soup Lα using Wilson’s algorithms with any α > 0.

isomorphism theorems, which are identities in law (i.e. exact relations on expectations) between the occupation time field and the (scalar) free field. A pedagogical reference to all the standard Markovian isomorphisms below is [Szn12b, Chapter 2].

2.6.1. BFS-Dynkin isomorphism theorem. Consider a standard Markov jump process (Zt)t≥0 of (symmetric) generator̃ A starting at some vertex a ∈V on the augmented graph̃ V . Suppose that (Zt)t≥0 is killed at the root vertex δ, denote by ϱ the killing time, and let S = S(ϱ) be the final

Ρ

0 1{Zt=i}dt). Theorem D (Standard Markovian BFS-Dynkin isomorphism theorem). Let ẼA

A Denote The Expec-

tation with respect to the process Z. For any smooth bounded function g with rapid decay,

Φ2)],

where b is the vertex last visited by Z before δ, ϕ is the Gaussian free field of generator̃ A with

Wbδ ẼA

a [⋅1{Zϱ−=b}]. See [Szn12b, Theorem 2.8] and [Dyn84, BFS82] for background and proof of this identity. 2.6.2. Second generalized Ray-Knight isomorphism theorem. Consider a standard Markov jump process (Zt)t≥0 of (symmetric) generator A starting at some vertex a ∈V on the graph V with

Ρ

0 1{Zt=i}dt. Denote by σ(γ) = inf{t > 0 ; Sa(t) > γ} the first instant the local time at the starting point a ∈V exceeds γ > 0, and S = S(σ) the final local times. Theorem E (Standard Markovian second generalized Ray-Knight isomorphism theorem). For any

Γ)2)],

where ϕ is the Gaussian free field with generator A and pinning ϕa = 0. We refer to [EKM+00] for background and proof of this identity. See also [ST16] for more connections to Vertex Reinforced Jump Process (reviewed in Section 2.9).

2.6.3. Eisenbaum’s isomorphism theorem. Consider a standard Markov jump process (Zt)t≥0 of (symmetric) generator̃ A starting at some vertex a ∈V on the augmented graph̃ V . Suppose that (Zt)t≥0 is killed at the root vertex δ, denote by ϱ the killing time, and let S = S(ϱ) be the final

Yinshan Chang, Yichao Huang∗, Dang-Zheng Liu, And Xiaolin Zeng

Theorem F (Standard Markovian Eisenbaum’s isomorphism theorem). For any smooth bounded

(Φ + S)2)],

where ϕ is the Gaussian free field with generator̃ A and pinning ϕδ = 0. We refer to [Szn12b, Thoerem 2.10] and [Eis95] for background and proof of this identity. 2.6.4. Standard Markovian loop soup isomorphism. Isomorphism theorems can be extended to the standard Markovian loop soup, which is somewhat already rooted in the original work of Dynkin [Dyn84] and developed by Le Jan [LJ10]. We will focus on the case with α = 1, although this identity can be generalized to any half integer-valued α.

Theorem G (Dynkin isomorphism for the loop soup). The occupation field of L1 of a loop soup with a symmetric generator A has the same distribution as the average of the squares of two independent Gaussian free fields of the same generator A. In other words, for any smooth bounded

(X2 + Y2))E−1

2 (xAx+yAy) det(A)dxdy. The proof of this theorem is due to Le Jan [LJ10] can be found in [Szn12b, Theorem 4.5] or [LJ24, Theorem 6.1]. In particular, in combination with Lemma C, this identity also applies to the occupation time field of the erased loops during the standard Markovian Wilson’s algorithm on the augmented graph̃ V with root δ.

2.7. Supersymmetric free field. We now recall some basic notions about supersymmetric field theories, following [Weg16] and [Efe99]. We will be mainly focusing on the case of the so-called (2,2)-supersymmetric free field, and drop the specification (2,2) when there is no ambiguity.

