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Solving Numerically a Problem Modelling Cancer Therapy

Solving Numerically a Problem Modelling Cancer Therapy

Maria Emilia Castillo 26, May 2010

Solving Numerically a Problem Modelling Cancer Therapy

Outline

The goals Model for cancer cells population growth Model without therapy Model with chemotherapy Toxicity Constraints Optimal Control Problem MCT Constraints Liang Model Future Work

Solving Numerically a Problem Modelling Cancer Therapy The goals

The goals

◮ To find a chemotherapy regime that minimizes the tumor

population at the end of a fixed chemotherapy period

Solving Numerically a Problem Modelling Cancer Therapy The goals

The goals

◮ To find a chemotherapy regime that minimizes the tumor

population at the end of a fixed chemotherapy period

◮ To learn how to use free derivative software to solve nonlinear

Optimal Control Problems with inequality constraints

Solving Numerically a Problem Modelling Cancer Therapy Model for cancer cells population growth Model without therapy

Most commons models to study growth population

dN(t) dt

= λN(t) „ 1 − N(t)

θ

« Logistic N(t) = N0 exp(−λt) Malthus

dN(t) dt

= λN(t) ln „

θ N(t)

« Gompertz

Solving Numerically a Problem Modelling Cancer Therapy Model for cancer cells population growth Model without therapy

dN(t) dt = λN(t) ln „ θ N(t) « .

Solving Numerically a Problem Modelling Cancer Therapy Model for cancer cells population growth Model without therapy

dN(t) dt = λN(t) ln „ θ N(t) « . ◮ N0 = 1

Solving Numerically a Problem Modelling Cancer Therapy Model for cancer cells population growth Model without therapy

dN(t) dt = λN(t) ln „ θ N(t) « . ◮ N0 = 1 ◮ λ = 9.9 × 10−4

Solving Numerically a Problem Modelling Cancer Therapy Model for cancer cells population growth Model without therapy

dN(t) dt = λN(t) ln „ θ N(t) « . ◮ N0 = 1 ◮ λ = 9.9 × 10−4 ◮ θ = 1012

Solving Numerically a Problem Modelling Cancer Therapy Model for cancer cells population growth Model without therapy

dN(t) dt = λN(t) ln „ θ N(t) « . ◮ N0 = 1 ◮ λ = 9.9 × 10−4 ◮ θ = 1012

Solving Numerically a Problem Modelling Cancer Therapy Model for cancer cells population growth Model with chemotherapy

[0, Tf ] =fixed period of chemotherapy v(t) =drug concentration    dv dt = u(t) − γv(t), v(0) = v0. u(t) =rate of increase of the drug concentration u(t) =          u1, if 0 = t1 ≤ t < t2; u2, if t2 ≤ t < t3; . . . . . . un, if tn ≤ t ≤ tn+1 = Tf ; ui ≥ 0

Solving Numerically a Problem Modelling Cancer Therapy Model for cancer cells population growth Toxicity Constraints

Toxicity Constraints

0 ≤ v(t) ≤ vmax ∀t ∈ [0, Tf ], AUC(t) = t v(s) ds, AUC(Tf ) ≤ vcum.

Solving Numerically a Problem Modelling Cancer Therapy Model for cancer cells population growth Toxicity Constraints

Theorem

0 ≤ v(t) ≤ vmax ∀t ∈ [0, Tf ]

• 0 ≤ v(ti) ≤ vmax

∀i = 1, . . . , n,

Solving Numerically a Problem Modelling Cancer Therapy Model for cancer cells population growth Toxicity Constraints

k = proportion of tumor cells killed per unit time per unit drug concentration vth = therapeutic drug concentration threshold

Solving Numerically a Problem Modelling Cancer Therapy Model for cancer cells population growth Toxicity Constraints

k = proportion of tumor cells killed per unit time per unit drug concentration vth = therapeutic drug concentration threshold Then, dN(t) dt = λN(t) ln

• θ

N(t)

• − k(v(t) − vth)+N(t)

N(0) = N0

Solving Numerically a Problem Modelling Cancer Therapy Model for cancer cells population growth Toxicity Constraints

                                    

dN(t) dt

= λN(t) ln

• θ

N(t)

• − k(v(t) − vth)+N(t)

N(0) = N0 dv dt = u(t) − γv(t), 0 ≤ v(ti) ≤ vmax ∀i = 1, . . . , n, AUC(Tf ) ≤ vcum.

Solving Numerically a Problem Modelling Cancer Therapy Optimal Control Problem

minimize{ˆ J(u) = N(Tf )} subject to                                     

dN(t) dt

= λN(t) ln

• θ

N(t)

• − k(v(t) − vth)+N(t)

N(0) = N0 dv dt = u(t) − γv(t), 0 ≤ v(ti) ≤ vmax ∀i = 1, . . . , n, AUC(Tf ) ≤ vcum.

