Mathematical modeling of fluid dynamics in in vitro gut fermentation systems: A new tool to improve the interpretation of microbial metabolism

In vitro systems are widely employed to assess the impact of dietary compounds on the gut microbiota and their conversion into beneficial bacterial metabolites. However, the complex fluid dynamics and multi‐segmented nature of these systems can complicate the comprehensive analysis of dietary compound fate, potentially confounding physical dilution or washout with microbial catabolism. In this study, we developed fluid dynamics models based on sets of ordinary differential equations to simulate the behavior of an inert compound within two commonly used in vitro systems: the continuous two‐stage PolyFermS system and the semi‐continuous multi‐segmented SHIME® system as well as into various declinations of those systems. The models were validated by investigating the fate of blue dextran, demonstrating excellent agreement between experimental and modeling data (with r2 values ranging from 0.996 to 0.86 for different approaches). As a proof of concept for the utility of fluid dynamics models in in vitro system, we applied generated models to interpret metabolomic data of procyanidin A2 (ProA2) generated from the addition of proanthocyanidin (PAC)‐rich cranberry extract to both the PolyFermS and SHIME® systems. The results suggested ProA2 degradation by the gut microbiota when compared to the modeling of an inert compound. Models of fluid dynamics developed in this study provide a foundation for comprehensive analysis of gut metabolic data in commonly utilized in vitro PolyFermS and SHIME® bioreactor systems and can enable a more accurate understanding of the contribution of bacterial metabolism to the variability in the concentration of target metabolites.


| INTRODUCTION
The gut microbiome is one component of a very complex ecosystem shaped through intricate interactions with the host and the exposome (e.g., diet, medication, antibiotics, circadian rhythms, physical activity, stress, etc.). 1,2ariations in its composition triggers distinct metabolic signatures. 3Gut microbiota and their metabolome thus play a key role in the host homeostasis and health. 1,3Human clinical trials are considered the gold standard to study how the gut microbiome affects this multifaceted equilibrium. 4et, this type of investigation is demanding and hindered by many impediments; they require formal ethical approbation for fecal collection or invasive procedures (e.g., gut biopsies, luminal brush, aspiration), and they lack representativeness, as end-point measurements do not always mirror the different segments of the small and large intestine. 4onetheless, fecal microbiota is considered the closest proxy for colonic microbiota, even if it blurs the differences between luminal and mucosal intestinal ecosystems or colonic regions. 5[8][9][10] The Polyfermentor Intestinal Model (PolyFermS) and the Simulator of the Human Intestinal Microbial Ecosystem (SHIME®) emerged as robust and validated alternatives to study the human gut microbiome in a host-uncoupled fashion. 11,12Gut fermentation systems rely on long-term cultivation of fecal microbiota (2 weeks minimum) under specific physicochemical conditions reproducing human colonic environment.While the PolyFermS system can reproduce multiple segments of the intestinal environment, it is commonly settled to mimic the composition of a single intestinal compartment. 11,13,14The advantage of this system lies in its ability to preserve the prolonged stability of an intestinal microbiota through its immobilization in polymer beads further maintained within an inoculum reactor.The stabilized microbiota is then transferred to a series of test reactors where it can be subjected to treatments.In turn, the SHIME® consists of five serial double-jacketed stirredtank bioreactors mimicking the conditions in the different parts of the gastrointestinal tract.It is used to characterize the long-term and regional effects of a supplement/treatment on the gut microbiota by gathering the complexity and dynamism of the human gastrointestinal tract from the stomach to the distal colon by using multiple bioreactors in series.Gastrointestinal transit in the SHIME® is mimicked by a semi-continuous flow-through as a close proxy for food intake.
Historically, gut fermentation systems aimed at deciphering how diets, food supplements, drugs, or probiotics modulated gut microbiota and made judicious use of metagenomic and metatranscriptomic approaches. 8For instance, in vitro gut systems were used to study the influence of food macro/micronutrients, [15][16][17][18] pre/probiotics, 14,[19][20][21] food additives, 22 food pollutants, 23 drugs, 24 microorganisms 25,26 on gut microbiota and their effect on health.Unfortunately, the application of metabolomic approaches to in vitro fermentation studies has lagged behind owing to the difficulty of modeling microbial metabolism in an ever-changing fermentation environment.8][29][30][31][32][33][34] It can be explained by the difficulty in following up given compounds during the fermentation process and the equivocality of the interpretation of metabolite changes associated with microbial metabolism.One characteristic of in vitro systems that hinders the study of microbiota metabolomics is the longitudinal (time-dependent) and dynamic operation of the gut fermentation systems in continuous or semi-continuous modes, to mimic a transit or a residence time, which greatly impacts the flow of metabolites and their dilution level.approaches).As a proof of concept for the utility of fluid dynamics models in in vitro system, we applied generated models to interpret metabolomic data of procyanidin A2 (ProA2) generated from the addition of proanthocyanidin (PAC)rich cranberry extract to both the PolyFermS and SHIME® systems.The results suggested ProA2 degradation by the gut microbiota when compared to the modeling of an inert compound.Models of fluid dynamics developed in this study provide a foundation for comprehensive analysis of gut metabolic data in commonly utilized in vitro PolyFermS and SHIME® bioreactor systems and can enable a more accurate understanding of the contribution of bacterial metabolism to the variability in the concentration of target metabolites.