First, let us recall some basics of Grassmann calculus. Consider bosonic variables (x,y,z,...) and fermionic variables (ξ,η,...). Recall that when a variable is fermionic, it anticommutes with other fermionic variables and commutes with bosonic variables. For example,

Ξx = Xξ,

... A bosonic variable commutes with all variables. A bosonic variable is not necessarily real, e.g. ξη is bosonic. In general, one can deal with fermionic variables by representing a function with its formal (infinite) Taylor series in the fermionic variables: this series is a polynomial in the fermionic variables by anticommutation. For a systematic treatment of elementary Grassmann calculus towards applications in statistical physics, one can consult [Weg16, Efe99].

It is a standard fact that for Grassmann variables, derivation and integration are essentially the same operation, modulo a sign convention. We adopt the following convention throughout this article, that integration with the differential to the left coincides with left differentiation, e.g.

∫Dξf(Ξ) = ∂

∂ξ f(ξ). This is the convention used in [BHS21, DSZ10]: see [Weg16, Chapter 3] for a quick reminder. Associate each vertex i in the graph V with a four vector Xi = (xi,yi,ξi,ηi), with x,y real variables and ξ,η fermionic variables. We usually denote this by Xi ∈R2∣2, the superscript 2∣2 refers to the numbers of each type of variable. To define the supersymmetric free field with a language that is familiar to probabilists, one can think of the variables Xi as spins, equipped

Reinforced Loop Soup Via Wilson’S Algorithm

where the ordering of ξ,η is important due to the anticommutation of fermionic variables. We sometimes drop the ⋅and write this inner product as XiXj when there is no ambiguity. An energy term (equivalently, action functional or Gibbs measure) associated to each configuration (Xi)i∈V of this system induces (formally) a randomized spin configuration. The Berezin integral form

Π Dxidyidξidηi,

which should be understood as the formal analog of the Lebesgue measure for R2. At this stage, the setup is purely formal, and performing Grassmann integrations does not yield in general quantities that have probabilistic interpretations. However, special choices of the energy term yield interesting field theory for probabilists. A prime example is the supersymmetric

(Ij)∈E(V )

(Xi −Xj)Wij(Xi −Xj). As with the usual scalar free field, one needs some extra pinning condition to make the integral

∫(R2∣2)∣V ∣E−1

2 X∆W XDX converge.3 Therefore, consider the augmented graph̃ V with extra vertex δ and its associated extra edge weights (Wiδ)i∈V , with at least one positive Wiδ > 0 to preserve the connectedness of the graph. We usually choose the pinning condition with̃ Xδ = (0,0,0,0) = 0, and it is a special feature of the supersymmetric field theory (see Lemma J) that the formal partition function is constant equal to 1, i.e.

̃ X∆̃W̃ Xd̃X = 1.

A systematic way to prove this is via the localization formula, see Corollary K below. The triviality of the partition function implies that there is no difference between the un- normalized and normalized expectations for the supersymmetric free field, and we denote them by J⋅K̃W ,̃Xδ=0. More specifically, for any smooth bounded function F with rapid decay,

̃ X∆̃W̃ Xd̃X

is called the supersymmetric free field expectation with pinning at the root vertex (or the boundary) δ. When all the edge weights (Wiδ)i∈V connecting to the root δ are constant equal to h > 0, we also call this the supersymmetric free field expectation on V with mass h > 0.

2.8. Supersymmetric hyperbolic sigma model. We briefly recall next the definition of the H2∣2-model following [Zir91, DSZ10] and its connections to the Vertex Reinforced Jump Process. Consider a superfield vi = (xi,yi,zi,ξi,ηi) ∈R3∣2 defined on vertices i ∈V , where the first three coordinates (xi,yi,zi) are real variables and the last two coordinates (ξi,ηi) are fermionic variables. The (symmetric) inner product on this space is defined as vi ⋅vj = xixj + yiyj −zizj + ξiηj + ξjηi.

This inner product is sometimes denoted by vivj when there is no ambiguity. We impose that this superfield vi lives on the supersymmetric hyperbolic space H2∣2 defined by the constraint that

I −Z2

i + 2ξiηi = −1. Under this constraint, by Taylor expansion of the square root at 1 + x2

I In The Fermionic

variables (see [DSZ10, Section 2] if one is not familiar with this standard procedure in Grassmann 3The pinning condition on the augmented graph̃ V is equivalent to a boundary condition on the graph V with boundary {δ}: we will use the pinning terminology in the sequel.

