Mechanistic inference of the metabolic rates underlying 13 C breath test curves

Carbon stable isotope breath tests offer new opportunities to better understand gastrointestinal function in health and disease. However, it is often not clear how to isolate information about a gastrointestinal or metabolic process of interest from a breath test curve, and it is generally unknown how well summary statistics from empirical curve ﬁtting correlate with underlying biological rates. We developed a framework that can be used to make mechanistic inference about the metabolic rates underlying a 13 C breath test curve, and we applied it to a pilot study of 13 C-sucrose breath test in 20 healthy adults. Starting from a standard conceptual model of sucrose metabolism, we determined the structural and practical identiﬁability of the model, using algebra and proﬁle likelihoods, respectively, and we used these results to develop a reduced, identiﬁable model as a function of a gamma-distributed process; a slower, rate-limiting process; and a scaling term related to the fraction of the substrate that is exhaled as opposed to sequestered or excreted through urine. We demonstrated how the identiﬁable model parameters impacted curve dynamics and how these parameters correlated with commonly used breath test summary measures. Our work develops a better understanding of how the underlying biological processes impact different aspect of 13 C breath test curves, enhancing the clinical and research potential of these 13 C breath tests.


Introduction
Carbon stable isotope breath tests offer new opportunities to better understand gastrointestinal function in health and disease [1]. These tests provide a dose of non-radioactive 13 C-labeled substrate, which is digested, absorbed, and metabolized, appearing on the breath as 13 CO 2 . As the range of labeled substrates that are commercially available grows, from whole-molecule labeling to position-specific (atom-level) labeling, 13 C breath tests can be developed to target a wide range of specific gut processes, such as digestion, absorption, or oxidation, with a correspondingly wide range of potential clinical applications.
Beyond one or two clinical tests with clear diagnostic criteria, the uses of 13 C breath tests have remained primarily limited to research, in part because standard methods to characterize 13 C breath test curves often reflect a complicated mix of both the underlying biological process of interest and other aspects of metabolism. For this reason, the most successful 13 C breath tests have relatively simple dynamics-such as the 13 C-urea breath for detection of Helicobacter pylori-or are carefully designed to ensure that the biological process of interest is the rate-limiting process. For example, the 13 C-octanoic acid breath test is based on the fact that medium-chain fatty acids are absorbed immediately on entering the duodenum, which causes gastric emptying to be the rate-limiting step [2]. In the 13 C-galactose test of hepatic function, the rate-limiting step is the hepatic clearance of galactose [3].
However, in other cases, it is not clear how to isolate information about a gastrointestinal or metabolic process of interest from a breath test curve (serial measurements of 13 CO 2 concentration in the breath over time). Breath test curves can be characterized by empirical curve fitting and by summary measures, often themselves derived from curve fitting results. However, these metrics do not necessarily correspond directly to biological processes. Two standard curve fitting models of the percent dose recovery rate (PDRr) as a function of time t, here denoted y(t), include the following two empirical curves, where a, b, c, m, k, and b are empirical constants estimated by optimization [4][5][6], yðtÞ ¼ mkb expðÀktÞ 1 À expðÀktÞ These methods can be applied to 13 C breath test curves, often with reasonable fits. However, the success of an empirical model only indicates that it is sufficiently flexible to capture features of the curve, not that it can offer mechanistic insight. Moreover, while many breath test curve summary measures have been proposed, e.g., cumulative percent dosed recovered (cPDR) by time t, peak PDRr, time to peak PDRr, and time to recover 50% of the dose (cPDR-50), their connection to specific processes of interest is weak. Nevertheless, certain summary measures are typically preferable to others for certain inferences, suggesting that there is some mechanistic connection between the biological rates and the dynamics captured by the summary measures. For instance, the time to peak PDRr has been shown to out-perform the time to recovery of 50% of the dose for gastric emptying [7]. In this analysis, we return to pharmacokinetic modeling fundamentals to develop a framework that can be used to make mechanistic inference about the metabolic rates underlying a 13 C breath test curve. We also demonstrate how these mechanistic parameters isolate aspects of breath test curve dynamics and how they correlate with standard curve summary measures. We illustrate this framework using 13 C sucrose breath test ( 13 C-SBT) curves from 20 healthy volunteers. The 13 C-SBT has been proposed as a potential measure of intestinal brush-border sucrase-isomaltase function to assess environmental enteric dysfunction (EED) [8]. EED is a syndrome characterized by villous atrophy (also called villous blunting), inflammation, and increased permeability of the small intestine [9,10] and is thought to be ubiquitous among the 2 billion children and adults who lack access to improved water or sanitation in low-and middle-income countries [11]. Prior studies have suggested that sucrase-isomaltase, which is excreted at villous tips, may be disrupted due to villous atrophy in EED [12], as well as in celiac disease (CD) and other gastrointestinal disorders with functional similarities to EED [13][14][15][16]. Accordingly, understanding how 13 C-SBT curves reflect healthy or disrupted sucrase-isomaltase activity may have important clinical and global public health applications, making it an appropriate motivating example for this theoretical work. Our work, which determines the identifiable mechanistic parameters underlying a 13 C-SBT curve is a first step toward developing a breath-test-based diagnostic and will be broadly applicable to breath tests developed for other 13 C substrates.

