# American Institute of Mathematical Sciences

June  2019, 24(6): 2577-2612. doi: 10.3934/dcdsb.2018266

## Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy

 Dep. Matemáticas, Estadística y Computación, Universidad de Cantabria, Avda. de los Castros, s/n, 39005 Santander, Spain

Received  November 2017 Revised  April 2018 Published  October 2018

We study a collection of problems associated with the optimization of cancer chemotherapy treatments, under the assumptions of Gomperztian-type tumor growth and that the drug killing effect is proportional to the rate of growth for the untreated tumor (Norton-Simon hypothesis). Classical pharmacokinetics and different pharmacodynamics (Skipper and Emax) are considered, together with a toxicity limit or the penalization of the accumulated drug effect. Existence and uniqueness of the optimal control is proved in some cases, while in others the total amount of drug is the unique relevant aspect to take into account and the existence of an infinite number of optimal controls is shown. In all cases, explicit expressions for the solutions are derived in terms of the problem data. Finally, numerical results of illustrative examples and some conclusions are presented.

Citation: Luis A. Fernández, Cecilia Pola. Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2577-2612. doi: 10.3934/dcdsb.2018266
##### References:

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##### References:
Optimization results for Skipper model on the left (see Table 1) and for the $E_{max}$ model on the right (see Table 2)
Some optimization results for $E_{max}$ model (see Table 2)
Tumor volume with trivial controls for Skipper model (see Table 1)
Tumor volume with trivial controls for $E_{max}$ model (see Table 2)
Switching times with $G = G_2$, $k_1 = 4$ and $k_2 = 0.25$
 Example Structure S_times $u_{sin}$ $ES_{52o2}$ $u_{max}/0$ 1.4e-01 $EM_{52o2}$ $u_{max}/0$ 2.3e+00 $EM_{52o3}$ $u_{max} / u_{sin}/$/0 5.6e-01 2.1e+00 3.4e-01 EL_{52o2} u_{max}/0 2.9e+00 EL_{52o3} u_{max}/ u_{sin}/$/0$ 6.8e-01 2.1e+00 4.0e-01 $ES_{62o2}$ $u_{max} / u_{sin}$ 1.6e-01 9.9e-02 $EM_{62o2}$ $u_{max} / u_{sin}$ 1.0e+00 5.6e-01 $EL_{62o2}$ $u_{max} / u_{sin}$ 1.4e+00 7.5e-01
