A PDE approach to studying evolutionary and spatial dynamics in cancer cell populations

Transcription

1 A PDE approach to studying evolutionary and spatial dynamics in cancer cell populations Tommaso Lorenzi LJLL, 30th June 2017

2 T. Lorenzi (University of St Andrews) June 30th 2 / 43

3 Synopsis Object of study : a theoretical framework based on PDE models to studying the dynamics of cancer cell populations Aim : use such mathematical models to address questions concerning the mechanisms which drive the dynamics of cancer Method : integrate numerical simulation with qualitative analysis to achieve biological conclusions with broad structural stability Ultimate goal : complement empirical research by offering alternative means of interpreting experimental data enabling extrapolation beyond empirical observation T. Lorenzi (University of St Andrews) June 30th 3 / 43

4 Plan of the talk PDE models to study : 1. the emergence of cytotoxic-drug resistance in cancer cell populations 2. the development of phenotypic heterogeneity in solid tumours 3. the formation of infiltrating patterns of cancer-cell invasion T. Lorenzi (University of St Andrews) June 30th 4 / 43

5 PDE model to study the emergence of cytotoxic-drug resistance in cancer cell populations T. Lorenzi (University of St Andrews) June 30th 5 / 43

6 In vitro experiments reveal a reversible drug-tolerant phenotype Sharma et al. Cell 141:69 80, T. Lorenzi (University of St Andrews) June 30th 6 / 43

7 Mutations VS epimutations T. Lorenzi (University of St Andrews) June 30th 7 / 43

8 Can these results be explained as the outcome of an evolutionary process driven by epimutations and natural selection? T. Lorenzi (University of St Andrews) June 30th 8 / 43

9 T. Lorenzi (University of St Andrews) June 30th 9 / 43

10 PDE model System under study : well-mixed in vitro population of cancer cells State of cancer cells : y = (y 1,y 2 ) Ω [0,1] [0,1] expression of a gene which controls drug resistance : y 1 [0,1] expression of a gene which controls cell proliferation : y 2 [0,1] Population density : n(t, y) Population size : ρ(t) = n(t,y) dy Ω Constant concentration of a cytotoxic drug : c T. Lorenzi (University of St Andrews) June 30th 10 / 43

11 PDE model n t (t,y) + α (v c(y) n(t,y)) = R c (y,ρ(t)) n(t,y) }{{}}{{} stress-induced proliferation/death epimutations + β n(t,y) }{{} random epimutations R c (y,ρ(t)) = proliferation and competition for resources {}}{ p(y,ρ(t)) cytotoxic action {}}{ c k(y) v c (y) = c a(y) T. Lorenzi (University of St Andrews) June 30th 11 / 43

12 Numerical solutions of the PDE model T. Lorenzi (University of St Andrews) June 30th 12 / 43

13 We consider the case in which Asymptotic analysis y R N with N 1 Epimutations are less frequent than proliferation/death: n t (t,y) + ε (v c(y) n(t,y)) = R c (y,ρ(t)) n(t,y) + ε n(t,y) Random epimutations occur on a timescale slower than that of stress-induced epimutations: n t (t,y) + ε (v c(y) n(t,y)) = R c (y,ρ(t)) n(t,y) + ε 2 n(t,y) Time rescaling to observe the effects of epimutations: ε n ε t (t,y) + ε (v c(y) n ε (t,y)) = R c (y,ρ ε (t)) n ε (t,y) + ε 2 n ε (t,y) T. Lorenzi (University of St Andrews) June 30th 13 / 43

14 Asymptotic analysis Method of proof : WKB method developed by Perthame & Barles (2008) and Lorz, Mirrahimi & Perthame (2011) T. Lorenzi (University of St Andrews) June 30th 14 / 43

15 Asymptotic analysis Theorem (Chisholm-L.-Lorz) With technical assumptions on the functions v c and R c, there exists a subsequence of ρ ε, denoted again as ρ ε, such that Moreover, weakly in measures, ρ ε (t) ρ(t) as ε 0. (1) n ε (t,y) ρ(t)δ(y y(t)) as ε 0. (2) T. Lorenzi (University of St Andrews) June 30th 15 / 43

16 PDE model to study the development of phenotypic heterogeneity in solid tumours T. Lorenzi (University of St Andrews) June 30th 16 / 43

17 Role of oxygen distribution in the development of intratumour phenotypic heterogeneity Grimes et al. J R Soc Interface 11: , 2014 T. Lorenzi (University of St Andrews) June 30th 17 / 43

18 Role of oxygen distribution in the development of intratumour phenotypic heterogeneity Zhang et al. Int J Mol Sci 16: , 2015 T. Lorenzi (University of St Andrews) June 30th 18 / 43

19 Role of oxygen distribution in the development of intratumour phenotypic heterogeneity Alfarouk et al. Evol Appl 6:46 53, 2013 T. Lorenzi (University of St Andrews) June 30th 19 / 43

20 Can spatial variations in the distribution of oxygen lead to the creation of distinct local niches and thus provide ecological opportunities for diversification? T. Lorenzi (University of St Andrews) June 30th 20 / 43

21 T. Lorenzi (University of St Andrews) June 30th 21 / 43

22 T. Lorenzi (University of St Andrews) June 30th 22 / 43

23 PDE model System under study : solid tumour spatial domain Ω R 3 State of cancer cells : 1. position in the tumour : x Ω 2. normalised expression level of a hypoxia responsive gene : y [0,1] y 0 : higher proliferation rates y 1 : lower proliferation rates T. Lorenzi (University of St Andrews) June 30th 23 / 43

