Modelling aspects of solid tumour growth

Philip K. MainiCentre for Mathematical Biology

Mathematical Institute;Oxford Centre for Integrative Systems Biology,

Biochemistry;Oxford

More precisely

• Using mathematical models to explore the interaction of a VERY SMALL subset of processes in cancer with a view to increasing our intuition in a very small way

and eventually …

Outline

• Acid-mediated invasion/Somatic evolution/therapeutic strategies

________________________________________

• Vascular Tumour Growth

• Colorectal Cancer

Cancer

Cell proliferation and cell death (apoptosis) are tightly controlled by genes to maintain homeostasis (steady state). Mutations in these genes upset the balance and the system moves out of steady state.

How can we control a growing population of cells?

The Warburg Effect

• Tumour cells undergo glycolytic (anaerobic) metabolism presumably because there is a lack of oxygen.

• But sometimes in the presence of sufficient oxygen they still do this – seems very strange because it is 20 times less efficient than aerobic metabolism

Acid Mediated Invasion Hypothesis

• A bi-product of the glycolytic pathway is lactic acid – this lowers the extracellular pH so that it favours tumour cell proliferation AND it is toxic to normal cells.

• Gatenby and Gawslinski (1996)

Gatenby-Gawlinski Model

Bifurcation parameter

Experimental results (Martin and Jain)

• Fasano et al, Slow and fast invasion waves (Math Biosciences, 220, 45-56, 2009)

Tumour encapsulation

• Predicts ECM density is relatively unchanged – inconsistent with other models but consistent with experimental observations.

Metabolic changes during carcinogenesis

K. Smallbone, D.J. Gavaghan (Oxford)R.A. Gatenby, R.J. Gillies (Moffitt

Cancer Research Inst)J.Theor Biol, 244, 703-713, 2007

Cell-environment Interactions

Nature Rev Cancer 4: 891-899 (2004)

DCIS Model

Model Development

• Hybrid cellular automaton:– Cells as discrete individuals

• Proliferation, death, adaptation

– Oxygen, glucose, H+ as continuous fields– Calculate steady-state metabolite fields after each generation

• Heritable phenotypes:– Hyperplastic: growth away from basement membrane– Glycolytic: increased glucose uptake and utilisation– Acid-resistant: Lower extracellular pH to induce toxicity

Cellular Metabolism

• Aerobic:• Anaerobic:

• Assume:– All glucose and oxygen used in these two processes– Normal cells under normal conditions rely on aerobic respiration

alone

ATP2acidlactic2glucose

ATP36CO6O6glucose 22

Two parameters:n = 1/18

1 < k ≤ 500c

cnc

c

kg

g

gh

ga

c

g

:H

)(:ATP

:oxygen

cellglycolytic

cellnormal:glucose

Automaton Rules

• At each generation, an individual cell’s development is governed by its rate of ATP production φa and extracellular acidity h

– Cell death• Lack of ATP:

• High acidity:

– Proliferation

– Adaptation

resistant-acid

normal

T

Ndea h/h

h/hp

)1/()( 00 aap adiv

0aa

Variation in Metabolite Concentrations

glucose

oxygen

H+

• For further details, see Gatenby, Smallbone, PKM, Rose, Averill, Nagle, Worrall and Gillies, Cellular adaptations to hypoxia and acidosis during somatic evolution of breast cancer, British J. of Cancer, 97, 646-653 (2007)

Therapeutics

• Add bicarbonate to neutralise the acid

(Natasha Martin, Eamonn Gaffney, Robert Gatenby, Robert Gillies)

Metastatic Lesions

Model Equations: Tumour Compartment

Model Equations: Blood Compartment

Equivalent dose less effective in humans

Analysis

• There are 3 timescales and lots of small and large parameters so can do asymptotics and obtain an approximate uniformly valid solution on which to do sensitivity analysis.

Sensitivity Analysis

Proton inhibitor + bicarbonate

Clinical Ideas

Effects of Exercise

• Periodic pulsing of acid may affect somatic evolution by delaying the onset of the invasive phenotype (hyperplastic, glycolytic and acid-resistant) (Smallbone, PKM, Gatenby, Biology Direct, 2010)

Cancer Growth

Tissue Level Signalling: (Tumour Angiogenesis Factors) Oxygen etc

Cells:Intracellular: Cell cycle,

Molecular elements

Partial Differential EquationsAutomaton Elements

Ordinary differential equations

• Tomas Alarcon

• Markus Owen

• Helen Byrne

• James Murphy

• Russel Bettridge

Vascular Adaptation

• Series of papers by Secomb and Pries modelling vessels in the rat mesentry – they conclude:

R(t) = radius at time t:

R(t+dt) = R(t) + R dt S

S = M + Me – s + C

M = mechanical stimulus (wall shear stress)

Me = metabolic demand

s = shrinkage

C= conducted stimuli: short-range (chemical release under hypoxic stress?)

long-range (mediated through membrane potential?)

• By varying the strengths of the different adaptation mechanisms we can hypothesise how defects in vasculature lead to different types of tumours: Conclude that losing the long range stimuli looks a reasonable assumption

• Tim Secomb has shown this more convincingly recently (PLoS Comp Biol 2009)

Potential uses of the model

• Chemotherapy

• Impact of cell crowding and active movement

• Vessel normalisation

Angiogenesis

• Recently, we have added in angiogenesis (Owen, Alarcon, PKM and Byrne, J.Math. Biol, 09) and gone to 3D (Holger Perfahl)

• Movie – both2_mov

An integrative computational model for intestinal tissue renewal

• Van Leeuwen, Mirams, Walter, Fletcher, Murray, Osbourne, Varma, Young, Cooper, Pitt-Francis, Momtahan, Pathmanathan, Whiteley, Chapman, Gavaghan, Jensen, King, PKM, Waters, Byrne (Cell Proliferation, 2009)

• CHASTE – Cancer, Heart And Soft Tissue Environment

• Modular

The effects of different individual cell-based approaches

• (to appear in Phil Trans R Soc A)

Conclusions and Criticisms• Simple multiscale model – gain some insight into why combination

therapies might work

• Heterogeneities in environment play a key role

• No matrix included! – Anderson has shown adhesivity could be important

• Cellular automaton model – what about using Potts model, cell centred, cell vertex models? – DOES IT MAKE A DIFFERENCE (Murray et al, 2009; Byrne et al, 2010)

• There are many other models and I have not referred to any of them! (Jiang, Bauer, Chaplain, Anderson, Lowengrub, Drasdo, Meyer-Hermann, Rieger, Cristini, Enderling, Meinke, Loeffler, TO NAME BUT A FEW)

Acknowledgements

• Colorectal: David Gavaghan, Helen Byrne, James Osborne, Alex Fletcher, Gary Mirams, Philip Murray, Alex Walter, Joe Pitt-Francis et al (EPSRC)

• Vascular: Tomas Alarcon, Helen Byrne, Markus Owen, Holger Perfahl (EU -5th and 6th frameworks)

Acknowledgements

• Natasha Martin, Kieran Smallbone, Eamonn Gaffney, David Gavaghan, Bobs Gatenby and Gillies

• Funded DTC (EPSRC), NCI (NIH)