 Example Structure S_times $u_{sin}$ $ES_{52o2}$ $u_{max}/0$ 1.4e-01 $EM_{52o2}$ $u_{max}/0$ 2.3e+00 $EM_{52o3}$ $u_{max} / u_{sin}/$/0 5.6e-01 2.1e+00 3.4e-01 EL_{52o2} u_{max}/0 2.9e+00 EL_{52o3} u_{max}/ u_{sin}/$/0$ 6.8e-01 2.1e+00 4.0e-01 $ES_{62o2}$ $u_{max} / u_{sin}$ 1.6e-01 9.9e-02 $EM_{62o2}$ $u_{max} / u_{sin}$ 1.0e+00 5.6e-01 $EL_{62o2}$ $u_{max} / u_{sin}$ 1.4e+00 7.5e-01
Parameters and numerical results with $G = G_1$, $k_1 = 4$
 Example $\xi$ $\lambda$ T $\hat{\alpha}$ $\int_0^T \bar{u}(t) dt$ $\int_0^T \bar{c}(t) dt$ $L_0/\theta$ $L(T)/L_0$ $ES_{41a1}$ 0.1 1 2.3e-03 2.4e+00 5.0e-02 1.7e-02 $ES_{41a2}$ 0.1 1 3.0e-02 9.5e-01 5.0e-02 3.8e-01 $ES_{41a3}$ 0.1 1 7.3e-02 0 5.0e-02 1.3e+00 $ES_{51a1}$ 0.1 0.27 1 9.0e-05 2.4e+00 5.0e-02 3.0e-01 $ES_{51a2}$ 0.1 0.27 1 3.0e-02 7.0e-01 5.0e-02 7.0e-01 $ES_{51a3}$ 0.1 0.27 1 6.4e-02 0 5.0e-02 1.3e+00 $ES_{61a1}$ 0.1 0.27 1 2.5e-02 2.4e+00 1.1e+00 5.0e-02 3.0e-01 $ES_{61a2}$ 0.1 0.27 1 3.0e-02 1.8e+00 9.5e-01 5.0e-02 3.8e-01 $ES_{61a3}$ 0.1 0.27 1 7.3e-02 0 0 5.0e-02 1.3e+00 $EM_{41c1}$ 0.1 1 1.2e-01 2.4e+00 5.0e-01 3.9e-01 $EM_{41c2}$ 0.1 1 1.4e-01 1.9e+00 5.0e-01 5.2e-01 $EM_{41c3}$ 0.1 1 1.5e-01 0 5.0e-01 1.0e+00 $EM_{51a1}$ 0.1 0.27 1 0 2.4e+00 5.0e-01 7.6e-01 $EM_{51a2}$ 0.1 0.27 1 5.9e-02 1.4e+00 5.0e-01 8.2e-01 $EM_{61b1}$ 0.1 0.27 1 1.4e-01 2.4e+00 1.1e+00 5.0e-01 7.6e-01 $EL_{41b1}$ 0.1 1 1.1e-01 2.4e+00 7.5e-01 6.8e-01 $EL_{41b2}$ 0.1 1 1.2e-01 0 7.5e-01 1.0e+00 $EL_{51a1}$ 0.1 0.27 1 0 2.4e+00 7.5e-01 8.9e-01 $EL_{51a2}$ 0.1 0.27 1 3.5e-02 1.5e+00 7.5e-01 9.1e-01 $EL_{61b2}$ 0.1 0.27 1 9.3e-02 2.4e+00 1.1e+00 7.5e-01 8.9e-01
 Example $\xi$ $\lambda$ T $\hat{\alpha}$ $\int_0^T \bar{u}(t) dt$ $\int_0^T \bar{c}(t) dt$ $L_0/\theta$ $L(T)/L_0$ $ES_{41a1}$ 0.1 1 2.3e-03 2.4e+00 5.0e-02 1.7e-02 $ES_{41a2}$ 0.1 1 3.0e-02 9.5e-01 5.0e-02 3.8e-01 $ES_{41a3}$ 0.1 1 7.3e-02 0 5.0e-02 1.3e+00 $ES_{51a1}$ 0.1 0.27 1 9.0e-05 2.4e+00 5.0e-02 3.0e-01 $ES_{51a2}$ 0.1 0.27 1 3.0e-02 7.0e-01 5.0e-02 7.0e-01 $ES_{51a3}$ 0.1 0.27 1 6.4e-02 0 5.0e-02 1.3e+00 $ES_{61a1}$ 0.1 0.27 1 2.5e-02 2.4e+00 1.1e+00 5.0e-02 3.0e-01 $ES_{61a2}$ 0.1 0.27 1 3.0e-02 1.8e+00 9.5e-01 5.0e-02 3.8e-01 $ES_{61a3}$ 0.1 0.27 1 7.3e-02 0 0 5.0e-02 1.3e+00 $EM_{41c1}$ 0.1 1 1.2e-01 2.4e+00 5.0e-01 3.9e-01 $EM_{41c2}$ 0.1 1 1.4e-01 1.9e+00 5.0e-01 5.2e-01 $EM_{41c3}$ 0.1 1 1.5e-01 0 5.0e-01 1.0e+00 $EM_{51a1}$ 0.1 0.27 1 0 2.4e+00 5.0e-01 7.6e-01 $EM_{51a2}$ 0.1 0.27 1 5.9e-02 1.4e+00 5.0e-01 8.2e-01 $EM_{61b1}$ 0.1 0.27 1 1.4e-01 2.4e+00 1.1e+00 5.0e-01 7.6e-01 $EL_{41b1}$ 0.1 1 1.1e-01 2.4e+00 7.5e-01 6.8e-01 $EL_{41b2}$ 0.1 1 1.2e-01 0 7.5e-01 1.0e+00 $EL_{51a1}$ 0.1 0.27 1 0 2.4e+00 7.5e-01 8.9e-01 $EL_{51a2}$ 0.1 0.27 1 3.5e-02 1.5e+00 7.5e-01 9.1e-01 $EL_{61b2}$ 0.1 0.27 1 9.3e-02 2.4e+00 1.1e+00 7.5e-01 8.9e-01
Parameters and numerical results with $G = G_2$, $k_1 = 4$ and $k_2 = 0.25$
 Example $\xi$ $\lambda$ T $\hat{\alpha}$ $\int_0^T \bar{u}(t) dt$ $\int_0^T \bar{c}(t) dt$ $L_0/\theta$ $L(T)/L_0$ $ES_{42a1}$ 0.1 1 1.1e-03 2.4e+00 5.0e-02 4.1e-01 $ES_{42a2}$ 0.1 1 3.0e-02 3.6e-01 5.0e-02 6.5e-01 $ES_{42a3}$ 0.1 1 2.9e-01 0 5.0e-02 1.3e+00 $ES_{52o1}$ 0.1 0.27 1 5.2e-07 2.4e+00 5.0e-02 5.2e-01 $ES_{52o2}$ 0.1 0.27 1 3.0e-02 3.4e-01 5.0e-02 7.1e-01 $ES_{52o3}$ 0.1 0.27 1 2.4e-1 0 5.0e-02 1.3e+00 $ES_{62o1}$ 0.1 0.27 1 1.7e-03 2.4e+00 1.1e+00 5.0e-02 5.2e-01 $ES_{62o2}$ 0.1 0.27 1 3.0e-02 4.6e-01 3.4e-01 5.0e-02 6.7e-01 $ES_{62o3}$ 0.1 0.27 1 2.9e-01 0 0 5.0e-02 1.3e+00 $EM_{42a1}$ 0.1 5 2.7e-03 1.2e+01 5.0e-01 1.5e-01 $EM_{42a2}$ 0.1 5 4.0e-03 1.0e+01 5.0e-01 1.7e-01 $EM_{42a3}$ 0.1 5 4.5e-01 0 5.0e-01 1.3e+00 $EM_{52o1}$ 0.1 0.27 5 4.4e-10 1.2e+01 5.0e-01 1.5e-01 $EM_{52o2}$ 0.1 0.27 5 4.0e-03 5.6e+00 5.0e-01 1.7e-01 $EM_{52o3}$ 0.1 0.27 5 4.5e-02 1.9e+00 5.0e-01 2.8e-01 $EM_{62o1}$ 0.1 0.27 5 4.2e-04 1.2e+01 2.0e+01 5.0e-01 1.5e-01 $EM_{62o2}$ 0.1 0.27 5 4.0e-03 4.7e+00 9.5e+00 5.0e-01 1.9e-01 $EL_{42a1}$ 0.1 5 5.2e-03 1.2e+01 7.5e-01 4.6e-01 $EL_{42a3}$ 0.1 5 2.4e-01 0 7.5e-01 1.1e+00 $EL_{52o1}$ 0.1 0.27 5 8.8e-10 1.2e+01 7.5e-01 4.6e-01 $EL_{52o2}$ 0.1 0.27 5 4.0e-03 6.9e+00 7.5e-01 4.7e-01 $EL_{52o3}$ 0.1 0.27 5 4.5e-02 2.2e+00 7.5e-01 5.6e-01 $EL_{62o1}$ 0.1 0.27 5 7.8e-4 1.2e+01 2.0e+01 7.5e-01 4.6e-01 $EL_{62o2}$ 0.1 0.27 5 4.0e-03 6.0e+00 1.2e+01 7.5e-01 4.8e-01
