SSP talk

690 views

Published on

Presentation on theoretical biology, risk taking in science to high school students at the Summer Science Program in Socorro, NM. June 2013

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
690
On SlideShare
0
From Embeds
0
Number of Embeds
473
Actions
Shares
0
Downloads
4
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

SSP talk

  1. 1. What the study of the stars can teach us about cancer: There’s no success like failure, and failure’s no success at all. Jacob G. Scott Key Factors in the Metas from Populatio Christopher McFarland1*, Jacob Scott2,3,*, David Bas 1Harvard-MIT Division of Health Science & Technology, 2H. Lee Moffitt Oncology, *contributed equally to this workoorly understood process that aths not explained by deterministic the genomic level and use explore this phenomenon Res Cells deriv Metastatic Three regi metastasis which only We further Feature of Model Observed Phenomenon Population size determined by fitness of cells Larger Tumors more likely to metastasize Cells can acquire passenger mutations that are slightly deleterious Many micrometastases never grown to macroscopic size Cells with more mutations are less likely to metastasize Stromal environment reduces efficacy of driver mutations Certain stromal conditions prohibit metastasis Metastases with same Key Factors in the Meta from Populat Christopher McFarland1*, Jacob Scott2,3,*, David B 1Harvard-MIT Division of Health Science & Technology, 2H. Lee Mo Oncology, *contributed equally to this work Background: Metastasis is a highly lethal and poorly understood process that accounts for the majority cancer deaths Patterns of metastatic spread are not explained by deterministic explain these patterns We develop a stochastic model at the genomic level and use population genetics techniques to explore this phenomenon R Cells Meta Three metas which We fu Feature of Model Observed Phenomenon Population size determined by fitness of cells Larger Tumors more likely to metastasize Cells can acquire passenger mutations that are slightly deleterious Many micrometastases never grown to macroscopic size Cells with more mutations are less likely to metastasize Stromal environment reduces efficacy of driver mutations Certain stromal conditions prohibit metastasis Radiation Oncology and Integrative Mathematical Oncology further, the stem compartment can differentiate at a rate β. Each population als growth (r) and death (d) rate, proportional to their population. FIG. 3: To capture the behavior of a putative compartment system in which there is only e from the stem compartment into the TAC compartment, death and growth in both, but by a common carrying capacity. We therefore write a system of ODEs as: Stem compartment : dN0 dt = growth r0N0(1 − N K ) − differentiation β0N0 − death d0N0 Differentiated : dN1 dt = growth r1N1(1 − N K ) + differentiation β0N0 − death d1N1 Where: Phys. Biol. 8 (2011) 015016 D Basanta et al Table 1. The four phenotypes in the game are autonomous growth (AG), invasive (INV), glycolytic (GLY) and invasive glycolytic (INV-GLY). The base payoff in a given interaction is r and the cost of moving to another location with respect to the base payoff is c. The fitness cost of acidity is n and k is the fitness cost of having a less efficient glycolytic metabolism. The benefits from having access to the vasculature as a result of angiogenesis are reflected by the parameter α. AG INV GLY INV-GLY AG 1 2 + α 2 1 1 2 − n + α 1 2 − n + α INV 1 − c 1 − c 2 1 − c 3 1 − c 3 GLY 1 2 − k + n + α 1 − k + α 2 1 2 − k + α 4 1 − k + α 2 INV-GLY 1 2 − k + n + α 1 − k + α 2 1 − c 3 − k + α 2 1 − k − c 6 + α 2 Table 2. List of variables used by the model. Value Affected phenotypes Meaning c INV, INV-GLY Cost of motility k GLY, INV-GLY Cost having a glycolytic metabolism n AG, INV Cost of living in an acid microenvironment to leaky or otherwise defective vascularization [9]. This is shown in the table by the fact that