Yinshan Chang, Yichao Huang∗, Dang-Zheng Liu, And Xiaolin Zeng

calculus), one can express zi as a function of the other coordinates (xi,yi,ξi,ηi). The convention

I

. The origin or the zero-vector 0 ∈H2∣2 is then defined to be 0 = (0,0,1,0,0) ∈H2∣2. The terminology “hyperbolic” refers to the fact that if we forget about the fermionic variables, the constraint x2 + y2 −z2 = −1 is that of a standard hyperbolic model, and this appellation is best justified when one switches to the so-called horospherical coordinates using [DSZ10, Appendix B].

However for us and as pointed out in [BHS21], all our results are coordinate-free, and the horospherical coordinates will not be used in this paper for simplicity. We refer to [DSZ10] for discussion on the hyperbolic nature of the H2∣2-model, and continue with R3∣2 coordinates.

Similarly to the supersymmetric free field theory above, we define the H2∣2 supersymmetric hyperbolic sigma model by defining its energy term (i.e. action functional) as

Wij(Xixj + Yiyj −Zizj + Ξiηj + Ξjηi + 1),

and the Berezin integral form on the H2∣2-model is

Zi

dξidηi. Due to the non-compactness of the space H2∣2 equipped with the above measure, we renormalize

The Integral ∫(H2∣2)∣V ∣E−1

2 v∆W vDµ(v) by adding a pinning at an extra vertex δ on the augmented graph̃ V . The localization formula of supersymmetric field theory yields that the partition function with pinning at δ is again 1 (see Corollary K):

̃ V∆̃W̃ Vdµ(̃V) = 1,

wherẽ v = (vi)i∈̃V is the augmented superfield oñ V = V ∪{δ}. More generally, we denote by ⟨⋅⟩̃W ,vδ=0 the formal expectation with respect to the above defined supersymmetric energy term: for smooth bounded functions F with rapid decay,

⟨F(̃V)⟩̃W ,Vδ=0 = ∫(H2∣2)∣V ∣F(̃V)1{̃Vδ=0}E−1

2̃ v∆̃W̃ vDµ(̃v). Remark 1 (“Equality in law”). We will often use some abuse of language by analogy to statistical physics. For example, the term “expectation” above is really just a (supersymmetric) integral, and one can realize the H2∣2-model via the supersymmetric free field using a “change of measure”.

All these terminologies should be understood in terms of the values of supersymmetric integrals against test functions: especially we will say two superfields are “equal in law” if for any smooth bounded function F with rapid decay, the expectations obtained by supersymmetric integration in the form of (3) or (5) of F in these two superfields are equal.

2.9. Vertex Reinforced Jump Process. We now introduce a random process with a rein- forcement mechanism which is central to this paper: the Vertex Reinforced Jump Process. This process was already introduced by [DV02], and has deep connections to the another random process with reinforcement called the edge reinforced random walk [Dav90]. This connection was at the heart of the modern proofs of the famous Coppersmith-Diaconis’ magic formula [CD87], and we recall some of the basic elements that are useful for this article.

Consider a finite connected graph V with edge weights W. At time s = 0, assign to each vertex i ∈V an initial positive local time ϑi > 0. Specify also a vertex i0 ∈V (we stress that this is not

Reinforced Loop Soup Via Wilson’S Algorithm

necessarily the root vertex), and start a continuous time jump process (Ys)s≥0 starting from i0, i.e. Y0 = i0. The process Y jumps from vertex i to another vertex j at time s with rate WijLj(s), where Lj(s) is the accumulated local time (or simply local time) of the process Y at time s, i.e.