Data
Twenty healthy adults were recruited to participate in a proof-of-concept 13 C-SBT study. Participants were recruited by advertisement in the Glasgow area, aged between 18-65 years with no history of gastrointestinal symptoms or disease. The cohort mean (standard deviation) BMI was 22.0 (3.5) kg/m 2 , mean age was 22.8 (4.6) years and maleto-female ratio was 10/10. Participants gave informed consent, and the study was approved by the University of Glasgow College of Medical, Veterinary & Life Sciences Research Ethics Committee (Application Number: 200170060). Plants that photosynthesize using a C 4 pathway are significantly more enriched with 13 C than those that use a C 3 pathway. Participants were instructed to follow a low 13 C diet by excluding plants C 4 plants and their derivatives [17] for 3 days prior to testing and to fast for 8 h prior to testing. A 20 g dose (0.116 mmol excess 13 C) of naturally enriched sucrose derived from sugar cane (0.01649 atom % excess; Tate and Lyle Europe, London, United Kingdom) was dissolved in 100 mL of water. (The dose of excess 13 C was calculated as the difference between the 13 C natural abundance of beet sucrose (a C 3 plant) and cane sucrose (a C 4 plant) [17]. Because of the C 4 -free diet run-in period, participants' baseline breath 13 CO 2 concentrations reflect a C 3 diet, so the dose of the tracer is calculated in terms of how enriched the tracer is for 13 C compared to the baseline diet). The small-volume, liquid administration of the dose was designed to minimize delay from gastric emptying. A baseline breath sample was collected immediately before participant ingested the tracer. Breath samples were then taken every 15 min for 8 h. Participants were provided a lunch 4 h into the test. Samples were collected in 12 mL Exetainer breath-sampling vials (Labco, United Kingdom) and analyzed by isotope ratio mass spectrometry (IRMS AP-2003, Manchester, United Kingdom). The analytical output of the IRMS is d 13 C, the relative difference in parts per thousand between R s =[ 13 C]/[ 12 C] in the sample and the internationally accepted calibration standard ratio R=0.0112372 [18], The units of d 13 C are per mil (%). A given d 13 C is converted to isotope abundance, expressed as ppm 13 C, as follows, We account for individual variation in baseline d 13 C by considering excess ppm 13 C over baseline, which was then converted to percent dose recovery rate (PDRr), PDRrðtÞ¼ 100Á ppm 13 C at time t Àppm 13 C at time 0 ð Þ Á V CO 2 10 6 Ádose of excess 13 C (mmol) ; where V CO 2 is a subject-specific estimate of CO 2 production (mmol/hour) based on the participant's estimated body surface area and sex [19]. The data have been included as supporting material.