AG cells interacting with other AG cells (assumed to produce only moderate amounts of HIF-1α) receive a benefit of α 2 from the moderate angiogenic vasculature. On the other hand, AG cells interacting with GLY cells produce, in combination, an optimal amount of HIF-1α and obtain in return the total benefit derived from functioning vascularity (α). Finally, as IDH-1 mutant GLY cells proliferate producing excessive amounts of HIF-1α, the benefit of angiogenesis is a reduced α 4 , consistent with the angiogenic vasculature being leaky and inefficient in this case. Another notable difference with the previous model is that the cost of motility is assumed to be smaller in the presence of acid-producing glycolytic phenotypes. This is represented by a cost of motility c 3 and represents the acid-mediated invasion [21–23] of glioma cells throughout the brain, particularly along the myelinated neuronal axons in the white matter of the brain along which glioma cells are known to quickly invade [24, 25]. This reduced cost of motility also quantifies and models the generally invasive characteristics of gliomas which are well known for their diffuse invasion that has been quantified in In one dimension, this becomes: ∂ ∂x   D ∂c ∂x + χc ∂a ∂x    = Dc ∂2 c ∂x2 − cχ(a) ∂2 a ∂x2 − χ ∂c ∂x ∂a ∂x (2.18) Additionally, we must consider the creation and dispersal of this chemoattractant, a. To do this, we assume Fickian diffusion as for the cells in the initial model as per 2.8, and a creation term that is linearly related to the death of the cells by a coefficient, ω. Further, we introduce a consumption term, the rate at which the chemoattractant is consumed by the cells, linearly related to the number of cells by a coefficient, µ. Therefore, we can write down a full model for both the cellular concentration, as derived above, and for the chemoattractant, a, thus: rate of change of glioma cell concentration ∂c ∂t = net dispersal of glioma cells ∇ · (Dc∇c) + net growth of glioma cells ρc(1 − c K ) − chemotaxis of glioma cells ∇ · (cχ(a)∇a) − death of glioma cells λc , (2.19) rate of change of chemotactic factor ∂a ∂t = net dispersal of chemotactic factor ∇ · (Da∇a) + creation of chemotactic factor λcω − consumption of chemotactic factor µca . (2.20) And again in 1-dimension: ∂c ∂t = D ∂2 c ∂x2 − cχ(a) ∂2 a ∂x2 − χ ∂c ∂x ∂a ∂x + c(1 − c Kc ) − λc, (2.21) ∂a ∂t = Da ∂2 a ∂x2 + λcωa − µca. (2.22) While the death term has remained a constant, λ, times the population, this addition does little to effect the overall dynamics. Only with very large parameter changes, likely large enough to be physically unrealistic, ?
  2. 2. Tortuous Path •Hawken ’94 •US Naval Academy ’98, Physics Major •Navy Nuclear Reactor Engineer ‘98-’03 •High School Physics Teacher Florida, ’03-’04 •Case Western Reserve MD 2004-2009 •Radiation Oncology Resident Tampa, FL, ’09-Present •Oxford D.Phil candidate - mathematics
  3. 3. Net worth. -300000 -225000 -150000 -75000 0 75000 1993 1995 1997 1999 2001 2003 2005 2007 2009 2011 2013 $$ Hawken Diploma MSBS Entrance to medical school MD
  4. 4. May 1783, John Goodricke - 100GBP
  5. 5. ~10-7 meters - 1,000 Angstroms ~1021 meters (106 light years)
  6. 6. Important to understand your limitations: Dogs are so cute when they try to comprehend quantum mechanics - I’m not. As Mr. Dlugozs will tell you, I am a terrible bench biologist - yes, I tried.
  7. 7. ~100 meters
  8. 8. MCAT
  9. 9. Radiation Oncology The study of cancer - a disease on the human length scale
  10. 10. Ptolemy’s geocentric solar system and crystalline heavens (c. AD 90 – c. AD 168)
  11. 11. measurement meta- phenomenological laws conserved laws (14 December 1546 – 24 October 1601) (December 27, 1571 – November 15, 1630) (25 December 1642 – 20 March 1727) mechanism (14 March 1879 – 18 April 1955 (29 April 1854 – 17 July 1912
  12. 12. Hippocrates - the four humors model of physiology
  13. 13. Mr. Dlugosz The Cell Cycle - IPMAT The cycle is broken in cancer!
