INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

ACID SECRETION FROM PARIETAL cells in the stomach is a highly regulated, complex, and dynamic process optimized to facilitate food digestion. Not only are there interactions between the central and enteric nervous systems (CNS and ENS, respectively), but a complex network of paracrine and endocrine cells is also involved. The overall goal is maintenance of stomach luminal pH within a strict range (i.e., pH homeostasis); food consumption and other deviations altering this range increase or decrease acid release.

Four cell populations and their secreted products form the core acid secretory process in humans. These four cell populations are gastrin (G)- and somatostatin (D)-secreting cells, enterochromaffin-like (ECL) cells, and parietal cells. G and ECL cell products stimulate acid secretion (positive feedback); D cells inhibit acid release (negative feedback). Inconsistencies in the integration of the feedback loops exist and may be due to species-specific differences. For example, the ability of gastrin to directly stimulate acid release in some species is in dispute (5, 96). Other inconsistencies involving effector and acid regulation may relate to the experimental approach, such as in vitro vs. in vivo studies. The basic requirement for acid secretion, however, appears to be conserved among the species (22, 61). The need for an integrative approach to study gastric acid secretion is clear. Mathematical modeling is a powerful tool that allows for exploration of the integrated system and its components in a systematic fashion. Furthermore, mathematical modeling is immune to inconsistencies that often arise from comparison of in vivo and in vitro studies.

Mathematical models based on acid secretion have appeared in the literature (17, 20, 58). De Beus et al. (17) explored the coupling of gastric acid release to bicarbonate secretion through extensive mathematical analyses of the cascade of molecular and ionic events constituting acid secretion. Licko and Ekblad (58) focused on gastric acid as a two-step, sequential process. They modeled formation of acid that contributes to an acid storage pool and translocation of stored acid into the lumen of the stomach. Both models provide insight into acid secretion but do not address regulation of acid secretion.

In this study, we use a novel mathematical model to describe the complex system of gastric acid regulation. Our model is unique, because we consider regulatory processes that have been identified experimentally as essential for proper maintenance of acid secretion. Our two main goals are to validate the model and perform new experiments. Validation involves comparing simulations during healthy and depletion situations with experimental data. The model can then be used to perform studies not yet experimentally reproducible. For example, we are able to preferentially deplete antral or corpus somatostatin.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

The stomach consists of many histologically distinct regions (Fig. 1A); however, we simplify the model by describing two main compartments: the antrum and corpus regions (Fig. 1B). The relevant biological processes affecting acid secretion occur here and include dynamic changes in cell populations; secretion of effectors, neurotransmitters, and acid; and release of gastric protective factors. We outline the components of the model and our assumptions; the mathematical details are described in the APPENDIX.

View larger version (19K): [in this window] [in a new window]   Fig. 1.   A: histology of the stomach. B: model diagram of effector regulation of gastric acid secretion. Model includes positive and negative effector feedback systems. Cells are assigned to their respective compartments. G cells in the antrum secrete gastrin (Gtn), an effector of gastric acid secretion. Gtn not only stimulates histamine (Hist) release from enterochromaffin-like (ECL) cells (EC) and gastric acid (H+) secretion from parietal cells (PC) but also stimulates somatostatin (SS) secretion. Greek symbols represent rates at which events occur: , transport rate; , death rate; A, washout rate of acid with gastric emptying. Also shown are central and enteric neural stimuli (CNS and ENS, respectively) supplied to the physiological system on feeding. Solid arrows, positive stimuli; dashed arrows, negative stimuli. Weight of arrows indicates relative intensity of stimulus.

Model

Cellular elements. The key cells involved in the model are found in gastric glands of the antrum and corpus (Fig. 2). Several studies have outlined gastric paracrine, endocrine, and exocrine cell development (40-47). These electron microscopy studies show that the lineage of these differentiated cells can be traced to undifferentiated stem cells abundant in the isthmus of gastric glands (41). Cells arising from stem cells are terminally differentiated and, with the exception of ECL cells, do not undergo mitosis (102, 103). We monitor seven cell populations in our model (Fig. 3): antral and corpus stem cells, antral G cells, antral and corpus D cells, ECL cells, and corpus parietal cells.

