Composite Gear Analysis Ansys

Abstract

Since the advent of Large Language Models (LLMs), research has primarily fo- cused on improving their instruction-following and deductive reasoning abilities. Yet a central question remains: can these models truly discover new knowledge, and how can we evaluate this ability? In this work, we address this gap by study- ing abductive reasoning—the process of generating plausible hypotheses to ex- plain observations. We introduce General Evaluation for Abductive Reasoning (GEAR), a new general-purpose, fully automated, transparent, and label-free evaluation paradigm that overcomes limitations of prior approaches. GEAR eval- uates a set of hypotheses using three metrics: consistency (each hypothesis cor- rectly explains the given observations), generalizability (consistent hypotheses make meaningful predictions on unseen inputs), and diversity (the set of hy- potheses covers many distinct predictions and patterns). Built this way, GEAR is scalable (no human gold answers needed), reliable (transparent, deterministic scoring aligned with classical abduction), and open-ended (scores improve only when models produce new, plausible hypotheses, unlike existing static bench- marks that saturate once accuracy is high). Using GEAR, we conduct a fine- grained study of nine LLMs on four popular abduction benchmarks (1,500 prob- lems), generating 50,340 candidate hypotheses. GEAR reveals model differences and insights that are obscured by prior gold-answer–based or purely human eval- uations. We further propose a momentum-based curriculum training strategy that dynamically adjusts GEAR-derived training data by learning velocity: it begins with what the model learns faster and shifts toward harder objectives such as gen- erating diverse hypotheses once the model is confident on foundational objec- tives (e.g., instruction following and consistency). Without gold-label supervi- sion, this strategy improves all three GEAR objectives—consistency, generaliz- ability, and diversity—and these gains transfer to established abductive-reasoning benchmarks. Taken together, GEAR provides a principled framework that not only evaluates abduction but also supplies label-free, scalable training signals that help LLMs produce more diverse and reliable hypotheses. Our code and data are available at: https://github.com/KaiyuHe998/GEAR-Abduction_ evaluation.

Ntroduction

In the current AI community, there are many competing definitions of abductive reasoning. The most widely adopted one is Harman’s view of abduction as inference to the best explanation (IBE) (Harman, 1965; Douven, 2021). Although this definition is simple and intuitive, it suffers from key limitations when applied to real-world settings, making benchmarks and evaluations built on it problematic.

First, IBE does not specify what counts as “best,” and the criteria vary across contexts. In some cases, simplicity is prioritized; in others, novelty or explanatory power is preferred. As a result, IBE-based benchmarks often select a single “gold” hypothesis according to annotators’ subjective

Arxiv:2509.24096V1 [Cs.Cl] 28 Sep 2025

Figure 1: Underdetermination in abductive reasoning: a logically sound, observation-consistent hypothesis need not coincide with the annotated “gold” hypothesis; both can fit the seen data yet disagree on held-out predictions. Single-gold labeling can mask the plurality of valid explanations.

judgments, yielding heterogeneous and unreliable labels (Zhao et al., 2023; Okasha, 2000; Cabrera, 2023). Second, multiple plausible hypotheses typically coexist. Real-world observations can often be explained in several ways, depending on how the data are conceptualized. This limitation appears across popular abductive benchmarks such as MINI-ARC (Kim et al., 2022), ACRE (Zhang et al., 2021), LIST FUNCTIONS (Rule, 2020), and ARC-2025 (Chollet, 2019). For example, in Figure 1, the annotated “gold” hypothesis and other valid alternatives may all be well supported by sound abductive reasoning. However, existing evaluation usually excludes these alternatives by enforcing agreement with a single annotated “gold” hypothesis under a vague “best” standard. Philosophical accounts of abduction emphasize that hypotheses are often evidentially underdetermined: given finite data and background assumptions, more than one hypothesis may be well supported (Quine, 2014; Stanford, 2023), the ability to generate other plausible, well-supported hypotheses should also count as abductive success. Scientific progress relies on maintaining a diverse pool of feasible hypotheses. Although many will ultimately be falsified, they guide the design of experiments to discriminate, verify, or refute competing explanations, thereby enriching our knowledge. Even in current mature fields, credible alternative hypotheses often remain.

To address these limitations, we use Peirce’s original definition of abduction, which frames it as a more general task of generating hypotheses from given observations (Frankfurt, 1958; Burks, 1946; Minnameier, 2004; Peirce, 1974). Building on this foundation, we propose GEAR, a new framework for systematically evaluating abductive reasoning. GEAR is grounded in three classical criteria for good scientific hypotheses: (1) Consistency. A hypothesis must not contradict observed facts, ensuring compatibility with existing evidence. (2) Generalizability. A good hypothesis ex- tends beyond the observed data by making testable predictions on unseen cases. In Popper’s terms, better hypotheses carry higher empirical content: they make riskier, more precise claims and thus invite more opportunities for refutation (Popper, 2005; 2014). We operationalize generalizability as the coverage of unseen inputs on which a hypothesis yields determinate predictions; larger coverage indicates greater generality. Accordingly, when two hypotheses fit the observed data, we prefer the broader (more falsifiable) one; if it withstands more severe tests, it is more strongly corroborated and more robust. (3) Diversity. A hypothesis should contribute a genuinely new perspective rather than echo existing ones, to avoid premature convergence on a single explanation. In the spirit of Chamberlin’s multiple working hypotheses and Platt’s strong inference (Chamberlin, 1965; Platt, 1964), we favor sets of hypotheses that articulate competing mechanisms testable by critical evi- dence. We quantify diversity with two complementary measures: γ-diversity, the average number of unique predictions per input across the hypotheses set (set-level variety), and β-diversity, the dissimilarity of prediction patterns between hypothesis pairs (dispersion). Higher diversity indi- cates broader causal coverage and multiple viewpoints on the observations; points of disagreement highlight decisive experiments and help accelerate scientific progress.

With GEAR, we re-examine four popularly used abduction benchmarks MINI-ARC, ACRE, LIST FUNCTIONS, and ARC-2025 across nine LLMs—spanning API-access models (GPT-o1 (Ope- nAI, 2024), GPT-4.1-mini (OpenAI, 2025a), GPT-o4-mini (OpenAI, 2025b)) and open-source mod- els (LLAMA-3.3-70B, LLAMA-3.1-8B (Grattafiori et al., 2024), QWEN-2.5-72B, QWEN-2.5- 7B (Qwen et al., 2025), GEMMA-2-9B (Team et al., 2024), NEXTCODER-7B (Aggarwal* et al., 2025))—and we find that— (1), Consistency remains hard, 70B-class models produce only 20% consistent hypotheses; (2) Consistency shows no significant correlation with the size of the initial observation set; (3), Model size is weakly related to abductive diversity—larger models do not nec- essarily generate more diverse hypotheses; and (4)existing gold-answer evaluations overlook the underdetermination inherent to abduction, around 80% of equally plausible hypotheses are labeled incorrect, and even the “accepted” hypotheses can differ substantially. Unlike prior frameworks that depend on gold answers or human raters, GEAR is label-free and fully automated, yielding dense, scalable signals that directly train models to generate consistent, generalizable, and diverse hypothesis sets. We convert these signals into optimization targets for preference-based RL and fine-tune base models with LoRA, so that improvements on GEAR ’s objectives are optimized end- to-end without gold supervision. To stabilize learning and broaden coverage across objectives, we introduce a momentum-based curriculum learning strategy that dynamically adjusts GEAR-derived training data by learning velocity: training begins with fast-to-learn, foundational objectives (in- struction following and consistency) and shifts toward harder reasoning objectives that foster diverse hypothesis generation as competence increases. This procedure raises GEAR scores and transfers as accuracy gains on standard abductive-reasoning benchmarks across multiple model families (e.g., Qwen-2.5-7B, Llama-3.1-8B, NextCoder-7B).

Related Work

Reasoning: evolution from non-defeasible to defeasible. Non-defeasible (deductive) reasoning preserves truth under added premises, whereas defeasible reasoning allows conclusions to be revised when new evidence appears (Yu et al., 2024). Abduction belongs to the latter: following Peirce, abduction proposes candidate hypotheses for observed facts, and deduction derives precise, testable predictions from a hypothesis (Frankfurt, 1958; Burks, 1946; Minnameier, 2004; Peirce, 1974). A key contrast with deduction is abduction’s reliance on broad background knowledge (commonsense and domain-specific), which naturally yields multiple distinct yet plausible hypotheses for the same observation (He & Chen, 2025). Early symbolic systems struggled here due to narrow knowledge bases, whereas LLMs pretrained on large corpora make such abductive tasks more tractable (Yang et al., 2023; He & Chen, 2025). Despite its central role in discovery, abduction remains under- studied compared with the extensive focus on deduction in AI (Niu et al., 2024; Liu et al., 2025; Yu et al., 2024; Huang & Chang, 2023; He & Chen, 2025).

Gold- and human-based evaluation for abductive reasoning. Current practice largely relies on two strands. Gold answer-based evaluation compares model outputs to a single reference either (i) at the hypothesis level using BLEU/ROUGE or embedding metrics such as BERTScore (Yang et al., 2024a; Qi et al., 2024; Movva et al., 2025; Bowen et al., 2024; Hua et al., 2025; Young et al., 2022), or (ii) at the behavior level by matching input–prediction pairs implied by the reference hypothesis (Sinha et al., 2019; Weston et al., 2015; Balepur et al., 2024; Shi et al., 2023; Wang et al., 2024; Rule, 2020; Chollet, 2019; Liu et al., 2024; Li et al., 2025; Chen et al., 2025; He et al., 2025). Despite scalability, single-reference matching is ill-suited to abduction: it rejects many plausible, logically sound alternatives that merely differ from the annotated answer; it is also costly (expert labeling) and unstable due to non-monotonic judgments and low inter-annotator agreement (Young et al., 2022). Complementarily, human evaluation is often used for qualities that are hard to algorithmically quantify (e.g., novelty, excitement) (Zhao et al., 2024; Qi et al., 2024; Yang et al., 2024b; Hu et al., 2024; Yang et al., 2025), but it is expensive, hard to reproduce or scale, and inherently subjective—particularly acute for abduction, where outcomes depend on rater expertise, instructions, and context, and small samples limit statistical power. In sum, both strands conflict with the essence of abductive reasoning: instead of testing agreement with a single “gold” explanation or subjective impressions, evaluations should assess a model’s capacity to propose multiple, novel, and plausible explanatory hypotheses when underlying causes are unknown.