S

0 1{Yτ =j}dτ. This process is naturally not Markov, since the jump rate depends on the past. Most of the modern understanding of this process is achieved with the following time change [ST15, Zen16]:

I ),

and consider the time-changed process (Zt)t≥0 defined by

{Zτ =I}Dτ = Li(D−1(T))2 −Θ2

i . for each vertex i ∈V . We will see below in Section 2.10 that the time-changed process (Zt)t≥0 acquires the so-called partial exchangeability property [DF84], thus is a mixture of standard Markovian jump processes in some random environment. This is an important philosophy throughout this article.

2.10. Connections between Vertex Reinforced Jump Processes and the supersym- metric hyperbolic sigma model. The H2∣2-model gains an extra probabilistic interpretation when looking at some marginal laws of the bosonic variables. The connection was unveiled by Sabot-Tarrès [ST15], and we only recall here some essential results that we use in this article.

Lemma H (Mixture of Markov jump processes). Recall the time-changed process (Zt)t≥0 in Section 2.9 starting from the vertex i0 ∈V with initial local times ϑ. It can be sampled in the following way. First, sample the environment u = {ui ∈R,ui0 = 0}i∈V ∖{i0} according to the following probability density function on R∣V ∣−1 with prescribed value ui0 = 0:

(7)

where the expression D(W,u) is a sum over the spanning trees T of the graph,

(Ij)∈E(T )

Wijeui+uj. Then, given a sample of the environment u, sample the Markov jump process with (static or time-independent) jump rate from a vertex i ∈V to another vertex j ∈V : 2Wijeuj−ui.

The law of (Zt)t≥0 is the same as the resulting continuous time jump process. We say that (Zt)t≥0 is a mixture of standard Markovian jump processes in the random environment (7). We also say that the above static Markov jump process is the quenched process (in the environment u). The fact that (7) is a probability density function is highly non-trivial without this lemma: a direct proof can be found in [LW17] and a supersymmetric proof in [DSZ10, Equation (5.1)]. The term D(W,u) is actually a determinant (see the discussion after [ST15, Theorem 2]), and (8) is obtained by the famous matrix tree theorem.

Yinshan Chang, Yichao Huang∗, Dang-Zheng Liu, And Xiaolin Zeng

As a consequence of the previous lemma, we get the following generalization to the Vertex Reinforced Jump Process on the augmented graph̃ V killed at the root vertex δ. All the notations are similarly defined on the augmented graph, and the mixing measure becomes

(9)

The law of (̃Zt)t≥0 is also that of a mixture of standard Markovian jump processes with jump rate

̃Wijeuj−Ui

starting at i0 ∈V and killed at the root δ ∈̃ V in the random environment defined by (9). Remark 2. Notice that u above is a vector indexed by V ∖{i0} and̃ u is a vector indexed by V . This is a slight abuse of notations due to the extra pinning condition.

We record a “change of starting point” formula in the sequel for the mixing measure. Lemma 3 (Shift in the mixing measure). Let a,b ∈V and consider the mixing measures dνW,1

B

with different starting points. If u′ and u are such that u′

B

(u′). As a consequence, if F is a function of the gradients (ui −uj)i∈V , then

A

(u′). Proof. This follows directly from the definition of the mixing measure (7).

□

The second result is the surprising link between the Vertex Reinforced Jump Process and the H2∣2 supersymmetric field theory. Theorem I (Marginal law of H2∣2 and the mixing measure of the Vertex Reinforced Jump Process).

Consider the following change of coordinates: for each i ∈̃ V , define

I + Ψiψi)Eti,

ξi = ψieti, ηi = ψieti. Then (̃t,̃s,̃ψ,̃ψ) is the horospherical coordinates of the H2∣2-model. The marginal law of̃ t of the H2∣2-model defined by (5) is equal to the law of the random environment̃ u defined with the mixing measure (9) and i0 = δ (in the sense of Remark 1).

This result is best understood when one uses the horospherical coordinates representation of the H2∣2-model as in [DSZ10]. Since we do not use this particular geometric input in the sequel, we refer to [DSZ10] and [ST15] for the background and proof of this fundamental result. We do not directly use Theorem I, but this philosophy will be important in the (self-contained) proof of the supersymmetric Bayes formula, Theorem 7 below.