View larger version (51K): [in this window] [in a new window]   Fig. 2.   Cross-sectional and schematic longitudinal section of a gastric gland. A: stained parietal cells in corpus region of a human stomach. Ellipse highlights a cross-sectional view of a typical corpus gland. B: schematic view of longitudinal section of a gastric gland; cross-sectional view fails to capture 3-dimensional nature of gland. It is assumed that the human gland is long and tubular. However, species-specific variations occur in arrangements of glands.

View larger version (24K): [in this window] [in a new window]   Fig. 3.   Ontogeny of G, D, ECL, and parietal cells from underlying antral (Asc) and corpus (Csc) stem cells. A: differentiation within antrum. B: differentiation in corpus region. Dashed lines, feedback control of stem cell differentiation [p(t)]. Alphanumeric and Greek symbols represent rates at which respective processes occur:  and , growth and death rates; T, differentiation of respective stem cells.

Feeding function. We model a standard American diet of three meals a day (at 0600, 1200, and 1800) using a sinusoidal function to describe the volume of food consumed (Fig. 4). The volume of food increases with each successive meal during the day and ranges between 0.0 and 1.0 liter, with 1.5 liters being the maximal capacity of the stomach. We rigorously test the model with other feeding functions in which the feeding intervals are varied (data not shown). The response is appropriate: there is strong correlation between the modality pattern of the feeding function and the effector, bicarbonate, and acid responses. This is critical for optimal food digestion.

View larger version (14K): [in this window] [in a new window]   Fig. 4.   Food input to virtual human gastric system. B, Ln, and D, breakfast (0700), lunch (1300), and dinner (1900), respectively. Amplitude of peak at each meal represents volume of food intake.

Neural elements. ENS and CNS neurotransmitters are secreted in response to food volume. There is a lack of kinetic data describing the influence of food volume on neural activity; we assume that neural activity increases in a Michaelis-Menten manner with food volume. Neuropeptides are metabolically degraded, and this degradation is governed by first-order kinetics (see APPENDIX). Food stimulates CNS activity, which is conducted via the vagus nerve to the ENS of the stomach, resulting in acid and gastrin release (16). Indirect action of the CNS on acid secretion has also been demonstrated; cholinergic neurotransmitters inhibit somatostatin secretion, promoting acid release (16, 73, 91). The physical (degree of distension) and biochemical (pH, neural, and effector concentrations) states of the stomach then feed back to the CNS, modulating its response (94, 95) (Fig. 1B).

Effector regulation of acid secretion. When food enters the lumen, alterations in stomach pH and volume, together with neural stimulation, lead to acid secretion. G cells within the antrum secrete gastrin, which is released into antral blood capillaries and diffuses into the corpus (Fig. 1B). In the corpus, gastrin directly stimulates parietal cells to secrete gastric acid (107) and stimulates ECL cells to release histamine in conjunction with ENS neurotransmitters (48, 82). Histamine acts in a paracrine manner in conjunction with gastrin and acetylcholine, enhancing acid secretion (16, 59), and also potentiates gastrin stimulation of parietal cells (96). To downregulate these processes in the antrum and corpus, D cells secrete somatostatin, a negative effector of gastric acid secretion (14, 63, 92). Gastrin, somatostatin, and histamine are released in a dose-dependent manner on appropriate stimulation (7, 54, 67). In addition to effectors described above, gastric acid secretion by parietal cells can also be directly stimulated by CNS activity in response to food, although not to the same extent (72).

Gastric protection. Gastric acid is corrosive to host cells; thus, to protect epithelia, bicarbonate ions are released into the mucus layer. Bicarbonate ions buffer secreted acid and increase pH at the mucus-epithelial interface to tolerable levels (78). Although gastric protection is important, it is not thoroughly described in our model, except to correct for acid levels buffered by bicarbonate ions (see APPENDIX).

