Dynamique des populations et résistance aux traitements : modèles mathématiques
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1 Dynamique des populations et résistance aux traitements : modèles mathématiques Alexander Lorz 1 R. Chisholm, J. Clairambault, A. Escargueil, M.E. Hochberg, T. Lorenzi, P. Markowich, B. Perthame, E. Trélat 1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie Team MAMBA, INRIA Rocquencourt Séminaire du Laboratoire Jacques-Louis Lions, April 11th, 2014 A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
2 Outline Outline 1 Motivation A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
3 Outline Outline 1 Motivation 2 A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
4 Outline Outline 1 Motivation 2 3 Back to cancer : cancer cells : healthy and cancer cells A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
5 Outline Outline 1 Motivation 2 3 Back to cancer : cancer cells : healthy and cancer cells 4 Heterogeneity in the model A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
6 Motivation Outline 1 Motivation 2 3 Back to cancer : cancer cells : healthy and cancer cells 4 Heterogeneity in the model A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
7 Motivation Motivation: Understand Darwinian evolution A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
8 Motivation Motivation: Understand resistance to therapy Many cancers escape therapy Cells adapt and become resistance to drug(s) Tumour as an ecological system, Darwinian evolution A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
9 Motivation Resistance to therapies A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
10 Motivation Darwinian evolution of a structured population density Population models are structured by a parameter representing a phenotypical trait. The length of the giraffe s neck spac or... A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
11 Motivation Darwinian evolution of a structured population density... The shape of beaks of Darwin s finches A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
12 ... Motivation Darwinian evolution of a structured population density The activity of ABC-transporters spac A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
13 Motivation Darwinian evolution of a structured population density We study the population dynamics under selection and mutations between the traits. Limited ressources Different traits have different capacities to use the environment Competition between the traits Selection The new born can have a trait slightly different from its parent one Mutations A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
14 Outline 1 Motivation 2 3 Back to cancer : cancer cells : healthy and cancer cells 4 Heterogeneity in the model A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
15 Darwinian evolution of a structured population density selection proliferation competition {}}{{}}{ t n(t,x) = n(t,x)[ p(x) d(x)i(t)] }{{ ) } x R d : phenotypic trait, n(t,x): density of trait x, R ( x,i(t) I(t) := n(t,x)dx R d I(t): total number of individuals, R(x,I): growth and death rates of trait x, A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
16 Example R = 4(x 0.3) 2 I n(t = 0,x) and R(x,I) t A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
17 Example R = 4(x 0.3) 2 I n(t = 1,x) and R(x,I) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
18 Example R = 4(x 0.3) 2 I n(t = 2,x) and R(x,I) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
19 Example R = 4(x 0.3) 2 I n(t = 3,x) and R(x,I) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
20 Example R = 4(x 0.3) 2 I n(t = 0,x) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
21 Example R = 4(x 0.3) 2 I n(t = 3,x) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
22 Example R = 4(x 0.3) 2 I n(t = 10,x) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
23 Example R = 4(x 0.3) 2 I n(t = 40,x) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
