Cancer can give you Maths
Philip K. MainiCentre for Mathematical Biology
Mathematical Institute;and
Oxford Centre for Integrative Systems Biology,Biochemistry
Oxford
• 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
mutations
Approx 1mm in diameter
• Nutrient required Hypoxic core TAF (tumour angiogenesis factors)
Avascular tumour Vascular tumour • Invasion Tumour produces proteases – digest ECM• Competition
Normal environment:
Normals
Tumour
Gatenby & Gawlinski Gap
Add H+
),,( 21
1
LNNfN
),( 22
2 LNhLDL
),()(. 21211
2
NNgNNDN
HDF
HHH
2
)( 2
2 dR
dHR
dR
d
R
DVHrr
H
v
T-tumour density
V-vascular density
concHH
Glycolytic pathway
Blood flow removal
Avascular Case: 2RRV vV elsewhere
Nondimensionalise:1rr Necrotic
core2122 '2" rrrrrhhr Proliferation zone,
T = constrrhr 22
Outside tumour
Assume necrosis arises when
constant
Using experimentally determined parameter values necrotic core arises at r = 0.1 cm [avascular case]
Thh
Tumour Growth No normal tissue
•Avascular tumour always reaches a benign steady state
•Vascular tumour is benign if invasive if
3
1
31
32
3
2 )( LRRRSdt
dR
Proliferation
necrotic core
(cf Greenspan 1972)
1Th1Th
ResultsThree regimes of growth:
•If rate of acid removal is insufficient, exponential growth followed by auto-toxicity benign tumourOccurs 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)
1Th1Th
Experimental results (Gatenby)
• 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)
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
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 respiration36 ATP / glucose
Cancer cellsAnaerobic respiration
2 ATP / glucose
Cell-environment Interactions
Nature Rev Cancer 4: 891-899 (2004)
DCIS Model
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
Cellular Metabolism
• Aerobic:• Anaerobic:
• Assume:– All glucose and oxygen used in these two processes– Normal cells under normal conditions rely on aerobic respiration
alone
ATP2acidlactic2glucose ATP36CO6O6glucose 22
Two parameters:n = 1/18
1 < k ≤ 500c
cnc
c
kg
g
gh
ga
c
g
:H
)(:ATP
:oxygen
cellglycolytic
cellnormal:glucose
Automaton Rules
• At each generation, an individual cell’s development is governed by its rate of ATP production φa and extracellular acidity h
– Cell death• Lack of ATP:
• High acidity:
– Proliferation
– Adaptation
resistant-acid
normal
T
Ndea h/h
h/hp
)1/()( 00 aap adiv
0aa
Somatic Evolution
P.C. Nowell, The clonal evolution of tumour cell populations, Science, 194 (4260), 23-28 (1976)
Variation in Metabolite Concentrations
glucose
oxygen
H+
t=10, normal epithelium t=100, hyperplasia
t=250, glycolysis t=300, acid-resistance
O2 diffusionlimit
basementmembrane
Typical Automaton Evolution
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.
• 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)
Cancer Growth
Tissue Level Signalling: (Tumour Angiogenesis Factors) Oxygen etc
Cells:Intracellular: Cell cycle,
Molecular elements
Partial Differential EquationsAutomaton Elements
Ordinary differential equations
• 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
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?)
• 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.
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
• Model predicts that the inhomogeneous oxygen concentration leads to lower tumour load but symmetry is broken.
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
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)
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)