Mathematical model
We use a standard conceptual model of carbon substrate (here, sucrose) metabolism as a starting point for a compartmental, ordinary differential equation model of the mass transfer of 13 C in a 13 C substrate breath test [20]. In this conceptual model (Fig. 1a), the ingested tracer enters the stomach and passes to the small intestine (gastric emptying). Sucrase-isomaltase, which is secreted in the brush-border at the villous tips, cleaves sucrose into glucose and fructose, facilitating active transport for glucose and fructose moieties into the blood, where the substrate moves to the liver via the hepatic portal vein. The substrates are oxidized through a series of intermediary metabolic processes and converted to bicarbonate. Bicarbonate kinetics are typically modeled with a fast pool and a slow pool [21]. Transfer to the slow pool, representing long-term processes such as lipid storage, is often considered to be irreversible on the time scale of a breath test. Plasma bicarbonate has two excretion pathways, urinary and pulmonary. Through the 13 C breath test, pulmonary excretion of 13 CO 2 is observed. Given the nature of the liquid sucrose tracer and the physiology of the stomach, we model the residence time in the stomach as a time-delay (although no time delay was needed for these participants) and all other processes as mass action. The fraction of the 13 C dose in each of the small intestine, liver, plasma bicarbonate, fast bicarbonate pool, and slow bicarbonate pool are denoted by x 1 through x 5 , respectively. Let q ij be the transfer rate coefficient (1/time) for the fraction of the dose moving from compartment x i to x j . Let q 30 be the pulmonary excretion and q 36 the urinary excretion rates of bicarbonate from plasma. Then, the ordinary differential equations are as follows.
The initial conditions of this model are x 1 ð0Þ ¼ 100 and all other x i6 ¼1 ð0Þ ¼ 0. Through the breath test, we have the following measurement equation for PDRr at time t, We denote the full parameter set as h. When we want to emphasize the dependence of the output on the parameters, we will write yðh; tÞ.
We note that this model implicitly assumes that all the tracer is absorbed in the small intestine and not transported to the large intestine, where it might be absorbed or metabolized by microbiota at a different rate. This assumption is reasonable for healthy populations, and, in other work in adults in a region where EED is highly prevalent, there was little evidence of gross malabsorption even with a much larger dose of sucrose [8]. However, preliminary evidence from breath tests in which sucraseisomaltase is experimentally inhibited suggests that there is a distinct signal in the breath curve denoting entrance of sucrose to the colon and metabolism by the microbiota, suggesting that the phenomenon could possibly occur in severely disordered individuals. In practice, we recommend analysis of only the portion of the breath curve prior to this signal.

Identifiability and model reduction
In the context of determining the health of the small intestine and ultimately diagnosing EED, the aim of the 13 C-SBT is to infer the value of q 12 , the rate of metabolism in the small intestine, given y(t). But, before estimating the value of a model parameter from observed data, we first need to determine whether that parameter is identifiable, that is, whether there is a unique value of the parameter associated with the best fit of the proposed mechanistic model to the available data. If a parameter is not identifiable, i.e., multiple values or a range of values of the parameter can explain the data equally well, then we can find a simpler, reduced model that similarly fits the data but whose parameters are all identifiable. For example, if our model was y ¼ ðm 1 þ m 2 Þx þ b for some (x, y)-pair data, parameter b would be identifiable, but parameters m 1 and m 2 would not be. However, we could reduce our model by defining a new paramter m :¼ m 1 þ m 2 . Unfortunately, identifiability and model reduction are not so Fig. 1 a. A conceptual model of 13 C mass transfer in a 13 C breath test, accounting for physical transport, tracer metabolism, bicarbonate kinetics, and excretion. b A structurally and practically identifiable model of 13 C mass transfer in a 13 C breath test. The parameters next to each arrow describe rate constants (1/time) straightforward for even modestly complex differential equation models.