24 Example R = 4(x 0.3) 2 I n(t = 100,x) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
25 Surface plot of n(t, x) from the side t A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
26 Surface plot of n(t, x) from the top x A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
27 BV-bound: I(t) t I t t n(t,x) = t n(t,x)r ( x,i(t) ) + n(t,x)r I ( x,i(t) )İ(t) t t n(t,x) = n(t,x)r 2( x,i(t) ) ( )İ(t) + n(t,x)r I x,i(t) d İ(t) = n(t,x)r 2( x,i(t) ) dx + dt }{{} İ(t) ( ) n(t,x)r I x,i(t) dx =: J(t) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
28 BV-bound: I(t) t I t t n(t,x) = t n(t,x)r ( x,i(t) ) + n(t,x)r I ( x,i(t) )İ(t) t t n(t,x) = n(t,x)r 2( x,i(t) ) ( )İ(t) + n(t,x)r I x,i(t) d n(t,x)r 2( x,i(t) ) dx + dt İ(t) ( ) n(t,x)r I x,i(t) dx İ(t) = }{{} =: J(t) J(t) J(t) d dt d dt J(t) J(t) K J(t) J(t = 0) e Kt n(t,x)r I ( x,i(t) ) dx A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
29 BV-bound: I(t) t I t t n(t,x) = t n(t,x)r ( x,i(t) ) + n(t,x)r I ( x,i(t) )İ(t) t t n(t,x) = n(t,x)r 2( x,i(t) ) ( )İ(t) + n(t,x)r I x,i(t) d n(t,x)r 2( x,i(t) ) dx + dt İ(t) ( ) n(t,x)r I x,i(t) dx İ(t) = }{{} =: J(t) J(t) J(t) d dt d dt J(t) J(t) K J(t) J(t = 0) e Kt J(t) = İ(t) + 2J(t) n(t,x)r I ( x,i(t) ) dx J(t) dt I M + 2J(t = 0) e Kt dt A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
30 Idea of proof 0 < I m I(t) = n(t,x)dx I M & BV-bound I(t) t I 1) max x R(x,I ) > 0 x x 2) max x R(x,I ) < 0 3) max x R(x,I ) = 0 x n (x) = I δ(x x ) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
31 Generalisation of BV-bound t n(t,x) = n(t,x)r ( x,i(t) ), x R d, t 0, β d dt I(t) = Q( I(t),ρ(t) ), ρ(t) := n(t,x)dx. with β small A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
32 Darwinian evolution of a structured population density selection proliferation competition {}}{{}}{ t n(t,x) = n(t,x)[ p(x) d(x)i(t)] }{{ ) } x R d : phenotypic trait, n(t;x): density of trait x, R ( x,i(t) I(t) := n(t,x)dx R d I(t): total number of individuals, R(x,I): growth and death rates of trait x, A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
33 Darwinian evolution of a structured population density selection + mutations proliferation competition {}}{{}}{ t n(t,x) = n(t,x)[ p(x) d(x)i(t)] + n(t,x) }{{ ) } x R d : phenotypic trait, n(t;x): density of trait x, R ( x,i(t) I(t) := n(t,x)dx R d I(t): total number of individuals, R(x,I): growth and death rates of trait x, A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
34 Darwinian evolution of a structured population density We study the large time behavior of the population density for rare or small mutations We expect: discrete set of traits, extinction and branching A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
35 Rescaled equation for mutations with small effects ǫ t n ǫ = n ǫ R ( x,i ǫ (t) ) + ǫ 2 n ǫ, t > 0, x R d, with I ǫ (t) = n ǫ (x,t)dx. R d we observe very small mutations for a long time A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
36 Rescaled equation for mutations with small effects 4 ǫ t n ǫ = n ǫ R ( x,i ǫ (t) ) +ǫ 2 n ǫ, }{{}}{{} concentrates expands A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
37 Example R = 4(x 0.3) 2 I + 0.5, ǫ = n(t = 0,x) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
38 Example R = 4(x 0.3) 2 I + 0.5, ǫ = n(t =.125, x) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
39 Example R = 4(x 0.3) 2 I + 0.5, ǫ = n(t =.25,x) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
40 Example R = 4(x 0.3) 2 I + 0.5, ǫ = n(t =.5,x) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
41 Example R = 4(x 0.3) 2 I + 0.5, ǫ = n(t =.75,x) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
42 Example R = 4(x 0.3) 2 I + 0.5, ǫ = n(t = 1,x) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
43 Rescaled equation for ǫ = 0 0 = nr ( x,i(t) ) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
