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 Growth can plateau in early stage

Introduction Tumor cells enter three main stages: 1. Avascular Stage 2. Vascular Stage 3. Metastasis

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

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

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

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

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

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

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

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.

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

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.

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

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.

References Beals, M., et al. Doubling time of Tumors Britton, Nicholas F. Essential Mathematical Biology. Springer: London p Obcemea, Ceferino. Chaotic Dynamics of Tumor Growth and Regenration. Proceedings of the Third ICCS, Perseus Books, Boston ODE software for Matlab.