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Modelling aspects of solid tumour growth

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Title: Modelling aspects of solid tumour growth


1
Modelling aspects of solid tumour growth
  • Philip K. Maini
  • Centre for Mathematical Biology
  • Mathematical Institute
  • Oxford Centre for Integrative Systems Biology,
  • Biochemistry
  • Oxford

2
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

3
Outline
  • Acid-mediated invasion/Somatic evolution/therapeut
    ic strategies
  • ________________________________________
  • Vascular Tumour Growth
  • Colorectal Cancer

4
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?

5
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

6
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)

7
Gatenby-Gawlinski Model
8
Bifurcation parameter
9
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12
Experimental results (Martin and Jain)
13
  • Fasano et al, Slow and fast invasion waves (Math
    Biosciences, 220, 45-56, 2009)

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17
Tumour encapsulation
  • Predicts ECM density is relatively unchanged
    inconsistent with other models but consistent
    with experimental observations.

18
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19
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

20
Cell-environment Interactions
Model
DCIS
Nature Rev Cancer 4 891-899 (2004)
21
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

22
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
23
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

24
Variation in Metabolite Concentrations
H
glucose
oxygen
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27
  • 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)

28
Therapeutics
  • Add bicarbonate to neutralise the acid
  • (Natasha Martin, Eamonn Gaffney, Robert Gatenby,
    Robert Gillies)

29
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30
Metastatic Lesions
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33
Model Equations Tumour Compartment
34
Model Equations Blood Compartment
35
Equivalent dose less effective in humans
36
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.

37
Sensitivity Analysis
38
Proton inhibitor bicarbonate
39
Clinical Ideas
40
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)

41
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42
Cancer Growth
  • Tissue Level Signalling (Tumour Angiogenesis
    Factors)
  • Oxygen etc
  • Cells
  • Intracellular Cell cycle,
  • Molecular elements

Partial Differential Equations
Automaton Elements
Ordinary differential equations
43
  • Tomas Alarcon
  • Markus Owen
  • Helen Byrne
  • James Murphy
  • Russel Bettridge

44
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46
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

47
  • 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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49
  • 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)

50
Potential uses of the model
  • Chemotherapy
  • Impact of cell crowding and active movement
  • Vessel normalisation

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52
Angiogenesis
  • Recently, we have added in angiogenesis (Owen,
    Alarcon, PKM and Byrne, J.Math. Biol, 09) and
    gone to 3D (Holger Perfahl)

53
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54
  • Movie both2_mov

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56
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)

57
  • CHASTE Cancer, Heart And Soft Tissue
    Environment
  • Modular

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62
The effects of different individual cell-based
approaches
  • (to appear in Phil Trans R Soc A)

63
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)

64
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)

65
Acknowledgements
  • Natasha Martin, Kieran Smallbone, Eamonn Gaffney,
    David Gavaghan, Bobs Gatenby and Gillies
  • Funded DTC (EPSRC), NCI (NIH)
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