Reinforced Loop Soup Via Wilson’S Algorithm

2.11. Localization formula. In supersymmetric field theories, a fundamental formula is the so-called (supersymmetric) localization formula. We will use especially the following formulation: Lemma J (Parisi-Sourlas formula). Recall the supersymmetric free field X defined in Section 2.7.

Let F((XiXj)i,j∈V ) be a smooth bounded function on the inner products (XiXj)i,j∈V with rapid decay. Then for the supersymmetric free field expectation with pinning at δ ∈̃ V , we have

(∆W +H)Xd̃X = F(0),

where H is the diagonal matrix with diagonal coefficients (Wiδ)i∈V . Many proofs of this lemma can be found in the literature, e.g. [DSZ10, Lemma 16]. An important consequence is the following [DSZ10, Proposition 2]: Corollary K (Trivialness of the partition functions). The partitions functions for the supersym- metric free field (2) and for the H2∣2-model (4) are both equal to 1, i.e.

J1K̃W ,̃Xδ=0 = ⟨1⟩̃W ,̃Φδ=0 = 1. Proof. Taking F ≡1 in the Parisi-Sourlas formula, we get the trivialness of the partition function for the supersymmetric free field (2).

For the H2∣2-model the argument is similar: define

+ X2

i for i ∈V (which is a function of the inner products) and let

Zi

be a rapidly decaying function on the inner products XiXj. Plugging this in Lemma J yields the triviality of the partition function for the H2∣2-model.

□

Another consequence of the localization formula is the invariance of the inner product vi ⋅vj and the Berezin integral form Dµ(v) under the Lorentz boost

(10)

θs(x,y,z,ξ,η) = (xcosh s + z sinh s,y,z cosh s + xsinh s,ξ,η). We refer to [DSZ10, Appendix B] for a proof. Notice that the Lorentz invariance is broken for the normalized supersymmetric expectation ⟨⋅⟩W,δ with pinning at the root vertex δ: indeed, the Lorentz invariance also acts on the boundary condition, so that on the augmented graph̃ V ,

∀S ∈R,

⟨F(̃v)⟩̃W ,vδ=θs(0) = ∫(H2∣2)∣V ∣F(̃v)1{vδ=θs(0)}e−1

̃ V∆̃W̃ Vdµ(̃V),

and similarly for expressions on the graph V .

Wilson’S Algorithm With Reinforcement

Consider a connected finite graph V with a distinguished vertex δ and edge weights Wij for i,j ∈V (we write V instead of̃ V for rooted graphs in sections dealing with Wilson’s algorithms). We describe below how an infinite chain of Vertex Reinforced Jump Process starting from δ can be used to generate a pair formed by a random spanning tree and a random collection of loops on the graph V . This process will be called Wilson’s algorithm with reinforcement, or the reinforced Wilson’s algorithm. We further show that, under certain random initial conditions, the pair of the random spanning tree and the random collection of loops on V obtained by Wilson’s algorithm with reinforcement can be obtained using a mixture of the standard Markovian Wilson’s algorithm in some random environment.

Yinshan Chang, Yichao Huang∗, Dang-Zheng Liu, And Xiaolin Zeng

3.1. Wilson’s algorithm generated by a single reinforced vertex jump process. Consider a Vertex Reinforced Jump Process on a connected finite graph V starting from a distinguished vertex δ. Suppose that the initial local time of each vertex i in V is ϑi and run a Vertex Reinforced Jump Process starting from δ, without being killed. It is known that this process is almost surely recurrent [ST15], therefore eventually visits every vertex in V infinitely many times.

We now describe our definition of Wilson’s algorithm with reinforcement, whose outcome is a pair formed by a random spanning tree and a random collection of loops on the graph V . The main difference with the standard Markovian Wilson’s algorithm is that the random walk will switch between two forms: a “visible” form where we perform loop erasure to construct the spanning tree and the loop soup, and an “invisible” form where we stop the loop erasure but only accumulate local times for the reinforcement mechanism.4 The reinforced Wilson’s algorithm gradually constructs a spanning tree as we will now explain.