K E Y W O R D S
accumulation, bacteria, digestion, dilution, gut microbiota, metabolism, metabolites Another one is the use of multi-compartmented bioreactors such as the SHIME®, used to recreate successive regional gut signatures, which disperse the metabolites across regions and thus influences the flux distributions.A third one relates to the mode of administration of the desired compound to the system (e.g., twice-daily, three-weekly, etc., initially mixed in the medium or directly added) which affects the overall fluid dynamics.Thus, to monitor the evolution of the concentration of a metabolite of interest attributable to chemo-biological causes, it becomes necessary to control and understand the variations in concentrations inherent to the in vitro system used.Therefore, to better decipher whether the variation in concentration is solely due to fluid dynamics or also implies microbial metabolism, we established here for the first time a mathematical modeling framework for fluid dynamics in two commonly used in vitro gut systems: the PolyFermS and the SHIME®.The computational modeling has been validated with the use of an inert transit marker (blue dextran) added to common configurations of both systems.The purpose of this mathematical modeling approach is to assist users and scientists using gut fermentation systems in better understanding the behavior and outcomes of a treatment compound (TrC) when added to a dynamic in vitro fermentation system, by establishing whether the observed concentration variations in the dynamic in vitro fermentation systems result solely from fluid dynamics or are influenced by both fluid dynamics and microbial metabolism (time-dependent and/or gut region-dependent).To demonstrate the potential of our approach, we conducted a case study to interpret the results obtained from the addition of procyanidin A2 (ProA2) as a TrC to PolyfermS and SHIME® systems.

| MATERIALS AND METHODS
The PolyFermS, with two different modes of TrC addition, and the SHIME®, under two common configurations, were first modeled considering configurations and parameters used in our lab and frequently encountered in the literature.Then, the quality of the modeling was validated with blue dextran.Finally, models were applied, as a proof of concept, to interpret ProA2 metabolomic data generated in the two systems challenged with proanthocyanidins (PACs)-rich cranberry extract.

| Description of the gut fermentation in vitro systems modeled
The PolyFermS modeled in this work consists of a twostage system with an inoculum reactor (IR) feeding a test reactor (TR).In a PolyFermS, both reactors are operated in a continuous mode, that is, fed by a culture medium and emptied at the same rate.Since the treatment is, in this system, only added to the TR, the fate of an undegraded inert compound was modeled only in this vessel.Two approaches of addition of the TrC to the PolyFermS were modeled (Figure S1): (1) continuously, through the fed medium, thereafter named the continuous PolyFermS system, or (2), regularly, by direct addition to the vessel, thereafter named the discrete PolyFermS system.The continuous PolyFermS system is the most described and used approach to TrC addition.The discrete PolyFermS is used in our group to allow repeatable addition of nonautoclavable and insoluble TrC to the system.
The SHIME® consists of multiple-stage bioreactors operated in semi-continuous conditions (3 cycles of 8 h mimicking 3 meals/day).Many configurations of the system are available according to the gut regions of interest and the desired statistical power aimed.In this work, two configurations were studied: (i) a short configuration (Figure S2A) including a stomach/small intestine reactor (S/SI), feeding a proximal colon (PC) connected to a transverse colon bioreactor (TC); and (ii) a full configuration (Figure S2B) including a stomach reactor (ST) feeding an ileum (IL), followed by a proximal (PC), a transverse (TC), and a distal colon (DC).With the short configuration, the SHIME® system is commonly operated as follows: (1) the meal is added to the S/ SI reactor, (2) following digestion in the S/SI reactor, pumps entirely transfer the volume of the meal to the following vessel (PC), concomitantly an equivalent volume is transferred between all the vessels (PC → TC, TC → waste), (3) the system is kept in a static condition until the next meal.The transfer system in the full configuration is similar to the short configuration: the first vessel (ST) is emptied following a digestion period to the second vessel (IL), which is itself emptied to the third vessel (PC) with concomitant transfers of equivalent volumes between the last vessels (PC → TC, TC → DC, DC → waste) to maintain constant volumes in the last three vessels.For this work, we have modeled a common approach consisting of adding the TrC only to the first meal of the day, the two remaining daily meals being free from TrC.Since the SHIME® is a semi-continuous system alternating between short phases of liquid transfer between the reactors and a longer static phase, fluid dynamics have been modeled only during the transfer cycles.Also, modeling was conducted only on the colonic bioreactors normally inoculated with a microbiota (short configuration: PC and TC, full configuration: PC, TC, and DC) as no microbial metabolites are observed in the uninoculated vessels (S/SI and ST).In the full configuration, the IL was also not modeled, despite being inoculated with a microbiota, since this vessel is completely emptied during the transfer phase, hence there is neither dilution nor accumulation of the inert TrC in this bioreactor (i.e., no fluid dynamics).

