What is Mathematical Biology and how useful is it?

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What is Mathematical Biology and how useful is it?

• Avner Friedman

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• What is life?
• What is mathematical biology?
• The role of oxygen in wound healing and tissue
  transfer
• The immune response to infection in the lung
• Cancer virotherapy
• Summary

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What is life?

• Unit of life is a cell. Processes of living.
  
• (according to F. Harold, The Way of the
  Cell, 2001)
• Flux of matter and energy
• Chemical activities absorb nutrients, produce
  biomass,
• eliminate waste products
• Adaptation
• Structure and function evolve to promote
  organism survival
• Organization
• A bacterial cell consists of 300 million
  molecules,
• assembled non-randomly
• DNA ? RNA ? Protein is strategically planned
  and executed
• Self-reproduction
• Autonomously, not by external forces

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What is Mathematical Biology?

• Talking to biologists and getting familiar with
  their experiments and data with respect to a
  biological process.
• Developing a mathematical model that describes
  the biological process (e.g., by differential
  equations).
• Simulating and comparing the numerical results
  with experimental results and keep revising
  until the fit is satisfactory.
• Using the model to make new biologically testable
  hypotheses.

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Experiments, data
Simulation
Mathematical model
Parameters estimation
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Wound healing as a function of tissue oxygen
tension A mathematical model

• R. Schugart, A. Friedman, R. Zao, C.K.
  Sen
• PNAS (2008)
• Chronic wounds represent a major public
• health problem worldwide affecting 6.5 million
• individuals in the U.S., with cost of 5-10
  billion
• each year.
• Wound healing represents a well-orchestrated
  reparative response that occurs after all
  surgical procedures or traumatic injuries.
  Angiogenesis plays a central role in wound
  healing. In this work the role of oxygen is
  investigated, and the use of oxygen intervention
  (hyperbaric chamber) is considered.

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• Hyperbaric Chamber
• Topical Oxygen Treatment

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(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
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Role of Oxygen
Ga
Moderate hypoxia and hyperoxia improve healing.
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A Mathematical Model of Ischemic Cutaneous Wounds
C. Xue, A. Friedman, and C. Sen
PNAS (2009) Experiment
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A more refined model is needed.

• Separating chemoattractant between VEGF and PDGF.
• Considering the partially healed tissue as
  viscoelastic material.
• Modeling ischemia in a circular geometry.

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Experimental and Simulation Results
C. Xue, A. Friedman, and C. Sen (2009)
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Modeling Oxygen Transport in Surgical Tissue
Transfer
A. Matzavinos, C.Y. Kao, J.E.F. Green, A.
Sutradhar, M. Miller, and A. Friedman
PNAS (2009)

• During surgery, a plastic surgeon must decide how
  large a flap (with one artery) can be lifted and
  transferred to another location, without
  developing fat necrosis.

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arterial pressure
venous pressure
oxygen diffusion and transport (tissue)
oxygen diffusion and transport (artery)
oxygen diffusion and transport (vein)
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Future Work

• The model represents initial step. Subsequent
  work will proceed jointly with animal
  experiments, in order to refine the model by
  including heterogeneity of the vasculature.
• The model will be coupled to that of ischemic
  cutaneous wounds.

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A model on the influence of age on immunity to
infection with Mycobacterium tuberculosis
A. Friedman, J. Turner, B. Szomolay Experimental
Gerontology Increasing susceptibility to many
infectious diseases is highly associated with the
loss or delay in the generation of antigen
specific CD4 T cells mediated immunity. For
tuberculosis, where antigen specific CD4 T cell
derived IFN-g is essential, such a loss is
associated with aging, and it can lead to a
significant failure to control infection.
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Macrophages

• Infected macrophages infected by 10 bacteria in
  the absence of IFN-? cannot control bacterial
  growth they burst releasing many bacteria.
• Activated macrophages infected by 5 bacteria in
  the presence of IFN-? they present antigen to T
  cells.
• Resting macrophages do not contain bacteria.

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The Model Variables
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Modeling the Immune Rheostat of Macrophages in
the Lung in Response to Infection

• J. Day, A. Friedman, and L. Schlesinger
• (PNAS, 2009)
• Alveolar macrophages are also called
  Alternatively Activated Macrophages (AAM). AAM
  form the first line of cellular defense.
• The macrophages in the lymph nodes are called
  Classically Activated Macrophages (CAM).
• CAM are more effective than AAM in combating
  infection.
• When infection in the lung occurs, there is time
  delay until CAM arrive and become more dominant
  than AAM switching time.

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• Using IFN-? as drug It decreases the switching
  time, the maximum bacterial load, and the
  residual bacteria.

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Virotherapy in Glioblastoma
A. Friedman, J.J. Tian, G. Fulci, E.A.
Chiocca, and J. Wang Cancer Research, 2006
Glioblastoma is a brain tumor, very invasive,
life expectancy 1 year
glioblastoma
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virus
cell
When the cell dies, a swarm of virus particles
burst out
b burst size replication number
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• Idea Use virus to destroy tumor cells
• Oncolytic virus Genetically altered virus which
  is
• Replication competent
• Infects tumor cells and reproduces in them
• Does not harm normal healthy cells
• Virotherapy Actively tested in clinical trials
  on various types of cancer
• Two important factors
• Safety
• Efficacy

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Factors to be considered

• The immune system cells which detect virus and
  virus-infected cells, and destroy them
• Cyclophasphamide (CPA) suppresses the innate
  immune response
• During infection, the population of immune cells
  increases dramatically. When the infection is
  gone, the population of immune cells returns to
  its normal size (quadratic clearance).

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Model Equations
uninfected cell
infected cell
necrotic cells
immune cells
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virus particles
radial velocity
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Tumor Radius
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• b large
• infected (uninfected )
• immune and kills infected cells and virus
• - then immune cells kill themselves
• immune
• In the meantime uninfected cells
• Remaining virus renew attack
• infected

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Conclusions

• OV hrR3 cannot eradicate glioma.
• If however b can be increased to 150 then the
  radius will shrink and become very small (even
  without CPA).
• CPA primary effect is in decreasing the density
  of uninfected tumor cells thus reducing the
  risk of secondary tumor.
• Protocols of CPA treatment (weekly, or
  double-dose biweekly) do not make a significant
  difference.

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Summary

• Mathematical models should relate to experiments.
• In PDE or ODE models the choice of parameters is
  crucial use experimental results as much as
  possible.
• Use sensitivity analysis.
• Simulations of the model must fit with
  experimental results.
• The model should then be used to suggest new
  hypotheses that are biologically testable.