Experiments

Parameter estimation. Once the model is developed and before simulations are performed, rates of each of the processes outlined in Figs. 1B and 3 must be estimated. Rate parameters are estimated from published experimental data and are presented in Table 1. Human-derived experimental data are used in estimations when possible. Animal data are used when no human data are available to derive magnitude estimates. In the absence of data, mathematical estimation is used. All parameters are evaluated using uncertainty analyses performed with C code based on Latin hypercube sampling (LHS) (9, 37, 38). To estimate cell population numbers, we perform immunohistochemistry on representative cross sections of human stomach mucosa (see below; see APPENDIX for details of parameter estimation).

Uncertainty and sensitivity analyses. There are variances in many of the parameter values due to extensive variability in data. Such variances require an evaluation of the uncertainty in the system. We employ the LHS method to assess effects of uncertainties in our parameter estimation on model outcomes. LHS allows for simultaneous random, evenly distributed sampling of each parameter within a defined range. A matrix consisting of m columns corresponding to the number of varied parameters and n rows for the number of simulations is generated; n solutions are generated that show uncertainty in model outcomes due to parameter variations. For our uncertainty analyses, we run 20 short-term simulations (18 degrees of freedom; 24 h) varying a given parameter by a factor of 1,000. This is repeated for each parameter in the system individually and in combination.

Immunohistochemistry. We obtained archived cross-sectional, biopsy, or surgical specimens of human stomach mucosa from individuals participating in a study on Helicobacter pylori colonization (University of Michigan Institution Review Board IRBMED no. 1999-0708). Biopsies were obtained from healthy regions of the stomach, and samples were fixed in 4% paraformaldehyde-PBS and embedded in paraffin. Sections were deparaffinized through an alcohol series and permeabilized in 3% H₂O₂ and 100% ethanol. Nonspecific binding sites were blocked with 20% goat serum-PBS and 0.1% Triton X-100 for 30 min before 2 h of incubation with a 1:200 dilution of rabbit anti-gastrin-releasing peptide (GRP) antibody specific for G cells or mouse anti-H⁺-K⁺-ATPase -subunit antibody (Medical and Biological Laboratories) specific for parietal cells. We incubated the samples in a 1:500 dilution of secondary anti-rabbit or anti-mouse IgG antibody for 30 min to conjugate secondary antibodies and achieved visualization in avidin-biotin complexes using the Vectastain Elite ABC kit and diaminobenzidine for substrate (Vector Laboratories, Burlingame, CA). Sections were also counterstained with hematoxylin and eosin. The stained cross sections were morphometrically analyzed by random selection of fields from which averages of each gastric cell type per gland could be assessed.

Computer simulations. Once we define the model and estimate parameters, we solve the system of ordinary differential equations (ODEs) to obtain temporal dynamics for each variable in our model. To this end, we use appropriate numerical methods for solving the system of ODEs over 24-h (short-term) and 5-day (long-term) periods. We use MatLab's ode15s solver for stiff systems (Math Works, Natick, MA) and compare results with those generated from a numerical algorithm using C code of a stiff adaptive solver based on the method of Rosenbrock and Storey (87) for consistency. Simulation results are also compared with similar experimental data.

Virtual depletions. To further validate our model, we perform virtual depletion experiments of different effectors of gastric acid secretion. We define our simulations as depletion experiments, because we set the appropriate variables (effectors) to be depleted to zero over a specific time frame. The system begins in steady state with wild-type conditions before depletion. An example of a depletion experiment involves neutralization of somatostatin. In contrast, in deletion experiments, a gene is disrupted or deleted; thus the system starts in a condition different from wild type. Virtual depletions are performed for gastrin, histamine, and somatostatin by using numerical methods and initial conditions (see above). We compare our results with published experimental depletion and deletion data. We then demonstrate the application of the model to address questions that are not easily addressed experimentally. For example, we independently deplete somatostatin in the corpus and antrum regions. Student's t-tests are used to evaluate significant differences between our virtual depletion simulations and appropriate controls.

