Cancer can give you Maths Philip K. Maini Centre 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+

T-tumour density V-vascular density Glycolytic pathway Blood flow removal Avascular Case: elsewhere Nondimensionalise: Necrotic core Proliferation zone, T = const Outside tumour

Assume necrosis arises when constant Using experimentally determined parameter values necrotic core arises at r = 0.1 cm [avascular case]

Tumour Growth No normal tissue Avascular tumour always reaches a benign steady state Vascular tumour is benign if invasive if Proliferation necrotic core (cf Greenspan 1972)

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)

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, , 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 cells Aerobic respiration 36 ATP / glucose Cancer cells Anaerobic respiration 2 ATP / glucose

Cell-environment Interactions Nature Rev Cancer 4: (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 Two parameters: n = 1/18 1 < k ≤ 500

Automaton Rules At each generation, an individual cell’s development is governed by its rate of ATP production φ a and extracellular acidity h –Cell death Lack of ATP: High acidity: –Proliferation –Adaptation

Somatic Evolution P.C. Nowell, The clonal evolution of tumour cell populations, Science, 194 (4260), (1976)

Variation in Metabolite Concentrations glucose oxygen H+H+

t=10, normal epitheliumt=100, hyperplasia t=250, glycolysist=300, acid-resistance O 2 diffusion limit basement membrane 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, (2007)

Cancer Growth Tissue Level Signalling: (Tumour Angiogenesis Factors) Oxygen etc Cells: Intracellular: Cell cycle, Molecular elements Partial Differential Equations Automaton 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, (2003) -- inhomogeneous media Alarcon, Byrne, PKM, Prog. Biophys. And Mol. Biol., 85, (2004) Alarcon, Byrne, PKM, JTB, 229, (2004) – cell cycle and hypoxia Ribba, Alarcon, Marron, PKM, Agur, BMB, 67, (2005) – doxorubicin Alarcon, Byrne, PKM, SIAM J. Mult. Mod. Sim, 3, (2005) Alarcon, Byrne, PKM, Microvascular Research, 69, (2005) – design principles Byrne, Alarcon, Owen, Webb, PKM, Phil Trans R Soc A, 364, (2006) --review Byrne, Owen, Alarcon, Murphy, PKM, Math Models and Methods, 16, (2006) – chemotherapy Betteridge, Owen, Byrne, Alarcon, PKM, Networks and Hetero. Media, 1, (2006) -- cell crowding Alarcon, Owen, Byrne, PKM, Comp and Math Methods in Medicine, 7, (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 (5 th and 6 th frameworks) IB, NCI Virtual Tumour)