Modelling aspects of solid tumour growth

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Modelling aspects of solid tumour growth

• Philip K. Maini
• Centre for Mathematical Biology
• Mathematical Institute
• Oxford Centre for Integrative Systems Biology,
• Biochemistry
• Oxford

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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

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Outline

• Acid-mediated invasion/Somatic evolution/therapeut
  ic strategies
• ________________________________________
• Vascular Tumour Growth
• Colorectal Cancer

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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?

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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

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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)

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Gatenby-Gawlinski Model
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Bifurcation parameter
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Experimental results (Martin and Jain)
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• Fasano et al, Slow and fast invasion waves (Math
  Biosciences, 220, 45-56, 2009)

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Tumour encapsulation

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

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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

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Cell-environment Interactions
Model
DCIS
Nature Rev Cancer 4 891-899 (2004)
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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

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Cellular Metabolism

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

Two parameters n 1/18 1 lt k 500
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Automaton Rules

• At each generation, an individual cells
  development is governed by its rate of ATP
  production fa and extracellular acidity h
• Cell death
• Lack of ATP
• High acidity
• Proliferation
• Adaptation

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Variation in Metabolite Concentrations
H
glucose
oxygen
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• 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)

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Therapeutics

• Add bicarbonate to neutralise the acid
• (Natasha Martin, Eamonn Gaffney, Robert Gatenby,
  Robert Gillies)

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Metastatic Lesions
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Model Equations Tumour Compartment
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Model Equations Blood Compartment
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Equivalent dose less effective in humans
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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.

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Sensitivity Analysis
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Proton inhibitor bicarbonate
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Clinical Ideas
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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)

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Cancer Growth

• Tissue Level Signalling (Tumour Angiogenesis
  Factors)
• Oxygen etc
• Cells
• Intracellular Cell cycle,
• Molecular elements

Partial Differential Equations
Automaton Elements
Ordinary differential equations
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• Tomas Alarcon
• Markus Owen
• Helen Byrne
• James Murphy
• Russel Bettridge

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Vascular Adaptation

• Series of papers by Secomb and Pries modelling
  vessels in the rat mesentry they conclude
• R(t) radius at time t
• R(tdt) R(t) R dt S

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• 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?)

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• 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)

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Potential uses of the model

• Chemotherapy
• Impact of cell crowding and active movement
• Vessel normalisation

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Angiogenesis

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

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• Movie both2_mov

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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)

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• CHASTE Cancer, Heart And Soft Tissue
  Environment
• Modular

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The effects of different individual cell-based
approaches

• (to appear in Phil Trans R Soc A)

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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)

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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)

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Acknowledgements

• Natasha Martin, Kieran Smallbone, Eamonn Gaffney,
  David Gavaghan, Bobs Gatenby and Gillies
• Funded DTC (EPSRC), NCI (NIH)