Optimal control problems for differential equations applied to tumor growth: state of the art

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Abstract

In this manuscript, we shall apply the tools and methods from optimal control to analyze various minimally parameterized models that describe the dynamics of populations of cancer cells and elements of the tumor microenvironment under different anticancer therapies. In spite of their simplicity, the analysis of these models that capture the essence of the underlying biology sheds light on more general scenarios and, in many cases, leads to conclusions that confirm experimental studies and clinical data. We focus on four applications: optimal control applied to compartmental models, brain tumors, drug resistance and antiangiogenic treatment.

Abstract

In this manuscript, we shall apply the tools and methods from optimal control to analyze various minimally parameterized models that describe the dynamics of populations of cancer cells and elements of the tumor microenvironment under different anticancer therapies. In spite of their simplicity, the analysis of these models that capture the essence of the underlying biology sheds light on more general scenarios and, in many cases, leads to conclusions that confirm experimental studies and clinical data. We focus on four applications: optimal control applied to compartmental models, brain tumors, drug resistance and antiangiogenic treatment.

1 Introduction

In this paper, we shall focus on topics where methods from optimal control theory will be applied to biomedical problems arising from cancer treatments. Specifically, we will focus on optimal control problems for systems of differential equations. Optimal control for partial differential equations, stochastic differential equations, discrete difference equations and so on are not considered in this document.

There exists a vast literature in which optimal control methods are applied to cancer treatment. The early literature on applications of optimal control to cancer treatment is almost exclusively on cancer chemotherapy (see references [14, 32, 45]). Much of this work is on cell-cycle nonspecific mathematical models and focusing on different growth models [32]. The monograph [31] by Martin and Teo is entirely devoted to such models, but including increasingly more comprehensive medical conditions such as single and multi-drug resistance.

Later, a greater emphasis in research has been put on cell-cycle specific models. In reference [10], the authors considered phase-specific administration of cytotoxic agents to increase the selectivity of therapy. Reference [46] models the proliferation cycle of leukemia. Eisen [14] also gives some introductory discussion of cell-cycle specific chemotherapy and compartmental models (distinguishing between proliferating and quiescent cells), but mostly in the context of pharmacokinetics and clearly the emphasis is more on modeling than on analysis.

After that, a great effort was done to study the development and analysis of compartmental models which describe and analyze the actions of drugs on specific compartments from a control theory point of view [49, 50, 51]. In these models, the emphasis is on the cancerous cells and side effects are only measured indirectly through the total dose of drugs administered.

Thus, from the mathematical point of view, the modeling of chemotherapy for cancer using the optimal control theory has 40 years of history. Different frameworks have been considered in the literature, depending on the study approach, from macroscopic levels [45] to microscopic levels [25,26], including systems of differential equations (v. [9,11,16,40] and their references) and equations in partial derivatives (v. [45] and their references). Normally, the control variable represents the dose of the drug and the state variables represent the tumor cells. In general, the main objective is to determine the optimal application of the drugs that minimizes the number of tumor cells at the end of treatment, while maintaining the toxic effects of the drug at an acceptable level. To reduce the toxicity of the anticancer drug, which also affects normal cells, it is customary to add some control restrictions, isoperimetric or state. There are different types of restrictions depending on the assumptions of the model, mainly punctual or integral. For example, a maximum rate is prescribed at which the chemotherapeutic drug is administered at each time or a limit is imposed on the total dose of the drug in the therapy interval (see [9, 12, 27]).

On the other hand, mathematical modeling of tumor growth in general is very important when it comes to having a model that fits the biology of the problem [35]. There are many mathematical models that deal with the different aspects of the growth of a tumor. They use ordinary differential equations [2, 6, 13, 54], partial differential equations [1, 3, 34], stochastic processes [38], game theory [52], etc.

In this review, we focus on 4 applications of the control theory to the growth of tumors:

  1. The first is referred to the application of the theory of optimal control to compartmental models.

  2. The second deals with the theory of optimal control of brain tumors.

  3. The third deals with a topic that is becoming more and more important: the resistance in tumors to different treatments and, therefore, to optimize such kind of treatments.

  4. The latest studies the application of optimal control to antiangiogenic and chemotherapeutic treatments.

Section 2 shows several compartmental models, which have been useful in the understanding of the application of optimal control theory to the growth of tumors.

