Modelling aspects of solid cancer growth

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Modelling aspects of solid cancer growth

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

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Colorectal Cancer

• Second leading cause of cancer deaths in the US

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Cell population dynamics in the colonic crypt

• Johnston, Edwards, Bodmer, PKM, Chapman, PNAS,
  104, 4008-4013 (2007)
• Crypt can be considered as 3-compartments
• Stem cells,
• Semi-Differentiated (transit-amplifying)
  Cells,
• Fully Differentiated Cells

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A previous model

• Tomlinson and Bodmer, PNAS, 92, 11130-11134
  (1995) proposed cell cycle synchrony and no
  feedback
• DOnofrio, Tomlinson, JTB, 244, 367-374 (2006)
  feedback, but still cell synchrony
• Both ignore the compounding effect of TA cells
  cycling faster than stem cells

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Continuous Model

• Interested on time scales greater than the cell
  cycle time continuous cell division

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Steady States

• For both these models, non-trivial steady states
  (corresponding to homeostatis) only occur if the
  parameters take specific values --- STRUCTURALLY
  UNSTABLE

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Need Feedback

• Wodarz 2007 (mutations) feedback in stem cell
  compartment
• Boman et al 2001 no feedback in stem cells so
  they tend to zero
• Komarova series on papers on mutations

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Model 1 Linear Feedback

• Assume that when the population of stem or TA
  cells increase, the per capita rate at which they
  differentiate increases in proportion

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Steady States

• Stem cells exhibit logistic growth nice bounded
  stable steady state solution
• For a very large region in parameter space this
  model predicts homeostasis
• Only a genetic hit which removes the feedback in
  the model will lead to unbounded growth

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Model 2 Saturating Feedback

• Assume maximum per-capita rate of differentiation

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Steady States

• Nice homeostatic steady state as long as net
  linear growth rate of stem cells lies in a
  certain region in parameter space. If a mutation
  moves us out of this region then the population
  grows unbounded, even in the presence of feedback

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Conclusions

• Developed a robust model for cell populations in
  the crypt
• Shown that the key parameters are the net
  per-capita growth rates of the stem cells and TA
  cells. So, the failure of programmed cell death
  or differentiation could lead to tumour growth,
  as well as increased proliferation rate
• Saturating feedback could explain the existence
  of benign tumours before carcinogenesis takes
  over early mutations could keep parameters
  below their critical values, later stage
  mutations could push them above their critical
  values.
• Evidence suggests that nearly all colorectal
  cancers go through benign stages, but not all
  develop into carcinomas

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Some bad things

• Continuum approximation individual-based model
  approach in IB
• Handling of mutations properly including mutant
  populations
• Space started on this reaction-advection
  equations

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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, to appear)

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• CHASTE Cancer, Heart And Soft Tissue
  Environment
• Modular

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Metabolic changes during carcinogenesis

• K. Smallbone, D.J. Gavaghan (Oxford)
• R.A. Gatenby, R.J. Gillies (Radiology, Arizona ?
  Moffitt Cancer Inst, Tampa)
• J.Theor Biol, 244, 703-713, 2007

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Introduction

• Carcinogenesis
• The generation of cancer from normal cells
• An evolutionary process selective pressures
  promote proliferation of phenotypes best-suited
  to their microenvironment

Normal cellsAerobic respiration 36 ATP / glucose
Cancer cells Anaerobic respiration 2 ATP / glucose
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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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Somatic Evolution

• P.C. Nowell, The clonal evolution of tumour cell
  populations, Science, 194 (4260), 23-28 (1976)

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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)
• More recently, we have been trying to write a
  continuum model for this

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Vascular Tumour Growth
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Vacular 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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• It has been proposed that this fingering is
  produced by some sort of symmetry-breaking
  bifurcation (Turing)
• However, we suggest that heterogeneities in the
  environment do it some nice work by Anderson et
  al on ECM-heterogeneities (adhesivity)

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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 (eye-ball norm) a reasonable
  assumption
• Tim Secomb has shown this more convincingly
  recently

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Illustrative potential uses of the model

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

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Acknowledgements

• Acid/somatic evolution Bob Gatenby, Kieran
  Smallbone, David Gavaghan, Bob Gillies (Funded
  EPSRC DTC NCI Virtual Tumour)
• Colorectal models (Matt Johnston, Walter Bodmer,
  Jon Chapman (EPSRC) and IB group)
• Vascular Tumour (EU RTN, Tomas Alarcon, Helen
  Byrne, Markus Owen)