Mathematics in Biology and Medicine

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Mathematics in Biology and Medicine
Summer Math Workshop for High School Middle
School Teachers
L.R. Ritter Southern Polytechnic State
University June 12, 2009
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Contents

• Some history and recent developments
• A predator-prey model with Matlab pplane
• Mathematics in Medicine an introduction to
  Atherosclerosis
• Chemo-taxis and the Keller-Segel Model/ Or the
  Dicty Story
• Angiogenesis and tumor growth
• Atherogenesis

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What are Mathematical Biology and Mathematical
Medicine
Mathematical biology is the study of biological
processes and biological laws, with an emphasis
on the construction and analysis of mathematical
models and mathematical logic.
Mathematical Medicine---sometimes referred to as
Theoretical Medicine---is mathematical biology as
it applies to applications in medicine.
By law, we mean a scientific generalization
based on empirical evidence.
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Some topics in Mathematical biology

• Ecological Models (large scale environment---organ
  ism interplay)
• Organism Models (such as predator/prey)
• Large and Small Scale Models (for example
  epidemiology)
• Cellular Scale (wound healing, tumor growth,
  atherosclerosis)
• Quantum/molecular Scale (DNA sequencing, and the
  study of neural networks)

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Some interesting current studies

• The effect of bacteria on wound angiogenesis (S.
  Afkhami, Virginia Tech)
• Zoonotic diseases carried by rodents seasonal
  fluctuations (L. Allen Texas Tech U.)
• Computational modeling of tumor development in
  bone tissue (P. Pivonka U. W. Australia)
• Hepatitis B in endemic China (L. Zou U. of
  Miami/Sichuan U.)

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Classical Predator/Prey (foxes and rabbits)
Suppose we have a population of foxes and a
population of rabbits, and we wish to build a
mathematical model to describe how the numbers of
specimen in each population will evolve over time
based on a few preliminary assumptions. Let us
make the following assumptions
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Classical Predator/Prey (foxes and rabbits)

• In the absence of foxes, rabbits will find
  sufficient food and breed without bound at a rate
  proportional to their population

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Classical Predator/Prey (foxes and rabbits)

• In the absence of foxes, rabbits will find
  sufficient food and breed without bound at a rate
  proportional to their population
• In the absence of rabbits, foxes will die out at
  a rate proportional to their population

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Classical Predator/Prey (foxes and rabbits)

• In the absence of foxes, rabbits will find
  sufficient food and breed without bound at a rate
  proportional to their population
• In the absence of rabbits, foxes will die out at
  a rate proportional to their population
• Each fox/rabbit interaction reduces the rabbit
  and increases the fox population (not necessarily
  equally)

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Classical Predator/Prey (foxes and rabbits)

• In the absence of foxes, rabbits will find
  sufficient food and breed without bound at a rate
  proportional to their population
• In the absence of rabbits, foxes will die out at
  a rate proportional to their population
• Each fox/rabbit interaction reduces the rabbit
  and increases the fox population (not necessarily
  equally)
• The environment doesnt change or evolve.

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Classical Predator/Prey (foxes and rabbits)
Fnumber of foxes and Rnumber of rabbits
D R A R B RF
Rabbit births
Change in rabbit pop.
Rabbit-Fox interaction
A and B are constant rates
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Classical Predator/Prey (foxes and rabbits)
Fnumber of foxes and Rnumber of rabbits
D F - C F D RF
Fox death
Change in Fox pop.
Rabbit-Fox interaction
C and D are constant rates
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Classical Predator/Prey (foxes and rabbits)
Fnumber of foxes and Rnumber of rabbits
D R A R B RF
D F - C F D RF
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Classical Predator/Prey (foxes and rabbits)
With 100 initial rabbits and 50 initial foxes.
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Classical Predator/Prey (foxes and rabbits)
With 100 initial rabbits and 50 initial foxes.
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Atherosclerosis
is a chronic condition involving the build up of
fatty deposits within the arterial wall. This can
result in a narrowing of the vessel lumen
restricting blood flow to vital organs such as
the heart and lungs. Cardiovascular disease
continues to be the principal cause of death in
the United States, Europe, and much of Asia.
Russell Ross (1999)
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The Human Large Muscular Artery
Intima Fine Collagen Fibrils, Smooth Muscle
Cells Media Smooth Muscle Cells, Elastic Lamina,
Collagen Adventitia Buddles of Collagen Fibrils
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Atherosclerosis
Normal Artery
Atherosclerotic Plaque
Ruptured Plaque with Severe Occlusion
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Initiation
Lipid Accumulation
DISEASE STAGES
Growth and Cap Formation
Plaque Rupture
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Mathematizing the Process