  14. 14. suggest we all adopt ideologies such side of the majority, but to escape j.stebbing Essay Phase itrialist There is a new breed of clinical trialist in cancer research. You might not have seen them yet—they will not be knocking down your door in the clinic. They do not know what HIPAA stands for. They do not know what to do in a code. They do not wear a white coat, you will be lucky if they wear a tie. They are not biologists—if you ask them to change the media, they will probably bring you some music you have not heard. They are the phase i trialists. What is a phase i trial? It is a preclinical trial, but one in which no cells, mice, or rats will be harmed. Before one begins killing cells in a dish, there is the step to decide how to treat those cells or mice in a sensible, yet new way. It is in this phase, before even stepping into a laboratory, in which we are now seeing an influx of other types of scientists—physicists, engineers, and mathematicians. Some of these folks have run out of problems in their field and have found fertile ground for their tools and physical science perspective in the dizzying biological complexity of cancer. Others have become frustrated by the esoteric nature of their first specialty—it takes a special mind to be happy studying things in other galaxies, or things so small that you need a super collider spanning three countries to learn anything new. And then, some are just naturally dreamers, or follow their hearts into a specialty that has affected their you turn research as app to gene energy people w be expla we can b or a com how a tu how a pe phase i. The bi right no and scie with a to bet grant m mathem to think they too until the from th Cancer and Society From experience, we believe that doctors are far less accepting of such ideas than patients or healthy non- physicians. Oncologists are more tolerant of the concept of ginger as a treatment for chemotherapy-induced nausea becausethese data come from a large trial with sound statistical analysis. However, its potential as an anticancer drug directly conflicts with the beliefs of most physicians, even though no precise mechanism of action has been confirmed for either potential use. We certainly do not suggest we all adopt ideologies such as those advocated by the authors of books such as How to Cure Almost Any Cancer at Home for $5·15 a Day, but perhaps many of us are guilty of intolerance of alternative therapeutic ideologies. Albert Einstein is quoted as saying that insanity was “doing the same thing over and over again and expecting a different result”, and perhaps some cancer researchers are guilty of this way of thinking. Marcus Aurelius once said ”the object of life is not to be on the side of the majority, but to escape finding o insane”. that if m researche philosoph professio worth of ginger m doctors m idea of th concentr Jonathan Imperial Co j.stebbing@ Essay Phase itrialist There is a new breed of clinical trialist in cancer research. You might not have seen them yet—they will not be knocking down your door in the clinic. They do not know what HIPAA stands for. They do not know what to do in a code. They do not wear a a dish, there is the step to decide how to treat those cells or mice in a sensible, yet new way. It is in this phase, before even stepping into a laboratory, in which we are now seeing an influx of other types of scientists—physicists, engineers, and you turn research? as appl to genet energy t people w be explai
  15. 15. Molecular Reductionism Qolism Cellular Organism the current i which invasi cancer progr There are these hypoth relevant to th First, this is a mathematica cancer resear type of insigh perhaps mos such quantit used. Second to experimen conceived. In experiments conducted in relevance of quantitative cult and the it exposes a g cal and expe Figure 2 | Cancer is multiscale. Changes at the genetic level lead to modified intracellular signal- lingwhichcauseschangesincellularbehaviourandgivesrisetocanceroustissue.Eventually,organs and the entire organism are affected. We propose that a focus on the cell as the fundamental unit PERSPECTIVES ~10-20 - 1020 meters
  16. 16. Each of these models explains only a small part of our experience on this earth - and with caveats... But can we just add them all up and recapitulate life?
  17. 17. Mechanistic Modeling: but at what scale? Bioinformatics 498 I. J. Radiation Oncology d Biology d Physics Volume 75, Number 2, 2009
  18. 18. Build models! What is Science? What do Scientists do? “All models are wrong, but some are useful” George E P Box (Statistician)
  19. 19. Nutrients (c) Signalling Proteases (m) Invasive front Angiogenesis Stem cell Inflammatory response Stromal cell Immune cell Normal cell Matrix adhesion Dm 2 m ni,j – m, m t – m , t Dc 2m – ni,jc – c. c t Tumour cells (n) Extracellular matrix (f) plement ch as M) s and ge over by a set of es a of cancer n intuitive ancer tion. behav- nt?Each riven by quantified escribes ucially, lattice s context fficients We can PERSPECTIVES
  20. 20. Cancer is not only a collection of mutated cells A complex system of many interacting cellular and microenvironmental elements that were once normal