Another topic of great importance is the study of the administration of chemotherapy in brain tumors [6, 20, 37]. Thus, in this review we study a model developed in the reference [40] based on a work by Ribba [37]. This revision can be found in Section 3.

Next topic is the drug administration in presence of multi-drug resistance cells. There exists a vast literature on optimal administration of chemotherapeutical drugs in the presence of drug resistance [7, 8, 18, 26, 33]. A model of sensitive and resistant cells to the treatment can be found in Section 4.

Optimal control is useful as a methodology in response to mathematical models for novel cancer treatments as anti-angiogenic treatments [27, 39]. A mathematical model with anti-angiogenic and chemotherapeutical treatment can be found in Section 5.

Finally, the review ends with some Conclusions in Section 6.

Fig. 1

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Fig. 1

A 2-compartment model with G2/M-specific cytotoxic agent.

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00029

2 Cell-cycle specific cancer chemotherapy

In this section, we analyze a set of differential equations for a class of cell-cycle specific compartmental models for cancer chemotherapy.

Each cell goes through a sequence of phases from birth to cell division: G1 →G0 →S→G2 →M. While the simplest mathematical models for the treatment of cancer with chemotherapy, applying the theory of optimal control, treat the complete cycle as a compartment, the multicompartmental models group the phases of the cell cycle with the purpose of representing the action of the different types of chemotherapeutic agents used. Cytotoxic agents generally act in the phase G2/M because the cell walls become thin and porous in mitosis M and, therefore, the cell is more vulnerable to attack. The cytostatic agents aim to synchronize the transitions of cells through the cell cycle causing a brief and invisible inhibition of the DNA synthesis in the S phase and the retention of the cells in the G1 phase. Recruitment agents cause the cells to leave the inactive G0 compartment where they are not vulnerable to attacks by any chemotherapeutic drug. This classification is general, being able to find agents that have more than one effect depending on their concentration [43]. Depending on the specific type of tumor that is considered, and the drugs that are administered, an appropriate grouping of the phases of the cell cycle is carried out.

2.1 A 2 compartment model with a cytotoxic agent

This model deals with the application of a single cytotoxic agent that is activated in the G2/M phase of the cell cycle. Taking into account the sensitivity of the drug to the phase, the cell cycle is divided into two compartments, one combining the second growth phase G2 and the mitosis M and the other formed by the rest of the phases. The state of the system is described by a two-dimensional vector where N1(t) represents the mean number of cancer cells in the first compartment at time t (composed of the phases G0, G1 and S) and N2(t) the average number of carcinogenic cells in the second compartment at time t (composed of G2 and M). The equation for the second compartment has the form

N˙2(t)=a2N2(t)+a1N1(t)

where ai > 0 is the average inverse transit time through the i-th compartment. The equation for the first compartment becomes

N˙1(t)=a1N1(t)+2a2N2(t).

In matrix form, the system given by the equations (1) and (2) can be expressed as N = AN where N = (N1,N2)T and A 22×2 with

A=a12a2a1a2.

Now, we consider a cytotoxic agent whose concentration in blood is represented by the function u acting in the G2/M phase. It is assumed that all cells are sensitive to treatment (homogeneous population). According to the log-kill hypothesis, it is assumed that the concentration of the drug u(t) kills a fraction of the output flow a2N2(t) of cells of the G2/M compartment and, therefore, the number of cells dead is given by fu(t)a2N2(t), ϕ being a parameter of constant chemotherapeutic death. The control set is a compact interval [0,umax], with umax the maximum dose rate or the maximum concentration. In the model, the u control always appears together with the constant ϕ and so, in order to keep the number of free parameters as low as possible, it is combined with the maximum dose rate in an amount represented as umax under the assumption that umax 1. If the concentration is high enough, then really umax = 1 is realistic: almost all the cancer cells in that compartment can be killed. The cells that die in G2/ M leave this compartment, that is, they are counted as outputs from the second compartment, but they no longer enter the first compartment. In this sense, it is considered to kill the cell even if apoptosis is not induced. The remaining fraction (1-u)a2N2 suffers cell division and therefore the controlled mathematical model becomes

N˙1(t)=a1N1(t)+2(1u)a2N2(t),N1(0)=N10,

N˙2(t)=a1N1(t)a2N2(t),N2(0)=N20.

