Topology Optimization Composite Ansys

Peter Bubenik And Tane Vergili

Abstract. Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules, including many of those that have been previously studied, and describe the relationships between them. In the cases where these classes are sets, interleaving distance induces a topology. We undertake a systematic study the resulting topological spaces and their basic topological properties.

Ntroduction

A standard tool in topological data analysis is persistent homology [30, 13, 29, 20, 41]. It is often applied as follows. One starts with some data, constructs an increasing family of complexes or spaces, and applies homology with coefficients in some fixed field to obtain a persistence module. Next, one computes a summary (e.g. barcode , persistence dia- gram , or persistence landscape [7, 11]) which determines this persistence module up to isomorphism. In practice, one computes these summaries directly from the increasing family of complexes or spaces. Nevertheless, the persistence module is the central algebraic object in this pipeline, and has been a focus of research.

A key discovery in the study of persistence modules is the notion of interleaving which provides a way of measuring the distance between persistence modules. For many persis- tence modules, this distance equals the bottleneck distance between the corresponding persistence diagrams [31, 12]. Interleavings and the resulting interleaving distance have been extensively studied both for the persistence modules considered here [31, 12, 1, 2, 18, 8, 5], for Reeb graphs [26, 35], for zig-zag persistence modules , for multiparameter persistence modules , and for more general persistence modules [9, 10, 27, 3, 33, 32].

For sets of persistence modules, the interleaving distance induces a topology. The main goal of the research reported here is to study the basic topological properties of the resulting topological spaces.

persistence modules is not a set, but a proper class. While it is possible the consider this class with the interleaving distance [9, 10, 8], here we want to work with actual topological spaces.

So to start, we consider various classes of persistence modules. These include classes that have been previously considered in theoretical work, such as pointwise finite-dimensional persistence modules , q-tame persistence modules , interval-decomposable persistence modules, ephemeral persistence modules , and constructible persistence modules [37, 25], as well as classes of persistence modules that arise in applications, such as those decomposable into finitely many interval modules, where each interval lies in some fixed bounded closed interval.

We determine various relationships between these classes, such as inclusion (Figure 1). We also identify pairs of classes where for each element of one, there is an element of the other that has interleaving distance 0 from the first (Section 3.3). We define and calculate an asymmetric distance we call enveloping distance that measures how far one needs to expand a given class to include another (Section 3.4). These two results are summarized in Figure 2.

Next, we determine which of these classes are sets and which are proper classes.

We

show that the classes of interval-decomposable persistence modules and q-tame persistence modules are not sets (Corollary 3.27), though the classes of pointwise finite-dimensional persistence modules and persistence modules decomposable into countable-many interval modules are sets (Propositions 3.22 and 3.25). We introduce a set of persistence modules containing these two sets that consists of persistence modules decomposable into a set of interval modules with cardinality of the continuum (Definition 3.28 and Proposition 3.29).

For the remainder, we restrict ourselves to the identified sets of persistence modules and the topologies induced by the interleaving distance (Figure 3). We identify which of the inclusions in Figure 3 are inclusions of open sets (Proposition 4.1).

We show that these topological spaces are large and poorly behaved in the following ways. They do not have the T0 or Kolmogorov property (Corollary 4.6), they are not locally compact (Corollary 4.9), and their topological dimension is infinite (Corollary 4.38). In fact, we prove the following.

Theorem 1.1 (Cube Theorem (Theorem 4.37)). Let N ≥1. There exists an ε > 0 such that there is an isometric embedding of the cube [0, ε]N with the L∞distance into each of our topological spaces of persistence modules.

On the other hand, our topological spaces of persistence modules do have the following nice properties. They are paracompact (Lemma 4.10), first countable (Lemma 4.21), and are compactly generated (Lemma 4.22).

We determine which of these topological spaces are separable (Theorems 4.18 and 4.20), as well as second countable and Lindel¨of (Lemma 4.23). We show that the space of point- wise finite-dimensional persistence modules is not complete (Theorem 4.24), but that the space of persistence modules that are both q-tame and that decompose into countably-many intervals is complete (Theorem 4.25). We prove a Baire category theorem for complete ex- tended pseudometric spaces (Theorem 4.35) that implies that this space is also a Baire space (Corollary 4.36).

We also identify the path components of the zero module in our topological spaces (Propo- sitions 4.14 and 4.16), and show that they are contractible (Proposition 4.17). Along the way, we observe the following mild strengthening of the structure theorem for persistent homology , which may be of independent interest.