(1) We start with the single-vertex graph T = {δ}: vertices of T will be called inactive. Define A to be the complement of vertices of T in V : this is the collection of active vertices. (2) We run a Vertex Reinforced Jump Process Y starting from the vertex δ with edge weights (Wij)i,j∈V and initial local times {ϑi}i∈V , and we denote by O = ∅the collection of loops at this stage.

(3) The process is in its invisible form when it is on a vertex inside T . When it jumps to another vertex inside T , we only update the local times according to the rules of the Vertex Reinforced Jump Process.

(4) The first time the process jumps to a vertex in A, declare the process to be visible. When the process is visible, it updates the local times according to the rules of the Vertex Reinforced Jump Process. Furthermore, for the visible part of the process, we perform the loop erasure procedure as in the standard Markovian Wilson’s algorithm until it jumps to a vertex in T .

(5) Upon returning to a vertex in T , the process is back to its invisible form, and we update T and the collection of loops O in the following way: the erased loops during the active form of the random walks will be added to the collection of loops O, and the remaining self-avoiding branch leading to T will be added to form an enlarged T .

(6) We repeat the steps (3) −(5) until the moment when T is a spanning tree of V . From this moment, the process will always be in its invisible form, so that T and O will not change.

In the sequel, we call the final random spanning tree Tϑ (as the final state of T ) the reinforced spanning tree and the final occupation time field̂ Lϑ

Of The Random Loop Process (As The Final State

of O) the reinforced loop soup occupation field, with initial local time conditions {ϑi}i∈V . Remark 4. We only need the occupation time field̂ Lϑ

Of The Reinforced Loop Soup In Our Main

Theorem 15, and the reconstruction of the reinforced loop soup Lϑ

From The Reinforced Wilson’S

algorithm is postponed to Section 7.1. 3.2. The standard Markovian Wilson’s algorithm generated by a single infinite Markov jump process. We first study the analog of the above-defined algorithm in the standard Markovian setting, i.e. without the reinforcement mechanism.

Lemma 5 (Standard Markovian Wilson’s algorithm using one infinite exploration). In the setting of a connected finite graph V with the usual notations, a variant of Wilson’s algorithm can be

Realized Using The Following Procedure:

4We choose the word “visible” to be coherent with terminologies in [LP16, Chapter 4] where the cycle-popping construction of Wilson’s algorithm is reviewed.

Reinforced Loop Soup Via Wilson’S Algorithm

(1) We start with the single-vertex graph T = {δ}: vertices of T will be called inactive. Define A to be the complement of vertices of T in V : this is the collection of active vertices. (2) We run a Markov chain X starting from the vertex δ, and we denote by O = ∅the collection of loops at this stage.

(3) The process is in its invisible form when it is on a vertex inside I. (4) The first time the process jumps to a vertex in A, declare the process to be visible. For the visible part of the process, we perform the loop erasure procedure until it jumps to a vertex in T .

(5) Upon returning to a vertex in T , the process is back to its invisible form, and we update the collection of loops L in the following way: the erased loops during the visible form of the random walks will be added to the collection of loops O, and the remaining branch leading to T will be added to form an enlarged T .

(6) We reiterate the steps (3) −(5) until the moment T is a spanning tree of V . From this moment, the process will always be in its invisible form, so that T and O will not change. Then the final random spanning tree T and the occupation time field of the final random loop process O have the same joint distribution as the uniform spanning tree and the occupation time field of the random loop process constructed from the standard Markovian Wilson’s algorithm of Section 2.3. In particular, the occupation time field of the final O is equal in law tô L1, the occupation field of the Poissonian loop soup with α = 1.

Notice that this is the algorithm in the above section except that no reinforcement mechanism is implemented: in particular no data is updated during the process. Proof. It suffices to check that this algorithm effectively pops all the possible cycles in the cycle-popping form of the standard Markovian Wilson’s algorithm in Section 2.3.2. Since the final random spanning tree T and the occupation time field of the random loop collection O are independent of the order in which the cycles are popped [LP16, Lemma 4.2], they have the same joint distribution as the pair described in Lemma B or Lemma A.