Table 1: Abductive reasoning begins with a set of observations O = {o1, o2, . . }, where each oi := (ini, outi) is an input–output pair (e.g., o1 = (floor, wet), o2 = (air, humid)). A hypothesis “it rained” can be represented as a function frain mapping inputs to outputs, e.g., frain(floor) = wet, frain(sky) = cloudy. Each hypothesis f has an input domain D; outside this domain, predictions may be undefined or uninformative (e.g., frain(Math)).

˜F

Trivial hypothesis that memorizes all seen observations

Gear

We introduce the General Evaluation for Abductive Reasoning (GEAR), an evaluation paradigm that scores hypotheses using reference-free, transparent criteria rather than agreement with a single gold answer. Unlike existing benchmarks that evaluate only a single generated hypothesis at a time, GEAR evaluates a set of hypotheses along three dimensions. An LLM exhibits stronger abductive proficiency when it can produce a hypothesis set that (i) correctly explains the given observations (Consistency), (ii) yields meaningful predictions on unseen inputs (Generalizability), and (iii) offers non-redundant alternatives rather than superficial variants (Diversity). Basic notation appears in Table 1.

Onsistency

Consistency is the most fundamental requirement of a hypothesis: it must not conflict with observed facts. Formally, a generated hypothesis f is consistent with the observation set O if ∀(ini, outi) ∈ O, f(ini) = outi. This criterion guarantees agreement with all known observations and underlies most existing evaluations of hypothesis generation, including gold answer–based evaluations.

Generalizability

Given several consistent hypotheses, a more general hypothesis is one that yields predictions on a broader set of unseen cases. A more general hypothesis confers two advantages: (1) it can be applied in more future situations, increasing its practical utility; and (2) because it makes predictions for more situations, it can be tested more extensively and—if it survives—becomes correspondingly more robust (Popper, 2005; 2014). Formally, for two hypotheses f1 and f2 with respective input domains D1 and D2 and a set-size measure M, if M(D1) > M(D2), then f1 is considered more general than f2, with a simple example.

For example, given three observations o1 = (1, 1), o2 = (10, 1), and o3 = (100, 1), a trivial lookup- table hypothesis ˜f that merely memorizes these pairs is consistent yet fails to generalize. In contrast, f1(n) = rev(n) (digit-reversal on integers), f2(x) = x/x for x̸ = 0 (undefined at x = 0), and the constant hypothesis f3(x) ≡1 are all consistent but differ in generalizability. Under the simple size measure M(D) = |D|, we have: {1, 10, 100} ⊂N0 ⊂R \ {0} ⊂R ⇒˜f ≺f1 ≺f2 ≺f3.

The effective domain D and its size measurement M are not static but depend on the problem con- text and representation. For instance, in arithmetic, f1 (reversal) is defined on integers, whereas in programming tasks the same hypothesis applies to strings, lists, and other finite sequences, making f1 more general than f2 in that setting. Since it is generally infeasible to determine a global do- main D across all conceivable contexts, GEAR evaluates generalizability relative to a pre-defined problem-specific sample space S shared across all hypotheses under comparison, together with its Figure 2: GEAR with a live example with |F| = 2, |S| = 3.

associated measurement M. This sample space serves as the operational domain for evaluation and provides the basis for comparing the generalizability of different hypotheses.

Ersity

Prior work typically measures hypothesis diversity with black-box text similarity metrics such as BERTScore (Zhang et al., 2019)—which capture surface-level semantic overlap rather than underly- ing explanatory mechanisms—or with subjective, annotator-based judgments. In contrast, following the Multiple Working Hypotheses view (Chamberlin, 1965; Platt, 1964), we define diversity directly from prediction patterns: distinct hypotheses should be separable by some input in S. Concretely, for two consistent hypotheses f1, f2 and an unseen input inx ∈S, if f1(inx)̸ = f2(inx), then f1 and f2 are separable on inx; the more such inputs exist, the more diverse the hypotheses are.

In light of classical ecological diversity theory—specifically β- and γ-diversity (Whittaker, 1960)—we adapt set-based diversity ideas to hypothesis sets, not as a one-to-one import from ecol- ogy but as a structural analogy over prediction patterns. Concretely, we define γ as the average number of unique predictions per input over a set of hypotheses F and β as the mean pairwise Jaccard dissimilarity between prediction sets on a shared sample space.

γ-diversity (average unique predictions per input). Let Pf = {(in, f(in)) : in ∈S}. Define

,

Under cardinality M1 = M2 = | · | and |F| = m, γ ∈[1, m], A hypothesis proposer that generates near-duplicate hypotheses will yield γ ≈1 because most hypotheses agree on almost all inputs. Conversely, a proposer that views the observations from diverse perspectives and produces genuinely novel hypotheses will achieve a larger γ, approaching γ ≈m when predictions are mutually distinct for every input. Notably, M1 is the size measure on the sample (input) space, whereas M2 is the size measure on the prediction space; the two need not be identical.

β-diversity (prediction-pattern dispersion). Measure pairwise dispersion via the Jaccard dissimi- larity between prediction sets, and then average across all pairs:

≤I

dJ(fi, fj). Example. Let S = {0, 1, 2}, Pf1 = {(0, 1), (1, 2), (2, 3)}, Pf2 = {(0, 1), (1, 2), (2, 2)}, and M = |·|. Generalizability: for each f, G(f) = |Pf|/|S|. Thus G(f1) = G(f2) = 3/3 = 1. Set coverage:

/|S| = 4/3. Diversity: The Intersection Size

is 2 and the union size is 4, hence the Jaccard distance dJ(f1, f2) = 1 −2

Pair, Β = 1

2. (If fewer than two consistent hypotheses are generated, we set β = 0.) An additional example from LIST FUNCTIONS is shown in Fig. While γ and β are mathematically related, neither uniquely determines the other. The formal rela- tionship and proofs are provided in Appendix D.

Because Consistency, Generalizability, and Diversity have precise mathematical definitions, they can be computed directly—without human labor or auxiliary black-box models—at evaluation time. Consequently, GEAR is (1) scalable: it runs automatically on any newly generated hypotheses; (2) reliable: all metrics are transparently defined and computed, including Diversity, which GEAR measures by underlying mechanisms rather than via black-box proxies; and (3) open-ended: GEAR evaluates how many genuinely different explanatory perspectives a model can produce and, because the hypothesis space is in principle unbounded, it imposes no upper limit on valid hypotheses.

Ata Preprocessing

Benchmarks. We use four widely used abductive benchmarks: MINI-ARC (Kim et al., 2022), ACRE (Zhang et al., 2021), LIST FUNCTIONS (Rule, 2020), and ARC-2025 (Chollet, 2019). These gold-answer benchmarks split observations into Otrain and Otest. Unlike the traditional set- ting—where models see only Otrain and are judged on Otest—GEAR measures how many con- sistent hypotheses a model can produce and how diverse they are. Because several datasets pro- vide only a few observations per problem (e.g., 2–3 train and 1 test), we pool all observations into Oall := Otrain ∪Otest to enable broader analyses. We choose these datasets because (i) hypotheses are expressible in formal languages (e.g., executable programs), allowing deterministic evaluation without extra adaptation; and (ii) each dataset provides sufficiently diverse observations to seed abduction.

Sampling initial observations. We study the effect of observation size by using n ∈{1, 2, 3, 4} observations. For ACRE, outputs are Boolean (on/off); with a single observation, label semantics may be ambiguous, so we use n ∈{2, 3, 4} and ensure at least one on and one off. When n observations are required, we sample n pairs from Oall without replacement to form On. We randomly select 100 problems per dataset. For three datasets we use n = 4 (total 3 × 100 × 4 = 1200), and for ACRE we use n = 3 (total 1 × 100 × 3 = 300), yielding 1,500 problems overall.

Generation protocol. Given On, we prompt with a dataset-agnostic template Pinit to obtain the first hypothesis f1. We then iterate with Piter, which lists previously generated hypotheses Ft−1 = {f1, . . , ft−1} and requests a new ft that is (i) consistent with On and (ii) distinct from all f ∈Ft−1.

Both Pinit and Piter are shared across datasets (see Pinit and Piter in the Appendix E). Stopping rule and “bad” hypotheses. Enumerating all potential hypotheses a model could gen- erate is infeasible. To keep generation finite and comparable across models, we adopt a quality- triggered early-stopping rule: generation for a problem stops once the model accumulates three “bad” hypotheses. A hypothesis is bad if it satisfies any of the following: (i) it cannot be parsed into executable code (format or syntax error); (ii) it is inconsistent with On (violates at least one given observation); or (iii) it lacks novelty relative to prior hypotheses. For (iii), let the shared sample

{In ∈S : ∃G ∈Ft−1 S.T. G(In) = F(In)}

We mark f as non-novel if cov(f | Ft−1) ≥0.8 (i.e., at least 80% of its predictions on S are duplicate relative to Ft−1). Compared with a hard quota on the number of hypotheses, this early-stopping rule (i) prevents unbounded generation and degenerate cycling; (ii) allows stronger models to produce more consistent and novel hypotheses before stopping; and (iii) reduces the chance that a fixed quota artificially limits performance.

Sample Space S

The sample space S is central to GEAR: a broader S exposes more inputs on which prediction patterns can be compared, thus providing a sharper lens for assessing generalizability and diversity. In practice, however, S must balance breadth with computational cost and with the effort required to construct valid, meaningful inputs. We therefore define S per dataset using simple, reproducible rules and a fixed random seed. And for simplicity, we use cardinality (| · |) as set-size measurement in our evaluation.