  21. 21. •More complex models are better •Something that looks similar is similar •Biological facts should drive derivation •Distill key components (Dialogue) •Focus on mechanism •Subsequent model refinement“All models are wrong, but some are useful” George E P Box (Statistician) “Models should be simple but no simpler” Albert Einstein Minimal modeling approach
  22. 22. We use a suite of mathematical and computational models to bridge a range of spatial and temporal scales. TIME/SPATIAL SCALE CELLULAR DETAIL Evolutionary Game Theory Reaction Diffusion Models Hybrid Cellular Automata Cellular Potts Model Immersed Boundary Model Hybrid Cellular Automata Non-spatial continuum Reaction Diffusion Network Theory
  23. 23. Dead P QProliferating Quiescent Dead Heterogeneous Population Spatial Constraints Nutrient/Growth Factor Constraints Stromal Constraints/Interactions
  24. 24. Time CellDensity dN(t) dt = λN(t) N(t), number of cells at time t λ, proliferation rate http://math.dartmouth.edu/~klbooksite/3.02/302.html
  25. 25. N(t) = Aeλt Doubling time Here λ =0.04, hence T=17.33 hrs T = ln2 λ
  26. 26. Time CellDensity dN(t) dt = λN(t) 1− N(t) K ⎛ ⎝⎜ ⎞ ⎠⎟ N(t), number of cells at time t λ, proliferation rate K, carrying capacity
  27. 27. Time CellDensity dN dt = −λN log N K ⎛ ⎝⎜ ⎞ ⎠⎟ N, number of cells at time t λ, proliferation rate K, carrying capacity
  28. 28. Cons Over simplification (no cycle) Proliferate with same rate at same time Non-spatial No mechanistic insight Pros Fit tumour growth data well Compartmental models Time CellDensity
  29. 29. We use a suite of mathematical and computational models to bridge a range of spatial and temporal scales. TIME/SPATIAL SCALE CELLULAR DETAIL Evolutionary Game Theory Reaction Diffusion Models Hybrid Cellular Automata Cellular Potts Model Immersed Boundary Model Hybrid Cellular Automata Non-spatial continuum Reaction Diffusion Network Theory
  30. 30. ∂N(x,t) ∂t = D ∂2 N(x,t) ∂x2 N, number of cells at time t, position x D, Diffusion coefficient the rate of change of cell number at position x and time t = change in cell number due to random dispersal
  31. 31. dN(x,t) dt = D ∂2 N(x,t) ∂x2 + λN(x,t) 1− N(x,t) K ⎛ ⎝⎜ ⎞ ⎠⎟ Sir Ronald Fisher 1890-1962 N, number of cells at time t, position x D, Diffusion coefficient λ, proliferation rate K, carrying cpacity the rate of change of cell number at position x and time t = change in cell number due to random dispersal + change in cell number due to cell proliferation
  32. 32. du dt = Du ∂2 u ∂x2 − σv, dv dt = Dv ∂2 v ∂x2 − u − v Alan Turing 1912-1954 u, activator v, inhibitor D, Diffusion coefficients , proliferation rateσ
  33. 33. We use a suite of mathematical and computational models to bridge a range of spatial and temporal scales. TIME/SPATIAL SCALE CELLULAR DETAIL Evolutionary Game Theory Reaction Diffusion Models Hybrid Cellular Automata Cellular Potts Model Immersed Boundary Model Hybrid Cellular Automata Non-spatial continuum Reaction Diffusion Network Theory
  34. 34. Phys. Biol. 8 (2011) 015016 D Basanta et al GF~0% GF~80% GF~100% starting t t t control raisereduce AG inv gly inv gly 0 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (INV-GLY). The base payoff in a given interaction is r and the cost of moving to another location with respect to the base payoff is c. The fitness cost of acidity is n and k is the fitness cost of having a less efficient glycolytic metabolism. The benefits from having access to the vasculature as a result of angiogenesis are reflected by the parameter α. AG INV GLY INV-GLY AG 1 2 + α 2 1 1 2 − n + α 1 2 − n + α INV 1 − c 1 − c 2 1 − c 3 1 − c 3 GLY 1 2 − k + n + α 1 − k + α 2 1 2 − k + α 4 1 − k + α 2 INV-GLY 1 2 − k + n + α 1 − k + α 2 1 − c 3 − k + α 2 1 − k − c 6 + α 2 Table 2. List of variables used by the model. Value Affected phenotypes Meaning c INV, INV-GLY Cost of motility k GLY, INV-GLY Cost having a glycolytic metabolism n AG, INV Cost of living in an acid microenvironment α AG, GLY, INV-GLY Benefit from angiogenesis moving cells incur since they cannot proliferate whilst moving [19, 20] or as the cost for degrading and detaching from the extra cellular matrix. The parameter k represents the cost of utilizing glycolysis as opposed to the more efficient oxidative phosphorylation. The parameter n represents the penalty that cells suffer for living in an acidic environment created by the glycolytic cells. GLY cells will suffer this penalty less as they are adapted to live in acidic environments. The parameter α represents the benefit of the surrounding vasculature. One way of envisioning variations in α is the increase in oxygen and nutrients resulting from an optimized vascularization resulting from the release of HIF-1α and downstream proteins. Table 2 lists all model variables. These variables are normalized and assumed to be in the range [0:1]. The payoff table 1 assumes that non-motile phenotypes (GLY and AG) will share existing resources with the cells they interact with, whereas motile phenotypes can chose whether to stay or move. In the case of INV cells, they will always move and leave existing resources for the cell it is interacting with unless the interaction happens with another INV cell, in which Phys. Biol. 8 (2011) 015016 (9pp) doi:10.1088/1478-3975/8/1/015016 The role of IDH1 mutated tumour cells