Theorem 1.2 (Structure Theorem (Theorem 3.8)). The radical of a q-tame persistence module is a countable direct sum of interval modules. Persistence modules and persistence diagrams Topological data analysis tends to focus on persistence diagrams rather than persistence modules. Readers more familiar with persistence diagrams may wonder why we work with persistence modules and what our Let us present three responses. First, persistent homology produces persistence modules.

In many but not all cases, these persistence modules may be represented by a persistence diagram. Mathematically, persistence modules are the fundamental object of study. Second, one of our main motivations was to develop a theory that could be extended to multipa- rameter persistence modules [14, 31] and generalized persistence modules [9, 27, 10]. In this more general setting there is no hope for an analog of the persistence diagram. Third, our results for persistence modules may be used to obtain results for persistence diagrams as corollaries.

To be more precise, consider persistence modules that are pointwise finite-dimensional (see Section 3.1) with the interleaving distance. This forms an extended pseudometric space that we label (pfd). If we take the quotient obtained by identifying persistence modules with zero interleaving distance, then we obtain an extended metric space that is isometric with a space of persistence diagrams with the bottleneck distance . This is the celebrated isometry theorem [16, 31, 18, 1, 12]. Call this extended metric space (pd).

Now (pd) inherits many of the properties of (pfd). Specifically, it is not totally bounded, any element of (pd) does not have a compact neighborhood, it is not path connected, the path component of the empty persistence diagram consists of persistence diagrams without points with infinite persistence, and this path component is contractible. Furthermore, (pd) is not separable and is not complete. In addition, for each N there is an ε > 0 such that there is an isometric embedding of the N-cube with diameter ε and the L∞distance into (pd). So the topological dimension of (pd) is infinite.

For the data scientist For the reader primarily interested in topological data analysis, we would summarize our results by stating that the extended metric space of persistence diagrams with the bottleneck distance is “big”. Say we fix c < d and restrict ourselves to persistence diagrams with finitely many points (ai, bi) each of which satisfies c ≤ai < bi ≤d.

This is a metric space. However, every neighborhood of every persistence diagram in this metric space is not compact. Also, the topological dimension of this metric space is infinite. In order to apply certain statistical and machine learning tools, one may be tempted to step.

Extended pseudometric spaces The results presented here for extended pseudometric spaces are straight-forward extensions of the standard results for metric spaces (Lemmas 4.10, 4.21, 4.22, and 4.23 and Theorem 4.35). However, in order to keep the material accessible to applied mathematicians without a background in point-set topology, we include the proofs.

Related work Mileyko, Mukherjee, and Harer consider the set of persistence diagrams with countably many points in R2 together with the topology induced by the p-Wasserstein distance for 1 ≤p < ∞. They show that the subspace consisting of persistence diagrams with finite distance to the empty persistence diagram is complete and separable. We show the corresponding space for the bottleneck distance (p = ∞) is complete (Theorem 4.25) but not separable (Theorem 4.20). In a subsequent paper with Turner they study geometric properties of the same set with a slightly different metric.

Blumberg, Gal, Mandell, and Pancia show that the set of persistence diagrams with finitely many points with the bottleneck distance is separable and that its Cauchy completion is separable. This completion is the set of persistence diagrams with the property that for every ε > 0 there are only finitely many points with persistence at least ε.

The authors have been informed of related work that is in preparation. Perea, Munch, and Khasawneh have characterized (pre)compact sets of persistence diagrams with the bottleneck distance. Their results imply that compact sets have empty interior. Cruz has results on metric properties for generalized persistence diagrams with interleaving distance.

Organization of the paper In Section 2, we provide background on persistence modules, indecomposable modules, interleaving distance, and pseudometric spaces. In Section 3, we define the classes of persistence modules that we consider, study the relationships between them, and identify which of them are sets. In Section 4, we study the basic topological properties of our topological spaces of persistence modules. Throughout, most of our argu- ments are elementary, except our proof of completeness which uses basic ideas from category theory. We also provide an appendix where we examine interleavings of interval modules.

Background

In this section we define persistence modules and interleaving distance, giving examples and basic properties. We also define extended pseudometric spaces and their induced topo- logical spaces.

2.1. Persistence modules Let k be a fixed field. A persistence module M is a set of k- vector spaces {M(a) | a ∈R} together with k-linear maps {vb

B ◦Vb

a. Equivalently, a persistence module is a functor M : R →Vectk, where R is the category whose set of objects is R and whose morphisms are the inequalities a ≤b, and Vectk is the category of k-vector spaces and k-linear maps.