We distinguish between structural identifiability, which asks whether a parameter can be determined given perfect measurement of the model output [22][23][24], and practical identifiability, which asks whether a parameter can be measured given actual data [25]. A parameter may be structurally but not practically identifiable because of excessive noise, insufficiently regular measurement, or other reasons; external factors are those controlled by the investigator (i.e., related to study design), while internal factors are intrinsic to the model (e.g., arise from practical indistinguishability of trajectories for a range of parameters) [26]. Structural identifiability analysis is often a useful first step because it can determine identifiable parameter combinations, whose values are identifiable even if the constituent parameters are not individually identifiable and which may aid in model reduction. After structural identifiability is determined, practical identifiability analysis determines the real-world identifiability and uncertainty in parameter estimates given a level of significance (e.g., 95% confidence). If practical identifiability determines systematic lack of identifiability (e.g., inability to observe certain characteristics of an output trajectory due to the time scales [27]), further model reduction is possible.
To determine the structural identifiability of our model parameters when observing y(t), we first found an inputoutput equation of the model [28][29][30]. An input-output equation is a monic polynomial equation in terms of the measured input and output variables and their derivatives. The coefficients of an input-output equation represent the identifiable parameter combinations of the model for the given measurement. We used Wolfram Mathematica v11.3 (Wolfram Research; Champaign, Illinois) to determine an input-output equation for the model and measurement in Eqs. (6) and (7).
To determine practical identifiability, we used profile likelihoods to determine how dependent the model fit to the data was on the specific values of the parameters. Profile likelihoods, which can be used to determine likelihoodbased confidence intervals, vary the fixed value of one parameter while determining the best fit (minimum negative log-likelihood) when fitting the remaining parameters [25,27,31]. If the likelihood-based confidence interval for a parameter has infinite width (for a given level of significance), then the parameter is not practically identifiable. Here, we used a normal log-likelihood as function of the parameters h where n is the number of data points and t i is the time at which measurement z i was taken. The variance r 2 is estimated as 1 nÀ1 P i ðyðh; t i Þ À z i Þ 2 and thus depends on h. Denote the maximum-likelihood estimate asĥ. Denote the maximum likelihood when the ith parameter is fixed to value h i as Lðĥ j6 ¼i ; h i Þ and call it the profile likelihood of h i . The likelihood based confidence interval at level of significance a is fh i : logðLðĥÞÞ À logðLðĥ j6 ¼i ; h i Þ\D a g, where D a is v 2 ða; dfÞ=2 where v 2 ða; dfÞ is the chi-squared distribution with a number of degrees of freedom equal to the number of parameters and a is the level of significance [25]. In layman's terms, the profile likelihood tracks how much worse the ''best'' fit of the model to the data is as we constrain one parameter away from its maximum-likelihood estimate, and the level of significance determines the level of ''how much worse of a fit'' corresponds to the bounds of our confidence interval for h i .
We made simplifying assumptions based on the profile likelihoods to arrive at a series of reduced models. For parameters that are structurally identifiable but not practically identifiable, the choices of simplifying assumptions may be partially subjective, with multiple potential reductions. We explain our specific assumptions and justifications in the Results.
After arriving at a reduced, practically identifiable model, we compared fits of the original model and each of the reduced models to each of the 20 breath test curves. The models were only fit to 5 h of data because the naturally occurring 13 C in the lunch given to participants at 4 h began to distort the curves after 5 h. We analyzed our final, reduced model by comparing the dynamics of the simulated breath curves as a function of each of the reduced model parameters in turn. We also calculated the correlations of the values of the reduced model parameters fit to each of the 20 breath tests with each of four breath test curve summary measures (cPDR at 90 min, peak PDRr, time to peak PDRr, and time to 50% cPDR) to assess how well these summary measures reflect the underlying mechanistic parameters. Model simulations and optimization were performed in R v4.0 (R Foundation for Statistical Computing; Vienna, Austria).
We also have information from the initial conditions of y(t) and its derivatives. The initial conditions of y and dy dt are 0, but the initial condition of d 2 y dt 2 is q 12 q 23 q 30 and the initial condition of d 3 y dt 3 is Àq 12 q 23 q 30 ða þ q 12 þ q 23 À q 43 Þ. Altogether, a minimal set of locally identifiable parameter combinations is then fq 12 ; q 23 ; q 30 ; q 35 þ q 36 ; q 34 ; q 43 ; g. Parameters q 12 , q 23 , q 30 , q 34 , and q 43 retain their original interpretations, but, because the parameters are only locally and not globally identifiable, we do not necessarily know which rate value belongs to which biological process. Parameter combination q 3À :¼ q 35 þ q 36 is the rate of the tracer leaving the body through urine or being sequestered in a bicarbonate pool. This last combination represents the fact that we cannot distinguish between tracer that has been sequestered in the slow bicarbonate pool and tracer that has been excreted in urine if we measure only pulmonary excretion. A (locally) structurally identifiable version of (6) is thus as follows.