Virtual ECL cell proliferation. To assess the role of gastrin in ECL cell proliferation, we induce prolonged hypergastrinemia (5 days) by increasing the maximal gastrin secretion rate due to CNS stimulation (K_NG1). The three curves are each fitted to a generic quadratic form (e.g., y = c + a₁t + a₂t², where y represents the number of ECL cells, c and t are the initial number of ECL cells and time, respectively, and a₁ and a₂ are parameters that we estimated). Because the confidence intervals for the three curves do not overlap, their trajectories are significantly different.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

We perform simulations that can be divided into two categories: 1) those designed to validate the model and 2) those that assess the importance of gastrin. We conduct simulations under normal conditions, and these serve as controls for subsequent simulated experiments. Simulations are performed over short-term (24-h) and long-term (5-day) time scales. Unless otherwise specified, we use the parameter values listed in Table 1. We first present baseline simulations comparing our results with published data and subsequently perform a series of virtual depletion experiments. Our simulation results represent effector levels in the extracellular spaces of the stomach mucosa. Experimentally, effector levels are typically determined from blood plasma measurements. Thus, in some cases, our simulation values are larger than those from published data and are accounted for by compartmental differences.

Baseline Conditions

View larger version (38K): [in this window] [in a new window]   Fig. 5.   Baseline simulations of effectors and gastric acid. A: simulation of CNS activity. B: simulation of ENS activity. C: virtual simulation of changes in plasma gastrin concentration due to release of gastrin by G cells. D: simulated gastrin release is in agreement with published plasma data (98). Upper and lower bounds of gastrin concentration are shown. E: simulated total somatostatin released by antral and corpus D cells. F: simulated somatostatin release (E) is consistent with experimental results (10). G: simulated histamine release from ECL cells. No human data on histamine diurnal changes have been reported. H: simulated gastric acid in corpus is consistent with experimental data from Feldman and Richardson (21) (I) showing upper and lower bounds of gastric acid. J: reciprocal behavior of gastrin and somatostatin (SOM) at 0700, 1300, and 1900.

Cell population sizes remain consistent over 24 h (Fig. 6), and this finding is consistent with biological evidence (62). Although cell numbers are in homeostasis during the short term, we capture changes in cell numbers in the long term (data not shown), and we show cell dynamics under altered conditions (see below).

View larger version (15K): [in this window] [in a new window]   Fig. 6.   Simulated cell populations. Cell population numbers remain in homeostasis over the 24-h time course. A: stem cells in corpus and antrum. B: antral endocrine cells include G and D cells. C: cells in corpus include D, ECL, and parietal cells.

Virtual Depletion Experiments

Gastrin depletion. Consistent with published data, we show a significant reduction of basal and stimulated acid secretion (P < 0.001; Fig. 7) (24, 34, 106). We suggest that this reduction is due to a decline in the secretion of downstream effectors (Fig. 7). In the long term, we also observe a decline in ECL cell populations that is consistent with similar experimental observations (24) (data not shown). This indicates a critical role for gastrin in maintenance of cell homeostasis.

View larger version (26K): [in this window] [in a new window]   Fig. 7.   Virtual depletion of gastrin. A: simulated gastrin. B: during gastrin depletion, somatostatin levels are lowered. C and D: histamine levels are reduced because of lack of gastrin stimulation and levels of gastric acid, respectively. Dashed lines, control; solid lines, histamine depletion. *P < 0.05; **P < 0.001.

Histamine depletion. During histamine depletion, we observe a significant reduction in acid levels (P < 0.001; Fig. 8), in agreement with experimental data (51). This reduced output is less dramatic than that observed during gastrin depletion (Fig. 7). Consistent with previously published data (51), basal acid levels are unaffected (Fig. 8). Gastrin is significantly elevated by 25% over wild-type conditions (P < 0.001). This is not as significant as the 300% increase in gastrin reported in mice with dysfunctional histamine receptors (51); however, this discrepancy is likely due to species-specific differences. In mice with deleted histamine receptors, acid secretion stimulated by gastrin is eliminated (51). On the contrary, in human studies using histamine antagonists, gastrin stimulates parietal cells to release acid (56). Hypergastrinemia continues during the duration of histamine depletion, resulting in a slow ECL cell overgrowth (data not shown).

View larger version (22K): [in this window] [in a new window]   Fig. 8.   Virtual histamine depletion. During histamine depletion, gastrin is elevated (A) and somatostatin concentration is significantly reduced (B). C: simulated histamine levels. D: lack of histamine significantly reduces gastric acid levels, although to a lesser extent than during gastrin depletion. Dashed lines, control; solid lines, histamine depletion. *P < 0.05; **P < 0.001.