Example 2.1. Let X be a topological space and f : X →R be a function. For each a ∈R

Fa := {X ∈X | F(X) ≤A} ⊂X

is called a sublevel set. Note that a ≤b implies Fa ⊂Fb so that we have an inclusion map

Ib

a : Fa֒ −→Fb for all a ≤b. This inclusion map induces a linear map

A) : Hn(Fa; K) →Hn(Fb; K)

on singular homology groups with a coefficients in k of degree n ≥0.

We Thus Have A

persistence module HF : R →Vectk given by HF(a) = Hn(Fa; k) and HF(a ≤b) = Hn(ib a). Example 2.2. Consider the half open interval [0, 2) in R and define the persistence module

Otherwise

where 1 is the identity map on k. For simplicity, we will abuse notation and denote this persistence module by [0, 2). Example 2.3. Replacing [0, 2) in the above with an arbitrary interval J ⊂R we obtain a persistence module that we call an interval module and we will also denote by J.

Example 2.4. A trivial but important example is the zero module, denoted 0, that has 0(a) = 0 for all a. A morphism of persistence modules M and N is a collection of linear maps {ϕa : M(a) → N(a) | a ∈R} such that the following diagram commutes for each pair a ≤b.

A

Equivalently, a morphism of persistence modules is a natural transformation ϕ : M ⇒N. We will often denote a morphism of persistence modules as ϕ : M →N. Such a morphism is an isomorphism if and only if each linear map ϕa is an isomorphism.

Example 2.6. It is a good exercise to check that because of the constraints due to the commutative squares in (2.5), there is a nonzero morphism from the interval module [a, b) to the interval module [c, d) only if c ≤a ≤d ≤b.

In the appendix, we present a more thorough discussion of interval modules (Section A.1) and maps between them (Section A.2). 2.2. Indecomposables Given two persistence modules M and N, their direct sum is the persistence module M ⊕N given by (M ⊕N)(a) = M(a) ⊕N(a) and (M ⊕N)(a ≤b) = M(a ≤b) ⊕N(a ≤b). In the same way we can define the direct sum of a collection of persistence modules indexed by an arbitrary set.

A persistence module is said to be indecomposable if it is not isomorphic to a nontrivial direct sum. For example, interval modules are indecomposable. However, not all indecom- posable persistence modules are interval modules (see [18, Theorem 2.5, Remark 2.6] for a discussion of examples due to do Webb , Lesnick, and Crawley-Boevey).

A special case of the following theorem follows from work of Gabriel , but the general case was proved by Crawley-Boevey . Theorem 2.7 (Structure Theorem). Let M : R →Vectk be a persistence module. If M(a) is finite dimensional for each a ∈R, then M is isomorphic to a direct sum of interval modules.

2.3. Interleaving distance Interleaving distance was introduced in and further studied in the context of multiparameter persistence in . Here we also adopt the categorical point of view from .

Definition 2.8. Let ε ≥0. An ε-interleaving between persistence modules M and N consists of morphisms ϕa : M(a) →N(a + ε) and ψa : N(a) →M(a + ε) for all a such that the following four diagrams commute for all a ≤b, where the horizontal maps are given by the respective persistence modules.

Ψa

Equivalently, we may describe this in terms of natural transformations. First, for x ∈R let Tx : R →R denote the functor given by Tx(a) = a + x. Next if x ≥0, let ηx : 1R ⇒Tx denote the natural transformation from the identity functor on R to Tx that has components (ηx)a : a ≤a + x. Then an ε-interleaving consists of natural transformations ϕ : M ⇒NTε and ψ : N ⇒MTε such that (ψTε)ϕ = Mη2ε and (ϕTε)ψ = Nη2ε. See [12, Section 3] for more details. We say M and N are ε-interleaved.

Remark 2.11. Two persistence modules are 0-interleaved if and only if they are isomorphic. If persistence modules M and N are ε-interleaved and N and P are δ-interleaved then M and P are (ε + δ)-interleaved.

Definition 2.12. Let M and N be two persistence modules. Then the interleaving distance

If no such ε exists, then dI(M, N) = ∞. Example 2.13. The interval modules [0, 2] and (0, 2) are not 0-interleaved. In fact, there are no nonzero maps between [0, 2] and (0, 2). However they are ε-interleaved for all ε > 0.

Thus, dI([0, 2], (0, 2)) = 0. Example 2.14. The interval modules M = [0, 1) and N = [0, ∞) are not ε-interleaved for any ε ≥0. Indeed, assume ϕ and ψ provide such an interleaving. Consider the following trapezoid.

N(0≤2+2Ε)

It decomposes into a commutative parallelogram and commutative triangle from (2.9) and (2.10) in two different ways. In either case, this diagram commutes.

