A mathematical model for the primary tumor of mcrc

Transcription

1 A mathematical model for the primary tumor of mcrc Marta Leocata Joint work with F.Flandoli, C. Ricci, M.C. Polito, V. De Mattei December 2, 2016 University of Pisa

2 Plan of the talk General Project; A mathematical model for mcrc before treatment; A mathematical model for mcrc during treatment; Monte Carlo simulation. 1

3 General Project

4 General Project

5 General Project Age Sex Number of lesions Albumina K-RAS B-RAF.

6 General Project FOLFOXIRI Age Sex Number of lesions Albumina K-RAS B-RAF PFS OS TTP. 5-FU 5-FU+BV 2

7 State of the art of the project

8 Main idea Age Sex Number of lesions Albumina K-RAS B-RAF TTP.

9 Main idea Age Sex Number of lesions Albumina K-RAS B-RAF. λ µ η C hypo V. 5-FU 5-FU+BV TTP

10 Main idea Age Sex Number of lesions Albumina K-RAS B-RAF.? λ µ η C hypo V. 5-FU 5-FU+BV TTP 3

11 A mathematical model for mcrc

12 A mathematical model for mcrc The model is characterized by the following quantities: N t = N hyp t + N norm t = number of cancerous cells; - Nt norm = number of cancerous normoxic cells; - Nt hyp = number of cancerous hypoxic cells; 4

13 A mathematical model for mcrc The model is characterized by the following quantities: N t = N hyp t + N norm t = number of cancerous cells; - Nt norm = number of cancerous normoxic cells; - Nt hyp = number of cancerous hypoxic cells; V t = intensity of VEGF field; A t = level of vascularization due to Angiogenesis. 4

14 Equation for N t We approximate the cancerous mass as a sphere: η thickness of proliferating boundary Hypoxic cells Remark: η = the thickness of proliferating boundary in term of cells.

15 Equation for N t We approximate the cancerous mass as a sphere: Angiogenesis η thickness of proliferating boundary Hypoxic cells Remark: η = the thickness of proliferating boundary in term of cells. 5

16 Equation for N t By this approximation we imagine that: d dt N t = λnt norm µn t 6

17 Equation for N t By this approximation we imagine that: d dt N t = λnt norm µn t ( ) 3 1 = λn t N 1/3 t /2η }{{} boundary term 6

18 Equation for N t By this approximation we imagine that: d dt N t = λnt norm µn t ( ) 3 1 = λn t N 1/3 t /2η }{{} boundary term ) 3 1 λa t N t (1 1 + N 1/3 t /2η }{{} Angiogenesis term + 6

19 Equation for N t By this approximation we imagine that: d dt N t = λnt norm µn t ( ) 3 1 = λn t N 1/3 + t /2η }{{} boundary term ) 3 1 λa t N t (1 1 + N 1/3 t /2η }{{} Angiogenesis term µn t }{{} death term 6

20 Equation for N t By this approximation we imagine that: d dt N t = λnt norm µn t ( ) 3 1 = λn t N 1/3 + t /2η }{{} boundary term ) 3 1 λa t N t (1 1 + N 1/3 t /2η }{{} Angiogenesis term µn t }{{} death term and N hypo t = N t N norm t = (1 A t ) N t ( N 1/3 t /2η ) 3 6

21 Equation for A t A t = space average of vascularization due to Angiogenesis. 7

22 Equation for A t A t = space average of vascularization due to Angiogenesis. d dt A t = C V A (V t V thrsld )(A t + A pre )(1 A t )1 Vt>V }{{ thrsld } growth term 7

23 Equation for A t A t = space average of vascularization due to Angiogenesis. d dt A t = C V A (V t V thrsld )(A t + A pre )(1 A t )1 Vt>V }{{ thrsld } 2A 10 t 1 Vt V }{{ thrsld } decresing term growth term 7

24 Equation for VEGF V t = space average of VEGF field. 8

25 Equation for VEGF V t = space average of VEGF field. d dt V t = ( C hypo V (N hypo t N t ) 2/3 }{{} growth term ) C A,V A t V t (1 V t ) }{{} Absorption term 8

26 Equation for VEGF: 2/3 formula

27 Equation for VEGF: 2/3 formula Pre-Angiogenesis: η

28 Equation for VEGF: 2/3 formula Pre-Angiogenesis: We imagine the sphere of Hypoxic η cells as a sphere uniformly charged of radius R η. 9

29 Equation for VEGF: 2/3 formula It induces a field : E(r) = with potential on the boundary C(R η)3 r 2 V (R) = C(R η) 2 VEGF work needed to move blood vessels from infinite distance to the sphere of hypoxic cells. V t = C(R t η) 2 C(N hypo t ) 2/3 10