44 Rescaled equation for ǫ = 0 0 = nr ( x,i(t) ) 1 d and x R(x,I) monotonic R(x,I) x A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
45 Rescaled equation for ǫ = 0 0 = nr ( x,i(t) ) 1 d and x R(x,I) monotonic R(x,I) x n = ρ(t)δ(x x) 0 = R ( x,i(t) ) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
46 WKB ansatz δ(x x) 1 2πǫ e x x 2 /2ǫ = e ( x x 2 ǫln(2πǫ))/2ǫ. A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
47 WKB ansatz δ(x x) 1 2πǫ e x x 2 /2ǫ = e ( x x 2 ǫln(2πǫ))/2ǫ. n ǫ (t,x) = e uǫ(t,x)/ǫ. A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
48 WKB ansatz δ(x x) 1 2πǫ e x x 2 /2ǫ = e ( x x 2 ǫln(2πǫ))/2ǫ. n ǫ (t,x) = e uǫ(t,x)/ǫ. { t u ǫ = u ǫ 2 + R(x,I ǫ (t)) + ǫ u ǫ, x R d, t 0, u ǫ (t = 0) = ǫln(n 0 ǫ) := u 0 ǫ. A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
49 Example R = 4(x 0.3) 2 I ln(max(10 11,n(t =.0625,x)) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
50 Example R = 4(x 0.3) 2 I ln(max(10 11,n(t =.125,x)) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
51 Example R = 4(x 0.3) 2 I ln(max(10 11,n(t =.25,x)) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
52 Example R = 4(x 0.3) 2 I ln(max(10 11,n(t =.5,x)) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
53 Example R = 4(x 0.3) 2 I ln(max(10 11,n(t =.75,x)) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
54 Example R = 4(x 0.3) 2 I ln(max(10 11,n(t = 1,x)) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
55 Passing to the limit t u = u 2 + R(x,I(t)) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
56 Passing to the limit t u = u 2 + R(x,I(t)) max R d u(t,x) = 0 supp n(t, ) {u(t, ) = 0} A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
57 Passing to the limit t u = u 2 + R(x,I(t)) max R d u(t,x) = 0 supp n(t, ) {u(t, ) = 0} open question: uniqueness A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
58 Challenges x(t) and I(t) are not continuous u(t,x) for t = t 1 < t 2 < t 3 A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
59 Results Theorem (A.L., S. Mirrahimi, B. Perthame) In R d, under concavity assumptions we obtain weak convergence in the sense of measures for a subsequence n ǫ : n ǫ (t,x) ǫ 0 ρ(t) δ ( x x(t) ), (initially one Dirac one Dirac for all times). There are many by many authors including: O. Diekmann, P.-E. Jabin, L. Desvillettes, St. Mischler, G. Barles, S. Genieys, M. Gauduchon, S. Cuadrado, J. Carillo, S. Mirrahimi, P. E. Souganidis A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
60 Can we get more on x(t)? A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
61 Canonical Equation u ǫ (t, x ǫ (t)) = 0 t u ǫ(t, x ǫ (t)) + D 2 xu ǫ (t, x ǫ (t)) x ǫ (t) = 0 D 2 xu ǫ (t, x ǫ (t)) x ǫ (t) = t u ǫ(t, x ǫ (t)) = x R( x ǫ (t),i ǫ (t)) ǫ x u ǫ A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
62 Canonical Equation Passing to the limit d dt x(t) = ( D2 1 u) R( x(t),ī(t)) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
63 Canonical Equation Passing to the limit d dt x(t) = ( D2 1 u) R( x(t),ī(t)) Let s illustrate that A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
64 Canonical Equation Passing to the limit d dt x(t) = ( D2 1 u) R( x(t),ī(t)) Let s illustrate that A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
65 Interplay of Canonical Equation and R ( x(t), Ī(t)) = 0 d dt x(t) = ( D2 1 u) R( x(t),ī(t)) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
66 with cut-off: Motivation The tail problem A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
67 with cut-off: Motivation The tail problem A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
68 with cut-off: Motivation The tail problem A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
69 with cut-off: Motivation The tail problem Further out: number of individuals < 1 should be zero A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
70 with cut-off: Motivation penalise the tails t n σ (t,x) = n σ (t,x)r ( x,i(t) ) + n σ (t,x) n σ σ 21 {n σ µ σ α } seems to cut exponential tails A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