Somatostatin depletion. Simulated basal effector levels are higher during somatostatin depletion than during control simulations (P < 0.001; data not shown). This is in agreement with data demonstrating the inhibitory role of somatostatin in regulating intragastric acidity using somatostatin receptor subtype 2 (sst-2)-deficient mice (66). We also observe increases in stimulated secretion of effectors inhibited by somatostatin. Histamine levels significantly increase by 15% (P < 0.001; data not shown), and gastrin levels are slightly elevated by 2.5%.

Antral vs. corpus somatostatin depletion. Of key interest is the difference between the contribution of somatostatin from the antral region and that from the corpus region. Using our human model of gastric acid secretion, we are able to preferentially deplete somatostatin in either region, an experiment that is impossible to perform. Given the known dissimilarities between antral and corpus D cells (see above), we expect the impact of antral and corpus somatostatin on system dynamics to differ significantly. Gastrin is significantly elevated by 15% during antral somatostatin depletion (P < 0.05), whereas it is unchanged during corpus somatostatin depletion (Fig. 9). Histamine levels are significantly increased (900%) during antral somatostatin depletion compared with corpus somatostatin depletion (7%). Our results indicate that antral somatostatin depletion has a much greater effect on acid output than somatostatin produced in the corpus.

View larger version (29K): [in this window] [in a new window]   Fig. 9.   Virtual depletion of somatostatin in antrum (A-D) and corpus (E-H). Antral somatostatin depletion has a more dramatic effect on the gastric system than does corpus somatostatin depletion. Breakfast, lunch, and dinner were provided at 0700, 1300, and 1900, respectively. Dashed lines, control; solid lines, histamine depletion. *P < 0.05; **P < 0.001.

Effect of Food

Importance of the CNS

20

17

1

1

View larger version (36K): [in this window] [in a new window]   Fig. 10.   Effect of variation of CNS parameters. A and B: effects of variation of maximal CNS activity due to food stimulation (Nmax1) on simulated gastrin and gastric acid, respectively. C and D: effects of variation of maximal gastrin secretion rate due to CNS stimulation (KNG1) on simulated antral gastrin and gastric acid, respectively. E: long-term effects of gastrin on ECL overgrowth over 1,200 h (5 days). Gastrin levels are elevated after increases in maximal gastrin secretion rate due to CNS stimulation (KNG1). F: long-term effects of gastrin on G cells over 1,200 h (5 days). There is no change in G cell population as KNG1 increases. *P < 0.05; **P < 0.001.

Importance of Neural-Independent Parameters

Effect of Gastrin on ECL Cell Growth

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

We present a model of human gastric acid secretion using a system of 18 nonlinear ODEs together with a food function. In this system, positive and negative effectors strictly maintain acid homeostasis, the degree of which depends on food input. We show that the model is valid by demonstrating consistency with experimental results under normal and depletion conditions. The key modulators of the system are food and neural input. We also demonstrate the significance of gastrin through depletion studies. This is further substantiated by using sensitivity analyses, and together these findings suggest that gastrin is an important signal transducer relaying information from the CNS to parietal cells. We also demonstrate that although gastrin is important, somatostatin activity is a key regulator of gastric acidity.

Maintenance of gastric acid levels is important for optimal function of the stomach (28). Food digestion and sterilization of the lumen require strict control of acid levels, implying that the system returns quickly to equilibrium if disturbed. We find that our model satisfies this requirement, thereby suggesting that the gastric system is stable. Although redundancies ensure that gastric acid release continues if one pathway is lost, these redundancies cannot fully explain gastric stability. Maintenance of stability requires extensive feedback mechanisms that act to achieve homeostasis during disturbances. We show that compensatory mechanisms are likely invoked to stabilize acid secretion during altered conditions, such as effector depletion.

