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Modelling Cancer Growth. Philip K. Maini, Centre for Mathematical Biology, Mathematical Institute, Oxford. mutations. Approx 1mm in diameter. Nutrient required Hypoxic core TAF (tumour angiogenesis factors) Avascular tumour Vascular tumour Invasion

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modelling cancer growth

Modelling Cancer Growth

PhilipK. Maini, Centre for Mathematical Biology, Mathematical Institute, Oxford



Approx 1mm in diameter

Nutrient required

Hypoxic core TAF (tumour angiogenesis factors)

Avascular tumour Vascular tumour


Tumour produces proteases – digest ECM


Normal environment:



Add H+

Gatenby & Gawlinski Gap



Metastatic tumor


Tomas Alarcón (UCL)Helen Byrne (Nottingham)EU RTN (5th Framework): “Using mathematical modelling and computer simulation to improve cancer therapy”Alarcón, Byrne, Maini, J. Theor. Biol, 225, 257-274 (2003) Prog. Biophys & Mol. Biol., 85, 451-472 (2004) J. Theor. Biol, 229, 395-411 (2004) SIAM Multiscale Mod & Sim.3, 440-475 (2005) Ribba, Marron, Agur, Alarcon, Maini Bull. Math. Biol., 67 79-99 (2005)

cancer growth

Cancer Growth

Tissue Level Signalling: (Tumour Angiogenesis Factors)

Oxygen etc


Intracellular: Cell cycle,

Molecular elements

Partial Differential Equations

Automaton Elements

Ordinary differential equations

tumour growth

Tumour Growth

Response to mechanical stimuli

Shrinkage (Pries et al 1988)

First, work out distribution of 02 (nutrient)

To do so, must consider vasculature: metabolic response

R = radius

= flow rate

H = haematocrit


P = pressure (transmural)

*Haematocrit Pries et al, 1994

*At a bifurcation:

(rat mesentry)

(Fung 1993)

algorithm for structural adaptation

Algorithm for structural adaptation

Flow rate


2. Given initial network configuration, compute flow rates

through and pressure drops across each vessel using Kirchoff’s


3. Compute distribution of haematocrit.

4. Update radius of each vessel.

5. Compute viscosities (using H and R from 3 and

4 respectively).

6. Repeat until steady state reached.

o 2 distribution
____________O2 distribution____________

(adiabatic approx)

P=O2 conc

ΚN for normal cell

Κc for cancer cell

0 o.w

Nw= normal to vessel wall

Ρb = О2level in blood

P = permeability

(at edge of domain, no flux

automaton rules

Automaton Rules

1. О2 distribution determined by BVP.

2. Cells attempt to divide at each time step.

3. Normal cell: if О2< threshold, cell dies

О2 > threshold, cell attempts to divide

Threshold = Ν1if more normal than cancer neighbours

= ΝT2if more cancer than normal neighbours

ΝT2> ΝT1

4. Cancer cell: if О2> threshold, cell attempts to divide

Threshold = СT1if more cancer than normal neighbours

= СT2if more normal than cancer neighbours

С T2> СT1


5. Cancer cell: if O2< threshold cell becomes quiescent If it remains quiescent for a certain length of time, it dies.6. Cells are sinks of O27. If O2 level is such that a cell may divide, sample neighbourhood for space. If more than one available space, go to the one with largest O2 (Patel et al 2001). If no space, die (Kansal et al, 2000)

cell dynamics
Cell Dynamics

NxN automaton elements.

State vector has 3 components:

  • Occupation: normal cell/cancer cell/vessel/empty
  • Cell status: proliferative/quiescent
  • Local О2 conc

We assume, for simplicity, vessel structures does not evolve.

  • Environmental heterogeneity decreases cancer cell growth but may contribute to metastasis
possible application

Possible application

Doxorubicin treatment of non-Hodgkin’s lymphoma (Ben Ribba, Zvia Agur, Tomas Alarcon, Philip Maini, K Marron)

Structural adaptation – vessels surrounded

by NHL leaky & unstable

Nutrient diffusion

-Drug pharmacokinetics in plasma

pharmacodynamics [kills proliferating cells]

tissue dynamics (adiabatic approx)

AIM – Explore different protocols of treatment

(presently a 21-day cycle is employed)

cell cycle dynamics

Cell-Cycle Dynamics


nutrient demand

hypoxia-induced quiescence

drugs work only on cells in a certain part of

their cell cycle.

Cell Cycle:

Cyclin-dependent kinases (CDK) }

cyclins }


In G1 CDK activity is low because its cyclin partners

are missing

At finish Cdhl (and Cdc 20) concs are high

degrade cyclins.

2 families of proteins


Tyson & Novak

  • Model for G1/S transition
e2f transcription factor take tyson and novak model incorporate inhibition by a kz term
E2F – transcription factor Take Tyson and Novak model:incorporate inhibition by a – Kz term

P27 conc in Cdhl



Growth regulation


[as m z ]

Cancer Cells

Hypothesis – growth regulation

is lost

Simulations show decrease in Cdk
  • This is observed experimentally
growth regulation of p 27

Growth regulation of p27?

Normals 

Cancer x

Growth factors p27

If growth is arrested, p27 is upregulated


Response to hypoxia (low O2)Expts on mouse embryo fibroblasts: hypoxiaNormal cells G1arrest Does not occur with p27 null mutants



x heterogeneities have a profound effect

on tumour dynamics

x effects of p27 – possible mechanism

x efficiency of drug treatments

Future Directions




role of acidity
Role of Acidity
  • Kieran Smallbone, David Gavaghan, Bob Gatenby, PKM

T-tumour density

V-vascular density

Glycolytic pathway

Blood flow removal

Avascular Case:



Necrotic core

Proliferation zone, T = const

Outside tumour


Assume necrosis arises whenconstantUsing experimentally determined parameter values necrotic core arises at r = 0.1 cm [avascular case]

Tumour Growth No normal tissue •Avascular tumour always reaches a benign steady state•Vascular tumour is benign if invasive if

(cf Greenspan 1972)

necrotic core




Three regimes of growth:

•If rate of acid removal is insufficient,

exponential growth followed by auto-toxicity

benign tumour

Occurs in avasculars and vasculars if

• vascular tumour displays sustained growth and invades

•Very small tumour – no growth (insufficient acid production to include normal cell death)

  • Kill tumour cells or cut them out
  • Anti-angiogenesis drugs – drug delivery
  • Treat normals?
  • Reduce vasculature in tumour

– tumour poisons itself

  • Destroy membrane pumps transporting H+ions from tumour
  • Increase acidity!