The two take-aways from the structural identifiability analysis are that the slow bicarbonate pool is indistinguishable from urinary excretion and that even if we determine the values of the underlying biological rates, we cannot determine their order (e.g., we will not know which of the biological processes corresponds to the slowest, limiting rate).

Practical identifiability
Even if the above set of parameters and parameters combinations are structurally identifiable, they may not be practically identifiable from real-world breath test curves.
To illustrate the practical identifiability of the model, we use data from the 13 C-SBT and determine the profile likelihoods of each of the structurally identifiable parameters. Each set of data, along with the corresponding best fit by the full, structurally identifiable model Eq. (11), are shown in Fig. 2. This plot also includes fits from a series of reduced models developed below. In Fig. S1, we demonstrate that all parameters in the full structurally identifiable model are practically unidentifiable given real data, with the possible exception of q 12 and q 23 . Moreover, none of the relationships with the other parameters along these profiles are indicative of practically identifiable parameter combinations, e.g., no two parameters vary together in a way that suggests that their sum or product is constant. However, we do see that parameter q 43 can be sent to 0 or 1 with negligible loss of fit, suggesting that the dynamics of the fast bicarbonate kinetics do not meaningfully impact the breath test trajectory. For simplicity, we reduce our model by assuming q 43 ¼ q 34 ¼ 0. Finally, we define j :¼ q 30 =ðq 30 þ q 3À Þ to be the fraction of bicarbonate that is exhaled on the breath; this parameterization is convenient because it demonstrates that although q 30 and q 3À can take multiple values depending on q 12 and q 23 , there is a fixed relationship between them.
The fit to each individual's breath test trajectory are shown in Fig. 2 (Reduced model 1). The fits from Reduced model 1 reproduce the Full model fits in nearly all cases, confirming that the dynamics of the fast bicarbonate pool can be neglected. We next compute the profile likelihoods for the four parameters in the model in Eq. (12) in Fig. S2. Parameter j is identifiable, which is sensible since it describes the asymptote of the cPDR. However, we find that q 12 , q 23 , and q 30 =j, which we know are only locally identifiable, are indeed interchangeable, so that they take 3 (possibly repeated) values between them. Only two local minima are observed for the three parameters for many plots, indicating that the likelihood wells for two of the rates have merged. Although for each individual's breath trajectory the likelihood profiles may prefer the two larger or two smaller values to be repeated (share the same value), we find that it is sufficient and more convenient to constrain the model so that the larger value is repeated. Thus, in our final model reduction in Eq. (13), without loss of generality, we assume that the first step is the slowest, i.e., we set q ¼ q 23 ¼ q 30 =j and pq ¼ q 12 where 0\p 1 is the ratio of the slower rate to the faster rates. It is important to note again that, because the parameters are only locally identifiable, we cannot determine which rate corresponds to which biological process. That is, we cannot determine which process has the limiting rate. In this reduced model, The fit to each individual's breath test trajectory are shown in Fig. 2 (Reduced model 2), and the confirmation that the parameters are globally practically identifiable is given in Fig. S3. The fits from Reduced model 2 reproduce the Full model fits in nearly all cases. This model also has a closed form solution. When p\1, The cumulative percent dose recovered is the integral of y(t), and it has a horizontal asymptote of 100j as t ! 1.
Practically, this reduced model indicates that a single breath test curve can be summarized as resulting from a faster, gamma-distributed process and a slower, exponential process (in some order, possibly with the slower, exponential process occurring in-between portions of the other process). The fraction of the tracer that will be exhaled as opposed to sequestered or otherwise excreted scales the overall PDR. This final, reduced model is summarized in Fig. 1b.

