modelling aspects of solid tumour growth
DESCRIPTION
Modelling aspects of solid tumour growth. Philip K. Maini Centre for Mathematical Biology Mathematical Institute; Oxford Centre for Integrative Systems Biology, Biochemistry; Oxford. More precisely. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/1.jpg)
Modelling aspects of solid tumour growth
Philip K. MainiCentre for Mathematical Biology
Mathematical Institute;Oxford Centre for Integrative Systems Biology,
Biochemistry;Oxford
![Page 2: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/2.jpg)
More precisely
• Using mathematical models to explore the interaction of a VERY SMALL subset of processes in cancer with a view to increasing our intuition in a very small way
and eventually …
![Page 3: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/3.jpg)
Outline
• Acid-mediated invasion/Somatic evolution/therapeutic strategies
________________________________________
• Vascular Tumour Growth
• Colorectal Cancer
![Page 4: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/4.jpg)
Cancer
Cell proliferation and cell death (apoptosis) are tightly controlled by genes to maintain homeostasis (steady state). Mutations in these genes upset the balance and the system moves out of steady state.
How can we control a growing population of cells?
![Page 5: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/5.jpg)
The Warburg Effect
• Tumour cells undergo glycolytic (anaerobic) metabolism presumably because there is a lack of oxygen.
• But sometimes in the presence of sufficient oxygen they still do this – seems very strange because it is 20 times less efficient than aerobic metabolism
![Page 6: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/6.jpg)
Acid Mediated Invasion Hypothesis
• A bi-product of the glycolytic pathway is lactic acid – this lowers the extracellular pH so that it favours tumour cell proliferation AND it is toxic to normal cells.
• Gatenby and Gawslinski (1996)
![Page 7: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/7.jpg)
Gatenby-Gawlinski Model
![Page 8: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/8.jpg)
Bifurcation parameter
![Page 9: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/9.jpg)
![Page 10: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/10.jpg)
![Page 11: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/11.jpg)
![Page 12: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/12.jpg)
Experimental results (Martin and Jain)
![Page 13: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/13.jpg)
• Fasano et al, Slow and fast invasion waves (Math Biosciences, 220, 45-56, 2009)
![Page 14: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/14.jpg)
![Page 15: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/15.jpg)
![Page 16: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/16.jpg)
![Page 17: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/17.jpg)
Tumour encapsulation
• Predicts ECM density is relatively unchanged – inconsistent with other models but consistent with experimental observations.
![Page 18: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/18.jpg)
![Page 19: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/19.jpg)
Metabolic changes during carcinogenesis
K. Smallbone, D.J. Gavaghan (Oxford)R.A. Gatenby, R.J. Gillies (Moffitt
Cancer Research Inst)J.Theor Biol, 244, 703-713, 2007
![Page 20: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/20.jpg)
Cell-environment Interactions
Nature Rev Cancer 4: 891-899 (2004)
DCIS Model
![Page 21: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/21.jpg)
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
![Page 22: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/22.jpg)
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
![Page 23: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/23.jpg)
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
![Page 24: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/24.jpg)
Variation in Metabolite Concentrations
glucose
oxygen
H+
![Page 25: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/25.jpg)
![Page 26: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/26.jpg)
![Page 27: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/27.jpg)
• 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)
![Page 28: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/28.jpg)
Therapeutics
• Add bicarbonate to neutralise the acid
(Natasha Martin, Eamonn Gaffney, Robert Gatenby, Robert Gillies)
![Page 29: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/29.jpg)
![Page 30: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/30.jpg)
Metastatic Lesions
![Page 31: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/31.jpg)
![Page 32: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/32.jpg)
![Page 33: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/33.jpg)
Model Equations: Tumour Compartment
![Page 34: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/34.jpg)
Model Equations: Blood Compartment
![Page 35: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/35.jpg)
Equivalent dose less effective in humans
![Page 36: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/36.jpg)
Analysis
• There are 3 timescales and lots of small and large parameters so can do asymptotics and obtain an approximate uniformly valid solution on which to do sensitivity analysis.