LIST FUNCTIONS: In the original dataset, Inputs are lists of integers with element domain {0, . . , 99} and length k ∈{0, . , 15}. After expansion, the full combinatorial space has size

P15

k=0 100k, which is infeasible to exhaust at evaluation time. We adopt stratified sampling by length: (i) include the unique empty list for k=0; (ii) include all 100 singletons for k=1; (iii) for each k ∈{2, . . , 15}, uniformly sample (without replacement) up to 1,000 lists from the 100k pos- sibilities. This yields |SLISTFUNC| = 1 + 100 + 14 × 1,000 = 14,101. (One could broaden the input domain beyond the original dataset—for example by allowing negative or floating-point val- ues, or even arbitrary real numbers—but we retain the original integer domain; in our experiments this range is already sufficient to probe diverse hypothesis behaviors.) ACRE: In the original dataset, each primitive entity is a triple ⟨color, shape, material⟩with color ∈{blue, brown, cyan, gray, green, purple, red, yellow} (8), shape ∈{cube, cylinder, sphere} (3), material ∈{metal, rubber} (2), so the vocabulary has 8 × 3 × 2 = 48 distinct entity types. An input is an ordered list (repetitions allowed) of c entities with c ∈{0, . , 8} since in the original dataset at most 8 entities are seen in one observation. We again use stratified sampling by c: include the empty list for c=0; include all 48 singletons for c=1; for each c ∈{2, . , 7}, uniformly sample up to 1,000 lists without replacement. This gives |SACRE| = 1 + 48 + 7 × 1,000 = 7,049.

(Broader stress tests are possible by extending the vocabulary with new colors, shapes, or materials, but we restrict ourselves to the original schema here; empirically this space is already rich enough

To Discriminate Among Hypotheses.)

MINI-ARC and ARC-2025: Unlike the previous two datasets, these benchmarks encode inputs as small integer grids (values in a fixed palette) that carry visual semantics. Naively enumerating grids within a size range produces overwhelmingly non-meaningful noise and is therefore counter- productive for abductive reasoning. Instead, we define S as the set of all unique input grids already present in the official splits (train/validation/test), after canonical serialization and deduplication of exact matches. This pragmatic choice keeps inputs semantically meaningful while avoiding sam- pling artifacts. It yields |SMINIARC| = 767, |SARC2025| = 4,826.

Reproducibility: All sampling steps use a fixed random seed and uniform sampling without re- placement within each stratum; we publish the materialized S for each dataset to ensure exact repro- ducibility of scores.

Ain Analysis: Llm Performance Under Gear

Across 9 LLMs we collected 50,340 hypotheses, of which 17,835 are consistent with the given observations. Among the remaining 32,505 inconsistent hypotheses, 4,346 failed to follow the required format/instructions (e.g., were not executable, or parsable), and 28,159 contradicted at least one observation. Figure 3 reports model-wise results.

Overall volume and consistency. Under the quality-triggered early-stopping rule, the total number of hypotheses generated per problem serves as a proxy for overall hypothesis-generation capacity. GPT-O4-MINI and GPT-O1 generate the largest number of consistent hypotheses and achieve the highest consistency rates. In panel (a) they clearly outperform the next tier, yielding on average ∼2–4 more consistent hypotheses per problem than other models, with correspondingly higher consistency in panel (c).

Effect of more initial observations. As the number of initial I/O pairs (observations) increases from 1 to 4, constraints tighten: both β-diversity and γ decline (panels (d)–(e)), while the consistency rate in panel (c) remains comparatively stable. Thus, early stopping is more likely to be triggered by the novelty threshold than by inconsistency when initial number of observations increases.

Instruction following. Most models adhere closely to the required output format (panel (b)). LLAMA-3.1-8B is an outlier, frequently appending free-form text or violating the code template, which harms parseability. In §5 we show that RL substantially close this gap.

Model size vs. abductive diversity. Among open-source models we observe only a weak link between parameter count and abductive diversity: LLAMA-3.3-70B trails smaller models such as GEMMA-2-9B and QWEN-2.5-7B on β-diversity (panel (d)), and QWEN-2.5-72B is only on par with those smaller models. This suggests model size does not fundemantally increases the abductive

O4-Mini-2025-04-16

Figure 3: Model performance under GEAR. Metrics are macro-averaged per problem: compute the value per problem, then take the unweighted mean. Panels (a)–(c) use all generated hypotheses; panels (d)–(f) restrict to consistent hypotheses, since diversity and generalizability are only meaningful for consistent sets.

reasoning ability of the LLMs, suggesting data, training procedure may matter more than raw size for abductive reasoning. Generalizability. Because the sample space S is instantiated pertain to each dataset’s distribution, consistent hypotheses achieve high defined coverage (about 95% on average; panel (f)). Corner cases remain: on mini-ARC and ARC-2025, some generated programs enter infinite loops or exhaust memory on some inputs. Since GPT-O1 and GPT-O4-MINI contribute a disproportionate share of consistent hypotheses on these harder datasets, their macro-averaged coverage appears lower—not because their hypotheses are intrinsically less general, but because the problems they solve are more challenging.

Generalizability. Because S pertains to each dataset’s distribution, consistent hypotheses achieve high coverage (≈95% on average; panel (f)). Corner cases remain in MINI-ARC/ARC-2025 (e.g., infinite loops). Since GPT-O1 and GPT-O4-MINI contribute a large share of consistent hypotheses on these harder datasets, their macro-averaged coverage appears lower—not due to intrinsically weaker generality, but because the solved problems are more challenging.

Simulation Study 1: Abductive Reasoning Is Defeasible

As noted in § 2, abduction is defeasible: a hypothesis generated by logical and sound abductive rea- soning need not coincide with the “gold answer.” Benchmarks that enforce a single gold hypothesis thus reward label matching rather than generating alternative, plausible explanations; and we also show that without extensive test cases previous benchmarks cannot even explicitly distinguish an alternative hypothesis from the gold-labeled one as intended.

Empirical illustration. We previously generated 17,835 hypotheses consistent with initial obser- vations On (n ∈{1, 2, 3, 4}, sampled from Oall). For each problem, fixing On and its consistent hypotheses set F, we sample m ∈{1, 2, 3, 4} hidden test pairs Om ⊂Oall \On and repeat each (n, m) five times. A hypothesis passes if it agrees with all test cases in Om. We report (a) pass rate and (b) γ-diversity of survivors, averaged over problems (Fig. 4a–b).

Pass rates decline as m increases but do not collapse; even at m=4, a nontrivial fraction remains valid. Survivors also retain diversity—for example, on ACRE at m=4 the passed set has γ ≈1.5, i.e., roughly 1.5 distinct predictions per input on the shared sample space. Because all candidates are LLM-generated, this is a conservative lower bound on plausible explanations. As available ob-

Figure 4: Simulation Study Results

servations and background knowledge broaden, underdetermination grows, and single-gold-answer metrics increasingly misclassify reasonable alternatives.

Simulation Study 2: Gear Is General

In this section, we show that GEAR is general rather than task-specific. Across varied abductive datasets, higher GEAR scores predict a greater chance that the set contains a hypothesis explaining unseen (hidden) observations, thereby aligning with existing evaluations of good abduction and indicating cross-task applicability.

For each problem, we sample m ∈{1, 2, 3, 4} hidden cases Om ⊂Oall \ On and draw a context Fc of size c ∈{0, 1, 2} from hypotheses consistent with On; we discard any context with a mem- ber already passing Om. Let Fconsistent be the full set of hypotheses consistent with On and from the remaining consistent pool Fconsistent \ Fc, we enumerate unordered pairs (fa, fb) and, for each f ∈{fa, fb}, compute marginal diversity gain for ρ∈{γ, β} as ∆ρ(f) = ρ(Fc∪{f}; S) −ρ(Fc; S),

|Pf |

|S| , the coverage of f over the sample space S, and an av-

We Label (Fchosen, Frejected) So

that Score(fchosen) > Score(frejected). We then test both on Om, declare a pass when a hypothe- sis explains all hidden cases, and aggregate over problems/contexts/pairs to report the odds ratio ORm = Pr(pass | fchosen)/ Pr(pass | frejected).

Figure 4(c) shows that ORm tends toward 1 as m increases, yet in most settings ORm > 1: the higher-GEAR-score candidate is more likely to pass the hidden cases, aligning with the idea that a more diverse prediction space on S increases the chance of covering the unseen evidence Om.

Training Data Preparation

Building on the simulation study above, we posit that learning better abduction—operationalized as generating a hypothesis set that aligns better with GEAR —which naturally improves the chance that at least one hypothesis fits unseen observations and thereby generalizes to downstream tasks, without requiring any gold hypotheses as supervision.

Training data are constructed from the previously generated 50,340 hypotheses (§5.1). Within each dataset (100 problems), we sample 50 for training, 10 for validation, and hold out 40 for final eval- uation. Similar to the preference-pair construction in § 5.3, for each training problem, let Fall be the set of hypotheses generated by nine LLMs (including both inconsistent and consistent hypotheses), and let Fc ⊆F be a small hypothesis context of size c ∈{0, 1, 2} sampled from Fc ⊆Fconsistent. We then enumerate all unordered hypothesis pairs (fa, fb) from Fall \ Fc. For each f ∈{fa, fb}, we assign a preference in three stages: (1) instruction following / format compliance (prefer parsable over non-parsable outputs; 159,981 pairs); (2) consistency (among parsable candidates, prefer those consistent with On; 429,980 pairs); (3) when both are parsable and consistent, GEAR score, where we score each candidate by the same score function Score(f) in § 5.3. We prefer the candidate with the higher Score(f) (249,561 pairs). In total, this yields 839,522 training preference pairs. We fine- tune the base models with Direct Preference Optimization (DPO) using LoRA (rank 128, α = 256) Table 2: Aggregated results across nine settings.

Gear Performance Improvement

Instruction Following Rate Consistency Generalizability

Ross-Task Generalization With Gear

Avg Train Pass Rate Avg Test Pass Rate Top-1 Acc. Top-2 Acc. Top-3 Acc.

Nextcoder-7B-Momentum-Curriculum

under 4-bit quantization with bfloat16 compute. For DPO, we randomly sample 51,200 pairs with a fixed 1:1:1 ratio across Parsing, Consistency, and GEAR preferences.

Omentum-Based Curriculum Learning

During fine-tuning we observed that the mixture of preference data materially affects outcomes, and different base models favor different mixtures (some benefit from earlier emphasis on format/con- sistency, others from earlier diversity). This led us to a momentum-based curriculum method. The intuition is to learn what improves fastest and is easiest first, then shift weight toward harder signals.