30 Parameters of the free model Parameters Meaning λ growth rate due to cell proliferation µ decay rate due to cell loss η thickness of proliferating boundary C hypo V C A,V C V A VEGF production rate from hypoxic cells absorption rate of VEGF from vasculature reaction rate of angiogenesis to VEGF 11

31 Discussion on Parameters of the free model: η 12

32 Discussion on Parameters of the free model: η Reasonable values for η? 12

33 Discussion on Parameters of the free model: η Reasonable values for η? Tumors of radius 1mm are usually avascular; Proportion of proliferating boundary with respect to the total (α) is not small. 12

34 Discussion on Parameters of the free model: η Reasonable values for η? Tumors of radius 1mm are usually avascular; Proportion of proliferating boundary with respect to the total (α) is not small. η = 103 (1 (1 α) 1/3 ) 12 Through Spherical Approximation 12

35 Discussion on Parameters of the free model: η Reasonable values for η are around

36 Discussion on Parameters of the free model: (λ, µ) 14

37 Discussion on Parameters of the free model: (λ, µ) Average of DT=

38 Discussion on Parameters of the free model Parameters Value λ 0.05 µ η 15 C hypo V 0.08 C A,V 0.01 C V A A pre 0.2 V thrshld

39 Simulation without therapy 16

40 Simulation without therapy. Comments Diamonds denote the range where angiogenesis is expected to start and the red star is that initial time; The second red star is when 10 9 cells are reached which should be close to 8 years; At the beginning Boundary term N t ; If N 1/3 t /η >> 1 Boundary term N 2/3 t ; If N t is large and there is Angiogenesis Exponential regime. 17

41 A mathematical model for mcrc during treatment

42 Treatment Following 1 we investigate the case of first line therapy based on 5-Fluoracile (5-FU) with or without Bevacizumab (BV). 8 weeks 8 weeks 12 cycles of 5-FU(+BV) 1 F. F. Kabbinavar et al., Addition of Bevacizumab to Bolus Fluorouracil and Leucovorin in First-Line Metastatic Colorectal Cancer: Results of a Randomized Phase II Trial, J. Clinical Oncology 23 (2005), n. 16,

43 Drug resistance due to mutations We introduce new quantities: 19

44 Drug resistance due to mutations We introduce new quantities: Nt R,norm Nt R,hypo Nt R = Nt R,norm resistant cells; = number of hypoxic 5-FU drug resistant cells; = number of normoxic 5-FU drug resistant cells; + N R,hypo t = total number of 5-FU drug 19

45 Drug resistance due to mutations We introduce new quantities: Nt R,norm Nt R,hypo Nt R = Nt R,norm resistant cells; = number of hypoxic 5-FU drug resistant cells; = number of normoxic 5-FU drug resistant cells; + N R,hypo t So we have two subpopulation: = total number of 5-FU drug N R t N S t = Nt R,norm = Nt S,norm + N R,hypo t + N S,hypo t 19

46 Drug resistance due to mutations We introduce new quantities: Nt R,norm Nt R,hypo Nt R = Nt R,norm resistant cells; = number of hypoxic 5-FU drug resistant cells; = number of normoxic 5-FU drug resistant cells; + N R,hypo t So we have two subpopulation: = total number of 5-FU drug N R t N S t = Nt R,norm = Nt S,norm + N R,hypo t + N S,hypo t How do we describe the growth of the two subpopulations? 19

47 Drug resistance due to mutations: growth of the two subpopulations Before treatment: 20

48 Drug resistance due to mutations: growth of the two subpopulations Before treatment: In red: 5-FU resistant cells

49 Drug resistance due to mutations: growth of the two subpopulations Before treatment: We conjecture that the fictitious proliferating boundary of 5-FU drug resistant is thinner than the case of sensitive cells In red: 5-FU resistant cells

50 Drug resistance due to mutations: growth of the two subpopulations Before treatment: We conjecture that the fictitious proliferating boundary of 5-FU drug resistant is thinner than the case of sensitive cells In red: 5-FU resistant cells η R < η S 20

51 Drug resistance due to mutations: growth of the two subpopulations After treatment: 21

52 Drug resistance due to mutations: growth of the two subpopulations After treatment: The approximation as two separate sphere becames reasonable

53 Drug resistance due to mutations: growth of the two subpopulations After treatment: The approximation as two separate sphere becames reasonable η R = η S 21

