Transcript Modelling aspects of solid tumour growth

Modelling aspects of solid tumour
growth
Philip K. Maini
Centre 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
DCIS
Model
Nature Rev Cancer 4: 891-899 (2004)
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:
glucose  6 O2  6 CO2  36AT P
glucose  2 lacticacid  2 AT P
• Assume:
– All glucose and oxygen used in these two processes
– Normal cells under normal conditions rely on aerobic respiration
alone
normalcell
g
glucose :  g  
kg glycolyticcell
oxygen: c  c
AT P:
 a  c  n( g  c)
H :
h   g  c
Two parameters:
n = 1/18
1 < k ≤ 500
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
 a  a0
pdea
h/hN

 h/hT
normal
acid - resistant
pdiv  (a  a0 ) /(1  a0 )
Variation in Metabolite Concentrations
H+
glucose
oxygen
• 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
Partial Differential Equations
Automaton Elements
Cells:
Intracellular: Cell cycle, Ordinary differential equations
Molecular elements
• 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)