Furthermore, The

bottom horizontal arrow is the identity on k and the top horizontal arrow is 0, which is a contradiction. In the appendix, we give a careful study of interleavings of interval modules (Section A.3).

We will make use of the following lemma without reference. Lemma 2.15 (Converse Algebraic Stability Theorem [31, Theorem 3.4]). Let ε ≥0. If for all α ∈A, the persistence modules Iα and Jα are ε-interleaved, then L

Α∈A Iα, L

α∈A Jα) ≤supα∈A dI(Iα, Jα). Proof For α ∈A, let ϕα and ψα be maps giving an ε-interleaving of Iα and Jα. Then L ϕα and L ψα provide the desired ε-interleaving.

Pseudometric Spaces

Definition 2.16. A pseudometric on a set X is a map d : X × X →[0, ∞) that satisfies

): D(X, Y) ≤D(X, Z) + D(Z, Y)

for all x, y, z ∈X. Note that we have omitted the condition d(x, y) = 0 implies x = y required of metric. More generally, an extended pseudometric on X is a map d : X × X → (extended) pseudometric space.

Theorem 2.17 ([15, 31, 12]). The interleaving distance is an extended pseudometric on any set of (isomorphism classes of) persistence modules. Remark 2.18. A proper class of persistence modules with the interleaving distance is not an extended pseudometric space since it is not a set. However it is a symmetric Lawvere space [9, 8, 10].

In an extended (pseudo)metric space, the condition d(x, y) < ∞defines an equivalence relation. As a result, such a space has a natural partition into (pseudo)metric spaces. In an (extended) pseudometric space one can consider equivalence classes of the equiv- alence relation x ∼y if d(x, y) = 0 to obtain an (extended) metric space. However, for persistence modules, one may be interested in distinguishing nonisomorphic modules with zero interleaving distance, so we will not apply this simplification.

Any extended pseudometric on a set induces a topology on it. Indeed, for any x ∈X and a real number r > 0 consider the open ball Br(x) centered at x with radius r, Br(x) := {y ∈X | d(x, y) < r}.

We call a set O open in X if for each x ∈O, there exists r > 0 such that Br(x) ⊂O. Then it is easy to check that the collection of all open sets is a topology on X. Note that each open ball Br(x) is also an open set in X and the collection of all open balls forms a base for this topology X since each open set O in X can be written as a union of open balls.

Example 2.19. Consider the interval module [0, 5) and let ε > 1. Then the ball Bε([0, 5)) contains the interval modules [−1, 6] and (1, 4). In the appendix, we study the interval modules in an ε-neighborhood of an interval module (Section A.4).

A sequence (xn)n≥1 in an extended pseudometric space X is said to converge to x ∈X if for all ε > 0 there exists N > 0 such that for all n ≥N, d(xn, x) < ε. The point x is called a limit of the sequence. Note that in an extended pseudometric space we no longer have unique limits, but we do have that if x and x′ are limits, then by the triangle inequality d(x, x′) = 0.

A sequence (xn)n≥1 in an extended pseudometric space is a Cauchy sequence if for all ε > 0 there exists an N > 0 such that for all n, m ≥N, d(xn, xm) < ε. If a subsequence of a Cauchy sequence has a limit x, then by the triangle inequality, x is also a limit of the Cauchy sequence.

(0)

Figure 1. Hasse diagram of sets and classes of persistence modules.

Sets And Classes Of Persistence Modules

In this section we define classes of persistence modules that contain many of the persistence modules considered in the literature. We study the relationships between these classes and determine which of them are in fact sets.

For the remainder of the paper, we will only consider isomorphism classes of persistence modules. That is, whenever we say ‘persistence module’, we really mean ‘isomorphism class of persistence modules’. This is standard when discussing both vector spaces and persistence modules.

3.1. Classes of persistence modules In this section, we consider the classes of persistence modules in Figure 1, which we now describe. • (pm) is the class of persistence modules.

• (id) is the class of interval-decomposable persistence modules: those isomorphic to α∈A Iα, where A is some indexing set, and each Iα is an interval module. • (cid), the countably interval-decomposable persistence modules, is the subclass of (id) where the index set A is countable.

• (cfid), the countably finite-interval decomposable persistence modules, is the subclass of (cid) in which each interval Iα is finite. • (fid), the finitely interval-decomposable persistence modules, is the class of persistence

Modules Isomorphic To Ln

k=1 Ik for some N, where each Ik is an interval module. • (ffid), the finitely finite-interval decomposable persistence modules, is the subclass of (fid) in which each Ik is a finite interval.