 Example $\xi$ $\lambda$ T $\hat{\alpha}$ $\int_0^T \bar{u}(t) dt$ $\int_0^T \bar{c}(t) dt$ $L_0/\theta$ $L(T)/L_0$ $ES_{42a1}$ 0.1 1 1.1e-03 2.4e+00 5.0e-02 4.1e-01 $ES_{42a2}$ 0.1 1 3.0e-02 3.6e-01 5.0e-02 6.5e-01 $ES_{42a3}$ 0.1 1 2.9e-01 0 5.0e-02 1.3e+00 $ES_{52o1}$ 0.1 0.27 1 5.2e-07 2.4e+00 5.0e-02 5.2e-01 $ES_{52o2}$ 0.1 0.27 1 3.0e-02 3.4e-01 5.0e-02 7.1e-01 $ES_{52o3}$ 0.1 0.27 1 2.4e-1 0 5.0e-02 1.3e+00 $ES_{62o1}$ 0.1 0.27 1 1.7e-03 2.4e+00 1.1e+00 5.0e-02 5.2e-01 $ES_{62o2}$ 0.1 0.27 1 3.0e-02 4.6e-01 3.4e-01 5.0e-02 6.7e-01 $ES_{62o3}$ 0.1 0.27 1 2.9e-01 0 0 5.0e-02 1.3e+00 $EM_{42a1}$ 0.1 5 2.7e-03 1.2e+01 5.0e-01 1.5e-01 $EM_{42a2}$ 0.1 5 4.0e-03 1.0e+01 5.0e-01 1.7e-01 $EM_{42a3}$ 0.1 5 4.5e-01 0 5.0e-01 1.3e+00 $EM_{52o1}$ 0.1 0.27 5 4.4e-10 1.2e+01 5.0e-01 1.5e-01 $EM_{52o2}$ 0.1 0.27 5 4.0e-03 5.6e+00 5.0e-01 1.7e-01 $EM_{52o3}$ 0.1 0.27 5 4.5e-02 1.9e+00 5.0e-01 2.8e-01 $EM_{62o1}$ 0.1 0.27 5 4.2e-04 1.2e+01 2.0e+01 5.0e-01 1.5e-01 $EM_{62o2}$ 0.1 0.27 5 4.0e-03 4.7e+00 9.5e+00 5.0e-01 1.9e-01 $EL_{42a1}$ 0.1 5 5.2e-03 1.2e+01 7.5e-01 4.6e-01 $EL_{42a3}$ 0.1 5 2.4e-01 0 7.5e-01 1.1e+00 $EL_{52o1}$ 0.1 0.27 5 8.8e-10 1.2e+01 7.5e-01 4.6e-01 $EL_{52o2}$ 0.1 0.27 5 4.0e-03 6.9e+00 7.5e-01 4.7e-01 $EL_{52o3}$ 0.1 0.27 5 4.5e-02 2.2e+00 7.5e-01 5.6e-01 $EL_{62o1}$ 0.1 0.27 5 7.8e-4 1.2e+01 2.0e+01 7.5e-01 4.6e-01 $EL_{62o2}$ 0.1 0.27 5 4.0e-03 6.0e+00 1.2e+01 7.5e-01 4.8e-01
Comparison Min, Max therapies with $G = G_1$, $k_1 = 4$, $\lambda = 0.27$, $\xi = 0.1$, $T = 2$ and $L_0 = 0.5\theta$
 Problem $\hat{\alpha}$ $L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ $(\widehat{OP}_5)$ 7.e-2 2.3e-01 2.9e+00 $(\widehat{OP}_{2MAX})$ 6.9e-01 2.9e+00 $(\widehat{OP}_6)$ 7.e-2 1.3e-01 4.7e+00 $(\widehat{OP}_{3MAX})$ 1.3e-01 4.7e+00 $(\widehat{OP}_5)$ 9.e-2 2.8e-01 2.6e+00 $(\widehat{OP}_{2MAX})$ 7.7e-01 2.6e+00 $(\widehat{OP}_6)$ 9.e-2 1.9e-01 3.8e+00 $(\widehat{OP}_{3MAX})$ 1.9e-01 4.5e+00
 Problem $\hat{\alpha}$ $L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ $(\widehat{OP}_5)$ 7.e-2 2.3e-01 2.9e+00 $(\widehat{OP}_{2MAX})$ 6.9e-01 2.9e+00 $(\widehat{OP}_6)$ 7.e-2 1.3e-01 4.7e+00 $(\widehat{OP}_{3MAX})$ 1.3e-01 4.7e+00 $(\widehat{OP}_5)$ 9.e-2 2.8e-01 2.6e+00 $(\widehat{OP}_{2MAX})$ 7.7e-01 2.6e+00 $(\widehat{OP}_6)$ 9.e-2 1.9e-01 3.8e+00 $(\widehat{OP}_{3MAX})$ 1.9e-01 4.5e+00
Comparison Min, Max therapies with $G = G_2$, $k_1 = 4$, $k_2 = 4$, $\lambda = 0.27$, $\xi = 0.1$, $T = 4$ and$L_0 = 0.05\theta$