54 Drug resistance due to mutations: transition of species Let p = the probability of mutation. In the infinitesimal interval [t, t + t] the number of proliferating cells is λnt S,norm t Thus the average number of mutated descendants is: pλnt S,norm t 22

55 Drug resistance due to mutations: Equation for N R t and N S t d dt NS t = λnt S,norm µ Nt S 23

56 Drug resistance due to mutations: Equation for N R t and N S t d dt NS t = λn t S,norm µ Nt S pλn t S,norm }{{} change of species term 23

57 Drug resistance due to mutations: Equation for N R t and N S t d dt NS t = λn t S,norm µ Nt S pλn t S,norm }{{} change of species term d dt NR t = λnt R,norm µ Nt R 23

58 Drug resistance due to mutations: Equation for N R t and N S t d dt NS t = λn t S,norm µ Nt S pλn t S,norm }{{} change of species term d dt NR t = λn t R,norm µ Nt R +pλn t S,norm }{{} change of species term 23

59 Drug resistance due to mutations: Equation for N R t and N S t d dt NS t = λn t S,norm µ Nt S pλn t S,norm }{{} change of species term d dt NR t = λn t R,norm µ Nt R +pλn t S,norm }{{} change of species term Then d dt N t = λp t Nt norm µn t where ( N S N t = t N R t ), N norm t = ( N S,norm) [ t Nt R,norm, P = 1 p p 0 1 ] 23

60 Action of 5-FU 5-FU acts as a control on our system. 24

61 Action of 5-FU 5-FU acts as a control on our system. It simply acts on N S t : d dt NS t = λn S,norm t (1 ut FU ) }{{} control term µn S t 24

62 Action of 5-FU 5-FU acts as a control on our system. It simply acts on N S t : d dt NS t = λn S,norm t (1 ut FU ) }{{} control term µn S t G 0 M G 1 S 5-FU acts on S G 2 where ut FU plasma describes concentration of 5-Fu in tissues, NOT in 24

63 Action of 5-FU: u FU t In every period of N0 5-FU therapy u FU t = 0. 25

64 Action of 5-FU: u FU t In every period of N0 5-FU therapy u FU t = 0. Administration of 5-FU

65 Action of 5-FU: u FU t In every period of N0 5-FU therapy u FU t = 0. Administration of 5-FU concentration during a single week? (1 + C FU ) exp( log(1+c FU) N FU t) where C FU = intensity of cell kill with respect cell proliferation; N FU = days of action of 5-FU. 25

66 Action of 5-FU: Plot of 1 u FU t

67 Action of 5-FU: Plot of 1 u FU t λ(1 u FU 0 ) = λc FU

68 Action of 5-FU: Plot of 1 u FU t λ(1 u FU 0 ) = λc FU (1 u FU N FU ) = 0 26

69 Action of Bevacizumab Bevacizumab acts as a control on our system. It acts on VEGF: ) d (N dt V t t = (C hypo 2/3 hypo V C V AA t N t ) C beva V ut beva V }{{} t (1 V t ) Control term 27

70 Action of Bevacizumab: u BV t In every period of No BEVA therapy u BV t = 0. 28

71 Action of Bevacizumab: u BV t In every period of No BEVA therapy u BV t = 0. where Administration of Bevacizumab concentration during two weeks? exp( log(2) N BV t) N BV = half life of Bevacizumab. 28

72 Action of Bevacizumab: Plot of u BV t 29

73 Action of Bevacizumab: Plot of u BV t u BV N BV = 1/2 29

74 Plot of VEGF and A t with therapy How do VEGF and A t react to therapy? Figure 1: On the right: plot of A t. On the left: plot of V t. 30

75 Parameters of the model with therapy Parameters Meaning Value η sens thickness of p.b. of sensitive cells 15 η res thickness of p.b. of resistant cells η sens /1.2 p probability of mutation 10 5 N start number of cells when therapy starts N FU number of days of action of 5-FU 6 C FU intensity of 5-FU action 20 N beva number of days of action of Bevacizumab 12 C beva V inhibition rate of bevacizumab on VEGF 1 31

76 Simulation with therapy (5-FU+ Bevacizumab) Figure 2: In green: resistant cells (N R t ). In blue: total cells (N S t + N R t ). 32

77 Computation of TTP How to evaluate differences between the two therapies? 33

78 Computation of TTP How to evaluate differences between the two therapies? s = Time of minimum volume. 33

79 Computation of TTP How to evaluate differences between the two therapies? s = Time of minimum volume. Clinical measuraments: in terms of diameter [D t > D s + 0.2D s ] Measuraments in our model: in terms of cells [?] 33