| Mathematical modeling of the gut fermentation systems
To model the evolution of the concentration of a treatment compound added to these dynamic in vitro gut fermentation systems, we developed an ordinary differential equations (ODEs) model for each configuration of the PolyfermS (continuous and discrete approaches of TrC addition) and SHIME® systems (short and full configurations).ODEs were numerically simulated using the ode function from the deSolve package (1.33) in R (4.1.3),using R Studio (2022.07.2 + 576), with parameters commonly used in our lab and found in the literature for the PolyFermS 35,36 and the short and full SHIME® 25,37 (Tables 1-3, respectively).Equations of the treatment compound concentration depending on time (C(t)) within the continuous PolyFermS, the continuous section of the discrete PolyFermS, the global discrete PolyFermS, and the short configuration of the SHIME® system were analytically obtained by solving the ODEs (see Supplementary Material and Methods).Maximum TrC concentration ( lim t → ∞ C(t)) and the time to reach 99% of maximum TrC concentration (t 99% ) were obtained from the analytical equations for both PolyFermS systems (see Supplementary Material) or computed from the ODE models for the SHIME®.

| Blue dextran monitoring in uninoculated PolyFermS systems for model validation
To validate the models developed for the PolyFermS systems, blue dextran was used as an inert marker in uninoculated water-filled systems configured with the same commonly used parameters employed to generate the model.Briefly, reactors (Multifors 2, INFORS HT, Switzerland) were filled with sterile distilled water: the IR with a volume exceeding that necessary to complete the experiment, and the TR with precisely 250 mL as useful volume (Table 1).As the system remained uninoculated, the IR was not fed nor emptied during the experiment and the flow rates in the TR corresponded to those commonly used and modeled (Table 1, Figure 1A,B).To reproduce standard conditions, bioreactors were heated at 37°C, impellers maintained permanent agitation (80 RPM), and N 2 flowed continuously (40 mL/min).
To mimic a continuous addition of the treatment compound through the medium (continuous PolyFermS system), a 1000 ppm blue dextran solution was prepared in sterile distilled water and stirred for 20 min before being used to feed the TR (Figure 1A,C).This solution was kept at 4°C and agitated across the duration of the experiment.For the regular direct addition of blue dextran to the vessel (discrete PolyFermS system), a 50 000 ppm blue dextran stock solution was prepared in sterile distilled water, stirred 20 min before use, and conserved at 4°C under agitation for the duration of the experiment.Every 12 h, 5 mL of blue dextran stock solution was pipetted directly to the TR (Figure 1B,C).For both approaches, 1 mL samples were sterilely collected every hour for the first 12 h, then after 18, 24, 30, 36, 42, 48, and 72 h.For the discrete PolyFermS system, except for the first injection (0 h), samples were collected before and after the addition of blue dextran to the TR vessel (at 12, 24, 36, 48, and 60 h).
Samples were conserved at 4°C until blue dextran quantification.For each PolyFermS system, three different TR units have been tested.

| Blue dextran monitoring in an uninoculated SHIME® system for model validation
Validation of the model developed for the short configuration of the SHIME® (ProDigest, Belgium) was also carried out with blue dextran in a water-filled uninoculated system (Figure 1D) with the same commonly used parameters employed to simulate the model (Table 2).To do so, S/SI, PC, and TC were filled with sterile distilled water at volumes and hydraulic residence times commonly used and thus modeled (Table 2 and Figure 1D).
In the first cycle of the day, a refrigerated stock solution of blue dextran, prepared as described above, was added to S/SI containing 100 mL of sterile water to reach a final concentration of 500 ppm and was homogenized by stirring.The S/SI was then emptied; its content (100 mL) was transferred through the system at flow rates indicated in Table 2 and Figure 1D.Following the transfer, the reactors remained in a static state.The total cycle time was of 8 h, including the transfer period (~ 25 min, Figure 1E).The next 2 cycles of the day followed in the same manner but without blue dextran addition (Figure 1E).This sequence was repeated for 10 days.Throughout the experiment, the fermenter units were agitated with magnetic stirrers (300 rpm) and the temperature was maintained at 37°C.Samples were collected in a sterile manner every 8 h immediately after transfer completion and conserved at 4°C until blue dextran quantification.

| Fermentation of a PACs-rich cranberry extract in a discrete PolyFermS system
An experiment was conducted in an inoculated discrete PolyFermS system treated by regular direct addition of a PACs-rich cranberry extract.For this experiment, the IR was inoculated with a consortium of eight bacteria mimicking the intestinal microbiota and already described in Becker et al. 38 Bacteria were grown according to Becker et al. 38 before being immobilized on gelan/xanthan beads and added to the IR following the procedure and the proportion described in Cinquin et al. 39 Bioreactors were filled with sterile MacFarlane medium 40 : IR with precisely 500 mL and TR with precisely 250 mL.IR was maintained in a continuous system fed with MacFarlane medium at a flow rate of 41.6 mL/h, while TR was maintained as previously described (Table 1).Fermentation conditions corresponded to those used for the uninoculated experiment except that pH was maintained at 6.45 by automatic addition of NaOH and CO

| SHIME® with PACs in an inoculated model
The SHIME® was configured as previously described (Table 2, Figure 1D) and vessels were filled with the SHIME® medium and inoculated by the addition of fecal microbiota from healthy adults.Consent for fecal donation was obtained under registration number 2019-312 (Université Laval, Canada).Procedure of the SHIME® setting and fermentation process followed the approach described previously in Roussel et al. 25 To evaluate the ProA2 bacterial metabolism, the system was treated for 14 days with 3.83 mg of ProA2/day/colon by introducing the PACs-rich cranberry extract into the SHIME® stomach compartment along with one meal per day.The system was inoculated and samples were collected as previously described 8 h after the completion of the transfer containing the PACs-rich cranberry extract just before the following transfer.

