Introduction To Tumor Growth
diffusion of nutrientsdevelopment of hypoxic zone
Avascular development of necrotic coreTumor release of macrophage chemoattractant
angiogenesis
VascularTumor
Macrophages
White blood cell – part of immune system
Main role: phagocytose pathogens
Attracted towards hypoxic region of tumor cells and promote proliferation of cancer cells and angiogenesis
The Plan
• Modify macrophages to release cytotoxic chemical once they get in the hypoxic zone to kill tumour cells
• Goal: new treatment which would target tumour cells
• Study effects of chemical concentration and macrophage concentration on the tumour
• Is this a possible treatment? Under which conditions?
Previous Results
Many models of tumor growth – most using a PDE approach
Owen, M.R. & Sherratt, J.A. 1998. Modelling the macrophage invasion of tumours: Effects on growth and composition. Journal of Mathematics Applied in Medicine & Biology, 15: 165-185.
Normal macrophages both promote proliferation and inhibit early growth of avascular tumors and therefore have no relevant effect on the overall growth.
Developing a new model
Involves chemotaxis and diffusion models
Using: c(r,t)=concentration of nutrients m(r,t)=concentration of macrophages R(t) = Radius of tumor at time t h(r,t)= concentration of chemoattractants x(r,t)=concentration of cytotoxic chemical
Equations
21R
Rdt
dR
2
2
21 r
mD
r
hm
rD
t
mmm
xdmr
xD
t
xx
2
2
Constant gradient
Chemotaxis + diffusion
Diffusion
Logistic growth
22
))((2
),( rtRkkR
cctrc outout
Assumptions
Logistic growth of radius and constant proliferating zone
Tumor only contains cancerous cells (no normal cells)
No angiogenesis yet
Macrophages do not proliferate and die when they reach the necrotic core
Toxic chemical concentration is considered to be uniformly released from hypoxic zone
No normal macrophages
Macrophage concentration is uniformly distributed around the tumor
Cytotoxic chemical is diffused: we model it using the solution of the diffusion equation
CA Rules
• Time and space are discrete• We define an (101*101)
matrix , and we assign a value to each element, representing its state.
• Determine the state of a cell at the next time step, from the state of its 8 neighbours and its own.
k1-1, k2+1
Neighbour
k1, k2+1
Neighbour
k1+1,k2+1
Neighbour
k1-1, k2
Neighbour
k1,k2
Cell that we consider
k1+1, k2
Neighbour
k1-1, k2-1
Neighbour
k1, k2-1
Neighbour
k1+1, k2-1
Neighbour
CA of Tumour Growth
k1-1, k2+1
Neighbour
k1, k2+1
Neighbour
k1+1,k2+1
Neighbour
k1-1, k2
Neighbour
k1,k2
Cell that we consider
k1+1, k2
Neighbour
k1-1, k2-1
Neighbour
k1, k2-1
Neighbour
k1+1, k2-1
Neighbour
• Matrix A• A(k1,k2) defines state of
tumour cell at the (k1,k2) element.
• A(k1,k2)=0: proliferating tumour cells
• A(k1,k2)=1: Hypoxic zone• A(k1,k2)=2 :Dead tumour
cells (Necrotic core)• A(k1,k2)=3: No tumour
cells (outside tumour)
CA Nutrient Concentration Gradient
New matrix called “c”
if dist(k1,k2)>R(i+1,2)c(k1,k2)=Cout;elseif dist(k1,k2)<Rn(i+1)
c(k1,k2)=0;elseif
dist(k1,k2)>Rn(i+1)&dist(k1,k2)<(Rn(i+1)+epsilon)
ADDelse if (Rn(i+1)>0) c(k1,k2)=Cout-k_c*(R((i+1),2)-
dist(k1,k2))^2;else c(k1,k2)=0;end
k1-1, k2+1
Neighbour
k1, k2+1
Neighbour
k1+1,k2+1
Neighbour
k1-1, k2
Neighbour
k1,k2
Cell that we consider
k1+1, k2
Neighbour
k1-1, k2-1
Neighbour
k1, k2-1
Neighbour
k1+1, k2-1
Neighbour
CA Macrophage Chemotaxis
C(k1-1, k2+1)
Neighbour
C(k1, k2+1)
Neighbour
C(k1+1,k2+1)
Neighbour
C(k1-1, k2)
Neighbour
C(k1,k2)
Cell that we consider
C(k1+1, k2)
Neighbour
C(k1-1, k2-1)
Neighbour
C(k1, k2-1)
Neighbour
C(k1+1, k2-1)
Neighbour
New matrix called “macro”: each cell has a value according to the number of macrophages in it
The concentration of chemoattractant is inversely proportional to the concentration of nutrient (because chemoattractant is released in the hypoxic zone)
=> Macrophage moves to neighbouring cell which has the least concentration of nutrient (the highest concentration of chemoattractant)
CA Activated Macrophage
New matrix called “macro_active”
Macrophage is activated (releases the cytotoxic chemical) when it reaches the hypoxic zone
=> in the code, that means when A(k1,k2)=1 and macro(k1,k2)>0
The core: A(k1,k2)=2
The crust: A(k1,k2)=0
Hypoxic zone:A(k1,k2)=1
CA Cytotoxic Chemical Release
• New matrix called “toxic”
• We suppose that the chemical diffuses rapidly compared to our time step: we model it using the solution of the diffusion equation : the Gaussian
CA: Final Results: Low threshold
Low threshold: necessary concentration of chemical to kill the tumour cell is low: tumour cells are killed easily
=>Tumour is destroyed.
CA: Final Results: Higher Threshold
Higher threshold
=>Tumour is not destroyed...
(hypoxic zone disappears=> no more cytotoxic diffusion!)
Discussion
Obtained results were not as expected: even if the radius of the tumour reaches steady state, it may not be destroyed... Conditions...
If we suppose that our macrophages can release a sufficient amount of cytotoxic chemical, it would be a possible treatment
Other variables that should be studied : amount of injected macrophages, how many injections, ...