![Page 37: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/37.jpg)
Sensitivity Analysis
![Page 38: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/38.jpg)
Proton inhibitor + bicarbonate
![Page 39: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/39.jpg)
Clinical Ideas
![Page 40: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/40.jpg)
Effects of Exercise
• Periodic pulsing of acid may affect somatic evolution by delaying the onset of the invasive phenotype (hyperplastic, glycolytic and acid-resistant) (Smallbone, PKM, Gatenby, Biology Direct, 2010)
![Page 41: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/41.jpg)
![Page 42: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/42.jpg)
Cancer Growth
Tissue Level Signalling: (Tumour Angiogenesis Factors) Oxygen etc
Cells:Intracellular: Cell cycle,
Molecular elements
Partial Differential EquationsAutomaton Elements
Ordinary differential equations
![Page 43: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/43.jpg)
• Tomas Alarcon
• Markus Owen
• Helen Byrne
• James Murphy
• Russel Bettridge
![Page 44: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/44.jpg)
![Page 45: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/45.jpg)
![Page 46: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/46.jpg)
Vascular Adaptation
• Series of papers by Secomb and Pries modelling vessels in the rat mesentry – they conclude:
R(t) = radius at time t:
R(t+dt) = R(t) + R dt S
![Page 47: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/47.jpg)
S = M + Me – s + C
M = mechanical stimulus (wall shear stress)
Me = metabolic demand
s = shrinkage
C= conducted stimuli: short-range (chemical release under hypoxic stress?)
long-range (mediated through membrane potential?)
![Page 48: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/48.jpg)
![Page 49: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/49.jpg)
• By varying the strengths of the different adaptation mechanisms we can hypothesise how defects in vasculature lead to different types of tumours: Conclude that losing the long range stimuli looks a reasonable assumption
• Tim Secomb has shown this more convincingly recently (PLoS Comp Biol 2009)
![Page 50: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/50.jpg)
Potential uses of the model
• Chemotherapy
• Impact of cell crowding and active movement
• Vessel normalisation
![Page 51: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/51.jpg)
![Page 52: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/52.jpg)
Angiogenesis
• Recently, we have added in angiogenesis (Owen, Alarcon, PKM and Byrne, J.Math. Biol, 09) and gone to 3D (Holger Perfahl)
![Page 53: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/53.jpg)
![Page 54: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/54.jpg)
• Movie – both2_mov
![Page 55: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/55.jpg)
![Page 56: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/56.jpg)
An integrative computational model for intestinal tissue renewal
• Van Leeuwen, Mirams, Walter, Fletcher, Murray, Osbourne, Varma, Young, Cooper, Pitt-Francis, Momtahan, Pathmanathan, Whiteley, Chapman, Gavaghan, Jensen, King, PKM, Waters, Byrne (Cell Proliferation, 2009)
![Page 57: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/57.jpg)
• CHASTE – Cancer, Heart And Soft Tissue Environment
• Modular
![Page 58: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/58.jpg)
![Page 59: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/59.jpg)
![Page 60: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/60.jpg)
![Page 61: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/61.jpg)
![Page 62: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/62.jpg)
The effects of different individual cell-based approaches
• (to appear in Phil Trans R Soc A)
![Page 63: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/63.jpg)
Conclusions and Criticisms• Simple multiscale model – gain some insight into why combination
therapies might work
• Heterogeneities in environment play a key role
• No matrix included! – Anderson has shown adhesivity could be important
• Cellular automaton model – what about using Potts model, cell centred, cell vertex models? – DOES IT MAKE A DIFFERENCE (Murray et al, 2009; Byrne et al, 2010)
• There are many other models and I have not referred to any of them! (Jiang, Bauer, Chaplain, Anderson, Lowengrub, Drasdo, Meyer-Hermann, Rieger, Cristini, Enderling, Meinke, Loeffler, TO NAME BUT A FEW)
![Page 64: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/64.jpg)
Acknowledgements
• Colorectal: David Gavaghan, Helen Byrne, James Osborne, Alex Fletcher, Gary Mirams, Philip Murray, Alex Walter, Joe Pitt-Francis et al (EPSRC)
• Vascular: Tomas Alarcon, Helen Byrne, Markus Owen, Holger Perfahl (EU -5th and 6th frameworks)
![Page 65: Modelling aspects of solid tumour growth](https://reader036.vdocuments.site/reader036/viewer/2022062721/56813679550346895d9e0731/html5/thumbnails/65.jpg)
Acknowledgements
• Natasha Martin, Kieran Smallbone, Eamonn Gaffney, David Gavaghan, Bobs Gatenby and Gillies
• Funded DTC (EPSRC), NCI (NIH)