| Blue dextran quantification, model fit, and ProA2 analysis
Blue dextran was quantified by spectrophotometry at 620 nm using a FLUOstar Omega microplate reader (BMG LABTECH, Germany) with a calibration curve of blue dextran prepared in sterile distilled water ranging from 10 to 1000 ppm.To assess the quality of the fit of the modeled data to the experimental ones (generated with the use of blue dextran), the coefficient of determination (r 2 ) was calculated using the function lm from the package stats (4.1.0)in R (4.1.3)using R Studio (2022.07.2 + 576).ProA2 was semi-quantified by ultraperformance liquid chromatography coupled with quadrupole time of flight mass spectrometer.The analysis was performed with an Acquity I-Class UPLC (Waters, Milford, MA) coupled with a Synapt G2-Si (Waters, Milford, MA) using a method described elsewhere. 41

| Continuous PolyFermS
In the continuous PolyFermS system, in which the treatment is added continuously through the feeding medium, the concentration of the treatment is modeled in a single reactor, the TR.Hence, the variation of the TrC concentration C in time can be expressed using one ODE: with a null initial concentration of TrC C(0) = 0 ppm.
The parameters of the model are summarized in Table 1.
Briefly, the first term represents the entry of the TrC in the bioreactor, with the TrC concentration C m in the medium feeding the TR at an inflow rate F, while the second term reflects the TrC exiting the TR through the outflow F to maintain a constant volume V in the bioreactor (Equation 1).Solving the ODE (see Supplementary Material, Section 1.1) leads to an exponential equation, describing the concentration of the TrC in the bioreactor as a function of time (Equation 2).
The theoretical maximum TrC concentration (reached when time tends to infinity ( lim t → ∞ C(t)), and the time t 99% needed to reach 99% of theoretical maximum TrC concentration were obtained from Equation ( 2) and are respectively given in Equations ( 3) and (4).

| Discrete PolyFermS
In the discrete PolyFermS, in which the TrC is periodically added directly to the TR, the general system consists of a semi-discrete model, that is, a model in which the dependent variable follows a continuous function before being regularly and abruptly modified by a discrete event. 42This model of PolyFermS is indeed characterized by a continuous washing of the TrC (its ODE corresponds to Equation 5) interrupted by the discrete addition of a new dose of TrC at regular intervals (Equation 6): with an initial concentration at the first continuous phase (i.e., for n = 0) corresponding to Solving the continuous part of the equation describes the dynamics of the TrC concentration with time in between new additions of TrC to the system and corresponds to the exponential Equation ( 7): In the discrete PolyFermS, when the TrC is directly added to the reactor, the continuous washing of the TrC (unique term in 5) is interrupted by the punctual and discrete addition of a new dose of TrC (Equation 6).The TrC concentration following the discrete events C t = n Δ t + is calculated by adding the TrC concentration introduced in the TR (second term in Equation 6) to the remaining concentration of TrC in the vessel immediately before the new TrC addition C(t = n Δ t).The general equation combining the continuous and discrete parts of the system can be obtained (see Supplementary Material and Methods, Section 1.2) and is given in Equation ( 8): with the index n being time-related with the floor function The concentration at the beginning of a cycle when n tends to infinity is given by Equation ( 9): The time to reach t 99% in the discrete PolyFermS is given by Equation ( 10): where n 99% is generated from Equations ( 8) and ( 9) as described in the Supplementary Material (Section 1.2) and corresponds to (1) ( (7) The short configuration of the SHIME® is composed of three compartments: S/SI, PC, and TC.TrC concentration dynamics in the PC (C PC (t)) and TC (C TC (t)) were modeled with the following set of ODEs: At the beginning of the experiment, the TrC concentrations in the proximal and transverse colons were null (C PC (0) = 0 ppm and C TC (0) = 0 ppm).The transfer of the S/SI content to the PC finishes before the concomitant transfer from the PC to the TC and from the TC to the waste as F S∕SI→PC > F PC→TC , and F PC→TC = F TC→W .This particularity is described by the condition added to the two first equations (Equations 12 and 13).The first equation (Equation 12) represents the emptying of the S/SI following meal intake three times per day.The flow F S∕SI→PC of this transfer is constant until the bioreactor is completely empty (V S∕SI = 0).Then, the flow rate F S∕SI→PC is to 0. The second equation (Equation 13) describes the variation in volume of the PC that corresponds to the volume entering the PC from the S/SI minus the volume exiting the PC toward the TC.Again, conditions are added to consider the differences in flow between F S∕SI→PC > F PC→TC .The last two (Equations 14 and 15) model, respectively, the concentration in the PC and the TC.As for the continuous PolyFermS system (Equation 1), the first term represents the entry of the treatment in the bioreactor, while the second term reflects the exit of the treatment to the next bioreactor or waste (W).As the treatment is added only once a day, the concentration in the S/SI is adjusted accordingly (with the index n and parameter R representing the number of meals per day, and the condition n mod R = 0 limiting the addition of treatment only to the first meal of each day, Equation 14).In Equation 15, V TC remains constant since F PC→TC = F TC→W , as previously mentioned.Finally, the simulation stops when the volume of the proximal colon (V PC ) reaches its initial volume, which corresponds to the end of the transfer cycle.The complex general equations of the system have been solved and are shown in Supplementary Material (Section 1.3).