71 The tail problem with cut-off: simplified version t n σ (t,x) = n σ (t,x)r ( x,i(t) ) + n σ (t,x) n σ σ 21 {n σ µ σ α } f = n σ (t,x) + n σ (t,x) n σ σ 21 {n σ µ σ α } solutions very non-unique variational solution maximal solution minimal solution on the right scale: interesting free boundary problem A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
72 Back to cancer Outline 1 Motivation 2 3 Back to cancer : cancer cells : healthy and cancer cells 4 Heterogeneity in the model A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
73 Back to cancer : cancer cells Biological problem a resistant sub-populations is selected can be viewed as Darwinian evolution this can be formulated in the frame work above t n = nr ( x,i(t) ) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
74 Back to cancer : cancer cells 1d model: cancer cells with therapy t n(x,t) = [ p(x) d(x)i(t) ] n(x, t) n(x,t): number of cells with trait x resistance phenotype x [0,1], x larger more resistant more resistant less proliferative (cost of resistance) two possible targets for therapy: slowing down proliferation or increasing cell death A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
75 Back to cancer : cancer cells 1d model: cancer cells with therapy t n(x,t) = [ ] p(x) d(x)i(t) µ(x)c n(x, t) n(x,t): number of cells with trait x resistance phenotype x [0,1], x larger more resistant more resistant less proliferative (cost of resistance) two possible targets for therapy: slowing down proliferation or increasing cell death cytotoxic drug c A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
76 8 2 Back to cancer : cancer cells 1d model: cancer cells with therapy t n(x,t) = [ ] p(x) d(x)i(t) µ(x)c n(x, t) level-sets of n(x,t)/ n(x,t)dx x t A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
77 Back to cancer : healthy and cancer cells How to avoid resistance? Drug resistance is a major obstacle for anti-cancer therapy using different drugs to reduce resistance. Cytotoxic drugs: killing cells Cytostatic drugs: slowing down cell proliferation Cytotoxic drugs + cytostatic drugs A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
78 Back to cancer : healthy and cancer cells Polychemotherapy cancer cells n C compare with effects on healthy cells n H t n C (x,t) = [ ] p C (x) d C (x)i C (t) c 1 µ C (x) n C (x,t), where I C = ρ c 1 : cytotoxic therapy A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
79 Back to cancer : healthy and cancer cells Polychemotherapy cancer cells n C compare with effects on healthy cells n H t n C (x,t) = [ ] 1 p C (x) d C (x)i C (t) c 1 µ C (x) n C (x,t), 1 + α C c 2 where I C = ρ c 1 : cytotoxic therapy c 2 : cytostatic therapy A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
80 Back to cancer : healthy and cancer cells Polychemotherapy compare with effects on cancer n C and healthy cells n H t n C (x,t) = [ ] 1 p C (x) d C (x)i C (t) c 1 µ C (x) n C (x,t), 1 + α C c 2 where space c 1 : cytotoxic therapy c 2 : cytostatic therapy A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
81 Back to cancer : healthy and cancer cells Polychemotherapy compare with effects on cancer n C and healthy cells n H t n C (x,t) = [ ] 1 p C (x) d C (x)i C (t) c 1 µ C (x) n C (x,t), 1 + α C c 2 t n H (x,t) = [ ] 1 p H (x) d H (x)i H (t) c 1 µ H (x) n H (x,t), 1 + α H c 2 where space c 1 : cytotoxic therapy c 2 : cytostatic therapy A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
82 Back to cancer : healthy and cancer cells Polychemotherapy compare with effects on cancer n C and healthy cells n H t n C (x,t) = [ ] 1 p C (x) d C (x)i C (t) c 1 µ C (x) n C (x,t), 1 + α C c 2 t n H (x,t) = [ ] 1 p H (x) d H (x)i H (t) c 1 µ H (x) n H (x,t), 1 + α H c 2 where I C = a CH ρ H + a CC ρ C c 1 : cytotoxic therapy c 2 : cytostatic therapy I H = a HH ρ H + a HC ρ C A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
