Dynamic Tumor Growth Modelling
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This document explores a tumor growth model using the method of characteristics to derive solutions for various growth rates. It discusses the dynamics of primary and metastatic tumors, incorporating mathematical equations like the von Foerster equation and various growth functions. Future work is outlined, aiming to fit data to models and analyze clinical data across multiple patients.
![Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work
Linear Separable Growth Rate Solution
Now we have Linear Separable Growth Rate
dx
= [k − E (t)] x
dt
Colony size distribution
m m t
m −α−1− k−E (t) α+ k−E (t) ( k−E (s)ds )t
ρ(x, t) = k−E (t) x e 0](https://image.slidesharecdn.com/talk-111011184147-phpapp02/85/Dynamic-Tumor-Growth-Modelling-11-320.jpg)
Dynamic Tumor Growth Modelling
- 1. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work Dynamic Tumor Growth Modelling Prof. Thomas Witelski, Matthew Tanzy, Ben Owens, Oleksiy Varfolomiyev NJIT, June 2011
- 2. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work Cancer background Current medical understanding of some types of cancer 1 Disease starts from a primary tumor which grows in one location 2 The primary tumor may shed cancer cells which get carried to other parts of the body by the circulatory system
- 3. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work Colony size distribution of metastatic tumors with cell number x at time t is governed by von Foerster Equation ∂ρ ∂ + (g (x) ρ) = 0 (1) ∂t ∂x g (x) is tumor growth rate determined by dx = g (x) (2) dt Initially no metastatic tumor exists ρ(x, 0) = 0 (3) Intitial tumor birth rate ∞ g (1)ρ(1, t) = β(x)ρ(x, t)dx + β(xp (t)), (4) 1 β(x) tumor birth rate
- 4. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work Linear Growth Function: g (x) = k − Ex Assuming that in the beginning the main contribution to the tumor growth is given by primary tumor (i.e. neglecting the integral term in the BC) we have ∂ρ(x, t) ∂ (g (x)ρ(x, t)) + =0 (5) ∂t ∂x ρ(x, 0) = 0 (6) g (1)ρ(1, t) = β(xp ), β(x) = mx α (7) dx = k − Ex = g (x) (8) dt
- 5. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work Solution by the Method of Characteristics dρ = Eρ (9) dt along the characteristics given by dx = k − Ex (10) dt Therefore ρ(x, t) = ρ0 (x(0)) e Et (11) along the characteristics K x(t) = x(0)e −Et + 1 − e −Et (12) E
- 6. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work Solution by the Method of Characteristics First we use the BC to solve for ρ0 (x(0)), i.e. α K K (K − E ) ρ0 e Et + 1 − e Et e Et = m e −Et + 1 − e −Et E E Finally, the colony size distribution of metastatic tumors with cell number x at time t is α mE −α e −Et (E −k)2 ρ (x, t) = k−Ex k+ Ex−k
- 7. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work Growth function: g (x) = Ex Figure: ρ(x) at t = 10 with all parameters at 1
- 8. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work Growth function: g (x) = k − Ex Figure: ρ(x) at t = 10 with E = 0.2, k = 0.3, and m = 0.1
- 9. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work Single primary tumor size growth (no drug is active) Growth function: g (x) = (k − E (t)e −rt )x Here no drug is active (i.e. g (x) = kx) Figure: ρ(x) at t = 3650 with k = 0.006, α = 0.663, and m = 5.3 × 10−8
- 10. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work Single primary tumor size growth (drug effect) At time t1 drug is activated −r (t−t1 ) In a region of constant E : x (t) = Ae kt+Ee /r Figure: ρ(x) at t = 3650 with k = 0.006, α = 0.663, E = 0.0083, r = 1 × 10−5 and m = 5.3 × 10−8
- 11. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work Linear Separable Growth Rate Solution Now we have Linear Separable Growth Rate dx = [k − E (t)] x dt Colony size distribution m m t m −α−1− k−E (t) α+ k−E (t) ( k−E (s)ds )t ρ(x, t) = k−E (t) x e 0
- 12. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work Logistic Tumor Growth Rate Solution Now we have Tumor Growth Rate given by Logistic Equation dx = (k − Ex) x dt Resulting colony size distribution e −kt ρ(x, t) = 2 (E x0 (e kt −1)+k )
- 13. Tumor Growth Model Method of Characteristics Solutions for various growth rates Future Work Future Work Fitting data to models, parameter estimation for multiple tumor and multiple patient data Incorporating limitations on measurable clinical data Comparing PDE, discrete population and polymerization models Solution of inverse problems for the birth rate Probability and statistics of different forms of clinical data