We did an ablation study on the effect of each preference category (See in Appendix C). This adap- tively reweights based on measured learning progress, avoiding hand-tuned fixed ratios and letting each base model gravitate to its preferred mixture.

Adaptive reweighting preferences. Unlike existing curriculum learning methods that either pre- scribe a static training curriculum—thereby overlooking model-specific competence differences (Bengio et al., 2009; Wang et al., 2019)—or adopt dynamic schemes that operate at the sample level (Zhou et al., 2021; Jiang et al., 2015; Sow et al., 2025), which are computationally expensive and often misaligned with the higher-level objective of producing diverse hypotheses, our approach is simple yet efficient: it dynamically delivers a skill-level curriculum—i.e., a goal-aligned schedule over preference types (Parsing/Format, Consistency, and GEAR/diversity) rather than over individ- ual instances—aligned with GEAR objectives.

For each preference type r, keep an Exponential Weighted Moving Average (EWMA) of its probe loss and convert recent improvement into sampling weight:

R

(then normalize across r and clip to [wmin, wmax]).

R

is the probe loss on the validation subset at epoch t;

R

is its EWMA; α ∈(0, 1) is the smoothing factor; m(t)

R

is the per-type sampling weight (normalized across r); ε>0 prevents zero weight;

R

is then normalized across r and clipped to [wmin, wmax]. Experimental settings. We use α = 0.1, ε = 0.1, mmax = 0.03, wmin = 0.8, wmax = 1.2, and update the sampling weights every 1,280 training examples. The same hyperparameters and schedule were used for all three fine-tuned models (no per-model tuning).

For evaluation, we use the original (non-sampled) splits with Otrain and Otest from held-out prob- lems, asking each model to generate three hypotheses per problem. As shown in Table 2, models trained from these GEAR-derived preferences—despite never seeing held out problems—achieve higher diversity scores (β, γ) and improved Top-1/2/3 (T3) accuracies, with the momentum-based curriculum consistently outperforming the fixed-ratio baseline across the reported settings.

Onclusion & Discussion

We present GEAR, a general evaluation framework for abduction that scores hypotheses by con- sistency, generalizability, and diversity. Across nine LLMs on four abduction benchmarks, GEAR reveals differences that gold- or human-centric evaluations miss; simulation study confirms abduc- tion is defeasible and shows that more diverse hypothesis sets are more likely to predict hidden observations. We convert GEAR into label-free training signals and propose a momentum-based DPO curriculum that adapts preference weights with learning progress, improving hypothesis diver- sity and downstream accuracy without gold supervision.

Although our experiments use programmable domains, GEAR is not limited to them. The frame- work hinges on four ingredients: (i) a size measure M, (ii) a sample space S, (iii) a deduction/exe- cution oracle, and (iv) a semantic equivalence predicate. In natural-language (NL) settings these re- main the same but grow harder: M should capture semantic coverage (e.g., topical/diversity weight- ings) rather than raw cardinality; S can be unlabeled yet must include many diverse probes to reveal prediction patterns; the deduction step becomes model- or tool-mediated and thus stochastic, mit- igated by calibrated decoding, tool use, and repeated sampling; and equivalence must be judged semantically or via canonicalization to executable meaning representations to curb polysemy. The primary bottleneck is therefore foundational NL tooling for reliable execution and equivalence un- der ambiguity. GEAR ’s applicability to NL tasks scales with the quality of these primitives; as they improve, the framework transfers with minimal changes—largely a swap of stronger semantic metrics and oracles.

Ethics Statement

We affirm adherence to the ICLR Code of Ethics. This work evaluates and trains language models on publicly available benchmarks (MINI-ARC, ACRE, LIST FUNCTIONS, ARC-2025) and model APIs; it does not involve human subjects, personally identifiable information, or sensitive attributes.

To minimize potential harms, we restrict experiments to benign, programmable tasks. Dataset li- censes and usage follow their original terms; no proprietary or private data are redistributed. There are no known conflicts of interest or third-party sponsorships that influenced results; any such re- lationships will be disclosed upon de-anonymization. Code and data used for evaluation will be released to facilitate auditing and responsible reuse.

Reproducibility Statement

We aim for full reproducibility. Datasets, problem sampling, and the construction of the evaluation sample spaces S are defined by simple, deterministic rules with fixed seeds; prompts for hypothe- sis generation (initial/iterative) are provided; and all scoring criteria (Consistency, Generalizability, β/γ Diversity) are formally specified. Training details (preference construction, DPO with LoRA, quantization, and the momentum-based curriculum schedule with hyperparameters) are described alongside exact settings. We will release code, prompts, and configuration files enabling end-to- end replication—from hypothesis generation to metric computation and fine-tuning—together with random seeds and logs upon acceptance.

Authors:

Peder EZ Larson 1, 2,* , Jenna ML Bernard1, James A Bankson 3, Nikolaj Bøgh 4, Robert A Bok1, Albert P. Chen 5, Charles H Cunningham 6,7, Jeremy Gordon1, Jan-Bernd Hövener 8, Christoffer Laustsen 4, Dirk Mayer 9,10, Mary A McLean11 12, Franz Schilling13, James Slater1, Jean-Luc Vanderheyden5, 14, Cornelius von Morze 15, Daniel B Vigneron1, 2, Duan Xu1, 2, and the HP 13C

94143, Usa.

Denmark. 5 GE Healthcare, Menlo Park, California, USA. 6 Physical Sciences, Sunnybrook Research Institute, Toronto, Ontario, Canada.

8 Section Biomedical Imaging, Molecular Imaging North Competence Center (MOIN CC), Medicine, Baltimore, MD, USA. Cambridge, United Kingdom.

14Jlvmi Consulting Llc, Dousman, Wi, Usa

#See Acknowledgements for a list of all HP 13C MRI Consensus Group Members This work was supported by the ISMRM Hyperpolarized Media MR Study Group, the ISMRM Hyperpolarization Methods & Equipment Study Group, and the Hyperpolarized MRI Technology Resource Center (NIH/NIBIB grant P41EB013598).

Abstract

MRI with hyperpolarized (HP) 13C agents, also known as HP 13C MRI, can measure processes such as localized metabolism that is altered in numerous cancers, liver, heart, kidney diseases, and more. It has been translated into human studies during the past 10 years, with recent rapid growth in studies largely based on increasing availability of hyperpolarized agent preparation methods suitable for use in humans. This paper aims to capture the current successful practices for HP MRI human studies with [1-13C]pyruvate - by far the most commonly used agent, which sits at a key metabolic junction in glycolysis. The paper is divided into four major topic areas: (1) HP 13C-pyruvate preparation, (2) MRI system setup and calibrations, (3) data acquisition and image reconstruction, and (4) data analysis and quantification. In each area, we identified the key components for a successful study, summarized both published studies and current practices, and discuss evidence gaps, strengths, and limitations. This paper is the output of the “HP 13C MRI Consensus Group” as well as the ISMRM Hyperpolarized Media MR and Hyperpolarized Methods & Equipment study groups. It further aims to provide a comprehensive reference for future consensus building as the field continues to advance human studies with this metabolic imaging modality.

Keywords: Hyperpolarized MRI, metabolic imaging, carbon-13, pyruvate, dissolution dynamic

Introduction

MRI with hyperpolarized 13C agents, also known as hyperpolarized (HP) 13C MRI, has shown great potential as a novel imaging modality, particularly for its ability to probe metabolic processes in real time. The first human studies with HP [1-13C]pyruvate were performed in 2011 in prostate cancer patients (1).

Since then, there have been over 60 papers published with imaging results of human subjects from 13 different sites, with applications including prostate cancer, brain tumors, breast cancer, kidney cancer, pancreatic cancer, metastatic disease, liver disease, ischemic heart disease, diabetes and cardiomyopathies. The vast majority of these studies used [1-13C]pyruvate (1–63), where [2-13C]pyruvate (64) and 13C-urea (56) have been demonstrated too.

As clinical HP 13C MRI advances, there is a growing need to build consensus for best practices, which are critical for comparing data across sites, performing multi-site trials,deploying methods to new sites, partnering with vendors, and potentially for obtaining broader regulatory approvals.

In March 2022, we initiated an effort to build consensus within the HP 13C MRI community with this opportunity in mind, and it was greeted with strong enthusiasm. The “HP 13C MRI Consensus Group”, containing over 55 members from 27 sites, identified the area of greatest need and opportunity for consensus building to be HP [1-13C]pyruvate human

●

Pyruvate is the most mature and widely used HP agent and has the most significant translational evidence emphasizing the potential clinical impact.

●

Clinical trials, particularly multi-site trials, have the strongest need for consensus methods to ensure that data can be combined across sites. This work is a Position Paper for which the goal is to describe current successful practices and study methods for HP [1-13C]pyruvate human studies along with justification to support those practices. This is divided into four major topic areas: (1) HP 13C-pyruvate preparation, (2) MRI system setup and calibrations, (3) data acquisition and image reconstruction, and (4) data analysis and quantification (Fig. 1). The current successful practices and study methods include a literature review of published peer-reviewed journal papers showing human HP [1-13C]pyruvate study data, up to September 2022 (1–63), as well as new unpublished information from surveys of HP 13C study sites. Based on this information, we also highlight the evidence gaps, strengths, and limitations of current practices which are summarized at the end of each section.

Figure 1: Illustration of the HP 13C MRI human study process, including the 4 major areas covered in this paper: Hyperpolarized 13C-pyruvate preparation, MRI system setup and calibration, Acquisition and Reconstruction, and Data Analysis and Quantification.

Figure 2: Anatomical targets of HP [1-13C]pyruvate MRI human studies published up to September 2022.

Hyperpolarized 13C-Pyruvate Preparation

This section covers the processes for creating the HP agent, 13C pyruvate, and will include many aspects and considerations that are needed to safely and effectively prepare doses for metabolic imaging studies in human subjects. These include material, personnel, equipment and facility, fluid path preparation, quality control, and release.

