Comparing ordinary and delay differential equations models for human gastric acid secretion

1. Comparing ordinary and delay differential equations models for human gastric acid secretion Denise Kirschner, PhD Department of Microbiology and Immunology University of Michigan Medical School Ann Arbor, Michigan USA

2. Outline of lecture • Context of problem- H. pylori • Review relevant gastric physiology • Present ODE model • Present delay and DDE model • Compare results of both • Discussion

3. Helicobacter pylori (identified in 1982) • Gram negative, spiral-shaped, motile bacteria • Strict human pathogen- colonizes the stomach • Different disease trajectories • Colonization/persistence – superficial gastritis (most common outcome)- acts like IM • Peptic ulcer disease (75% correlated) • Duodenal ulcers (95% correlated) • Lymphomas/ carcinomas • pH-dependent growth, virulence is adherence and motility • Some countries 100% infected, USA:50%, Italy:80% • Treatment: antibiotics for 6 weeks • How does this pathogen survive in the hostile environment of the stomach? • pH, shedding of mucus, sloughing of cells, peristalsis

7. Introduction to gastric acid secretion • Gastric acid is important for two reasons: • Activation pepsin • Sterilization of the stomach • Maintenance of pH homeostasis is critical for proper function and protection of the stomach. • Gastric acid secretion is diurnal.

8. Goals • Study how Helicobacter pylori, a strict human stomach pathogen, affects the gastric acid secretion system to allow its infection and persistence • * Develop a virtual human model of gastric acid secretion • *Incorporate a delay to reduce system

9. The stomach: corpus and antrum regions Corpus Antrum

10. Gastric glands mucosa mucus gland Taken from H.F. Helander (1992)

11. Key cell types in the gastric system • Antrum • G cells – secrete Gastrin (+) • D cells – secrete Somatostatin (-) • Corpus • D cells • ECL cells – secrete histamine (+) • Parietal cells – secrete acid

12. Development of antrum and corpus cells Antrum Corpus

13. Gastric acid secretion regulation by gastric cells

14. Gastro-protective mechanisms: bicarbonate secretion

15. Methods for ODE model system • Developed a system of ordinary differential equations. • Estimated parameters • estimated numbers for each cell population • acquired parameter values from literature with preference given to estimates from human studies • studied the effect of these parameters using sensitivity and uncertainty analyses (latin hypercube sampling/partial rank correlation) • Analyzed the system of differential equations using three approaches: Matlab, Mathematica and code we wrote based on a finite differencing schemes. • Compared simulations results with published experimental data • Performed virtual deletion experiments

16. Food function profile F(t) Breakfast: 7:00h Lunch: 13:00h Dinner: 19:00h The daily food function profile is representative of the volume of food eaten during each meal. (1 liter maximal volume)

17. Neural stimulation equations

18. Cell dynamics: stasis in the short-term

19. Hormonal dynamics: gastrin, an inducer of gastric acid secretion Taken from Smith et al., 1990.

21. Hormonal dynamics: somatostatin, an inhibitor of gastric acid secretion Taken from Burhol et al., 1984

22. Acid dynamics: gastric acid Taken from Feldman and Richardson, 1986.

23. Hormonal dynamics: histamine, an inducer of gastric acid secretion- no data available

24. Ion dynamics: bicarbonate, a gastro-protective mechanism

25. Partial reciprocity of gastrin and somatostatin: positive and negative regulators of gastric acid secretion Taken from Zavros et al, 1999 • Negative Feedback

26. Stability: attracting limit cycles • A stable period 3- cycle is observed which is a function of food intake

27. Results of virtual deletion studies • Virtual gastrin deletion study • basal and stimulated acid concentrations are lower than controls during deletion simulations and qualitatively compare with literature (e.g. Wada et al, 1997). • Virtual histamine deletion study • Basal acid concentrations remain normal but stimulated acid levels are drastically reduced during deletion simulations when compared to controls. This results is demonstrated by Kobayashi et al (2000). • Virtual total somatostatin deletion study • Both basal and stimulated gastrin, histamine, and acid levels are increased during deletion simulations when compared to controls. Martinez et al (1998) demonstrate this in studies done in somatostatin deficient mice.

28. Findings • The system is robust • Cellular and physiological homeostasis observed • Simulation results correlate with experimental data • pH homeostasis is maintained during the course of the virtual experiments • We will now extend the model to assess the interaction dynamics between H. pylori and the virtual host. • Our findings may provide more insight into how H. pylori alters the gastric physiological environment to favor its persistence within its human host.

29. Discussion • Gastrin is the key regulator of gastric acid secretion • Model is complex, can we reduce? • 18 non-linear ODEs, 1 forcing function • A delay exists between the signals received in the corpus region, and the transference of information to the antrum. • Can we introduce a continuous delay in the system and still capture the qualitative behavior?

30. Review: stomach anatomy

31. Continuous Delay functions We explored three different delay functions: • **The total amount of antral gastrin produced in the past minutes; • The average amount of antral gastrin produced in the past minutes; • The percentage of the total amount of antral gastrin produced in the past minutes (p1+p2=1)

32. Delay physiology • . Amount of gastrin released by G cells in the antrum diffuses gradually into the corpus and is then available to the D and parietal cells only after a certain time period • Thus, the total amount of gastrin released in the previous minutes that is already located in the corpus region is effectively inducing the secretion of somatostatin and acid • This is physiologically relevant as it in part describes the Hill kinetics (i.e., a critical concentration of gastrin is required before a “surge” of its affect is observed)

33. Figure 2: DDE diagram

34. Figure 3: ODE vs DDE baseline

35. Figure 4: phase portraits ODE vs DDE

36. Figure 5: phase portraits DDE with different delays

37. Figure 6: Antral Somatostatin depletion

38. Figure 7: Corpal Somatostatin depletion

39. Figure 9:Antral Gastrin depletion

40. Conclusions: • the temporal behavior of the DDE model closely reproduces that of the ODE model • the stability of the ODE system is also observed in the DDE model at a delay length of tau= 30 minutes • virtual depletion experiments further validate that the DDE model replicates the behavior of the ODE system

41. TheKirschner Lab Brian M. Murphy, PhD David Gammack, PhD Simeone Marino, PhD Suman Ganguli, PhD Ping Ye, MS Seema Bajaria, MS Ian Joseph Benjamin H. Singer Stewart Chang Karyn Sutton Jim Zakowski Christian Ray Vanessa Pherigo Acknowledgments $$ from NIH and The Whitaker Foundation

42. New equations