Modelling Cancerous Tumour Dynamics

1
Modelling Cancerous Tumour Dynamics

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

2

• 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

3
mutations

Approx 1mm in diameter
4
Multicellular spheriod

• Limited size due to limited oxygen (nutrient)
  diffusion
• Necrotic core dead cells --- starts to make
  chemicals

5

• ANGIOGENESIS new blood vessels
• new blood supply and nutrients for the tumour

6

• 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
7
Acellular gap at the tumor-host interface in head
and neck cancer
8
Role of Acidity

• Kieran Smallbone, David Gavaghan, Bob Gatenby, PKM

9
(No Transcript)
10

T-tumour density V-vascular density
Glycolytic pathway
Blood flow removal
Avascular Case
elsewhere
Nondimensionalise
Necrotic core
Proliferation zone, T const
Outside tumour
11
(No Transcript)
12
Assume necrosis arises when
constantUsing experimentally
determined parameter values
necrotic core arises at
r 0.1 cm avascular case
13
New explanation for formation of dead (necrotic)
core

• As the mass grows, it produces so much acid it
  poisons itself. Now it needs a blood supply to
  pump away the acid.

14
Vascular Case
elsewhere
15
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
16
(No Transcript)
17
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)

18
Experimental results (Gatenby)
19
PH profiles in 6 directions
20
(No Transcript)
21
Multiscale Modelling

• WHY???????????????
• Effects of combination treatment
  anti-angiogenesis plus chemotherapy

22
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)
23
Cancer Growth

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

Partial Differential Equations
Automaton Elements
Ordinary differential equations
24
(No Transcript)
25
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
26
schematic
27
Tyson Novak

• Model for G1/S transition

28
(No Transcript)
29
(No Transcript)
30
(No Transcript)
31
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
32
Results
33
(No Transcript)
34
(No Transcript)
35
(No Transcript)
36
(No Transcript)
37

• Simulations show decrease in Cdk
• This is observed experimentally

38
Growth regulation of p27?
Normals ?

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

Cancer x
39
Response to hypoxia (low O2)Expts on mouse
embryo fibroblasts
hypoxiaNormal cells G1
arrest Does not occur with p27 null mutants
40
(No Transcript)
41
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?)

42
(No Transcript)
43
(No Transcript)
44

• 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.

45
(No Transcript)
46
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

47
(No Transcript)
48
(No Transcript)
49

• Model predicts that the inhomogenesis oxygen
  concentration leads to lower tumour load but
  symmetry is broken (other models, assuming
  homogeneity require symmetry-breaking mechanisms)

50
(No Transcript)
51
Conclusion

• Environmental heterogeneity decreases cancer cell
  growth but may contribute to metastasis
• Testable predictions on effects of
  anti-angiogenesis treatments, and on aspects of
  abnormal structural adaptation in blood vessels
  (note, the base model we use here is empirical
  and has no mechanism!!!)

52
(No Transcript)
53
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)

54
(No Transcript)
55
(No Transcript)
56

• Model can predict the effects and efficiency of
  different drug application protocols

57
(No Transcript)
58
Metabolic changes during carcinogenesis

• K. Smallbone, D.J. Gavaghan (Oxford)
• R.A. Gatenby, R.J. Gillies (Radiology, Arizona)

59
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
60
Cell-environment Interactions
Model
DCIS
Nature Rev Cancer 4 891-899 (2004)
61
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

62
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
63
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

64
Typical Automaton Evolution
t10, normal epithelium
t100, hyperplasia
O2 diffusion limit
basement membrane
t250, glycolysis
t300, acid-resistance
65
Variation in Metabolite Concentrations
H
glucose
oxygen
66
Accumulation of Heritable Changes
Hyperplasia ? Glycolysis ? Acid-resistance
67
In conclusion

• Results
• Hypoxia common in premalignant lesions such as
  DCIS
• Glucose supply not a limiting factor over
  length-scales of carcinogenesis
• Upregulation of glycolysis represents an
  adaptation to hypoxia
• Further work
• Model carcinogenesis as competing populations
  using PDEs
• Less accurate, easier mathematics
• Compare to stochastic (CA) model

68
(No Transcript)
69
Summary

• Heterogeneity can have profound affects on tumour
  dynamics (structural adaptation in vessels)
• Possible mechanisms for hypoxia-induced
  quiescence
• Acid-mediated hypothesis of tumour invasion
• Selection and evolution of tumour phenotypes

70
(No Transcript)