It is helpful to understand that the specifications of a dose of 13C pyruvate suitable for in vivo MR HP metabolic imaging were shaped in part by early preclinical studies performed by GE HealthCare summarized in Ref. (65). In short, the safety of the two novel drug components, 13C pyruvate and the electron paramagnetic agent (EPA) AH111501, were demonstrated in those studies. The more precise formulation of the dose suitable for human use was then determined from clinical studies (66) that included two Phase 1 clinical trials in young and elderly healthy volunteers without hyperpolarization of the 13C nuclei and another Phase 1/2a dose escalation and imaging feasibility study with HP 13C pyruvate in 31 prostate cancer patients at the With the exception of the first HP 13C imaging clinical trial, which utilized a prototype device in a cleanroom (1), all HP 13C studies performed in humans to date have utilized the SPINlab polarizer (manufactured by GE HealthCare). Consequently all doses of the HP 13C pyruvate delivered by SPINlab have been produced using the “SPINlab Pharmacy Kit” that serves as the container-closure system for the various drug components (13C pyruvic acid and EPA mixture, dissolution medium, and neutralization and dilution medium) during sample polarization, dissolution and quality control (QC) processes. Thus many aspects of the HP sample preparation considerations discussed below are related to the SPINlab instrument and the consumables designed to be used with it (67).

General Considerations

While more than 860 patients or healthy subjects having been injected with HP 13C pyruvate as of January 2022 without reports of any serious adverse events (68), HP 13C pyruvate injection remains an investigational MR contrast agent and can only be administered by those with Investigational New Drug (IND) exemption from the Food and Drug Administration (FDA) in the USA, a Clinical Trial Application (CTA) in Canada, approval from National Research Ethics Committee Services in the UK, or approval from the relevant local regulatory body. Thus, methods and processes involved to produce a dose should have patient safety as the first priority. Since utilizing dissolution dynamic nuclear polarization (dissolution-DNP) for human use is still a relatively new development, there are no existing published regulatory guidelines specifically for this method.

There are two major production styles that determine how various sites approach the agent preparation. In the US, the most common approach is to rely on a sterilizing filter (“Terminal Sterilization”) to ensure sterility of the final product, akin to PET tracer production, where a starting molecule with a radioisotope is processed using various other ingredients to make the final, desired and injectable contrast agent within a necessarily short amount of time (69). For these sites, sterilization of the components and accessories upstream of this filter are not required, although many of them were manufactured and tested following Good Manufacturing Practice (GMP) or Good Laboratory Practice (GLP) requirements. The filling process is usually performed under an ISO 5 laminar flow hood, but a clean room or an isolator is not required.

This approach is typically accompanied by testing the integrity of the sterilizing filter prior to release of the dose for injection. Typically, post release endotoxin and sterility tests are performed using an aliquot reserved from each released dose.

In the UK and EU, the most common approach is to more-closely follow sterile pharmaceutical compounding guidelines (70), where all components and ingredients are required to be sterile or manufactured under GMP guidelines and are assembled and filled within a clean room environment or an isolator system (“Sterile Preparation”). Typically a batch of Pharmacy Kits for HP 13C pyruvate injection are prepared together. The sterility of the final dose is also ensured by batch validation testing, in addition to the sterility of the ingredients and the sterile compounding process. The endotoxin and sterility testing are performed for the process validation but are not performed for each injected dose.

Some institutions fill and assemble the Pharmacy Kit required for a specific study on the same day or the day prior to polarization, dissolution, and patient administration, but others have also demonstrated the feasibility of preparing a batch of kits, keeping them in a -20ºC freezer and using them over a period of a few months.

Beyond the obvious requirements that the process and the facility has to ultimately produce a dose that is safe to inject into a human, regulatory authorities will also focus on the question “Are you in control of your processes?”. To be in control of your process requires an in-depth and broad understanding of all processes involved in pre, post, and during the production process.

Personnel

It is typical and may be required to have licensed personnel involved in the production process depending on local regulations.Typically a pharmacist, radiopharmacist or other similarly qualified person (QP), in charge of the facility where the Pharmacy Kit filling and preparation is taking place, is responsible for the overall process and the release of the injectable dose.

Qualified cleanroom technicians are often involved in the Pharmacy Kit filling under the supervision of the pharmacist or QP. As is required for pharmaceutical compounding or PET tracer production, training requirements and training records for all personnel need to be maintained and available for audit by the FDA or equivalent.

Equipment And Facility

The facility and all equipment need to have standard operating procedures (SOPs) that describe how equipment is used, maintained, and calibrated to comply with relevant legislation. Currently, almost all the filling of the Pharmacy Kit takes place within a compounding laminar flow hood or isolator (typically ISO 5). At some sites, the filling is conducted within a cleanroom, while at others, it is conducted in a dedicated non-cleanroom space, reflecting differences in cleanroom approach and specifications between regulators worldwide (71). Some equipment or facilities, such as the compounding hood or cleanroom, may require external certified laboratories for testing.

Material Handling

Material handling guidelines (69,70) require SOPs detailing a system to track all of the materials involved in the HP production process for a particular patient dose, similar to current good manufacturing practice (cGMP) requirements for material handling for drug compounding. This includes acceptance standards, storage conditions, amount used in the patient dose for each ingredient and materials used in the assembly of the fluid path and Pharmacy Kit. Currently some users choose to open and inspect and sometimes modify the Pharmacy Kits upon arrival, but some users keep them in the sealed packaging until they are required for dose preparation.

Pharmacy Kit Filling And Assembling

As required by an IND or its equivalent, the preparation of the doses of HP 13C agent are detailed in the Chemistry, Manufacturing, and Control (CMC) section of an applicable regulatory submission; an example of this has been made available (72). It describes the processes of filling the Pharmacy Kit with the different components that make up the final drug product, and of assembling the final kit for either storage or immediate use in the polarizer. Special attention should be given to the laser welding process in order to satisfy installation qualification (IQ) and operational qualification (OQ). Typically, the final developed process is validated by process qualification (PQ) runs, during which 3 or more Pharmacy Kits are filled and used and the final HP 13C products are tested for endotoxin and sterility and to confirm that they meet the dose specifications for injections (usually including pyruvate concentration, residual EPA concentration, pH, liquid state polarization level and dose temperature). The data from 3 consecutive PQ runs are submitted as part of the IND submission (or its equivalent), and are often also reviewed by the Institutional Review Board (IRB) where the studies are conducted.

Quality Control And Dose Release

The quality control (QC) and dose release can be separated into two aspects: one is the QC and release of the filled Pharmacy Kit, and second is the QC and release of the HP 13C agent for injection, after polarization and dissolution. For institutions filling a batch of kits and storing them to use over a period of time, typically the batch can be released based on initial validation, environmental monitoring data from the day of kit production, and if filters are used during preparation of any of the components, filter integrity testing. But in some cases one or more kits are used for validation before the batch of kits are released for future use. For institutions that fill only the kits required for specific studies shortly before the experiment, the filled kits often do not go through separate release tests before they are used.

The quality control of the HP 13C pyruvate solution post dissolution is primarily performed to ensure that the agent meets the dose specifications (Table 1) before it is administered to the subject. These specifications target both safety (pH, residual EPA, temperature) and efficacy (pyruvate concentration, polarization, volume). Typically, the pyruvate concentration, residual EPA concentration, pH, dose temperature, dose volume, and liquid state polarization are measured by the QC accessory associated with the SPINlab polarizer. Some users perform a secondary measurement for one of the parameters, such as pH, using a different instrument or pH paper. For sites that do not go through a separate release testing process for batch filled kits, the integrity of the sterilization assurance filter, a part of the Pharmacy Kit, is typically tested as a part of the dose release. It is also common for these users to preserve an aliquot of the final HP 13C pyruvate solution for post-release endotoxin and sterility testing. This testing cannot be completed fast enough to test an individual dose prior to injection, but this is why other processes such as PQ runs and validation testing are done to minimize the chance a subject could be injected with a contaminated dose.

The Final Dose Release And Injection

should be done under the supervision of a licensed professional, based on local regulations.

Some Key Challenges

Many of the challenges associated with HP 13C pyruvate preparation can be attributed to the conditions required for the dissolution-DNP method of high magnetic field (~3-7 T) and very low temperature (~1 K) during polarization, with pressurized and superheated water necessary for the rapid dissolution event. These extreme conditions are quite challenging for the design of the container-closure and fluid path system. In particular, the cryogenic temperature in the polarizer requires special attention to any moisture or ambient (moist) air introduced into that portion of the fluid path, which can form an ice block at ~1 K. This ice can lead to flow restriction during the dissolution event and reduce the strength of the laser welded bond between the cryovial and its cap. This can ultimately produce failures in the dissolution step, including variations in final pyruvate concentration and pH that may fail to meet QC release criteria as well as fluid path ruptures that provide no available dose and result in polarizer down-time.

The polarization of the HP 13C pyruvate sample decays quickly over the span of a few minutes after dissolution, and thus the process of dissolution, QC for release, and injection should be completed as fast as possible to preserve the high polarization level achieved. Any delays in the preparation process, such as transportation time or equipment malfunction, can significantly reduce the final polarization and result in lower quality imaging data.

Current Practices

A summary of data collected from all sites performing clinical trials with HP 13C-pyruvate is shown in Fig. 3 and Table 1, including the specification of the final dose and how the quality control and release of the final dose are performed. There is a split in the Production Style, described in the General Considerations section above, with 8/13 sites using Sterile Preparation versus 5/13 using Terminal Sterilization. While many of the dose specifications show notable differences in acceptable ranges, all of these variations listed in tables have been successfully and safely been used to perform HP 13C pyruvate studies in humans. Their differences depend on the institutions’ preferences, resources and their particular regulatory situation. There is high similarity in pyruvate ranges, temperature ranges, EPA limits, and volume limits. There is modest variability in pH ranges and large variability in the endotoxin test limit. There is a 3-fold difference in acceptable polarization levels, which are measured to ensure a futile dose is not injected since the polarization is directly proportional to SNR. This reflects the decision by several sites to believe that useful data can be still be obtained with suboptimal polarizations.