in secondary glioblastomas: an evolutionary game theoretical view David Basanta1, Jacob G Scott1, Russ Rockne2, Kristin R Swanson2 and Alexander R A Anderson1 1 Integrated Mathematical Oncology, H Lee Moffitt Cancer Center and Research Institute, Tampa, FL 33612, USA 2 Pathology and Applied Mathematics at the University of Washington, Seattle, WA 98104, USA E-mail: david.basanta@kclalumni.net and jacob.scott@moffitt.org Received 17 September 2010 Accepted for publication 10 January 2011 Published 7 February 2011 Online at stacks.iop.org/PhysBio/8/015016 Abstract Recent advances in clinical medicine have elucidated two significantly different subtypes of glioblastoma which carry very different prognoses, both defined by mutations in isocitrate dehydrogenase-1 (IDH-1). The mechanistic consequences of this mutation have not yet been fully clarified, with conflicting opinions existing in the literature; however, IDH-1 mutation may be used as a surrogate marker to distinguish between primary and secondary glioblastoma multiforme (sGBM) from malignant progression of a lower grade glioma. We develop a mathematical model of IDH-1 mutated secondary glioblastoma using evolutionary game theory to investigate the interactions between four different phenotypic populations within the tumor: autonomous growth, invasive, glycolytic, and the hybrid invasive/glycolytic cells. Our model recapitulates glioblastoma behavior well and is able to reproduce two recent experimental findings, as well as make novel predictions concerning the rate of invasive growth as a function of vascularity, and fluctuations in the proportions of phenotypic populations that a glioblastoma will experience under different microenvironmental constraints. 1. Introduction Our ability to tease apart pathologic differences in cancers began with microscope and differential staining and has progressed to the current age of molecular medicine. The mantra of clinical medicine in the molecular age is ‘personalized medicine’—the hope that one day we will be able to perfectly understand each person’s tumor at the molecular and mechanistic level in order to prescribe the perfect treatment. While we have made many advances in subtyping many different cancers and even designed molecularly targeted therapies, the results so far have been disappointing. One cancer that has remained particularly resistant to our therapies is glioblastoma multiforme (GBM), which carries a prognosis of less than a year and certain mortality. It has been understood for several years that there are different subtypes of glioblastoma characterized by mutation pattern and cell of origin [1], but this knowledge has not altered our treatment strategy, only our ability to prognosticate outcome. That these subtypes all end up looking the same under the microscope and end up behaving very similarly as aggregates is an example of convergent evolution—genotypically different cells with similar phenotypic characteristics. Most recently, two significantly different classes of glioblastoma have been identified which carry very different prognoses [2–4]. These two groups of glioblastoma are, for the most part, differentiated by mutations found in a single coding region of an enzyme involved in the Krebs cycle, isocitrate dehydrogenase 1 (IDH1). This mutation is present in the majority of secondary glioblastomas (sGBM) and low grade gliomas (LGGs), many of which progress to become 1478-3975/11/015016+09$33.00 1 © 2011 IOP Publishing Ltd Printed in the UK IOP PUBLISHING Phys. Biol. 8 (2011) 015016 (9pp) doi:10.1088 The role of IDH1 mutated tumour c secondary glioblastomas: an evolut game theoretical view David Basanta1, Jacob G Scott1, Russ Rockne2, Kristin R Swanson2 and Alexander R A Anderson1 1 Integrated Mathematical Oncology, H Lee Moffitt Cancer Center and Research Institute, Tampa, FL 33612, USA 2 Pathology and Applied Mathematics at the University of Washington, Seattle, WA 98104, USA E-mail: david.basanta@kclalumni.net and jacob.scott@moffitt.org Received 17 September 2010 Accepted for publication 10 January 2011
  35. 35. LETTER TO THE EDITOR Production of 2-hydroxyglutarate by isocitrate dehydrogenase 1–mutated gliomas: an evolutionary alternative to the Warburg shift? Neuro-Oncology Neuro-Oncology Advanc Neuro-Oncology NEURO-ONCOLOGY Neuro-Oncology Advance Access published July 22, 2011 015016 D Basanta et al Invasion BeforeBevacizumabAfterBevacizumab Loss of PET signal ~ Less Glycolysis starting t t t reduce 0 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 3. Plot of k = 0.1, n = 0.2, c = 0.1 and α = 0.3, 0.32, 0.35. The first panel shows two interesting dynamics: with increasing benefi vasculature (increasing α), we see a more rapid progression as well as a higher overall proportion of cells with the GLY phenotype. Also decreasing α promotes the INV phenotype (stars) which is recapitulated in recurrent glioblastoma after bevacizumab treatment. The two panels below the control one show what happens after bevacizumab has been administered after 600 time steps without assuming wheth the main effect would be a normalization of the angiogenic vasculature (which would increase α, shown in the second row) or the reduc of the existing vasculature (which would have a