• Given c < d, (ffid[c,d]) is the subclass of (ffid) in which each Ik ⊂[c, d]. • (pfd), the pointwise finite dimensional persistence modules, is the class of all persis- tence modules M with each M(a) finite dimensional.

• (qtame), the q-tame persistence modules, is the class of all persistence modules M

Where Each A < B The Linear Map Vb

a : M(a) →M(b) has a finite rank. • (eph), the ephemeral persistence modules, is the class of all persistence modules M

Where For Each A < B The Linear Map Vb

a : M(a) →M(b) is zero. • (0) is the class consisting of only the zero persistence module. Remark 3.1. The class (fid) is a slight generalization of the class of constructible persistence modules. A persistence module M is said to be constructible if there exists a finite

• For T < A1, M(T) = 0,

• for ai ≤s ≤t < ai+1, M(s ≤t) is an isomorphism where i ∈{1, . . , n −1} , and • for an ≤s ≤t, M(s ≤t) is an isomorphism.

Nclusions

Lemma 3.2. Let M be an ephemeral module. Then M ∼= L

Α∈A Mα, Where Each Mα ∼= [R, R]

for some r ∈R. Proof Let M ∈(eph). For r ∈R, let Mr be the persistence module with Mr(x) = M(r) if x = r and otherwise Mr(x) = 0. Then M ∼= ⊕r∈RMr. Furthermore each M(r) has a basis, so Mr decomposes over this basis into [r, r] interval modules.

□

Proposition 3.3. The diagram in Figure 1 is a Hasse diagram for the poset structure of these classes of persistence modules under the inclusion order. Proof By Theorem 2.7, (pfd) is in (id). By Lemma 3.2, (eph) ⊂(id). It is easy to check that all of the other arrows indicated in the diagram are inclusions and that in fact all of the inclusions are proper. With the observation that if A ⊂B, C ⊂D and A̸ ⊂D then B̸ ⊂C, it remains to check the following cases.

(1) (Eph)̸ ⊂(Pfd): L∞

k=1[0, 0] is in (eph) but not in (pfd).

(2) (Eph)̸ ⊂(Cid): L

r∈R[0, 0] is in (eph) but not in (cid). (3) (ffid[c,d])̸ ⊂(eph): [c, d] is in (ffid[c,d]) but is not in (eph). (4) (fid)̸ ⊂(cfid): [0, ∞) is in (fid) but is not in (cfid).

(5) (Pfd)̸ ⊂(Cid): L

r∈R[r, r] is in (pfd) but is not in (cid).

(6) (Cfid)̸ ⊂(Qtame): L∞

k=1[0, 1) is in (cfid) but is not in (qtame).

K=1[0, 1

k) is in (qtame) but is not in (id) .

□

1In particular, the multiplicity of [ai, aj) can be calculated using the inclusion/exclusion formula rank M(ai ≤aj−1) −rank M(ai ≤aj) −rankM(ai−1 ≤aj−1) + rank M(ai−1 ≤aj) , which is an example of M¨obius inversion .

Almost Inclusions

Definition 3.4. Say that a class of persistence modules A almost includes in a class of persistence modules B if for each A ∈A there exists an element B ∈B such that dI(A, B) = 0.

Lemma 3.5. A finite sequence of inclusions and almost inclusions is an almost inclusion. Proof This follows from the triangle inequality.

□

Lemma 3.6. M is an ephemeral persistence module if and only if dI(M, 0) = 0. That is, (eph) almost includes in (0). Proof Let M be an ephemeral persistence module. Then M and 0 are ε-interleaved for all ε > 0 by the zero maps.

Next assume dI(M, 0) = 0. Consider a < b. Let ε = b−a

Since M And 0 Are Ε-Interleaved,

the map M(a < b) factors through 0, and is thus the zero map.

Ψ A+B

Therefore M is an ephemeral persistence module.

□

For a persistence module M, define the radical of M by (rad M)(a) = P

C

a) . Note that rad M ⊂M and inherits the structure of a persistence module. Proposition 3.7. Let M be a persistence module. Then dI(M, rad M) = 0. Proof Let ε > 0. For all a ∈R, let ϕa = M(a < a + ε) : (rad M)(a) →M(a + ε), and let ψa = M(a < a + ε) : M(a) →(rad M)(a + ε). Then by the functoriality of M, this is an ε-interleaving of rad M and M. Therefore dI(rad M, M) = 0.

