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

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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

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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

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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, USA50, Italy80
• 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

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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.

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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

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The stomach corpus and antrum regions
Corpus
Antrum
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Gastric glands
mucosa
mucus gland
Taken from H.F. Helander (1992)
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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

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Development of antrum and corpus cells
Antrum
Corpus
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Gastric acid secretion regulation by gastric cells
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Gastro-protective mechanisms bicarbonate
secretion
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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

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Food function profile F(t)
Breakfast 700h Lunch 1300h Dinner 1900h
The daily food function profile is representative
of the volume of food eaten during each meal. (1
liter maximal volume)
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Neural stimulation equations
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Cell dynamics stasis in the short-term
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Hormonal dynamics gastrin, an inducer of gastric
acid secretion
Taken from Smith et al., 1990.
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Hormonal dynamics somatostatin, an inhibitor
of gastric acid secretion
Taken from Burhol et al., 1984
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Acid dynamics gastric acid
Taken from Feldman and Richardson, 1986.
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Hormonal dynamics histamine, an inducer of
gastric acid secretion- no data available
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Ion dynamics bicarbonate, a gastro-protective
mechanism
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Partial reciprocity of gastrin and somatostatin
positive and negative regulators of gastric acid
secretion
Taken from Zavros et al, 1999

• Negative Feedback

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Stability attracting limit cycles

• A stable period 3- cycle is observed which is a
  function of food intake

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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.

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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.

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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?

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Review stomach anatomy
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Continuous Delay functions
We explored three different delay functions

1. The total amount of antral gastrin produced in
   the past minutes
2. The average amount of antral gastrin produced in
   the past minutes
3. The percentage of the total amount of antral
   gastrin produced in the past minutes (p1p21)

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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)

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Figure 2 DDE diagram
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Figure 3 ODE vs DDE baseline
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Figure 4 phase portraits ODE vs DDE
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Figure 5 phase portraits DDE with different
delays
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Figure 6 Antral Somatostatin depletion
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Figure 7 Corpal Somatostatin depletion
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Figure 9 Antral Gastrin depletion
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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

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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
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New equations