83 Back to cancer : healthy and cancer cells Parameter functions: proliferation/death Cost of resistance Cytotoxic drugs Adapted to cancer cells p > 0, p < 0. µ > 0, µ < 0. µ H < µ C α H < α C. A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
84 Back to cancer : healthy and cancer cells Effect of therapy on R: p C (x) c 1 µ C (x) space c 1 = A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
85 Back to cancer : healthy and cancer cells Effect of therapy on R: p C (x) c 1 µ C (x) space c 1 = A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
86 Back to cancer : healthy and cancer cells Effect of therapy on R: p C (x) c 1 µ C (x) space c 1 = A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
87 Back to cancer : healthy and cancer cells Effect of therapy on R: p C (x) c 1 µ C (x) space c 1 = A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
88 Back to cancer : healthy and cancer cells Preliminaries: initial conditions 0.5 n H (t=0,x) x A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
89 Back to cancer : healthy and cancer cells Cytotoxic agents only (c 1 = 0, 1.75, 3.5 with c 2 = 0) n C (t=2000,x) n H (t=2000,x) 60 c 1 =0, c 2 =0 25 c 1 =0, c 2 =0 c 1 =1.75, c 2 =0 c 1 =1.75, c 2 =0 c 1 =3.5, c 2 =0 c 1 =3.5, c 2 = x cancer cells x healthy cells c 1 = 0 selection for strong proliferative potential. c 1 increases both the number of cancer cells and healthy cells become smaller + selection for resistance. A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
90 Back to cancer : healthy and cancer cells Cytostatic agents only (c 2 = 1, 3, 7 with c 1 = 0) n C (t=2000,x) n H (t=2000,x) 30 c 1 =0, c 2 =1 30 c 1 =0, c 2 =1 c 1 =0, c 2 =3 c 1 =0, c 2 =3 c 1 =0, c 2 =7 c 1 =0, c 2 = x cancer cells x healthy cells c 2 increases the dynamics of healthy cells is kept unaltered while the proliferation of cancer cells is reduced + the dynamics of cancer cells is slowed down. A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
91 Back to cancer : healthy and cancer cells Simultaneous action of cytostatic and cytotoxic agents (c 1 = c 2 = 0, 1, 1.5, 2) cancer cells A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
92 Back to cancer : healthy and cancer cells Simultaneous action of cytostatic and cytotoxic agents (c 1 = c 2 = 0, 1, 1.5, 2) healthy cells A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
93 Back to cancer : healthy and cancer cells Action of cytostatic and cytotoxic agents A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
94 Back to cancer : healthy and cancer cells Simultaneous action of cytostatic and cytotoxic agents cytostatic agents slow down the growth of resistant cells cytotoxic agents kill the cancer cells A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
95 Heterogeneity in the model Outline 1 Motivation 2 3 Back to cancer : cancer cells : healthy and cancer cells 4 Heterogeneity in the model A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
96 Heterogeneity in the model Resistance to therapies A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
97 Heterogeneity in the model Resistance to therapies In ecology, competitive exclusion principle : In a ecological niche only one single species can exist Why is there such a large phenotypical heterogeneity in tumour cells? A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
98 Heterogeneity in the model Resistance to therapies In ecology, competitive exclusion principle : In a ecological niche only one single species can exist Why is there such a large phenotypical heterogeneity in tumour cells? A large phenotypical heterogeneity in tumour cells can make them more resistant. A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
99 Heterogeneity in the model for a spherical tumor Collaboration with J. Clairambault, A. Escargueil, T. Lorenzi, B. Perthame Nutrients and cytotoxic agents diffuse in the tumour through the boundary. A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