negative effect on α, shown in the third row). changes that we see in glioblastoma patients after failure of bevacizumab (a monoclonal antibody to VEGF-α). The recapitulation of known behaviors allows some measure of confidence in our model and gives some credence to predictions that the model can make. Now we can begin to make observations about the mechanisms driving the behaviors that were, otherwise, obscured by the biological complexity. A recurring theme observed in the time-dependent behavior of our model suggests an underlying mechanism driven by interactions between different phenotypes. Specifically, the emergence of the invasive phenotypes is always preceded by a rise in the glycolytic fraction. This rise in the glycolytic fraction is preceded by an overgrowth of AG cells. cells grow into a viable proportion, the damage that they to the local environment with their excessive acid product begins to promote the benefit of cells that can move to a n place (INV). We see this sequence reproduced in nearly areas of the parameter space, and certainly in all the areas are relevant to glioma. Further, these results agree nic with earlier work done by this group suggesting that glycolytic phenotype is necessary to bring about the emerge of invasion [18]. In addition to this sequence, there was an interest dynamic that emerged in some areas of the parameter spa Figure 5 shows an example of two types of oscillatory behav that our model can produce. Even though neither manag
  36. 36. We use a suite of mathematical and computational models to bridge a range of spatial and temporal scales. TIME/SPATIAL SCALE CELLULAR DETAIL Evolutionary Game Theory Reaction Diffusion Models Hybrid Cellular Automata Cellular Potts Model Immersed Boundary Model Hybrid Cellular Automata Non-spatial continuum Reaction Diffusion Network Theory
  37. 37. Symmetric Division Rate and live in a ‘continuous’ milieu described completely by the experimenter end result is a model that looks complicated, but is entirely described by a minimum of parameters - allowing for emergent phenomena and the subtype driven by PDGF overexpression. For NF1-driven cancers, we investigated bi-allelic loss of NF1 and a dominant negative mutation of TP53 as the necessary driver mutations that must be accumulated in a single cell to initiate tumorigenesis. For PDGF-driven cancers, the necessary driving alterations are those leading to PDGF overexpression and bi-allelic loss of INK4A/ARF. We did not include the accumulation of passenger mutations in this model since those alterations, by definition, do not influence of cells carrying alterations in similarly leads to increased pro on the background of either T INK4A/ARF2/2 NF12/2 cells fitness RARF 6RNF1,mut and div In contrast, cells mutated in NF a fitness detriment, RNF1,wt0. cell divisions beyond the norm Figure 1. A mathematical model of the cell of origin of PDGF- and NF1-driven gliomas. Initially, th cells (blue) and 2z+1 21 wild-type transit-amplifying non-self-renewing (TA) cells (purple). At each time ste probability a, the SR cell divides symmetrically and one daughter cell replaces another randomly chosen SR divides asymmetrically and one daughter cell remains a SR cell while the other daughter cell becomes comm This new TA cell divides symmetrically z times to give rise to successively more differentiated cells (progres becoming terminally differentiated. This restriction of the stochastic process ensures that the total number of homeostatic conditions in the healthy brain. In the figure, the darkening purple gradations refer to successively clarify a single time step of the stochastic process. We investigate the dynamics of only one cell cluster since the given by the probability per cluster times the number of clusters; hence, a consideration of all clusters does n cell of origin of brain cancer. doi:10.1371/journal.pone.0024454.g001 Rounds of Transient Amplification Vascular Density 0.01 0.05 0.1 !!!!! !! !! !! ! !!!! ! !!!!! ! !!!!!! ! !!!!!! ! !!!!!!!
  38. 38. Loss of homeostasis Symmetric division rate - 0.5 Rounds of Amplification -11 Vascular density - .05
  39. 39. 1 5 10 15 20 0 0.5 1 1.5 2 2.5 x 10 5 Divisions per progenitor cell Cells s/a 0.1 s/a 0.3 s/a 0.5 1 5 10 15 20 0 0.5 1 1.5 2 x 10 5 Divisions per progenitor cell s/a 0.1 s/a 0.3 s/a 0.5 1 5 10 15 20 0 0.5 1 1.5 2 2.5 x 10 5 Divisions per progenitor cell Cells s/a 0.1 s/a 0.3 s/a 0.5 Vascular Density Quantifying the unmeasurable 0.01 0.05 0.1 Background: Metastasis is a highly lethal and poorly understood process that accounts for the majority cancer deaths
  40. 40. We use a suite of mathematical and computational models to bridge a range of spatial and temporal scales. TIME/SPATIAL SCALE CELLULAR DETAIL Evolutionary Game Theory Reaction Diffusion Models Hybrid Cellular Automata Cellular Potts Model Immersed Boundary Model Hybrid Cellular Automata Non-spatial continuum Reaction Diffusion Network Theory
  41. 41. 90% of cancer death is from metastatic disease Yet from the clinicians perspective, metastasis is a binary event - the least understood process
  42. 42. M0
  43. 43. M1
  44. 44. Which patients will end up as M1 vs. M450?? We now have targetted local therapies (like SBRT) that ablate these tumors...