Figure 3: Hyperpolarized agent preparation methods reported by sites currently performing HP

In House

Table 1: HP 13C-pyruvate preparation parameters, methods, and dose specifications used for quality control testing and release as well as validation. These were obtained from a survey of all sites performing clinical trials with HP [1-13C]pyruvate. The parameters used for product release are noted in bold text, otherwise these parameters are measured for batch validation or other QC measurements. The endotoxin and sterility testing are performed during process validation of the batch and/or post-injection, and largely depends on the agent production approach.

Summary

The overall safety record of HP 13C-pyruvate has been very strong, and the SPINlab hyperpolarizer has proven to provide high polarizations at human sized doses while meeting numerous QC and release criteria. A weakness remains the failure modes of the SPINlab Phamacy Kits (e.g. ice blocks, path ruptures), which are placed under extreme requirements particularly during dissolution. The preparation process still requires a high degree of expertise.

Therefore, there is a significant need to improve the reliability, robustness, and ease of operation for generating HP 13C-pyruvate doses for human studies. Furthermore, there is a divide between manufacturing and sterile compounding style preparation as well as other site-specific practices, resulting in variations in SOPs and justification required to relevant regulatory bodies. There have also been no comparisons between these approaches. It is also unclear what release criteria and QC parameters are truly required to ensure patient safety.

However, all of the reported methods are acceptable and approved by the appropriate regulatory authorities, and have led to the rapid expansion of successful human studies in recent years.

Mri System Setup And Calibrations

This section covers the MRI system setup, including the imaging system, RF coils, phantoms, and prescan calibration methods.

Imaging System

The main prerequisite for a given MRI scanner to be capable of supporting studies with HP 13C is its “broadband” capability to transmit and receive radiofrequency (RF) signal at the frequency of 13C, which is around 4 times lower than 1H. This does not come as a default on clinical MR devices. The transmit power of the broadband amplifier should also be sufficient to support the intended flip angle and RF pulse shape with the employed transmission RF coil(s) for 13C. Most studies to date use relatively low flip angles (< 90 degrees) for HP 13C in order to preserve polarization for time-resolved imaging. The capability to receive 13C signal on multiple channels is also desirable to increase SNR, as discussed further in the “RF coils” section.

The choice of magnetic field strength is primarily dependent on the metabolites’ frequency separation due to chemical shift dispersion and 1H imaging. High field strengths do not enhance hyperpolarized 13C signal as they do for 1H because the signal strength in a HP experiment relies on manipulating the population of quantum energy states outside of the MRI scanner.

However, the injected HP 13C-pyruvate and its metabolic products have greater frequency separation at higher fields, and it may thus be easier to separate and quantify these resonances at higher fields. This comes at the cost of a reduction in the achievable T2* and often reduced T1. As the initial polarization is independent of the imaging field strength it has been proposed that the increased T2* at 1.5T can potentially be exploited to increase SNR by adapting the acquisition bandwidth or reduce off-resonance imaging effects in cases when the decay of the transverse magnetization is dominated by T2* (73). In practice, 3T has been used in all published human 13C-pyruvate studies surveyed (Supporting Table S1), and comprises the majority of scanners currently in use for human studies (Table 3). A field strength of 3T is well-suited for 1H MRI anatomical reference and correlative imaging.

Stronger and more rapidly slewing magnetic field gradients support more rapid spatial encoding, particularly for metabolite-specific single-shot imaging using echo-planar imaging (EPI) or spiral imaging (See “Acquisition and Reconstruction”). Although the spatial resolution acquired for HP 13C imaging is typically much coarser than for 1H MRI, the factor of ~4 in gyromagnetic ratio leads to the same reduction factor in performance of the gradient system, so 13C experiments are potentially more limited by gradient hardware performance. To date, all human studies have used the commercially-available integrated gradient systems provided in clinical MRI scanners.

Optimization of scanner design has understandably focused on minimization of artifacts in 1H MRI, where devices such as room lights, the gradient amplifiers, and the motors driving the patient bed are checked to ensure that they do not produce RF interference at the 1H frequency, but artifacts may arise at other frequencies. Eddy current compensation is also not always appropriately adjusted for nuclei at other frequencies (74). In order to optimize for 13C, many sites have performed checks on phantoms for RF interference, gradient artifacts, and eddy currents (74), including the use of post-hoc gradient impulse response function characterisation and correction, and some vendors have fixed these issues as well.

Rf Coils

For HP 13C imaging studies in humans, RF coils for both 1H and 13C nuclei are needed, with 1H MRI providing an anatomical reference for registration and optional additional multiparametric MRI readouts. At the Larmor frequency of 13C nuclei, the relative contributions from coil noise compared to sample noise increase compared to 1H (73,75), although sample noise still is likely the dominant contributor for human-sized coils at 32.1MHz - the resonance frequency of 13C nuclei at 3T.

The key requirement for human 13C-pyruvate RF coils are that the coil geometry and sensitive volume must cover the volume of interest in the subject. Table 2 and Figure 4 shows coil configurations that have been used and optimized for applications in different anatomic regions.

Volume resonators are most commonly used for transmit, as they surround the subject to

Provide B1 Transmit Across The Fov (B1

+). While 1H relies on a large birdcage (“body”) coil built into the scanner, 13C transmit coils must be placed inside the bore. This takes up valuable space within the magnet, and also has led to the use of designs with relatively inhomogeneous

B1

+. Many human studies have used Helmholz pair resonators for transmit, including the “clamshell coil”, which has a notably inhomogeneous B1

+ Profile But Has Been Used Because Of

relatively easy integration into the scanner bore. B1

+ Variation Results In Variations In The Flip

angles that control the use of the hyperpolarized magnetization and creates errors in common HP metrics (9,76). The exception are head coils, where birdcage designs with highly

Homogeneous B1

+ can be placed around the head while easily fitting inside the bore. As with 1H MRI, higher SNR can typically be achieved by smaller receive coil elements, such as surface coils or phased arrays, and the majority of 13C receive coils used have layouts similar to 1H phased arrays.

RF coil quality control is important to ensure proper functioning of the coils to provide consistent imaging quality, especially with limited natural abundance 13C signal in vivo. It typically involves 1) a physical integrity check of the coil cables and connectors and 2) phantom SNR tests to check the coil’s performance and to monitor it over time (see Phantoms below). An useful reference for RF coil quality control is outlined in the MRI accreditation program of the American College of Radiology (77) and can be adapted for 13C coils.

Notably, configurations for brain and prostate studies used dual-tuned 1H/13C coil designs, which greatly simplify workflow and registration of 1H and 13C images, as no switching of coils is needed.

(1)

Table 2: RF coil configurations reported for human HP [1-13C]pyruvate studies.

Tx = Transmit

coil, RX = receive coil. The commonly used “clamshell” TX coil is a Helmholz pair design. For 1H RF configurations, all used the Body coil for TX unless otherwise noted, and “repositioned” indicates the 13C coil was removed for 1H imaging. One representative reference is listed for each configuration. The RF coil configurations reported in the reviewed papers are shown in Supporting Table S1.

Figure 4: Examples of RF coil configurations used for human HP [1-13C]pyruvate brain studies. (A,B) 13C Clamshell TX (Helmholz pair) and 2× 4-channel paddle RX arrays. (C) 13C Birdcage volume TX and 32-channel RX array (RX array slides into TX coil). (D) 13C Birdcage volume TX and 24-channel RX array, combined with a 1H 8-channel RX array. Image reproduced with permission from Ref (16).

Phantoms

Since hyperpolarized magnetization is non-renewable, phantoms containing 13C nuclei are important to: 1) test the multi-nuclear capabilities of the imaging system, including all parts of the signal excitation and receive chain; 2) perform calibration measurements before a scan with hyperpolarized nuclei; and 3) perform necessary pre-scan adjustments (see “Prescan Calibration” section). The phantoms currently in use are listed in Table 3. Their composition must provide sufficient 13C signal, with additional considerations of conductivity, stability, chemical shift(s) present, potential for dynamic imaging, and cost. The phantom geometries are typically either compact, in order to be used alongside the subject during a HP scan, or large enough to mimic the inner volume of a RF coil for system testing.

One popular compact design contains enriched 13C-urea at high concentration, typically 8 M, which provides a single resonance, placed inside a small container ~1 mL. The most common recipe mixes 13C-urea in a 90% water/10% glycerol solution, with glycerol used to increase the urea solubility and doping with a Gd-based contrast agent to shorten T1 which increases the potential SNR per unit time. For example, when Dotarem is added at a 3:1000 volume ratio the 13C-urea T1 is around 500 ms and T2 is around 100 ms. However, when testing pulse sequences influenced by T1 and T2, doping should be used carefully. This phantom is suitable for frequency calibration, transmit gain calibration, sequence testing, and as a fiducial marker when placed next to a patient. However, enriched 13C-urea has a relatively high cost compared to natural abundance compounds.

For larger volumes (>100 ml), the phantoms most often used contain undiluted ethylene glycol, glycerol, or dimethyl silicone. These compounds have sufficiently high carbon concentrations to provide sufficient 13C signal even with the 1.1% natural abundance of 13C. These larger phantoms matching the inner volume of an RF coil are useful for coil testing, including transmit

+) And Receive (B1

-) coil profile mapping, as well as to mimic acquisitions using in vivo FOV requirements. In this case, size and conductivity should match the expected subject size in order to mimic coil loading and get a realistic estimation of B1+. Large-volume natural abundance urea phantoms have also been used by some sites, but suffer from higher conductivity compared to biological tissues. Typically, it is easier to increase the conductivity and hence coil loading of the non-conductive phantom by adding NaCl to match physiological loading (16,78).

Dynamic phantoms that aim to mimic metabolite kinetics have also been developed (79–81), and have the potential to more closely mimic the HP experiment, but so far these are not widely used.

Prescan Calibration

Prior to performing an MRI acquisition, the so-called prescan procedure is used to set the shim parameters to maximize B0 homogeneity over the field of view (FOV) or a specific region of interest (ROI), the scanner center frequency (CF), the RF transmit gain, and the receiver gain.

While this calibration procedure is usually automated for 1H, the lack of sufficient natural abundance 13C signal prevents use of automated methods. (Although natural abundance 13C lipid signal has been detected, there are so far no reports on using this signal for prescan.) Table 3 shows current practices across sites.