□

3.3. Reinforced Wilson’s algorithm as a mixture of standard Markovian Wilson’s algorithms in random environment. This proof of the following lemma is essentially the same as the proof of Lemma H originally stated for Vertex Reinforced Jump Processes.

Lemma 6 (Mixture of standard Markovian Wilson’s algorithms in random environment). The randomized reinforced Wilson’s algorithm is a mixture of standard Markovian Wilson’s algorithm for a Markov chain in a random environment u, with jump rates

Wijeuj−Ui,

and the law of the random field u is given by the mixing measure dνW,ϑ

Δ

(u) defined in (7). Proof. By Lemma H, the Vertex Reinforced Jump Process is a mixture of standard Markovian jump

Δ

(u) defined in (7) and the above jump rate. Conditioning on the environment u and performing the standard Wilson’s algorithm, which outputs the quenched uniform spanning tree and random loop process. In particular, the quenched Wilson’s algorithm can be performed using the form of Lemma 5, as all the standard Markovian Wilson’s algorithms yield the same uniform tree and random loop process in law. Since our reinforced Wilson’s algorithm of Section 3.1 is the same as that of Lemma 5 in the quenched sense, this finishes the proof that the reinforced Wilson’s algorithm of Section 3.1 is the annealed version of Lemma 5, integrated in the random environment u with the mixing measure dνW,ϑ

Yinshan Chang, Yichao Huang∗, Dang-Zheng Liu, And Xiaolin Zeng

The rules we defined for the reinforced Wilson’s algorithm in Section 3.1 are such that the process is partially exchangeable in the sense of [DF84], which guarantees the mixture of Markov process representation of Lemma 6.

A Supersymmetric Bayes Formula

In this section, we introduce a supersymmetric Bayes formula which will be used multiple times in the sequel. We first introduce some shorthand notations. Recall from Section 2.10 that the Vertex Reinforced Jump Process on V can be realized by first sampling a random environment u on the graph V with mixing measure (7), then sample the standard Markovian jump process with the following jump rate from a vertex i ∈V to another

Vertex J ∈V :

2Wijeuj−ui. This motivates the following notation: introduce the square matrix Au of size ∣V ∣with entries

∑K∈V ∖{I} Wikeuk−Ui

i = j . Then the generator of the quenched process, that is the standard Markovian jump process in

The Environment U, Has Generator 1

2e−uAueu, where eu is the diagonal matrix of size ∣V ∣with diagonal entries (eui)i∈V . Similarly, on the augmented graph with root vertex δ, the mixing measure of the random environment̃ u induces the following square matrix̃ Au of size ∣̃V ∣with

∑K∈̃V ∖{I} Wikeuk−Ui

i = j . The matrix Au (resp.̃ Au) is not symmetric, but we can introduce the following symmetric square matrix Bu = euAueu (resp.̃ Bu = ẽũAuẽu). Now it makes sense to speak about the supersymmetric free field in Section 2.7 with the matrix W replaced by the matrix Bu (resp.̃ Bu), for which the (unpinned) supersymmetric free field expectation will be denoted J⋅KBu (resp.

J⋅K̃Bu) as in Section 2.7. Now we introduce the shorthand notations:

Jf(X)Kau = Jf(Eux)Kbu,

JF(̃X)K̃Au = JF(ẽũX)K̃Bu. Finally, recall the Lorentz boost θs with s ∈R defined in Section 2.11: in particular, θs(0) = (sinh s,0,cosh s,0,0) for the (2,2)-supersymmetric free field and θs(0) = (sinh s,0,0,0) for the H2∣2-model. We are now ready to announce the supersymmetric Bayes formula.5 Theorem 7 (Supersymmetric Bayes formula). Recall that J⋅K denotes the supersymmetric free field expectation defined in Section 2.7 and above, ⟨⋅⟩denotes the H2∣2 expectation defined in

A

denotes the mixing measure defined in Section 2.10. Then the following H2∣2 expectations can be realized as supersymmetric free field expectations in random environments: 5See [DR06] on this terminology, which we borrow here by analogy without using any Bayesian analysis.

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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