Model dynamics
To understand how each of the 3 parameters q, pq, and j impact the dynamics of the breath curve, we first take the mean values of each parameter across the 20 trajectories ( q= 1.97, pq= 0.32, j=0.82). Then we vary one parameter while keeping the other two constant. The range of dynamics is shown in Fig. 3a-c. In Fig. 3d-f, we scale the curves in Fig. 3a-c appropriately (by ð pqðp À 1Þ 2 Þ=ðpqð p À 1Þ 2 Þ in (a), by p=p in (b), and by j=j in (c)). The scaled figures highlight that q impacts the rate of increase, pq controls the rate of decline, and j is a vertical scaling factor. Breath test curve simulations demonstrating that the dynamics of the early portion of the curve do not constrain the later curve. Here q and j are chosen for each value of pq in a way that preserves the dynamics up to 90 min These curves further suggest that the early part of the curve may not be informative for pq. Indeed, in Fig. 4, we see that we can choose values of q and j such that the early part of the curve (first 90 min) does not constrain the later part of the curve. This result suggests that certain summary measures may not be informative for certain underlying processes.

Summary measures
Here, we compare the model parameters q, pq, and j estimated for each of the 20 breath test curves to the estimated values of each of the four breath-test curve summary measures: cPDR at 90 min, peak PDRr, time to peak PDRr, and time to recover 50% of the dose (50% cPDR). Best fit lines and correlation coefficients are given in Fig. 5. Note that only those 11 breath curves that achieved 50% dose recovery within 5 h were included in the correlation analysis of the time to 50% cPDR measure. Parameter q, which controls the early phase of the curve, was most strongly correlated with the time to peak PDRr (R ¼ À0:75) and was moderately correlated with cPDR at 90 min (R ¼ 0:56). No summary measure was strongly correlated with pq, the parameter that controls the late phase of the curve, and even the weak correlations were drive by one or two points. Parameter j, the fraction of the dose that will be recovered, which acts as a vertical scaling parameter, was strongly correlated (R ¼ À0:78) with time to recovering 50% of the dose; other correlations were weaker and driven by one or two points.

Discussion
Our work provides an alternative, mechanistic modeling approach to empirical curve fitting and summary measures when analyzing 13 C substrate breath test curves. A threeparameter model-based on an underlying assumption of substrate passage through an exponential, rate-limiting process; a gamma-distributed process; and a scaling factor representing the fraction of the tracer that would be recovered-can be fit to 13 C breath curve data such that each parameter has a uniquely identifiable value. Although this work was illustrated using a 13 C-sucrose tracer, it is broadly applicable to most 13 C substrate breath tests, since the metabolic pathways are broadly similar. One important conclusion of our identifiability analysis is that a single breath test curve alone cannot necessarily resolve all the underlying metabolic processes that occur as part of substrate metabolism. Indeed, we found that a 3-parameter model with a simplifying assumption about the fast processes fit nearly all 13 C-SBT curves as well as the full 6-parameter model. This result should be expected as the 3-parameter empirical models in Eq. (2) are typically flexible enough to capture breath test curve dynamics. This limitation of not being able to fully resolve the underlying transport and metabolism is not a limitation of the mechanistic approach and is instead a limitation inherent to the breath test itself: the breath test curve does not contain enough information on its own to support inference about all aspects of the metabolism, and, indeed, we should not expect it to. To further disaggregate key metabolic processes, other data-such as serial measurements of plasma 13 C-bicarbonate or multiple breath tests repeated with different substrates on the same individualwould be required. On the other hand, if the goal is to translate characteristic curve dynamics into interpretable clinical information about an individual's underlying health or disease state, our mechanistic approach ensures that the amount of information that can meaningfully extracted from the breath curve is maximized.