80 Computation of TTP How to evaluate differences between the two therapies? s = Time of minimum volume. Clinical measuraments: in terms of diameter [D t > D s + 0.2D s ] Measuraments in our model: in terms of cells [?] D t > D s + 0.2D s V t > (1, 2) 3 V s N t 1.7 N s n t := N t N s N s

81 Computation of TTP 12 cycles of therapy T 0 Computation of N Computation of N 1. We compute N k at times T 0 + k 8. Where T 0 = time when therapy starts; 2. N k0 = inf k {N k }; 3. We compute n k = N k N k0 N k0 for k > k 0 ; 4. We compute the first k 1 such that n k

82 Computation of TTP 12 cycles of therapy T 0 Computation of N Computation of N 1. We compute N k at times T 0 + k 8. Where T 0 = time when therapy starts; 2. N k0 = inf k {N k }; 3. We compute n k = N k N k0 N k0 for k > k 0 ; 4. We compute the first k 1 such that n k1 0.7 TTP 34

83 Simulation with Therapy Figure 3: Simulation with Therapy= 5-FU. TTP=

84 Simulation with Therapy Figure 4: Simulation with Therapy= 5-FU+BEVA TTP=9.3 36

85 Monte Carlo Simulations

86 Monte Carlo Simulations GOAL: We want to evaluate the model 37

87 Monte Carlo Simulations GOAL: We want to evaluate the model Compute Kaplan-Meier curves. 37

88 Monte Carlo Simulations GOAL: We want to evaluate the model Compute Kaplan-Meier curves. We randomize some of the coefficients: 37

89 Monte Carlo Simulations GOAL: We want to evaluate the model Compute Kaplan-Meier curves. We randomize some of the coefficients: Parameters Value or distribution η res η sens /unif[1, 2] p N start 10 unif[4,6] 2 10 unif[8,11] N FU unif[1, 7] C FU unif[5, 25] N beva unif[1, 14] C beva V unif[0, 5] 37

90 Monte Carlo Simulations GOAL: We want to evaluate the model Compute Kaplan-Meier curves. We randomize some of the coefficients: Parameters Value or distribution η res η sens /unif[1, 2] p N start 10 unif[4,6] 2 10 unif[8,11] N FU unif[1, 7] C FU unif[5, 25] N beva unif[1, 14] C beva V unif[0, 5] 37

91 Monte Carlo Simulation: Kaplan-Meier curves 38

92 Monte Carlo Simulation: Kaplan-Meier curves Remark:The result is in agreement with trial of Kabbinavar et al., restricted to patients with one metastatic disease site. 38

93 Link with prognostic factors Prognostic factors? Parameters of our model 39

94 Link with prognostic factors Prognostic factors Three factors:? Parameters of our model 39

95 Link with prognostic factors Prognostic factors Three factors:? Parameters of our model Age; Prior Adjuvant chemotherapy; Baseline albumin. 39

96 Link with prognostic factors: Age We conjecture: Age λ µ 40

97 Link with prognostic factors: Age We conjecture: Age λ µ Table 1: Change of values for some class of younger patients. Parameters Value or distribution λ 0.08 µ 0.02 Median TTP (5-FU) = 5.6. Median TTP (5-FU+BEVA) =

98 Link with prognostic factors: Prior adjuvant chemotherapy (PAC) We conjecture: PAC N start 41

99 Link with prognostic factors: Prior adjuvant chemotherapy (PAC) We conjecture: PAC N start Table 2: Change of distribution for a class of patient who received a prior adjuvant chemotherapy. Parameters N start Value or distribution 10 unif[8,9] Median TTP (5-FU) = 9.3. Median TTP (5-FU+BEVA) =

100 Link with prognostic factors: Baseline Albumina We conjecture: Albumina N FU C FU N bevat C beva V 42

101 Link with prognostic factors: Baseline Albumina We conjecture: Albumina N FU C FU N bevat C beva V Table 3: Change of distribution for a class of patient with b.a. 3.5 g/dl. Median TTP (5-FU) = 7.5. Median TTP (5-FU+BEVA) = 7.5. Parameters Value or distribution N FU unif[1, 4] C FU unif[5, 20] N beva unif[1, 7] C beva V unif[0, 1] 42

102 Next step FOLFOXIRI Age Sex Number of lesions Albumina K-RAS B-RAF. λ µ η C hypo V. 5-FU 5-FU+BV PFS OS TTP

103 Next step FOLFOXIRI Modelize: 1.Metastasis 2.Multidrug resistance Age Sex Number of lesions Albumina K-RAS B-RAF. λ µ η C hypo V. 5-FU 5-FU+BV PFS OS TTP 43

104 Thank you for your attention! 44