24 PDE model Population density at position x : n(t,x,y) 1 Density of cancer cells at position x : ρ(t,x) = n(t,x,y)dy 0 Mean phenotype at position x : µ(t,x) = Concentration of oxygen at position x : s(t,x) 1 1 ρ(t, x) y n(t,x,y)dy 0 T. Lorenzi (University of St Andrews) June 30th 24 / 43

25 PDE model n(t,x,y) = R (y,ρ(t,x),s(t,x)) n(t,x,y) t }{{} proliferation/death R (y,ρ(t,x),s(t,x)) = competition proliferation for space {}}{{}}{ p(y,s(t,x)) dρ(t, x) p(y, s(t, x)) = f (y) }{{} proliferation in hypoxic conditions + r(y, s(t, x)) }{{} proliferation in oxygenated environments f (y) = ζ [1 (1 y) 2] s(t,x) ( and r(y,s(t,x)) = γ s 1 y 2 ) α s +s(t,x) T. Lorenzi (University of St Andrews) June 30th 25 / 43

26 PDE model β s s(t,x) }{{} diffusion 1 = η s 0 r( y,s(t,x) ) n(t,x,y)dy }{{} consumption + λ s s(t,x) }{{} natural decay Dirichlet boundary conditions on Ω T. Lorenzi (University of St Andrews) June 30th 26 / 43

27 Formal analysis Proposition Let s(x) be the long-term limit of s(t, x). Under biologically consistent assumptions, ρ(t,x) t ρ(x) = 1 d [ ζ 2 ] A s (x) + ζ +A s (x) (3) and where µ(t,x) µ(x) = ζ t ζ +A s (x), (4) s(x) A s (x) = γ s α s +s(x). Remark : A rigorous asymptotic analysis for a similar problem has been developed by Mirrahimi & Perthame (2015) and Jabin & Schram (2016) T. Lorenzi (University of St Andrews) June 30th 27 / 43

28 Numerical solutions of the PDE model T. Lorenzi (University of St Andrews) June 30th 28 / 43

29 Numerical solutions of the PDE model Data obtained from the 3D-IRCADb-01 database T. Lorenzi (University of St Andrews) June 30th 29 / 43

30 Numerical solutions of the PDE model T. Lorenzi (University of St Andrews) June 30th 30 / 43

31 PDE model to study the formation of infiltrating patterns of cancer-cell invasion T. Lorenzi (University of St Andrews) June 30th 31 / 43

32 Infiltrating patterns of cancer-cell invasion T. Lorenzi (University of St Andrews) June 30th 32 / 43

33 Computational results from an individual-based model T. Lorenzi (University of St Andrews) June 30th 33 / 43

34 Can a minimal fluid mechanical PDE model reproduce such patterns? T. Lorenzi (University of St Andrews) June 30th 34 / 43

35 T. Lorenzi (University of St Andrews) June 30th 35 / 43

36 PDE model System under study : dividing cells embedded in a system of non-dividing cells Cell state : spatial position x = (x 1,x 2 ) Ω R 2 Local density of dividing cells : ρ C (t,x) 0 Local density of non-dividing cells : ρ A (t,x) 0 Local pressure : p(t,x) = K γ ( ρa + ρ C ) γ with γ 1 T. Lorenzi (University of St Andrews) June 30th 36 / 43

37 PDE model ρ A t (t,x) µ A (ρ A (t,x) p(t,x) ) = 0, }{{} mechanical motion ρ C t (t,x) µ C (ρ C (t,x) p(t,x) ) ( ) = G C p ρc (t,x) }{{}}{{} mechanical motion proliferation (5) The growth rate G C is such that G C ( ) 0, G C (P C ) = 0 T. Lorenzi (University of St Andrews) June 30th 37 / 43

38 Numerical solutions of the PDE model T. Lorenzi (University of St Andrews) June 30th 38 / 43

39 Numerical solutions of the PDE model µ C µ A µ C > µ A T. Lorenzi (University of St Andrews) June 30th 39 / 43

40 Travelling-wave solutions We search for 1D travelling-wave solutions that satisfy σ ρ A µ ( A ρa p ) = 0, (6) σ ρ C µ ( C ρc p ) = GC (p) ρ C with the additional conditions ρ C > 0 on (,0], (7) ρ A > 0 on [0,r] (8) and p(x) x P C (9) T. Lorenzi (University of St Andrews) June 30th 40 / 43

41 Numerical solutions of the PDE model in 1D for µ A > µ C P C p ρ C ρ A 0 0 r x 0 0 r x T. Lorenzi (University of St Andrews) June 30th 41 / 43

42 Travelling-wave solutions Theorem (L.-Lorz-Perthame) There exist σ > 0 and r > 0 such that the solution of (6) exists, satisfies conditions (7)-(9), with ρ C non-increasing and ρ A that satisfies r ρ A (x) dx = M A > 0. 0 The pressure p has a kink at x = 0 with sgn ( [p ] ) = sgn(µ C µ A ) and µ C p (0 ) = µ A p (0 + ). Remark : Such travelling-wave solutions are unstable if µ C > µ A. T. Lorenzi (University of St Andrews) June 30th 42 / 43

43 A PDE approach to studying evolutionary and spatial dynamics in cancer cell populations Tommaso Lorenzi LJLL, 30th June 2017