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

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

Outline of lecture: 

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

Helicobacter pylori (identified in 1982): 

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

Introduction to gastric acid secretion: 

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.

Goals: 

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

The stomach: corpus and antrum regions: 

The stomach: corpus and antrum regions Corpus Antrum

Gastric glands: 

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

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 Key cell types in the gastric system

Development of antrum and corpus cells: 

Development of antrum and corpus cells Antrum Corpus

Gastric acid secretion regulation by gastric cells: 

Gastric acid secretion regulation by gastric cells

Gastro-protective mechanisms: bicarbonate secretion: 

Gastro-protective mechanisms: bicarbonate secretion

Methods for ODE model system: 

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

Food function profile F(t): 

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)

Neural stimulation equations: 

Neural stimulation equations

Cell dynamics: stasis in the short-term: 

Cell dynamics: stasis in the short-term

Hormonal dynamics: gastrin, an inducer of gastric acid secretion: 

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

Slide21: 

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

Slide22: 

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

Slide23: 

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

Slide24: 

Ion dynamics: bicarbonate, a gastro-protective mechanism

Partial reciprocity of gastrin and somatostatin: positive and negative regulators of gastric acid secretion: 

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

Stability: attracting limit cycles: 

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

Results of virtual deletion studies: 

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.

Findings: 

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.

Discussion: 

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?

Review: stomach anatomy: 

Review: stomach anatomy

Continuous Delay functions: 

Continuous 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) We explored three different delay functions:

Delay physiology: 

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)

Figure 2: DDE diagram: 

Figure 2: DDE diagram

Figure 3: ODE vs DDE baseline: 

Figure 3: ODE vs DDE baseline

Figure 4: phase portraits ODE vs DDE: 

Figure 4: phase portraits ODE vs DDE

Figure 5: phase portraits DDE with different delays: 

Figure 5: phase portraits DDE with different delays

Figure 6: Antral Somatostatin depletion: 

Figure 6: Antral Somatostatin depletion

Figure 7: Corpal Somatostatin depletion: 

Figure 7: Corpal Somatostatin depletion

Figure 9: Antral Gastrin depletion: 

Figure 9: Antral Gastrin depletion

Conclusions:: 

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

Acknowledgments: 

Acknowledgments The Kirschner 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 $$ from NIH and The Whitaker Foundation

New equations: 

New equations

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