| SHIME®: Full configuration
As for the short configuration, at the beginning of the experiment, TrC concentrations in the colonic compartments are null (C PC (0) = 0 ppm, C TC (0) = 0 ppm, and C DC (0) = 0 ppm).With the full configuration of the SHIME® system, the TrC concentration was modeled in the three distinct colon regions (PC, TC, and DC).As stated before, the concentration was neither modeled in the ST nor in the IL because these compartments are entirely transferred to the proximal colon at each cycle and there is no dilution or accumulation of the treatment in these bioreactors.The model consists of the following set of ODEs.
As for the PolyFermS systems and for the short configuration of the SHIME® system, each equation depicts the dynamics of the concentration of the treatment in each bioreactor, with the first term representing the inflow from the previous bioreactor, and the second term the outflow to the next bioreactor.As for the previous short (11) configuration system, a condition is added to the concentration of the treatment in the ileum (C IL , see Equation 16), since the treatment is added in only one of the three meals per day (with the index n and the parameter R) and the simulation stops at the end of the transfer cycle.

| Validation of the ODE models with blue dextran
To validate the models generated, experiments using blue dextran as a transit marker were conducted in uninoculated water filled PolyFermS (for both approaches of TrC addition) and short configuration SHIME® systems.The quality of model fit was evaluated with the r 2 , which represents the proportion of variation in the experimental data explained by the model.Within the PolyFermS, the r 2 between the experimental and theoretical points was 0.998 and 0.959 for the continuous (Figure 2A) and discrete approaches of TrC addition (Figure 2B), respectively.For the two systems, we calculated the theoretical maximum concentration (reached when time tends to infinity, represented by the horizontal dashed line in Figure and the time needed to reach 99% of this theoretical maximum concentration (represented by the vertical dashed line in Figure 2).For the short configuration SHIME® system, the r 2 was calculated within each colonic compartment, PC and TC, and was 0.861 and 0.959, respectively (Figure 3).As for the PolyFermS, the theoretical maximum concentration and the time needed to reach 99% of this theoretical maximum concentration were calculated and are represented in Figure 3 by dashed lines.Due to the complexity of the equations of the SHIME® containing an exponential integral function, simulated values were obtained computationally.

| Simulation of the full SHIME® configuration ODE model
The ODEs generated to model the full SHIME® system have been simulated computationally and the fate of an inert marker in an uninoculated system was plotted for each of the three colonic compartments (PC: proximal colon, TC: transverse colon, DC: distal colon, Figure 4).

| Application of the ODE to the interpretation of metabolomics data: the PACs-rich cranberry extract experiments
Synthetic microbiota and fecal suspensions were, respectively, inoculated and stabilized in a discrete  PolyFermS and a short configuration system.Then the systems were supplemented with a PACs-rich cranberry extract.The relative abundance of ProA2 (area under the peak), an A-type procyanidin dimer particularly abundant in cranberry and degraded by the gut microbiota, 43,44 was monitored in the systems' vessels.To use the models with the most commonly employed metabolomic data in the field, we performed semi-quantification based on untargeted metabolomic results, and relative concentrations were used in order to compare the model to the experimental results.Then, the ProA2 experimental relative concentration was compared to the theoretical concentration calculated by the systems' models to assess whether its variation in time was solely due to the fluid dynamics within the in vitro systems or if ProA2 metabolism by the gut microbiota could be observed.For the discrete PolyFermS (Figure 5A), the simulated and experimental relative concentrations reached in the TR directly following the first addition (t = 0 h) were both set to correspond to a relative concentration of 100%, in order to compare simulated and experimental data on the same scale.In this system, the experimental ProA2 relative concentrations were globally lower than the simulated relative concentrations (Figure 5A).We observed a sharp decrease in ProA2 experimental concentration in the first hour followed by a series of fluctuations leading to a global increase between 2 h and 6 h.Between 6 and 12 h, the experimental concentration decreased slightly more compared with the simulated one and then the increase and decrease rates remained similar.For the SHIME® (Figure 5B) system, the experimental and simulated relative concentrations reached in the PC right after the first transfer were set to 100%.In the PC, the ProA2 experimental relative concentration remained under 100% for the duration of the experiment and remained close to 0% within the TC, indicating, again, that the concentration variation was due to the metabolism by the gut microbiota.The PAC experiments were conducted as a proof of concept to demonstrate the usefulness of our model in metabolite analyses.However, it is important to note that no biological replicates were carried out.Therefore, the results should be interpreted with caution.