  45. 45. Game  1 Game  2 Jacob Scott, Maciunas STEMM Keynote 2010
  46. 46. The cutoff frequency .[o is then defined as 1 ./o= ~ (17) Values afro for each segment are listed in Table 1. 5 Computational procedure A digital computer program was written in FOR- TRAN to operate on the branching configuration multiple branching parallel. Transmiss node calculated fr backward towards pedance of the wh final result is a com ching configuration and transmission p every node. Hence ejection waveform flow waveforms m 53 52/~ 55 51~5136 3~ ~7 7i ~4 the branching struc 0s 6o flow throughout th ---r-~ pheral resistance v57 35 47 ~564~)]~0__.~33]1~ sistances and visco The input data t dimensions and o~4' _ . ?2 42 i!/6; 2,1 ,e sS/.~ 9 8~8~70 15 49 61, ,, 3, ,, o2 :,, I , }1 3200[ !/ .o, ,,oo,!, ,o8 ' ~'~, ,03 ~q ~o9 10cm ! ~L ' I 800 pH} c 0 113 ,,8 y ' , , s 12it 125 humanarterialtree Fig. 1 Schematic representation of the human arterial tree with all lengths drawn to scale. Segment numbers correspond to arteries listed in Table l 2 r 1-0 . = -1.ot i Fig. 2 Input impedanc pedance in each simultaneous re ascending aorta culatedfrom mo Medical Biological Engineering Computing November Jacob Scott, Maciunas STEMM Keynote 2010
  47. 47. Simple  experiment  and  ODE  model    to   begin  a  conversa:on Can  likely  measure  f(t)  in  a  mouse  model  by   injec:on  a  bolus  of  tumor  cells  into  a  tail   vein  and  measuring  CTCs  at  several  :me   points tumor Other  organ •C  is  number  of  CTCs •Alpha  and  beta  are  constants •T(t)  is  a  func:on  describing  tumor  size •z(t)  is  rate  of  tumor  cell  intravasa:on   “shedding  rate” •f(t)  is  the  rate  of  filtra:on  or  CTC  arrest Jacob Scott, Maciunas STEMM Keynote 2010
  48. 48. Jacob Scott, Maciunas STEMM Keynote 2010
  49. 49. Jacob Scott, Maciunas STEMM Keynote 2010 What would we need for Kirchoff’s rules?
  50. 50. !#$%'#()*'+, -+'.# !./*+ 0, -1'((*+'#(2+3%,',* -3#* β η γ Unifying metastasis — integrating intravasation, circulation and end-organ colonization Jacob Scott1,2 , Peter Kuhn3 and Alexander R. A. Anderson1 !#$%'$()(*+'+,$($+'-,./.01'/(23,'+#($-$(-3+(+,/+2($-+(4+#5%+4+,$(.6('1%'5/$1,0( $54.5%('+//#(7898#:(1,(;$1+,$#(-3+(#;5%%+2(1,$+%+#$(1,($-+('1%'5/$.%(;-#+(.6(4+$#$#1#=( 9+'-,15+#($-$(2.(,.$(#./+/(%+/(.,((/..2(#4;/+(//.?(#5#$,$1/(1./.01'/(1,$+%%.0$1.,( +.,2(#14;/('.5,$1,0(898#= for gy, A. In patients with advanced primary cancer, circulating tumour cells (CTCs)1 can be found throughout the entire vascular system2 . When and where these CTCs form metastasis is not fully understood, and is currently the subject of intensive biological study. Paget’s well-known seed–soil hypothesis3 suggests that the ‘soil’ (the site of a metastasis) is as important as the ‘seed’ (the metastatic cells) in the determination of successful metastasis. The mechanism by which seeds are disseminated to specific soil has, to date, been a ‘known unknown’. We think that it is during this poorly understood phase of metastasis that we stand to answer important questions4 . We hypothesize that the rich variety of possible meta- static disease patterns not only stems from the physical aspects of the circulation but also from CTC hetero- geneity (FIG. 1). These seeds represent many different populations that are derived from a diverse population of competing phenotypes within the primary tumour5 . Because such seeds need to pass through a system of physical and biological filters in the form of specific organs, the circulatory phase of metastasis could be modelled as a complex deterministic filter. In theory, until the evolution of a suitable seed, any number of CTCs could flow through the circulation and arrest at end organs without metastases forming. As tumour heterogeneity is thought to expand as the tumour pro- gresses, it follows that at some point a seed will come into existence that is suited to a specific soil within that patient’s body. If this seed is to propagate it must find its soil, a process that we hypothesize is governed by solvable physical rules that relate to the dynamics of do not fit a model that is defined only by physical flow and filtration. To begin the process of physical interrogation, we propose a model that represents the human circulatory system as a directed and weighted network, with nodes representing organs and edges representing arteries and veins.The novelty is only fully realized when combined withaheterogeneousCTCpopulation(drivenbyprimary tumour heterogeneity) modulated by the complex organ filter system (with physiologically relevant connections) under dynamic flow. Four important biological processes emerge from this representation. First, the shedding rate, which is defined as the rate at which the tumour sheds CTCs into the vasculature. Second, CTC heterogeneity, which is defined as the distribution of CTC phenotypes present in the circulation. Third, the filtration fraction, which is defined as the proportion (and type) of CTCs that arrest in a given organ. Fourth, the clearance rate, which is defined as the rate at which cancer cells are cleared from the blood and/or organ after arrest. Each of these biological processes is probably disease- and even patient-specific, and each is extremely poorly understood. Using this representation to motivate the develop- ment of a mathematical model we can define both the concentration of CTCs and their phenotypic distribu- tion at any given point in the network, as well as organ- specific filtration values. To parameterize this model, characterization and enumeration of CTCs taken from a single patient at different time points and from differ- ent points in this network will need to be undertaken. A complete understanding of the model will also pro- Unifying metastasis — int intravasation, circulation end-organ colonization Jacob Scott1,2 , Peter Kuhn3 and Alexander R. A. Anders !#$%'$()(*+'+,$($+'-,./.01'/(23,'+#($-$(-3+(+,/+2($ $54.5%('+//#(7898#:(1,(;$1+,$#(-3+(#;5%%+2(1,$+%+#$(1,($-+('1% 8@A Nature Reviews Cancer | AOP, published online 24 May 2012; doi:10.1038/nrc3287
  51. 51. Jacob Scott, Maciunas STEMM Keynote 2010
  52. 52. Jacob Scott, Maciunas STEMM Keynote 2010 Matrices are important - pay attention in AMH!!
