Cancer can give you Maths

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Cancer can give you Maths

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

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• Very brief overview of cancer growth
• First, mutations lead to cells losing appropriate
  signalling responses for PROLIFERATION (cell
  division) and APOPTOSIS (cell suicide)
• Result a growing mass of cells

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mutations

Approx 1mm in diameter
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• Nutrient required
• Hypoxic core TAF (tumour
  angiogenesis factors)
• Avascular tumour Vascular tumour
• Invasion
• Tumour produces proteases digest ECM
• Competition
• Normal environment

Tumour
Normals
Add H
Gatenby Gawlinski Gap
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T-tumour density V-vascular density
Glycolytic pathway
Blood flow removal
Avascular Case
elsewhere
Nondimensionalise
Necrotic core
Proliferation zone, T const
Outside tumour
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Assume necrosis arises when
constantUsing experimentally
determined parameter values
necrotic core arises at
r 0.1 cm avascular case
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Tumour Growth No normal tissue
Avascular tumour always reaches a benign
steady stateVascular tumour is benign if
invasive if
(cf Greenspan 1972)
necrotic core
Proliferation
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Results

• Three regimes of growth
• If rate of acid removal is insufficient,
• exponential growth followed by auto-toxicity
• benign tumour
• Occurs in avasculars and vasculars if
• vascular tumour displays
  sustained growth and invades
• Very small tumour no growth (insufficient acid
  production to include normal cell death)

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Experimental results (Gatenby)
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• PROBLEM THE GAP PREDICTED BY THIS MODEL IS TOO
  BIG!!!!!
• Introduce quiescent cells (it is known that
  excess acid induces quiescence). These cells
  produce very little acid (Smallbone, Gatenby, PKM
  in prep)

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Metabolic changes during carcinogenesis

• K. Smallbone, D.J. Gavaghan (Oxford)
• R.A. Gatenby, R.J. Gillies (Radiology, Arizona)
• 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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Variation in Metabolite Concentrations
H
glucose
oxygen
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Typical Automaton Evolution
t10, normal epithelium
t100, hyperplasia
O2 diffusion limit
basement membrane
t250, glycolysis
t300, acid-resistance
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Cellular evolution was demonstrated. 1 of 3
spheroids in 15 days and 3 of 3 in 30 days
demonstrated proliferating clusters of GLUT1
positive clusters of cells in normoxic regions.
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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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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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• Vessels source of nutrient (oxygen) satisfy
  Pries-Secomb ??????
• Viscosity Fahraeus-Linqvist effect
• Cells to divide or not to divide?
  Thresholds/cell cycle
• Competition acid etc

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Structural adaptation in normal and cancerous
vasculature

• (PKM, T. Alarcon, H.M. Byrne, M.R. Owen, J.
  Murphy)
• Blood vessels are not static they respond to
  stimuli mechanical and metabolic. Other stimuli
  are
• Conducted stimuli downstream (chemical
• ATP? released under hypoxic stress)
• upstream (along vessel wall changes in membrane
  potential through gap junctions?)

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• Model includes the production of VEGF by cells in
  response to low levels of oxygen (hypoxia). VEGF
  is an angiogenesis factor it produces more
  blood vessels.

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Results

• No VEGF production necrotic cores
• VEGF production extensive hypoxic regions
  within the tumour but few necrotic regions
• Downstream signalling tumours with smaller
  hypoxic regions, more homogeneous distribution of
  oxygen
• Upstream signalling VEGF more concentrated
  around the hypoxic regions

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• Model predicts that the inhomogeneous oxygen
  concentration leads to lower tumour load but
  symmetry is broken.

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References

• Alarcon, Byrne, PKM, JTB, 225, 257-274 (2003) --
  inhomogeneous media
• Alarcon, Byrne, PKM, Prog. Biophys. And Mol.
  Biol., 85, 451-472 (2004)
• Alarcon, Byrne, PKM, JTB, 229, 395-411 (2004)
  cell cycle and hypoxia
• Ribba, Alarcon, Marron, PKM, Agur, BMB, 67, 79-99
  (2005) doxorubicin
• Alarcon, Byrne, PKM, SIAM J. Mult. Mod. Sim, 3,
  440-475 (2005)
• Alarcon, Byrne, PKM, Microvascular Research, 69,
  156-172 (2005) design principles
• Byrne, Alarcon, Owen, Webb, PKM, Phil Trans R Soc
  A, 364, 1563-1578 (2006) --review
• Byrne, Owen, Alarcon, Murphy, PKM, Math Models
  and Methods, 16, 1219-1241 (2006) chemotherapy
• Betteridge, Owen, Byrne, Alarcon, PKM, Networks
  and Hetero. Media, 1, 515-535 (2006) -- cell
  crowding
• Alarcon, Owen, Byrne, PKM, Comp and Math Methods
  in Medicine, 7, 85-119 (2006) vessel
  normalisation

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Summary

• Simple model for acid-mediated invasion
• Hybrid model for somatic evolution
• Multiscale model
• effects of heterogeneity
• structural adaptation in vessels
• drug delivery (NOT COVERED TODAY)

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Acknowledgements

• Acid/somatic evolution Bob Gatenby, Kieran
  Smallbone, David Gavaghan, Mike Brady, Bob
  Gillies (Funded EPSRC DTC)
• Multiscale modelling Tomas Alarcon, Helen Byrne,
  Markus Owen, James Murphy, Russel Betteridge
  (Funded EU RTN (5th and 6th frameworks) IB, NCI
  Virtual Tumour)