| DISCUSSION
In the last few years, a growing interest in the metabolic capacity of the gut microbiota to degrade unabsorbable dietary compounds has emerged.A demonstration of this interest is the shift from the debatable concept of enterotype, which classifies and groups gut microbiota from individuals according to their microbial composition, 45  to the concept of metabotype, stratifies individuals within the population according to the specific metabolites produced by their gut microbiota, hence taking into account gut microbial composition and functions. 46n vitro fermentation systems, such as the PolyFermS and the SHIME®, are useful systems to study the metabolic capacity of the gut microbiota, allowing to overcome problems associated with in vivo studies on humans, such as the invasive sampling methods and the ethics considerations.Moreover, these dynamic systems allow for real-time tracking of the metabolism of dietary compounds by the gut microbiota across various segments of the gastrointestinal tract, effectively simulating the human digestive process.It is, however, important to remind that most of these systems predominantly emphasize the interplay between microbiota and treatment compounds, while overlooking the input from the host.Still, these dynamic systems permit to easily monitor, in real time, the metabolism of dietary compounds by the gut microbiota in the different sections of the gastrointestinal tract, and thus to mimic the human digestive system.However, the interpretation of the metabolomic data is much more complex than for static systems, since the concentration of the treatment compound varies due to both the fluid dynamics (treatment injection, dilution, accumulation, etc.) and the metabolism by gut microorganisms.Hence, to discriminate between these two processes and to correctly interpret the results, it is important to isolate the fluid dynamics effect of the system by itself on the TrC concentration in the absence of microbial metabolization.The model curves generated can then be used as a baseline to evidence and determine if there is a contribution of microbial metabolism on the fate of molecules.In this study, two modes of addition of the treatment compound into the PolyFermS and a short configuration of the SHIME® were modeled, and experimentally cross-validated using blue dextran as an inert marker before being tested in full-set systems with PACs as TrC.
The PolyFermS systems are simpler than the SHIME® systems since the treated bioreactor (TR) is continuously fed and nonsegmented.The ODE systems generated for the PolyFermS were solved, general equations of the systems were generated, and important differences were observed between the two modes of TrC addition.In the continuous PolyFermS, TrC accumulation in the vessel evolved according to a strictly increasing function asymptotically tending to a stable concentration of TrC (Equation 2) corresponding to the TrC concentration in the feeding medium (Equation 3).When the TrC was repeatedly added directly to the discrete PolyFermS system, the continuous phase of the general semi-discrete model describing the TrC concentration (Equation 7) corresponded to an exponential decay function interrupted by the cyclic additions of TrC.For this general model of  9).It is interesting to note that, for the same operating parameters and an equivalent TrC concentration in the culture medium or in the vessel after direct addition (~1000 ppm in our blue dextran experiment), the TrC reaches t 99% faster when the compound was directly added than when it was added through the medium (Figure 2A,B).This elevated was however temporary since, when the TrC was directly added, the washout induced by the continuous flow of feeding medium and waste reduced the TrC concentration.Therefore, for a cycle time of 12 h, in the discrete PolyFermS modeled, the TrC minimum concentration reached at the end of every cycle (immediately before a new addition of TrC) was systematically lower when compared with the TrC concentration at the same time point in the continuous PolyFermS.This highlights the significant cyclic variations in the concentration of TrC to which microorganisms are exposed within a discrete PolyFermS.Important and regular variations of nutrients 47 or antibiotics 48 have been shown to affect bacterial growth and physiology.In the PolyFermS, the choice of the modality of addition is therefore able to modify the behavior of the bacterial community studied.
For the SHIME®, solving the sets of ODEs generated for each compartment led to complex functions.In the common systems modeled, as the TrC was added to only one of the three daily meals, its washout between different additions to the system was observed in every vessel (Figures 3 and 4).The amplitude of TrC concentration variation was especially high -but decreasing with time -in the first compartment (PC), since it is the first to receive the TrC from the S/SI compared to the other compartments (TC and DC).Hence, the microbiota would be subject to greater variation of TrC concentration in the PC than in the following compartments (TC and DC), where the bacterial community face a more gradual TrC concentration increase.Further, as expected, the maximum concentration was higher and was reached faster in the first compartment (PC) than in the second compartment (TC).
To our knowledge, few studies have attempted to model the fate of a TrC within a PolyFermS or a SHIME®.In a study designed to evaluate the persistence of bacteriophages in a SHIME® system similar to the full SHIME® presented here, Verthé et al. 49 calculated the theoretical dilution of inert particles in each of the three colonic vessels in order to monitor phages population.As this previous study added a single dose of phage directly to the first colonic vessel (the PC in the full SHIME® presented in this study) and only on the first day of the experiment, they applied to all the three colonic vessels exponential decay function: [X ] = X 0 × e −kt , where X 0 is the initial concentration and k is a constant representing the flow of the feed rate/volume of the vessel.The same function has also been used by Fernandez et al. 50to evaluate the bacterial population in a single-stage fermenter with immobilized cells as in the PolyFermS system.Again, this function was modeling the fate of an inert molecule added once at the beginning of the experience.This negative exponential function corresponds to the one calculated here for the continuous phase observed in the discrete PolyFermS (Equation 7) and would describe the model adequately if no TrC addition were performed.However, this single function can neither be applied to describe discrete systems in which the concentration of TrC is abruptly modified by its addition nor for systems in which doses of TrC are continuously added and thus accumulating.Furthermore, it can hardly account for the concomitant transfer characterizing the SHIME® systems, for which an increase in TrC concentration is expected in the colonic vessels following the first transfers.Therefore, this model is not appropriate for both the complex SHIME® and PolyFermS dynamics systems presented in this study.Since food components are ingested regularly, frequent addition of the TrC to in vitro systems is likely to better represent the fate of food components, the metabolites generated, and their long-term impact on the microbiota.Adaptation of the modeling to this increased complexity is thus required to perform meaningful metabolomics studies, in order to fully understand whether the concentration changes are caused by the metabolism of the gut microbiota, or only by the fluid dynamics of the system.Recently, Li et al. 51 have modeled the accumulation of catechin in the short SHIME® system fed up with catechin up to three times a day for 35 days but have limited their analysis to the PC.ODE systems generated in this work have allowed to conduct analytical and simulation analyses seizing the complexity generated by either repeated TrC addition and the numerous vessels of a multi-segmented system.