100 Heterogeneity in the model for spherical tumour t n(t,r,x) = [ { proliferation death }}{{}}{ p(x)s(t,r) d(x)ρ(t) ] n(t,r,x), n(t,r,x): cancer cells density at radius r [0,1] with resistance phenotype x [0, 1] ρ(t): total number of cancer cells s(t, r): nutrients c(t, r): cytotoxic therapy p(x),d(x),µ(x) given, σ s,σ c,γ s,γ c constants A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
101 Heterogeneity in the model for spherical tumour t n(t,r,x) = [ { proliferation death }}{{}}{ p(x)s(t,r) d(x)ρ(t) effect of therapy {}}{ µ(x)c(t,r) ] n(t,r,x), n(t,r,x): cancer cells density at radius r [0,1] with resistance phenotype x [0, 1] ρ(t): total number of cancer cells s(t, r): nutrients c(t, r): cytotoxic therapy p(x),d(x),µ(x) given, σ s,σ c,γ s,γ c constants A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
102 Heterogeneity in the model for spherical tumour t n(t,r,x) = [ { proliferation death }}{{}}{ p(x)s(t,r) d(x)ρ(t) [ σ s s(t,r) + }{{} γ }{{} s + p(x)n(t,r,x)dx diffusion degradation } {{ } consumption effect of therapy {}}{ µ(x)c(t,r) ] n(t,r,x), ] s(t,r) = 0, n(t,r,x): cancer cells density at radius r [0,1] with resistance phenotype x [0, 1] ρ(t): total number of cancer cells s(t, r): nutrients c(t, r): cytotoxic therapy p(x),d(x),µ(x) given, σ s,σ c,γ s,γ c constants A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
103 Heterogeneity in the model for spherical tumour t n(t,r,x) = [ { proliferation death }}{{}}{ p(x)s(t,r) d(x)ρ(t) [ σ s s(t,r) + }{{} γ }{{} s + p(x)n(t,r,x)dx diffusion degradation [ σ c c(t,r) + γ c + } {{ } consumption µ(x)n(t,r,x)dx effect of therapy {}}{ µ(x)c(t,r) ] n(t,r,x), ] s(t,r) = 0, ] c(t,r) = 0. n(t,r,x): cancer cells density at radius r [0,1] with resistance phenotype x [0, 1] ρ(t): total number of cancer cells s(t, r): nutrients c(t, r): cytotoxic therapy p(x),d(x),µ(x) given, σ s,σ c,γ s,γ c constants A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
104 Heterogeneity in the model for spherical tumour t n(t,r,x) = [ { proliferation death }}{{}}{ p(x)s(t,r) d(x)ρ(t) [ σ s s(t,r) + }{{} γ }{{} s + p(x)n(t,r,x)dx diffusion degradation [ σ c c(t,r) + γ c + } {{ } consumption µ(x)n(t,r,x)dx effect of therapy {}}{ µ(x)c(t,r) ] n(t,r,x), ] s(t,r) = 0, ] c(t,r) = 0. boundary data: s(t,r = 1),c(t,r = 1) n(t,r,x): cancer cells density at radius r [0,1] with resistance phenotype x [0, 1] ρ(t): total number of cancer cells s(t, r): nutrients c(t, r): cytotoxic therapy p(x),d(x),µ(x) given, σ s,σ c,γ s,γ c constants A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
105 Heterogeneity in the model nutrient profile s(t,r) r A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
106 Heterogeneity in the model Without cytotoxic therapy phenotype averaged over radius r phenotype at radius r n(t,r,x)dr/ρ(t) n(t,r,x)/ n(t,r,x)dx x x decreases of heterogeneity r A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
107 Heterogeneity in the model With cytotoxic therapy phenotype averaged over radius r phenotype at radius r n(t,r,x)dr/ρ(t) n(t,r,x)/ n(t,r,x)dx x x decreases of heterogeneity r A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
108 Heterogeneity in the model Definition of heterogeneity averaged dominant trait X(t) := 1 ρ(t) xn(t,r,x)dxdr heterogeneity or trait variance xn(t,r,x)dx h(t) := ρ(t,r) X(t) 2 dr where ρ(t,r) = n(t,r,x)dx A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
109 Heterogeneity in the model Future Work analytic results e.g. reduction of heterogeneity by chemotherapy include growing or shrinking tumour model validation with data therapy optimisation: polychemotherapy, adaptive therapy A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
110 Heterogeneity in the model THANK YOU Gatenby R.A., A change of strategy in the war on cancer, Nature, 459 (2009) A. Lorz (LJLL, UPMC & INRIA MAMBA) Dynamique des populations Apr 11th, / 62
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