Maximizing B0 homogeneity is independent of the nucleus and is therefore performed prior to 13C imaging using the 1H water signal and existing shimming tools, such as by a standard automated process (“Auto Shimming”) or using high order shimming routines. Similarly, the 13C CF can be calculated from the 1H CF using a predetermined scaling factor that depends on the target chemical shift (82). Another common approach used is to have a small, high-concentration 13C phantom, e.g. 8M 13C-urea, integrated in the RF coil or placed next to the scan subject (1). The reference frequency can also be based on real-time measurements after the HP injection but prior to imaging (83). Both the CF and B0 shimming are critical when using spectrally-selective RF pulses, as inmetabolite-specific imaging methods, where the desired excitation bandwidths are typically very narrow and frequency offsets can lead to a failure mode that is only apparent after injection.

The calibration of the RF transmit power is typically performed on a small, high-concentration 13C phantom placed near the region of interest during the scan or on a large 13C phantom of similar size and coil loading as the subject, prior to the subject scan. Reference power is often done by sweeping the power in a pulse-acquire sequence (53,62), or the Bloch-Siegert method (52,84). When using a small phantom, the location of the phantom, B1

+ Inhomogeneity As Well

as any shielding effects, e.g., when the phantom is integrated into a coil (1), may degrade the accuracy. Other methods include real-time Bloch-Siegert method measurements after the HP injection (83), and using the stronger natural abundance 23Na signal that is close enough to the 13C resonance frequency to be detected by 13C coils (82).

The receiver gain is predetermined, either systematically based on independent phantom measurements and assuming the dose and polarization of the HP compound is known prior to injection, or based on past HP imaging studies.

Power [Kw]

Phantom(s) - during study Phantom(s) - before study 13C Frequency

8

13C-bicarbonate doped with dimethyl silicone, various

Power [Kw]

Phantom(s) - during study Phantom(s) - before study 13C Frequency

Maximum Values

Table 3: Summary of the imaging systems, phantoms, and prescan procedures used at sites currently performing HP 13C-pyruvate human studies. These were obtained from a survey of all sites performing clinical trials with HP [1-13C]pyruvate. *Previously performed studies with a Siemens 3T Tim Trio. The imaging systems, phantoms, and prescan procedures reported in the reviewed papers are shown in Supporting Table S1.

Summary

Commercially available 3T MRI systems are by far the most commonly used for human HP 13C-pyruvate studies, although a systematic investigation of the impact of B0 has only recently been investigated (73). The multi-nuclear RF transmit and receive chain has proven sufficient for current acquisition strategies, although many sites have observed artifacts due to RF interference, gradient interference, and residual eddy currents when operating at the 13C frequency. A variety of 13C RF coils, tailored for numerous anatomical targets, have been successfully demonstrated, with the main limitation that most transmit coils take up a lot of additional space inside the bore and provide relatively inhomogeneous B1

+ Profiles. The

phantoms used have converged into generally 2 categories - small phantoms containing 13C-enriched compounds that can be used during the study and human-sized phantoms containing compounds with high carbon concentrations but without 13C enrichment that are used to test and calibrate the coils. There are no standardized compositions or geometry, and dynamic phantoms that recapitulate in vivo kinetics would be desirable but are still an emerging area. Prescan calibration procedures were not well defined in most publications, so we surveyed individual sites to determine current practices. Calibration procedures for the B0 field (13C CF and shimming) for most sites take advantage of 1H signal and methods, while methods

For Calibration Of B1

+ is more variable across sites, likely a reflection of remaining challenges in how to perform this calibration. Standardization of both phantoms and calibration procedures would synergistically improve the robustness and reproducibility of HP 13C studies.

Acquisition And Reconstruction

Data acquisition strategies in human HP [1-13C]pyruvate MRI studies must account for multiple chemical shifts, efficiently utilize the non-renewable HP magnetization, and acquire data quickly relative to metabolism and relaxation decay processes. These studies require spectral encoding to separate metabolites, necessitating pulse sequences that efficiently encode up to 5D data (3 spatial + 1 spectral + 1 temporal dimension). RF pulses must efficiently sample without immediately saturating the non-renewable HP magnetization, and sequences must acquire data quickly and be robust to both experimental and physiologic variation (e.g. B1

+ Inhomogeneity,

variation in perfusion) to ensure reproducibility and minimize scan-to-scan variability. This section covers current successful practices for data acquisition in human [1-13C]pyruvate studies, and accompanying 1H imaging, from different anatomic regions, including scan parameters and image reconstruction.

Acquisition And Reconstruction Methods

The acquisition methods used in human [1-13C]pyruvate studies can be classified into 3 categories: 1) MR spectroscopy or MR spectroscopic imaging (“MRS/I”), 2) chemical shift encoding methods, and 3) metabolite-specific imaging (Fig. 5).

Mrs/I Methods Specifically

resolve a spectrum that can be analyzed to extract expected as well as unexpected resonances, making this approach very robust. It was used in many initial studies (1).

Chemical Shift

encoding methods, most commonly the Iterative Decomposition of water and fat with Echo Asymmetry and Least-squares estimation (IDEAL) method, use imaging sequences acquired with multiple TEs and rely on a model-based separation of expected chemical shifts (85).

Metabolite-specific imaging methods use specialized RF pulses that are spatially and spectrally selective to excite individual metabolites which are then typically imaged with fast k-space trajectories such as echo planar imaging (EPI) or spirals (86).

Their Application To Different

organ systems is described below. The image reconstruction methods used in human [1-13C]pyruvate studies have typically been conventional methods (e.g. FFT, non-uniform FFT, or equivalent). The incorporation of accelerated imaging and advanced reconstruction methods including parallel imaging (4,57,87) and compressed sensing (7) has also been applied in human studies for improved spatial resolution, temporal resolution and coverage, but have the potential for additional artifacts as well as SNR losses due to ill-conditioning of the reconstruction (e.g. g-factor).

The Majority Of

published studies do not use accelerated imaging indicating the resolution and coverage achievable without acceleration is currently adequate for successful data collection. Performing coil combination, even with fully sampled data has also been shown to have specific challenges for HP human images: using naive sum-of-squares methods suffer from high noise amplification in the relatively low SNR regime of HP [1-13C]pyruvate (compared to 1H), motivating several HP 13C-specific methods that include data-driven coil sensitivity estimation which have shown obvious improvements over sum-of-squares (11).

More recently denoising techniques have been applied as post-processing of human HP data(41,42,44). The techniques applied are based on spatial-temporal singular value decomposition for unsupervised estimation of signal and noise components. They have shown improvements in apparent SNR in the brain and liver, while care must be taken to choose parameters such as the rank threshold to avoid oversmoothing and overfitting to the estimated signal components.

Prostate Studies

Prostate cancer was the first human application of HP [1-13C]pyruvate (1), and data was acquired with MRS/I methods: 1D dynamic MRS, single-slice 2D dynamic echo-planar spectroscopic imaging (EPSI), and single time point 3D EPSI. Advances in imaging strategies led to the development and application of new acquisition schemes, including undersampled 3D EPSI with compressed-sensing (7), model-based chemical shift encoding methods that use a priori information (47,59), and metabolite-specific EPI (10), all of which can provide volumetric whole-organ coverage and dynamic acquisitions.

The pyruvate bolus arrival in the prostate can vary by ± 10 s between patients, necessitating dynamic imaging to reliably and consistently capture the pyruvate bolus (18). For this reason, all currently ongoing studies acquire dynamic data. While MRS/I, chemical shift encoding, and metabolite-specific imaging can all achieve dynamic imaging, chemical shift encoding and metabolite-specific imaging provide greater dynamic and volumetric coverage (85). For scan prescriptions, the FOV is designed to provide full prostate coverage and typically to match the orientation of the anatomic imaging used for registration. Flip angles used in current studies are constant through time, as quantification with a variable-through-time flip scheme is highly sensitive to bolus timing (8) and errors in the RF transmit (B1 +) field (76).

Heart Studies

Data acquisition methods for 13C imaging in the heart must be designed to meet the demands of significant cardiac motion and blood flow. To cope with the periodic cardiac motion, most human heart studies to date used gating to the diastolic window, the longest cardiac cycle interval, which has reduced motion (2,22,28,30,35,36,38,45,52). The duration of the diastolic window limits the available data sampling time, making cardiac acquisitions the most time-constrained of the HP 13C MRI applications. The most common acquisition approach is metabolite-specific imaging with spiral k-space trajectories (2). Their single-shot imaging capability makes these methods particularly robust to motion effects. Furthermore, spiral k-space trajectories provide rapid k-space coverage and relatively benign flow and motion artifacts. The majority of studies have used 2D multi-slice acquisitions, but 3D encoding has also been used successfully (35).

Brain Studies

For HP 13C MRI of the human brain, the majority of studies have also used 2D (slice selective) acquisitions (10–12,14,16,28,33,40,41,44,51,53,60), with a trend toward volumetric coverage using 2D multi-slice metabolite-specific imaging. 3D metabolite-specific imaging of the whole brain, with phase encoding of the slice direction (34,57), has been shown to provide similar SNR efficiency (88) compared with multislice imaging. A number of studies have employed MRS/I (5,6,29,31–33,50,55) resulting in a spectrum from each voxel, which has the advantage of not requiring a priori information about which peaks to encode. This was important in early brain studies when it was not known which peaks would be detectable. Chemical shift encoding, using a set of images with different echo times and an iterative reconstruction of the individual resonances (i.e. the IDEAL approach (85)), has also been used (12,49,54), with the drawback that coverage in the slice direction was limited due to the time required to acquire multiple echo time images.

Abdomen And Breast Studies

The fundamental approaches to data acquisition and reconstruction in the abdomen and breast are largely similar to the aforementioned applications, but demand attention to particular challenges associated with these anatomic regions, especially relating to respiratory motion.

Although it has been shown that a basic 2D MRSI approach based on phase encoding and FID readout can be successfully applied for HP 13C imaging in breast (15) and kidney (13), major advantages in terms of spatiotemporal resolution and coverage have been realized using tailored approaches based on metabolite-specific imaging (43,62) and chemical shift encoding (43), which have facilitated multi-slice or 3D dynamic acquisitions over large FOVs in the abdomen (4,37,46).