Blue dextran has been identified as one of the best inert markers to use in bioreactor modeling 52,53 and has already been used to follow and model the intestinal transit in humans 54 as well as in upper in vitro digestive systems (namely, the TNO Gastro-Intestinal Model (TIM) 55,56 ) and in other simple in vitro intestinal bioreactors. 57Here we used it for the first time to validate fluid dynamics in complex PolyFermS and SHIME® systems.
For the PolyFermS, the quality of the fit of the experimental data-generated with the addition blue dextran to the systems)-to the simulated data was excellent, with a r 2 of 0.998 for the continuous PolyFermS and of 0.959 for the discrete PolyFermS (Figure 2).Hence, for the two systems, the modeling explained more than 98% of the variation, confirming that the models accurately depict the experimental systems.
For the short configuration SHIME® system, the r 2 calculated in the PC and TC was 0.861 and 0.959, respectively (Figure 3), thus slightly lower than those obtained for the PolyFermS.These differences are probably the result of an increased error potential induced by the complexity of the systems: multiple pumps transferring liquid volumes from many compartments might have accumulated slight variations amplified over the longer experience duration (3 days vs 10 days).proportion of the variation in the experimental data (>85%) is explained by the models, thus validating our proposed models.Since our models have been experimentally validated, we developed and simulated a model for a full configuration of the SHIME® system, demonstrating the strength and the polyvalence of this approach (Figure 4).
Finally, to illustrate the purpose of the modeling framework, we used the models to interpret metabolomics data obtained from the fermentation of a PACs-rich cranberry extract in the discrete PolyFermS system inoculated with a synthetic gut microbiota, and in the short configuration SHIME® system inoculated with human gut microbiota.While reaching the colon, (poly)phenols such as PACs are catabolized into smaller bioavailable and bioactive metabolites, considered key markers of human health, 46,58 and proposed as potential prebiotics. 59The concept of prebiotics relies on the capacity of the gut microbiota to use nutritionally specific dietary compounds to confer positive health effects. 60Hence, the catabolism of these molecules by the microbiota is essential in the demonstration of this potential.In the PolyFermS system, the experimental relative concentration of ProA2 was lower than the simulated relative concentration of an inert compound for all the experiments, indicating that the variation in concentration was not solely due to the fluid dynamics (Figure 5A).Since the experimental concentration was lower than expected with the model, we can conclude that the TrC, in this case, ProA2, was metabolized by the synthetic gut microbiota.The sharp decrease in the first hour of the fermentation indicates that a large portion of the treatment is metabolized very quickly.The following increase in concentration can be explained by the catabolism of bigger PACs molecules present in the PACs-rich cranberry extract into ProA2.Between 6 and 12 h, slight decrease in concentration compared with the model was associated with further degradation of the ProA2 in addition to its dilution.In addition, as ProA2 can be released by catabolism of larger PACs, ProA2 acts as a metabolite and a substrate.Hence, its metabolism can be masked by its production.Since we did not collect samples in the next hours following the subsequent addition of the PACs-rich cranberry extract, we could not determine if the same early sharp decrease was reproduced.Nevertheless, the measured concentration decreased more than the simulated one between 6 and 12 h after each addition, indicating that ProA2 was metabolized during the entire experiment.For the SHIME® system, ProA2 was extensively metabolized by the gut microbiota in the PC, since the concentration after 8 h of fermentation was under 100% every day (Figure 5B).As for the TC, only very low concentrations were detected all along the experiment.However, since the samples were collected right after the transfer of the meal containing the PACs-rich cranberry extract to the PC, only a small proportion of the TrC went directly to the TC.Hence, the concentration measured at this time is mainly the residual concentration from the previous transfers.Thus, we can conclude that the metabolism of ProA2 by gut microbiota was almost complete and there was no accumulation in the PC, nor in the TC.Some considerations should be taken into account when using models to analyze the fate of an added TrC to a bioreactor.First, it is difficult to account for the variation in volume resulting from the addition of acid and base added to the system to maintain the pH at biological values.Depending on the compounds added to the bioreactor, and owing to the biological activity of intestinal microbiota, these variations can be important and are able to induce significant shifts from the model.Modern fermentation systems generally allow to calculate in real time the volume of acid and base added to the system and thus to adjust the calculations if these volumes should prove to be significant.However, volume variations due to pH control are inconsistent during the course of the fermentation and acid/base additions are thus difficult to integrate to a system of ODEs and to compute.Alternatively, a good knowledge of the pH variations generated by the microbiota in the presence of the compounds added would allow to prepare an acid or base solution of appropriate concentration making it possible to reduce the volumes added to the vessels and thus the variations from the model.Another variation on the system volumes capable of affecting the model resides in the additions of the TrC as well as sampling volumes and recurrence.Indeed, an important volume of liquid TrC added at high frequency directly to the bioreactor will affect the model and its nature.The same is true for the sampling events that can recurrently subtract a significant part of the vessel content and thus affect its volume and the general nature of the model.When possible, as it is for the PolyFermS for which every vessel produces an outflow, sampling can be done by collecting effluent, thus without affecting the bioreactor volume. 14In any case, as TrC addition and sampling-induced variations are related to the experimental design and are well controlled, they can easily be taken in account in the ODEs and be integrated to the numerical simulations.Caution should also be used when interpreting the data as, in addition to microbial degradation, events of chelation and physicochemical degradation can be involved and contribute to the deviation from the model; appropriate controls should be used.Further, it is to be noted that modeling was conducted for an undegraded TrC and therefore tested in uninoculated models.It therefore constitutes a baseline of inert fluid dynamics of which a deviation indicates a degradation of any kind.This has to be taken into account especially for the model curves obtained for the SHIME® TC DC that depend on the concentration in the PC.Important TrC degradation in the PC would necessarily have an impact on the TrC concentration in the following vessels.
Despite its inherent limitations, the modeling of fluid dynamics in different in vitro fermentation systems constitutes a useful tool to improve the interpretation of metabolomics data from these complex systems.In addition to the interpretation of the metabolomics data throughout the fermentation process, the modeling can be used to understand the effects of parameter changes on the fluid dynamics, such as the flow, the volume of the reactors, the time between two additions of TrC, etc.Ultimately, parameters specific to bacterial metabolism and to the abundance of microbial species involved in TrC degradation could be integrated into models in order to gather new knowledge on the trophic dynamics of compounds degradation and metabolite appearance.