Solving Numerically a Problem Modelling Cancer Therapy Optimal Control Problem

Numerical method

◮ Numerical computation of the functional based on Runge

Kutta (4th order) approximation of the state equation

◮ Software for optimizing: NOMAD which implements the

MADS algorithm (Mesh Adaptive Direct Search)

◮ Direct search ◮ Costly functions ◮ No derivatives ◮ Many kinds of constraints

Solving Numerically a Problem Modelling Cancer Therapy Optimal Control Problem

Numerical Problem

◮ Tf = 84 days ◮ n = 16 ◮ λ = 9.9 × 10−4 days−1. ◮ θ = 1012 ◮ vth = 10[D] ◮ γ = 0.27 days−1, the half life of the drug is 2.5 days. ◮ k = 8.4 × 10−3 ◮ N0 = 1010. ◮ v0 = 0. ◮ vmax = 50 [D]. ◮ vcum = 2.1 × 103 [D] days.

Solving Numerically a Problem Modelling Cancer Therapy Optimal Control Problem

Numerical Results

N(Tf ) = 56943

Solving Numerically a Problem Modelling Cancer Therapy Optimal Control Problem MCT Constraints

MCT Constraints

N(τi+1) ≤ ǫN(τi) ∀i = 0, . . . , M − 1. with 0 ≤ ǫ ≤ 1

Solving Numerically a Problem Modelling Cancer Therapy Optimal Control Problem MCT Constraints

Numerical Problem with MCT

◮ Tf = 84 days ◮ n = 16 ◮ λ = 9.9 × 10−4 days−1. ◮ θ = 1012 ◮ vth = 10[D] ◮ γ = 0.27 days−1, the half life of the drug is 2.5 days. ◮ k = 8.4 × 10−3 ◮ N0 = 1010. ◮ v0 = 0. ◮ vmax = 50 [D]. ◮ vcum = 2.1 × 103 [D] days. ◮ ǫ = 0.5

Solving Numerically a Problem Modelling Cancer Therapy Optimal Control Problem MCT Constraints

Numerical Results with MCT

N(Tf ) = 99386

Solving Numerically a Problem Modelling Cancer Therapy Optimal Control Problem Liang Model

Liang Model

w(t) = AUC(t) Martin   

dw(t) dt

= v(t) w(0) = 0 w(Tf ) ≤ vcum Liang   

dw(t) dt

= v(t) − ηw(t) w(0) = 0 0 ≤ w(t) ≤ vcum ∀t ∈ [0, Tf ]

Solving Numerically a Problem Modelling Cancer Therapy Optimal Control Problem Liang Model

Numerical Results for Liang Model

◮ Tf = 84 days ◮ n = 16 ◮ λ = 9.9 × 10−4 days−1. ◮ θ = 1012 ◮ vth = 10[D] ◮ γ = 0.27 days−1, the half life of the drug is 2.5 days. ◮ k = 8.4 × 10−3 ◮ N0 = 1010. ◮ v0 = 0. ◮ vmax = 50 [D]. ◮ vcum = 100 [D] days. ◮ ǫ = 0.5 ◮ η = 0.4

Solving Numerically a Problem Modelling Cancer Therapy Optimal Control Problem Liang Model

Liang Model

N(Tf ) = 481

Solving Numerically a Problem Modelling Cancer Therapy Future Work

Future Work

◮ Comparison with gradients methods.

Solving Numerically a Problem Modelling Cancer Therapy Future Work

Future Work

◮ Comparison with gradients methods. ◮ Use another methods: Condor.

Solving Numerically a Problem Modelling Cancer Therapy Future Work

Future Work

◮ Comparison with gradients methods. ◮ Use another methods: Condor. ◮ Increase the number of therapy intervals to get a comparison

between therapies with continuous dosis and discrete dosis.

Solving Numerically a Problem Modelling Cancer Therapy Future Work

Future Work

◮ Comparison with gradients methods. ◮ Use another methods: Condor. ◮ Increase the number of therapy intervals to get a comparison

between therapies with continuous dosis and discrete dosis.

◮ Add the space variables, even in simple cases as radial growth.

Solving Numerically a Problem Modelling Cancer Therapy Future Work

References

• R. B. Martin and M. E. Fisher and R. F. Minchin and K. L. Teo, A mathematical model of cancer

chemotherapy with an optimal selection of parameters, Mathematical Biosciences, 99, Num. 2, pp. 205–230, 1990

• R. B. Martin, Optimal control drug scheduling of cancer chemotherapy, Automatica, Vol. 28, No. 6, pp.

1113–1123, 1992 Yong Liang, Kwong-Sak Leung and Tony Shu Kam Mok, A Novel Evolutionary Drug Scheduling Model in Cancer Chemotherapy, IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE, Vol. 10, NO. 2, pp. 237–245, APRIL 2006