During virtual depletion of histamine, we observe a compensatory mechanism whereby the D cell population declines, promoting elevation of gastrin levels by reducing somatostatin production. When gastrin levels rise, gastric acid secretion is stimulated. This may account for the higher acid levels during histamine depletion (Fig. 8D) than during gastrin depletion (Fig. 7D). During virtual gastrin depletion, gastric acid secretion is reduced. We argue that, by boosting ECL cell numbers, histamine levels would increase, thereby restoring acid levels. On the contrary, this is not observed in gastrin knockout mice (34, 106). We therefore suggest that gastrin, and not histamine, plays a pivotal compensatory role. Furthermore, we suggest that gastrin levels may be useful as indicators of gastric health status.

We also demonstrate the significance of the negative-feedback loop involving somatostatin in acid release. Therefore, intact negative regulation is critical for proper function of the gastric system. We demonstrate that acid levels are sensitive to variations in somatostatin dissociation from G and ECL cells. Variations in these parameters may have dramatic and even detrimental effects on acid secretion given the prominence of somatostatin in inhibition of acid release. This may partially explain why these parameters do not vary significantly unless manipulated experimentally (108). Although somatostatin is important in inhibition of acid, it may also play a compensatory role controlling gastrin levels, thereby modulating acid secretion. During long-term absence of somatostatin, higher gastrin levels may promote ECL cell overgrowth. On the other hand, chronic elevations in somatostatin levels lower gastrin levels and may lead to loss of mucosal integrity. Therefore, we suggest that somatostatin levels are strictly controlled to maintain acid and cell homeostasis.

We demonstrate a technique for exploring gastric acid secretion and its regulation by gastric effectors. Mathematical models are not only applicable to long- and short-term studies, but they allow for rapid assessment of global effects. We are able to quickly assess critical elements in the system using virtual depletion/deletion analyses. Although our model is a simplification of human gastric acid secretion and its regulation, we include cells and effectors that are conserved among different species. Species-specific differences do not significantly affect our results, because we capture qualitatively and, to some extent, quantitatively the dynamical behavior of the human system.

Our model is a powerful tool for analyzing gastric effector and acid secretion, but there are limits to its potential. One of our immediate goals is to model acid secretion to assess important effectors in the system. To do this, we neglect many of the complex cellular events that contribute to the secretion of effectors and acid. Therefore, we cannot precisely reproduce some of the dynamical behaviors in secretion that are observed experimentally. For example, the secretion of somatostatin is biphasic, and this may be due to intracellular calcium release, which is intimately coupled to somatostatin release (18). We also observe differences in some of the cell dynamics compared with in vivo and in vitro data. We attribute these discrepancies, such as the elevation of immature cell populations, to model simplifications. In the case of cell dynamics, we do not account for intermediate cell stages observed during stem cell differentiation. Although cell and signal transduction dynamics are not modeled rigorously, this does not detract from the qualitative significance of our results. Inclusion of some of these events may render the model too complex for study.

Although the main purpose of the stomach is food digestion, pathogens can be ingested with food. Many of these pathogens are acid intolerant; thus acid secretion mechanisms help maintain a sterile environment. However, H. pylori has adapted to persist in this hostile environment. Most infections are asymptomatic and persist for the lifetime of the host; other outcomes such as peptic ulcer and gastric carcinomas occur less often. One key application of this model is to study colonization by this pathogen. For example, one effect of bacterial colonization is a significant elevation of gastrin levels (101, 111), the significance of which is not fully understood. Recent studies in mice have demonstrated that this response is not specific to H. pylori but, rather, involves mixed flora that colonize the mouse stomach (111). It is therefore possible that elevated gastrin levels during colonization may be host induced and may represent an effort toward bacterial clearance by increasing gastric acidity. With this model, we have another tool for exploring not only host-bacterial interactions but also the potential importance of compensatory mechanisms in bacterial persistence.

Another model application is designing therapeutic strategies to diminish effects of gastric ailments associated with H. pylori. A recent study has suggested that prolonged use of proton pump inhibitors by individuals infected with cytotoxin-associated gene A-positive H. pylori strains accelerates progression of gastric mucosal atrophy (27). Furthermore, although they are effective in reducing acid output, long-term administration of proton pump inhibitors may predispose individuals with gastrinomas to ECL cell overgrowth (23). Our model could be used to identify targets for reducing acid secretion without harmful side effects.