| CONCLUSION
In vitro gut fermentation systems have demonstrated their importance in the study of the intestinal microbiota.At the same time, metabolomics approaches are emerging as an essential method to understand the intricate relationships existing between the microbiota and the fermentable compounds reaching the gut.The quality of the metabolomics analysis and the conclusions drawn rely, however, on the comprehensive understanding of the system fluid dynamics.The present study modeled the fate of an inert compound within two commonly used in vitro gut fermentation systems: the SHIME® and the PolyFermS; relying on ODE systems of equations provided a solid ground to improve the analysis of metabolite fate and flux in such systems.Indeed, utilizing the models outlined in this study allows one to readily determine if a compound introduced to the fermentation systems has undergone metabolism or if the concentration shifts are purely a result of fluid dynamics.It is, however, essential to understand the particularity of every experimental system to adjust the model accordingly.

T A B L E 1 aF I G U R E 1
Notations and definitions used in the PolyFermS models.Initial concentration of the TrC in the TR at the beginning of a continuous phase in the discrete PolyFermS (corresponds to C t = n Δ t + ), after the nth TrC addition -μg/mL (ppm) n Index of the direct additions of TrC to the reactor (TR) (the discrete event) --Δ t + Time directly after the addition of the treatment compound to the reactor (TR) Interval of time between each addition of the treatment compound to the reactor (TR) 12 h This concentration is obtained by considering the blue dextran concentration in the feeding medium (1000 ppm) entering the TR at 18.7 mL/h and the continuous addition of water (diluting the blue dextran contained in the medium) from IR entering TR at 2,1 mL/h.b General inflow F is calculated by adding the inflow of the medium entering TR to the inflow of the water from IR entering TR.Schematic description of the water-filled and uninoculated systems used to validate the model.The common parameters used to simulate the systems' models and for the validation (flows, vessels volumes, and time scales) as well as the blue dextran (BD) concentrations used are represented.(A) Continuous PolyFermS, (B) discrete PolyFermS, (C) daily time scale for both continuous and discrete PolyFermS systems, (D) SHIME® short configuration, (E) time scale for the SHIME®.IR, inoculum reactor; PC, proximal colon; S/SI, stomach/small intestine; TC, transverse colon; TR, test reactor.

T A B L E 2 nT A B L E 3
Notations and definitions used in the model for the short configuration of the SHIME® system.Index of the meal added to the S ∕ SI --Parameters V TC Volume of the transverse colon 400 mL C S∕SI Concentration of the treatment compound in the stomach/small intestine 500 or 0 μg/mL (ppm) F S∕SI→PC Flow rate for the transfer of the stomach/small intestine to the proximal colon 4 mL/min F PC→TC Flow rate for the transfer of the proximal colon to the transverse colon 3 mL/min F TC→W Flow rate for the transfer of the transverse colon to the waste 3 mL/min R Number of meals per day 3 -Notations and definitions used in the model for the full configuration of the SHIME® system.

F I G U R E 2
Representation of the TrC concentration obtained experimentally (with blue dextran, blue line) or simulated (black line) within the TR of A) the continuous or B) the discrete PolyFermS.Horizontal and vertical dashed lines indicate lim t → ∞ C(t) and t 99% , respectively.r 2 indicates the coefficient of determination calculated between experimental and simulated data.

F I G U R E 3
Representation of the experimental (blue dextran, blue line) and simulated (black line) data of TrC concentrations in the two colonic compartments of the short configuration SHIME® system.Horizontal and vertical dashed lines indicate lim t → ∞ C(t) and t 99% , respectively.r 2 indicates the coefficient of determination calculated between the experimental and the simulated data.PC, proximal colon; TC, transverse colon.

| 13 of 18 LESSARD
-LORD et addition to the discrete PolyFermS, initial TrC concentration at each new cycle increases before reaching stable maximum (Equation

F I G U R E 5
Experimental and simulated relative concentrations of ProA2 in A) the PolyFermS and B) the two colonic compartments of the short SHIME® system.Dashed lines represent direct PACs-rich cranberry extract (every 12 h) to the TR of the discrete PolyFermS.PC, proximal colon, TC, transverse colon.