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Modeling Tumor Growth

Modeling Tumor Growth. Katie Hogan 7 December 2006. 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

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Modeling Tumor Growth

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  1. Modeling Tumor Growth Katie Hogan 7 December 2006

  2. 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

  3. Introduction • Tumor cells enter three main stages: • Avascular Stage • Vascular Stage • Metastasis

  4. 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

  5. Avascular Growth Use basic logistic model, N: size of tumor, measured as total volume K: carrying capacity a: growth constant

  6. Figure 1. Logistic growth curve with a=0.35 and K=40.

  7. Figure 2. Logistic growth curve with a=0.8 and K=40.

  8. Avascular Growth Gompertz equation: N: size of tumor, measured as total volume K: carrying capacity a: growth constant b: decay rate

  9. Figure 3. Gompertz model with a=0.3 and b=0.5.

  10. Figure 4. Gompertz model with a=0.8, b=0.5.

  11. 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 .

  12. 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

  13. 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.

  14. 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

  15. 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.

  16. References Beals, M., et al. Doubling time of Tumors. 1999. <http://www.tiem.utk.edu/~gross/bioed/webmodules/tumorgrowth.html> 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. <http://www.uvm.edu/~pduval/iccs2000/chaotictumor.htm> ODE software for Matlab. <http://math.rice.edu/~dfield/dfpp.html>

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