□

Theorem 3.8. Let M ∈(qtame). Then rad M ∈(qtame) and rad M ∈(cid). Proof Let M ∈(qtame). Since rad M is a submodule of M, it follows that rad M ∈(qtame) as well. By [17, Corollary 3.6], rad M ∈(id). We will strengthen this to show that rad M ∈ (cid).

Since Rad M ∈(Id), Rad M ∼= L

α∈A Iα. For q, r ∈Q with q < r, let Aq,r = {α ∈A |

Q, R ∈Iα}, And Let A′ = S

q

□

Combining the previous two results we have the following. Corollary 3.9. Let M ∈(qtame). Then there exists N ∈(cid) such that dI(M, N) = 0. That is, (qtame) almost includes in (cid).

3.4. Enveloping distance In this section, we define a non-symmetric distance between classes of persistence modules and calculate its value for most of the pairs in Figure 1. Definition 3.10. Let A and B be classes of persistence modules. We define the enveloping distance from A to B as follows.

E(A, B) = inf(r | ∀B ∈B and s > r, ∃A ∈A such that A, B are s-interleaved) If there is no such r, we set E(A, B) = ∞. For example, as we will demonstrate later in this section, E((0), (ffid[c,d])) =

And

E((ffid[c,d]), (0)) = 0. We will use the following basic fact about interleavings. Lemma 3.11 ([31, 12]). If persistence modules A and B are s-interleaved and persistence modules B and C are t-interleaved, then A and C are (s + t)-interleaved.

The enveloping distance has the following properties. Lemma 3.12. E(A, A) = 0 and E(A, C) ≤E(A, B) + E(B, C). Proof For reflexivity, each persistence module is s-interleaved with itself for all s ≥0. The triangle inequality follows from Lemma 3.11.

□

Definition 3.13. In the case that E(A, B) = ∞, we write that E(A, B) = ∞−if ∀B ∈B ∃s and A ∈A such that A, B are s-interleaved. From now on we reserve E(A, B) = ∞for the case that this condition is not satisfied.

Lemma 3.14. If A (almost) includes in B then E(B, A) = 0. Proof This follows immediately from the definitions.

□

Corollary 3.15. E((0), (eph)) = 0 and E((eph), (0)) = 0. Lemma 3.16. If A (almost) includes in B, E(B, C) = ∞, and C (almost) includes in D, then E(A, D) = ∞.

Proof Assume E(A, D) < ∞. Then there is some s ≥0 such that for all D ∈D there exists an A ∈A such that D and A are s-interleaved. Let ε > 0. Let C ∈C. Since C (almost) includes in D, there is a D ∈D such that C and D are ε-interleaved. By our first observation, there is an A ∈A such that D and A are s-interleaved. Since A (almost) includes in B, there is a B ∈B such that A and B are ε-interleaved. Therefore by Remark 2.11, C and B are (s + 2ε)-interleaved. So for all C ∈C there is a B ∈B such that C and B are (s + 2ε)-interleaved. Thus E(B, C) < ∞.

□

Proposition 3.17. (1) We have the following enveloping distances: E((0), (ffid[c,d])) =

D−C

and E((ffid[c,d]), (ffid)) = ∞−. Also, E((0), (ffid)) = ∞−, E((eph), (ffid)) = ∞−

And E((Eph), (Ffid[C,D])) = D−C

2 . (2) In addition, E((cfid), (fid)) = ∞and E((qtame), (cfid)) = ∞. (3) With the exception of (0) ⊂(eph), (0) ⊂(ffid[c,d]), (ffid[c,d]) ⊂(ffid) and the possible exception of (pfd) ⊂(qtame), all of the other inclusions A ⊂B in Figure 1 have enveloping distance E(A, B) = ∞. Also E((qtame), (cid)) = ∞.

D−C

2 . • (ffid[c,d]) ⊂(ffid): For all M ∈(ffid[c,d]) and N ∈(ffid), dI(M, N) ≤dI(M, 0) + dI(0, N) < ∞. Let z ≥0. For all M ∈(ffid[c,d]), there are no nontrivial maps from M to (d, d + 2z]. Thus dI(M, (d, d + 2z]) ≥dI((d, d + 2z], 0) ≥z.

• The other three cases follow from the same arguments.

(2)

• (cfid) to (fid): Consider [0, ∞).

(3)

• (ffid) ⊂(fid): Consider [0, ∞).

• (Ffid) ⊂(Cfid): Consider L∞

k=1[0, k). • (cfid) ⊂(cid): Consider [0, ∞).

• (Id) ⊂(Pm): Consider Q∞

k=1[0, ∞). • (eph) ⊂(id), (eph) ⊂(qtame), (pfd) ⊂(id), (qtame) ⊂(cid), and (qtame) ⊂ (pm) follow from Lemma 3.16.

