Modelling Cancer Growth

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Modelling Cancer Growth

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

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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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Acellular gap at the tumor-host interface in head
and neck cancer
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Hepatocytes
Metastatic tumor
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Tomas Alarcón (UCL)Helen Byrne (Nottingham)EU
RTN (5th Framework) Using mathematical
modelling and computer simulation to improve
cancer therapyAlarcón, Byrne, Maini, J.
Theor. Biol, 225, 257-274 (2003)
Prog. Biophys Mol. Biol., 85,
451-472 (2004)
J. Theor. Biol, 229, 395-411 (2004)
SIAM Multiscale Mod Sim.3,
440-475 (2005) Ribba, Marron, Agur, Alarcon,
Maini Bull. Math. Biol., 67 79-99 (2005)
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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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Tumour Growth

• First, work out distribution of 02 (nutrient)
• To do so, must consider vasculature metabolic
  response
• R radius
• flow rate
• H haematocrit
• Tw WSS
• P pressure (transmural)
• Haematocrit Pries et al,
  1994
• At a bifurcation

Response to mechanical stimuli
Shrinkage (Pries et al 1988)
(rat mesentry)
(Fung 1993)
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Algorithm for structural adaptation

• Prescribed
• 2. Given initial network configuration, compute
  flow rates
• through and pressure drops across each vessel
  using Kirchoffs
• law.
• 3. Compute distribution of haematocrit.
• 4. Update radius of each vessel.
• 5. Compute viscosities (using H and R from 3 and
• 4 respectively).
• 6. Repeat until steady state reached.

Flow rate
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____________ O2 distribution____________
(adiabatic approx)
PO2 conc
?N for normal cell
?c for cancer cell
0 o.w
Nw normal to vessel wall
?b ?2 level in blood
P permeability
(at edge of domain, no flux
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Automaton Rules

• 1. ?2 distribution determined by BVP.
• 2. Cells attempt to divide at each time step.
• 3. Normal cell if ?2lt threshold, cell dies
• ?2 gt threshold, cell
  attempts to divide
• Threshold ?1 if more normal than cancer
  neighbours
• ?T2 if more cancer than normal
  neighbours
• ?T2gt ?T1
• 4. Cancer cell if ?2gt threshold, cell attempts
  to divide
• Threshold ?T1 if more cancer than normal
  neighbours
• ?T2 if more normal than cancer
  neighbours
• ? T2 gt ?T1

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5. Cancer cell if O2lt threshold
cell becomes
quiescent If it
remains quiescent for a certain length of time,
it dies.
6. Cells are sinks of O2 7. If O2 level is
such that a cell may divide, sample neighbourhood
for space. If more than one available space,
go to the one with largest O2 (Patel et al 2001).
If no space, die (Kansal et al, 2000)
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Cell Dynamics

• NxN automaton elements.
• State vector has 3 components
• Occupation normal cell/cancer cell/vessel/empty
• Cell status proliferative/quiescent
• Local ?2 conc
• We assume, for simplicity, vessel structures does
  not evolve.

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Conclusion

• Environmental heterogeneity decreases cancer cell
  growth but may contribute to metastasis

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Possible application

• Doxorubicin treatment of non-Hodgkins lymphoma
  (Ben Ribba, Zvia Agur, Tomas Alarcon, Philip
  Maini, K Marron)
• Structural adaptation vessels surrounded
• by NHL
  leaky unstable
• Nutrient diffusion
• -Drug pharmacokinetics in plasma
• pharmacodynamics kills proliferating
  cells
• tissue dynamics (adiabatic approx)
• AIM Explore different protocols of treatment
• (presently a 21-day cycle is employed)

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Cancer-Proliferation
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Cell-Cycle Dynamics

• Why?
• nutrient demand
• hypoxia-induced quiescence
• drugs work only on cells in a certain part of
• their cell cycle.
• Cell Cycle
• Cyclin-dependent kinases (CDK)
• cyclins
  
• 
  
• In G1 CDK activity is low because its
  cyclin partners
• 
  are missing
• At finish Cdhl (and Cdc 20) concs are high
  
• degrade
  cyclins.

2 families of proteins
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schematic
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Tyson Novak

• Model for G1/S transition

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E2F transcription factor Take Tyson and Novak
modelincorporate inhibition by a Kz term
P27 conc in Cdhl
oxygen
Normals
Growth regulation
hypoxia
as m z

Cancer Cells
Hypothesis growth regulation
is lost
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Results
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• Simulations show decrease in Cdk
• This is observed experimentally

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Growth regulation of p27?
Normals ?

• Growth factors p27
• If growth is arrested, p27 is upregulated

Cancer x
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Response to hypoxia (low O2)Expts on mouse
embryo fibroblasts
hypoxiaNormal cells G1
arrest Does not occur with p27 null mutants
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CONCLUSION

• x heterogeneities have a profound effect
• on tumour dynamics
• x effects of p27 possible mechanism
• x efficiency of drug treatments
• Future Directions
• VEGF
• HIF-1
• Elasticity

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Role of Acidity

• Kieran Smallbone, David Gavaghan, Bob Gatenby, PKM

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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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Vascular Case
elsewhere
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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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Include normal tissue Normal cells die if
So, tumour will only advance if
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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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PH profiles in 6 directions
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Acid moves in direction of arrows
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Therapies

• Kill tumour cells or cut them out
• Anti-angiogenesis drugs drug delivery
• Treat normals?

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Therapies?

• Reduce vasculature in tumour
• tumour poisons itself
• Destroy membrane pumps transporting H ions from
  tumour
• Increase acidity!

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