• Identify chemical and cellular species
• Identify what influences each species
• Translate each influence into a mathematical
  expression by using
• existing math models,
• data fitting,
• educated (hopeful) guesses.

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CHEMOTAXIS
chemotaxis    (ke'mo-tak'sis, kem'o-)  n.  
The characteristic movement or orientation of an
organism or cell along a chemical concentration
gradient either toward or away from the chemical
stimulus.
The American Heritage Dictionary of the English
Language, Fourth Edition.
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Chemo taxis From Bizarre Bugs to Human Cells
http//dictybase.org/Multimedia/development/develo
pment.html
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Math Medicine
Mathematical models of chemo-taxis have been used
to help us understand the process of
Angiogenesis Angiogenesis is the process by
which a small tumor can tap into ones vascular
system to obtain nutrients for growth and
possibly spread to other parts of the body.
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Angiogenesis
Initially, a tumor obtains nutrients from the
surrounding tissue.
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Angiogenesis
Initially, a tumor obtains nutrients from the
surrounding tissue.
After reaching a critical size, the core of the
tumor will die.
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Angiogenesis
The tumor sends out a chemical message to the
nearby capillaries.
VEGF Vascular Endothelial Growth Factors
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Angiogenesis
Endothelial cells respond and branch out toward
the tumor.
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Angiogenesis
The tumor accesses the blood supply. This allows
it to grow and can possibly lead to metastasis!
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Numerical Study of Angiogenesis Model
Numerical Simulation using Stokes-Lauffenburger
Model of Angiogenesis (1991)
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Numerical Study of Angiogenesis Model
Numerical Simulation using Anderson Chaplain
Model of Angiogenesis (1998)
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Numerical Study of Angiogenesis Model
Numerical Simulation Using Model presented by
Mantzaris, Webb and Othmer (2004)
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Our Model of Atherogenesis
We define a number of generalized cellular and
chemical species I-Immune cells D-Debris (the
lesion itself) C-chemical agents for signaling
cells (chemo-attractant) L-LDL molecules Lox-LDL
molecules that have become oxidized R-Free
radical molecules (oxygen)
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Our Model of Atherogenesis
Cells are influenced by Diffusion, chemo-taxis,
natural turnover, and reproduction Chemicals are
influenced by Diffusion, chemical reactions
(e.g. oxidation), natural decay, and removal or
production by cells
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Lesion Equation Example
D D a I Lox b I D
Decrease due to normal immune function
Change in the amount of lesion
Increase due to foam cell formation
a and b are constant rates
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Chemical Reaction Example LDL Oxidation
Chemical Equation
Mathematical Representation
D Lox rate L R c I Lox
Increase due to oxidation
Decrease due to uptake by immune cells
Change in oxidized LDL
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Initial Conditions Domain is Seeded with
Debris
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Immune Cells At time (a) t0, (b) t0.1, (c )
t0.2, (d) t0.26
0.1
0.5
4
90
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Analysis of the Model Shows

• The suggested modeling approach is able to
  capture several well documented aspects of lesion
  formation (cell aggregation and build up).
• Stability analysis of the model results in
  physically relevant stability criteria.
• Early result show the potential for suggested
  experimentation.
• More information would help us to better hone
  this model and increase its usefulness as a tool
  for in-silico experimentation.