□

Remark 3.18. Together with Corollary 3.15, Lemma 3.14, and Lemma 3.16, this proposi- tion implies all of the pairwise enveloping distances between the sets and classes of persis- tence modules in Figure 2, except E((pfd), (qtame)). For example, E((id), (qtame)) = 0, E((cid), (pfd)) = 0, and E((cid), (qtame)) = 0 by Lemmas 3.14 and 3.5, and E((fid), (cfid)) = ∞by Lemma 3.16.

We end this section by showing that E((pfd), (qtame)) = 0. First we give a definition. Definition 3.19. Let M be a persistence module. Let p ≥0. We define the p-persistent

Submodule Of M By

M(p)(a) = im M(a −p ≤a). For a ≤b, there is an induced map between objects M(p)(a) and M(p)(b) given by M(a ≤b). Since M is a persistence module, so is M(p), and since M(p)(a) is a sub-vector space of M(a) for all a, M(p) is a submodule of M.

Proposition 3.20. Let M be a persistence module and let p ≥0. Then M and M(p) are p-interleaved. Proof For a ∈R, define ϕa : M(a) →M(p)(a+p) by ϕa = M(a ≤a+p), and ψa : M(p)(a) → M(a + p) by ψa = M(a ≤a + p). Then all the arrows in diagrams (2.9) and (2.10) are maps in M and hence commute.

□

Corollary 3.21. E((pfd), (qtame)) = 0. Proof Let M be a q-tame persistence module. Let p > 0.

Then By Definition, M(P) Is

a pointwise finite-dimensional persistence module. By Proposition 3.20, M and M(p) are p-interleaved. Thus, by definition, E((pfd), (qtame)) = 0.

∞

Figure 2. Diagram of sets and classes of persistence modules.

Solid Ar-

rows indicate inclusions, dashed arrows indicate almost inclusions, and dotted arrows do not indicate any relationship. Annotations of arrows indicate en- veloping distance from the source to the target, given in Definitions 3.10 and 3.13.

3.5. Sets of persistence modules Next we consider whether the classes defined above are sets or proper classes. We will use the following notation. Let R := R ∪{±∞} and N := N∪{∞}. Given a set X, let P(X) denote its power set. Let I be the set of all intervals

,

inf I ∈I, sup I ∈I. Proposition 3.22. The class (cid) is a set.



where m(i) denotes the multiplicity of the direct summand Iα. This map is an injection, hence (cid) is a set.

□

Corollary 3.23. Therefore the classes (cfid), (fid), (ffid), (ffid[c,d]), and (0) are also sets. Lemma 3.24. Each interval appears only finitely many times in the direct-sum interval- module decomposition of a pointwise finite-dimensional persistence module.

□

Proposition 3.25. The class (pfd) is a set.

Α∈A Iα Where Iα Is An Interval And A Is A

set. By Lemma 3.24, we can define the following map.



where m(i) denotes the multiplicity of the direct summand Iα. This map is an injection, hence (pfd) is a set.

□

Proposition 3.26. The class (eph) is not a set.

Proof For A Cardinal C, Let Fc = L

α∈c[0, 0]. That is, Fc is the k-vector space generated by c. For c̸ ∼= d, Fc̸ ∼= Fd. Thus we have an injection from the proper class of cardinals into (eph).

□

Corollary 3.27. Since (eph) is not a set, neither are (id) (qtame) and (pm). 3.6. Interval-decomposable persistence modules of arbitrary cardinality Motivated by the desire to have a set of persistence modules that contains all of the sets of persistence modules in Section 3.5 and the proofs of Proposition 3.22 and 3.25, we make the following definition.

Definition 3.28. Given a cardinal κ, let (κ-id) denote the class of persistence modules

Isomorphic To L

α∈A Iα where Iα is an interval module and the cardinality of A is at most κ. As a special case, and to avoid confusion with our previously defined notation, let (rid) denote the class of interval-decomposable persistence modules with at most the cardinality of R-many summands.

By definition, (cid) ⊂(rid) and by Lemma 3.24, (pfd) ⊂(rid). Proposition 3.29. For any cardinal κ, the class (κ-id) is a set. Proof The proof is the same as the proof of Proposition 3.22, replacing N with κ.

Topological Properties

Since we are interested in studying topological spaces of persistence modules, we will for the most part restrict ourselves to the sets in Figure 3. We will consider the basic topological properties of these sets with the topology induced by the interleaving metric.

