Modeling Tumor Growth

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Modeling Tumor Growth

• Katie Hogan
• 7 December 2006

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Introduction

• Cell growth typically well controlled
• Mutations in oncogenes can cause cancerous cells
  to form and grow out of control, forming a tumor
• Tumor cells multiply when somatic cells cannot
• Can produce own growth factors
• Growth can plateau in early stage

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Introduction

• Tumor cells enter three main stages
• Avascular Stage
• Vascular Stage
• Metastasis

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Definitions

• Somatic cells normal body cells
• Diffusion-limited phase (a.k.a. avascular stage)
  stage in which tumor cells acquire nutrients
  through diffusion from outside the tumor
• Vascular stage (a.k.a. angiogenesis) stage in
  which cells stimulate blood-vessel production by
  secreting a TAF
• TAF tumor angiogenesis factor
• Metastasis stage in which cells break free from
  normal controls and begin to spread uncontrollably

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Avascular Growth

• Use basic logistic model,

N size of tumor, measured as total volume K
carrying capacity a growth constant
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Avascular Growth

• Gompertz equation

N size of tumor, measured as total volume K
carrying capacity a growth constant b decay rate
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Vascular Growth

• Tumors have to have a blood supply to grow beyond
  the diffusion-limited state.
• In the model, the del operator is a vector
  differential operator, where
  .

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Vascular Growth

• c tumor cell concentration
• D constant diffusion coefficient
• h(c) rate of decay of TAF
• f(c)g(n) is the rate of intake by the cells, n,
  which make up new blood vessels

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Metastasis

• Main characteristic of malignant tumors is the
  fact that they metastasize.
• All tissues (cancerous or not) secrete growth
  inhibitors
• Tumors also produce their own growth-promoters.
• The following model represents the activator and
  inhibitors reacting and diffusing within a tumor.

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Metastasis

• u activator concentration
• v inhibitor concentration
• u column vector of concentrations, u (u,v)T
• f (f,g)T column vector of the reaction
  kinetics
• D diagonal matrix of diffusion coefficients of
  u,v

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Metastasis

• Model is too complicated for analytical means
  but
• For certain reaction kinetics, there is a range
  of unstable eigenvalues (their real part is
  positive).
• Within the unstable range, there is a maximum
  that represents a growing mode which will
  eventually dominate over time.

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References

• Beals, M., et al. Doubling time of Tumors. 1999.
  lthttp//www.tiem.utk.edu/gross/bioed/webmodules/t
  umorgrowth.htmlgt
• Britton, Nicholas F. Essential Mathematical
  Biology. Springer London. 2003. p235-249.
• Obcemea, Ceferino. Chaotic Dynamics of Tumor
  Growth and Regenration. Proceedings of the Third
  ICCS, Perseus Books, Boston 2001.
  lthttp//www.uvm.edu/pduval/iccs2000/chaotictumor.
  htmgt
• ODE software for Matlab. lthttp//math.rice.edu/df
  ield/dfpp.htmlgt