The significant respiratory motion encountered in these regions can directly blur 13C images, and has further favored these rapid acquisition strategies. Motion also degrades B0 homogeneity, which can shift frequency-selective excitation profiles and introduce artifacts into rapid imaging readouts. This makes accurate determination of the acquisition center frequency and shimming essential in these regions which often cover large FOVs. (See “Prescan Calibration” section for more information). In some studies, breath-holding was used to minimize motion effects and enforce frame-to-frame data consistency (42). A pragmatic and reasonably effective approach for dealing with respiratory motion during 13C data acquisition is an initial breath-hold (as long as can be tolerated), followed by free-breathing (46,62).

1H Imaging

Collection of 1H imaging data is essential both for prescribing the 13C acquisition and for interpretation of the resulting 13C data. Multi-planar 1H scouts are acquired prior to 13C acquisition to enable graphical prescription of the 13C imaging region. All human HP 13C-pyruvate imaging studies acquire conventional MRI scans (e.g. T1- and T2-weighted volumes) for anatomic reference, aiming to cover at least the full 13C FOV. Acquiring these anatomic scans as close as possible to the time of 13C imaging (immediately before or after) minimizes potential misregistration between the data sets. Depending on the application, other advanced 1H sequences are also acquired (e.g. diffusion-weighted imaging for cancer imaging).

When contrast-enhanced data is acquired, it is done after 13C imaging, as paramagnetic contrast agents will accelerate 13C relaxation.

Reported Study Parameters

Figures 5 and 6, and Supporting Table S2 shows the reported acquisition study parameters for human HP [1-13C]pyruvate studies published as of September 2022. Figure 5 shows a mixture of MRS/I, metabolite-specific imaging, and chemical shift encoding methods have been successfully used, where spectroscopy-based methods have become less prevalent in recent studies. Figure 6 shows the acquisition timing, including the important start time and interval/temporal resolution, is quite variable across studies.

Figure 5: Acquisition methods used in published HP [1-13C]pyruvate human studies published up to September 2022, classified into: MR spectroscopy and spectroscopy imaging (MRS/I); chemical shift encoding methods, such as IDEAL, that use multiple TEs and model-based reconstructions; and metabolite-specific imaging methods that use spectrally-selective excitation to image a single resonance at a time.

Figure 6: Temporal acquisition characteristics reported in HP [1-13C]pyruvate human studies published up to September 2022. (a) Reported referencing of acquisition start times.

(B)

Acquisition start times reported when using dynamic imaging and when timing was reported relative to the end of the injection. (c) Temporal resolutions. “Not Applicable” indicates dynamic imaging was not used.

Summary

Three general categories of acquisition strategies have been used successfully for human HP 13C-pyruvate studies: MRS/I, model-based chemical shift encoding (e.g. IDEAL) methods, and metabolite-specific imaging methods. These have enabled successful studies in the prostate, heart, brain, abdomen, and breast. Recent studies increasingly have used the imaging-based strategies of metabolite-specific imaging and chemical shift encoding which are the fastest methods, although a heads-to–head comparison between techniques has not been performed.

Metabolite-specific imaging is quite popular because of its speed and compatibility with single-shot imaging, but is sensitive to B0 field variations and thus requires careful calibrations. Nearly all studies surveyed acquired data dynamically, allowing measurement of the bolus and metabolite kinetics. The exact timings and associated flip angles vary quite widely across reported studies, with no consensus yet as to how to choose these parameters. Image reconstruction is typically done directly using Fourier Transform methods, and accelerated imaging strategies are uncommon.

Data Analysis And Quantification

This section covers the analysis of data from human HP [1-13C]pyruvate studies, including modeling and metrics, visualization, as well as considerations for how to store data and metadata. Depending on study design, the analysis may need to give quantitative or semi-quantitative output reflecting a biological process or may just reflect a contrast between different regions of interest for quantitative evaluation.

Metrics

Figure 7: HP [1-13C]pyruvate raw data (A) have typically been quantified using four categories of metrics depending on the acquisition. Data acquired as a single time point are often quantified using normalized metabolite images or metabolite ratios (B). Dynamic data can be quantified using normalized metabolite images or metabolite ratios (B), or with metabolite timings such as time-to-peak (TTP) or pharmacokinetic (PK) models (C). The latter two require the data to be time-resolved. [1-13C]alanine and 13C-bicarbonate are analyzed similarly to [1-13C]lactate but omitted here for display.

Metabolite images are commonly used as summary metrics for HP MRI data, often including some form of normalization as well as summed over time as an area under the time curve (AUC) (17). These are analogous to the visual evaluation that is most used for routine clinical work (89,90). In these metabolite images, we expect that the [1-13C]pyruvate AUC signal is predominantly weighted towards perfusion and uptake, while [1-13C]lactate, [1-13C]alanine and 13C-bicarbonate AUCs represent metabolic conversion. The strength of this approach lies in its simplicity and relatively few underlying assumptions. Limitations to the use of single-metabolite images or AUCs include sensitivity to inhomogeneous coil profiles (57,87,91), the acquisition strategy and acquisition parameters, pyruvate polarization and concentration level, and signal relaxation rates (92). Further, the reader must be careful to interpret all the images in conjunction to better understand the underlying biology; for example, increased [1-13C]lactate in the presence of decreased [1-13C]pyruvate delivery can have a very different meaning compared to increased [1-13C]lactate with increased [1-13C]pyruvate delivery.

In an attempt to address variations in coil sensitivity, polarization level, and pyruvate delivery, AUC images are often computed by normalizing to a specified parameter, such as the maximum pyruvate or average lactate signals, or presented as a ratio such as lactate/pyruvate or divided by “total Carbon” - the sum total of HP 13C signal observed across all metabolites. The AUC ratios between metabolites and pyruvate are proportional to the corresponding forward kinetic rates (81,93), but are not directly comparable to rate constants when magnetization loss rates (e.g. relaxation and losses due to signal excitation) differ between studies. Similarly, the ratios between the produced metabolites (e.g. bicarbonate/lactate) can reflect the balance between downstream metabolic pathways (12,55). Care must be taken to consider how AUC images are calculated and normalized before comparing values between studies.

To further quantify the interpretation, pharmacokinetic (PK) modeling approaches were developed to compute the apparent kinetics of pyruvate-to-metabolite exchange (92,94–99). These yield semi-quantitative to quantitative apparent rate constants, given in s-1. Some models require a vascular input function, while others avoid this requirement (95). PK models can explicitly account for acquisition-specific details such as excitation angle and repetition time, and thus may reduce the effects of these details on quantification. An input-less model, provided in the Hyperpolarized-MRI-Toolbox (https://github.com/LarsonLab/hyperpolarized-mri-toolbox) (100) and thus frequently employed for human data, has been shown to fit well and robustly to prostate and brain data (8,20). PK models are quantitative in nature, arguably provide more relevant biological information (8,20), and appear to be reproducible across sites (51). However, rate constants derived from PK models are still apparent rates, and likely do not reflect a single biological characteristic.

Some additional considerations include whether complex or magnitude data is used, as the noise behaviors will impact the analysis differently. Additionally, cut-off thresholds or other criteria may be used to identify and avoid voxels with insufficient SNR before analysis to improve robustness (20,41).

Regardless of the analysis approach, the underlying biology is not always clearly represented by the data; instead, the metrics may be influenced by perfusion, barrier permeability, intercellular shuttles, enzyme activities, co-substrate concentrations, or combinations thereof, depending on the organ and disease of interest (19,43,94,101–103). This may be addressed by incorporating complementary information. As an example, HP 13C pyruvate data is influenced by perfusion, and thus addition of perfusion MRI could be important for interpretation (98,104,105).

All the methods outlined above have been explored in clinical studies, described in Supporting Table 3 and summarized in Figure 8. As of September 2022, approximately 52% of studies involving human subjects report rate constants derived from a PK model with a few different models reported. A nearly equal fraction (51%) of the studies report AUC ratio values.

Approximately 66% of these studies report metabolite-specific images or AUC values. About 40% report SNR values; this metric is particularly frequent in manuscripts that describe technical developments for clinical HP MRI. Approximately 16% of these studies summarize model-free metrics, and 10% report measurements from a single timepoint. Most studies report a combination of quantities.

Figure 8: Reported metrics used for analysis in HP [1-13C]pyruvate human studies published up to September 2022.

Visualization

A wide variety of approaches have been used for visualizing data from human HP 13C-MRI studies. The challenges and practical considerations are: 1) choosing the appropriate metrics to display, 2) how to encode the parameters (e.g. the colormap), and 3) choosing how to provide anatomical context and other multi-parametric data. The choice of visualization also depends on the goal which could be for diagnostic interpretation, but also quality control, reproducibility among readers and publication.

Metrics

The choice of HP 13C metrics is described in detail above. At this stage in HP 13C development where there is no standardized metric, often a combination of metabolite images and ratios or PK model parameters are shown.

Parameter Encoding

The mapping function chosen should provide an adequate, often quantitative, impression of the parameter mapped. There is a consensus in the visualization field that perceptually uniform maps are best suited to visualize continuous parameters, like the greyscale typically used by radiologists as well as other monochrome (black to blue) and color ranges (fire-type, rainbow-type) (106,107). Multi-color heatmaps have been the most frequently employed method for HP 13C data, while greyscale has infrequently been used but it ensures there is no coloring-based bias as well as facilitating later reuse (Fig. 9a). Among the color schemes employed in the clinical HP 13C literature, fire-type scheme seems to be the most common [similar to “Plasma” or “Inferno” in matplotlib.org]. Next most commonly employed is the rainbow-type scheme [similar to “Rainbow” in matplotlib.org].

Anatomical Context

HP MRI faces the challenge that it does not necessarily depict the anatomical features, similar to PET, and thus requires an anatomical reference. Most often, a grayscale anatomical image is overlaid with a HP colormap (Fig. 9c,d). This approach is very intuitive, but can skew perception as the grey-scale anatomical reference may affect the brightness of the HP data (e.g. signal in the skull). This bias does not occur when showing adjacent maps (Fig. 9a, b). Here, anatomical outlines may help to provide reference (Fig. 9b).

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