Our work, determining the identifiable parameters underlying the dynamics of 13 C-SBT curves in healthy adults, is only the first step in the development of a 13 C-SBT clinical diagnostic test. Analysis of 13 C-SBT curves in individuals with disordered or inhibited intestinal surcraseisomaltase activity is necessary to determine how breath test curves and our model parameters can distinguish between healthy individuals and those with gut dysfunction. Our next work will analyze 13 C-SBT curves in individuals given Reducose, an intestinal sucrase-isomaltase inhibitor, followed by analysis of 13 C-SBT curves in individuals suspected to have EED [8]; further methodological work may be needed to understand the practical identifiability of model parameters estimated from diseased individuals' curves, particularly p and j, when the tests ends prior to reaching peak PDRr or if the tracer enters the colon. Although a 13 C-SBT diagnostic could be based on standard summary measures, our model-based approach to evaluate breath curves likely has advantages over conventional approaches, because model parameters, unlike conventional summary measures, have mechanistic interpretations. Notably, summary measures, while attractive for their simplicity, appeared to be only somewhat correlated with the actual mechanistic rates underlying the breath curve. Some metabolic processes, including the ratelimiting step, appear to be poorly captured by all of the summary measures investigated. The summary measures such cPDR at 90 min, peak PDRr, and time to 50% cPDR, were all associated with the scaling parameter j, reflecting the fraction of the plasma bicarbonate excreted on the breath (as opposed to excreted in urine or sequestered). However, in most 13 C breath test applications, this scaling factor is not the metabolic process of interest, meaning that natural variation in this parameter may reduce the discriminatory power of these summary statistics. Indeed, we expect j not only to vary between individuals but within individuals day-to-day, based physical activity in the past few days [32,33]. The rate q was inversely correlated with time to peak PDRr, and so this summary measure may be useful if the fast process is shown to be the process of interest. Even so, the model we presented here can be used to estimate these parameters directly, making breath curves more readily interpretable.
Our results also have implications for breath test administration. For instance, it may also be necessary to adjust the length of test duration depending on which underlying rate is of interest, e.g., longer tests may be needed to measure the limiting rate more accurately. By understanding the dynamics as a function of underlying metabolic rates, we can better design our testing procedures and our analysis plans.
Our work may need to be adjusted for other specific applications, which is why we presented the model reduction approach in full, so that it can be adapted for other applications as needed. One limitation of this analysis is that we assumed instantaneous gastric emptying because of the application of the small, liquid tracer. This work may need to be adapted to account for gastric emptying in the case of other tracer formats, e.g., when administered as part of a meal. In this work, we also used a naturally enriched sucrose tracer, with 13 C occurring in both the fructose and glucose moieties, which means that some of the estimated rates reflect a combination of the individual rates of transport and metabolism of these molecules. In future work, we will explore the use of tracers in which only one sugar moiety is labeled in order to understand the impact of intermediary metabolism on the digestion of 13 C sucrose isotopologues to 13 CO 2 . Additionally, to enhance the interpretation of breath test curves and develop a clinically meaningful diagnostic for the health of an underlying process, it will also be important to determine which aspects of the 13 C metabolic pathway affect the mechanistic parameters we have identified here as capturing the breath test dynamics. This determination can be accomplished using multiple experiments designed to isolate different aspects of the metabolism. There is also a need to understand which aspects of the metabolism are folded into a single parameter and to characterize within-person (i.e., day-to-day) and between-person variation in these rates. This information would improve our ability to design breath tests to isolate specific aspects of the metabolism and to develop clinically meaningful thresholds for parameter values. Finally, we emphasize that sucrose may not ultimately be the optimal tracer for the clinical detection of EED or other metabolic disorders. For example, sucrase-isomaltase activity may not be sufficiently impacted by the disorder of interest, or the dynamics of the breath curve may be dominated by other processes that are not relevant to the disease, such as tissue uptake, energy generating processes, or bicarbonate kinetics. Indeed, tests using a variety of other substrates are under development. Fortunately, the mathematical framework we developed here can serve as a starting point for analysis of 13 C breath test curves regardless of the specific substrate of interest.

Conclusion
We developed a new approach to making biological inferences from 13 C breath test curves and connected specific aspects of curve dynamics to underlying biological rates. A better understanding of how underlying biological processes impact different aspects of the breath curve enhances the clinical and research potential of the 13 C-SBT and other breath tests like it.