 Problem $\hat{\alpha}$ $L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ $(\widehat{OP}_5)$ 1.e-3 3.8e-01 7.9e+00 $(\widehat{OP}_{2MAX})$ 6.7e-01 7.9e+00 $(\widehat{OP}_6)$ 1.e-3 3.7e-01 9.6e+00 $(\widehat{OP}_{3MAX})$ 3.7e-01 9.6e+00 $(\widehat{OP}_5)$ 5.e-3 5.4e-01 4.9e+00 $(\widehat{OP}_{2MAX})$ 1.5e+00 4.9e+00 $(\widehat{OP}_6)$ 5.e-3 7.2e-01 4.3e+00 $(\widehat{OP}_{3MAX})$ 9.3e-01 6.8e+00
 Problem $\hat{\alpha}$ $L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ $(\widehat{OP}_5)$ 1.e-3 3.8e-01 7.9e+00 $(\widehat{OP}_{2MAX})$ 6.7e-01 7.9e+00 $(\widehat{OP}_6)$ 1.e-3 3.7e-01 9.6e+00 $(\widehat{OP}_{3MAX})$ 3.7e-01 9.6e+00 $(\widehat{OP}_5)$ 5.e-3 5.4e-01 4.9e+00 $(\widehat{OP}_{2MAX})$ 1.5e+00 4.9e+00 $(\widehat{OP}_6)$ 5.e-3 7.2e-01 4.3e+00 $(\widehat{OP}_{3MAX})$ 9.3e-01 6.8e+00
Comparison Min, Max therapies with $G = G_2$, $k_1 = 4$, $k_2 = 4$, $\lambda = 0.27$, $\xi = 0.1$, $T = 8$ and $L_0 = 0.5\theta$
 Problem $\hat{\alpha}$ $L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ $(\widehat{OP}_5)$ 1.e-3 3.6e-01 1.9e+01 $(\widehat{OP}_{2MAX})$ 4.0e-01 1.9e+01 $(\widehat{OP}_6)$ 1.e-3 3.6e-01 1.9e+01 $(\widehat{OP}_{3MAX})$ 3.6e-01 1.9e+01 $(\widehat{OP}_5)$ 1.e-2 4.1e-01 1.4e+01 $(\widehat{OP}_{2MAX})$ 7.6e-01 1.4e+01 $(\widehat{OP}_6)$ 1.e-2 5.8e-01 1.1e+01 $(\widehat{OP}_{3MAX})$ 7.6e-01 1.4e+01
 Problem $\hat{\alpha}$ $L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ $(\widehat{OP}_5)$ 1.e-3 3.6e-01 1.9e+01 $(\widehat{OP}_{2MAX})$ 4.0e-01 1.9e+01 $(\widehat{OP}_6)$ 1.e-3 3.6e-01 1.9e+01 $(\widehat{OP}_{3MAX})$ 3.6e-01 1.9e+01 $(\widehat{OP}_5)$ 1.e-2 4.1e-01 1.4e+01 $(\widehat{OP}_{2MAX})$ 7.6e-01 1.4e+01 $(\widehat{OP}_6)$ 1.e-2 5.8e-01 1.1e+01 $(\widehat{OP}_{3MAX})$ 7.6e-01 1.4e+01
Catalog for Skipper model ($G = G_1$) under $({\bf{H}}_{\bf{1}}))$
 Problem Optimal control $(OP_1)$ $u_{max}/0$ (*) $(OP_2)$ $u_{max}/0$ $(OP_3)$ $u_{max}/0$ (*)
 Problem Optimal control $(OP_1)$ $u_{max}/0$ (*) $(OP_2)$ $u_{max}/0$ $(OP_3)$ $u_{max}/0$ (*)
Catalog for $E_{max}$ model ($G = G_2$) under $({\bf{H}}_{\bf{1}}))$
 Problem Optimal control $(OP_1)$ $u_{sin}$ $(OP_2)$ $u_{max}/0$   $u_{max}/u_{sin}/0$ $(OP_3)$ $u_{max}/u_{sin}$
 Problem Optimal control $(OP_1)$ $u_{sin}$ $(OP_2)$ $u_{max}/0$   $u_{max}/u_{sin}/0$ $(OP_3)$ $u_{max}/u_{sin}$
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