(Ffid[C,D])

Figure 3. Sets of metric spaces, each with the topology induced by the in- terleaving metric. 4.1. Open subsets In this section we consider which of the inclusion maps in Figure 3 are inclusions of open subsets. Recall that in a pseudometric space X, a subset A ⊂X is said to be open if for all a ∈A, there exists ε > 0 such that Bε(a) ⊂A.

Proposition 4.1. Among the inclusion maps in Figure 3, only the inclusions (ffid)֒ →(fid) and (cfid)֒ →(cid) are inclusions of open subsets. Proof Let M ∈(ffid) and N ∈(fid) \ (ffid). Then N is isomorphic to a direct sum of interval modules, at least one of which is unbounded. It follows that dI(M, N) = ∞. Thus (ffid) is an open subset of (fid). The same argument shows that (cfid) is an open subset of (cid). For each of the following inclusions A ⊂B we show that for all M ∈A and for all ε > 0, there is an N ∈B \ A such that dI(M, N) < ε. Therefore A is not an open subset of B.

• (ffid[c,d]) ⊂(ffid). Let N = M ⊕[d, d + 2ε).

□

Remark 4.2. While (ffid[c,d]) is not an open subset of (ffid), if we restrict (ffid) to direct sums of interval modules whose intervals are contained in an open interval (c, d), then we obtain an open subset of (ffid).

Separation

Proposition 4.3. Any set of ephemeral persistence modules with the interleaving distance has the indiscrete topology. Proof Let S be a set of ephemeral persistence modules. By Lemma 3.6, each M ∈(eph) has dI(M, 0) = 0. So for M, N ∈S, by the triangle inequality, dI(M, N) = 0. Thus for all M ∈S and for all ε > 0, Bε(M) ⊇S.

□

Lemma 4.4. Let M be a persistence module let r ∈R. Then dI(M, M ⊕[r, r]) = 0. A topological is said to be a T0-space (or a Kolmogorov space), if for any pair of distinct elements in the space there exists at least one open set which contains one of them but not the other.

Proposition 4.5. Let c < d. Then (ffid[c,d]) is not a T0-space. Proof Apply Lemma 4.4 to M = [a, b) where c ≤a < b ≤d, and r = c+d

Then M′ =

M ⊕[r, r] ∈(ffid[c,d]) and there does not exist an open neighborhood U of M that does not contain M′ and vice versa.

□

Since (ffid[c,d]) is a subspace of any the other spaces in Figure 3, we obtain the following. Corollary 4.6. None of the spaces in Figure 3 are T0. 4.3. Compactness Let X be an extended pseudometric space. Then a subset S ⊂X is totally bounded if and only if for each ε > 0, there exists a finite subset F = {x1, x2, . . , xn} ⊂

Such That S ⊂∪N

i=1Bε(xi). Such a union is called a finite ε-cover. Lemma 4.7. The space (ffid[c,d]) is not totally bounded.

D−C

2 . Therefore (ffid[c,d]) does not have a finite ε-cover.

□

An open cover of a topological space X is a collection of open sets O = {Oi}i∈I of X such that ∪i∈IOi = X. A topological spaces is compact if every open cover has a finite subcover. We say that a topological space is locally compact if each point has a compact neighborhood, where by a neighborhood of a point p ∈X we mean a subset V ⊂X such that there exists an open set p ∈U ⊂V .

Proposition 4.8. Any of element in (ffid[c,d]) does not have a compact neighborhood.

Proof Let M ∼= Lq

j=1 Ij with Ij ⊂[c, d]. Suppose that M has a compact neighborhood, K. Then there exists a real number ε > 0 such that M ∈Bε(M) ⊂K.

Minj Diam Ij. Choose An Interval

I of diameter δ contained in [c, d]. Consider for n ∈N, the persistence modules Mn =

So That The Set {Mn}N∈N Is Contained In

Bε(M), and hence in K. Let M0 = M. Then by the algebraic stability theorem , dI(Mp, Mq) ≥

(N) | N ∈K} Of K. It Does Not Have A Finite

subcover, since there does not exist a persistence module N such that B δ

□

Corollary 4.9. All of the spaces in Figure 3 are not locally compact. An open covering O = {Oi}i∈I of X is locally finite if every x ∈X has a neighborhood which has a nonempty intersection with only finitely many of the open sets {Oi}. Given an

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).

Related Journal Articles & DOI Links

Selected peer-reviewed publications relevant to 12 Lead ECG Acquisition. Click the DOI to access the full paper (may require institutional access).

Why Choose Us?

Bangalore guidance for robotics, Spectre and autonomous systems projects.