  53. 53. “Whereas a good simulation should include as much as possible, a good model should include as little as possible.” Jacob Scott, Maciunas STEMM Keynote 2010
  54. 54. Lung    outflow ∂CL ∂t = inflow IL − filtering ηLCL buildup ∂OL ∂t = arresting ηLCL − clearing γLOL Liver    outflow ∂CLi ∂t = inflow *A αLiCLi + inflow *B αGCG − filtering ηLiCLi + shedding β buildup ∂OLi ∂t = arresting ηLiCLi − clearing γLiOLi Make everything as simple as possible, but no simpler Jacob Scott, Maciunas STEMM Keynote 2010
  55. 55. Brain Liver Gut Bone Venous Arterial Portal System CTCflow Lung Breast Primary Seeding Primary Tumor (a) Primary Seeding cartoon Brain Liver Gut Bone Venous Arterial Portal System CTCflow Lung Breast Primary Tumor Secondary Tumor Secondary Tumor Secondary Seeding (b) Secondary Seeding cartoon A Time (cell cycles) Logtumourmass B Removal rate λ Returnprobabilityp growthacceleration(logscale) primary seeding secondary seeding Figure 2: Simulating the dynamics of primary seeding.(A) shows the total tumour b for three different conditions where the removal rate was fixed at λ = 10−5 and return proba was taken to be p = 10−2,10−3 and 10−4 respectively. (B) illustrates the model dynamics the parameters λ and p are varied systematically, and shows that accelerated tumour growth , 20130011, published 20 February 2013102013J. R. Soc. Interface Jacob G. Scott, David Basanta, Alexander R. A. Anderson and Philip Gerlee growth secondary metastatic deposits as drivers of primary tumour A mathematical model of tumour self-seeding reveals Open Access
  56. 56. science society B iology has long been the stepchild of the natural sciences. Compared with mathematical proofs, physical formulae and the molecules of chemistry, biology, like life itself, has often seemed unquantifiable, unpredictable and messy. Yet, scientists have striven gallantly to pin biology down through the application of the of inspiration for mathematicians. “In my 40 plus years of research, I have found that problems in biomathematics almost always uncover unexplored and undeveloped areas of mathematics,” he said. “These are areas that mathematicians have not even thought about exploring. New mathematics.” involves events a gene expression o take place in nano nisms or body-wi minutes, hours o between people tions that last mon understand these s different layers an Jost points out, ha mathematical biolo T he applicatio logy itself i back at least on the inheritanc nineteenth centur the theory of Men foundation of mod ally reproducing o Biology is the new physics The increasing use of mathematics in biology is both inspiring research in mathematics and creating new career options for mathematical biologists Philip Hunter EMBO reports VOL 11 | NO 5 | 2010350 “For many years the inspiration for innovation in applied mathematics has come from physics, but in my opinion, in this century it will come from the bio- logical sciences, broadly defined,” Mackey explained, adding that this switch has been taking place slowly over several decades. While physics has stagnated, waiting for new theoretical insights to make progress against fundamental problems such as quantum gravity, Mackey argued, theoreti- cal biology has emerged as a new source complexes; able to accou tle difference infection am ous approac growing use biological an capable of a the systems u of a protein, essence, inf geometry wi changing, co Other tech information tions betwee a huge ran The human continue to the extent that it might even become the main driving force behind innovation and development in mathematics

×