The matrix expression for the system has the form [41]

N˙(t)=(A+uB)N(t),N(0)=N0,

where A and B are

A=a12a2a1a2,B=02a200.

Consider now as a functional objective

J=rN(T)+0TqN(t)+su(t)dtmin

where T is the final fixed time, r = (r1,r2) is a positive vector of weights and q = (q1,q2) is a vector row of weights not negative. The penalty term rN(T) = r1N1(T)+r2N2(T) represents an average weight of the total number of cancer cells at the end of treatment and the term qN(t) = q1N1(t)+q2N2(t) measures the volume of the tumor during treatment. Side effects of the treatment are only indirectly included in the model through the minimization of the total dose, 0Tu(t)dt. The positive coefficient s in this integral is redundant since it could be absorbed by the other weights. Recall that the number of cancer cells that do not undergo cell division at time t and that are consider eliminated is given by u(t)a2N2, that is, u(t) is proportional to the fraction of inefficient divisions. Since the drug kills healthy cells at a similar rate, the integral 0Tu(t)dt represents the cumulative negative effects of the treatment on the normal tissue or its toxicity. The aim is to minimize tumor volume in both compartments at the end of treatment, but taking into account the tumor volume in both compartments during treatment, while minimizing side effects. Thus, the following optimal control problem arises:

Problem 1

For a fixed time T > 0, minimize J subject to the system of equations

N˙(t)=(A+uB)N(t),N(0)=N0,

for all measurable functions Lebesgue (respectively, piecewise continuous) u : [0,T] [0,umax] with A and B given by equation (6), [41, 47].

The necessary conditions for optimality for Problem 1 are given by the Pontryagin maximum principle [28, 36, 42]. The main necessary conditions for optimality is the statement that optimal controls minimize (respectively, maximize) the Hamiltonian function

H=H(λ,N,u)=qN+su+λ(A+uB)N,

over the control set [0,umax] pointwise along the optimal controlled trajectory (N*,u*) and the multipliers l.

The conditions of the Pontryagin maximum principle for our optimal control problem are

Theorem 1

If u* is an optimal control with corresponding trajectory N* then there exist a continuous function λ : [0,T] (ℝ2)* called the adjoint function, such the following conditions are satisfied:

  1. Nontriviality: λ(t) ≠ 0 for all t ϵ [0,T].

  2. Adjoint equations and transversality conditions: the multiplier λ is a solution to the terminal value problem

    λ˙=HN(λ,N,u)=qλ(A+uB),λ(T)=r,

    i.e,

    λ˙1=q1+λ1a1λ2a1,λ1(T)=r1,λ˙2=q22(1u)λ1a2+λ2a2,λ2(T)=r2.

  3. Minimum condition: the optimal control minimizes the Hamiltonian H pointwise over the control set [0,umax] along the optimal controlled trajectory and the multiplier λ(t), i. e,

H(λ(t),N(t),u(t))=minu[0,umax]H(λ(t),N(t),u(t)),

and the minimum value is constant over the interval [0,T],

H(λ(t),N(t),u(t))=const.

The switching function Φ for this problem is

Φ(t)=Hu=s+λ(t)BN(t).

Then, the optimal controls are given by

u(t)=umaxifΦ(t)<0,usingifΦ(t)=0,0ifΦ(t)>0.

For control-affine problems as is our case, the best candidates for optimal controls turn out to be finite concatenations of bang and singular control. The precise concatenation sequences need to be determined through an analysis of the switching function. Then, it is straightforward, although somewhat complicated and tedious, to calculate the singular control [22]. It is given by

using=1a1s2a2q1N1(t)+12a1(a1q2+a2q1)N2(t).

However, singular controls are not necessarily minimizing, but they can also be maximizing. Now, if we use the Legendre-Clebsch condition, a higher-order necessary condition for optimality of singular controls [42], allows us to distinguish between these two classes. For the case of a minimizing control u, the Legendre-Clebsch condition states that

ud2dt2Hu(λ(t),N(t),u(t))0,tI.

In our problem, such Legendre-Clebsch condition becomes

ud2dt2Hu(λ(t),N(t),u(t))=4a1a2s>0.

Hence, the Legendre-Clebsch condition for minimality of a singular control is violated. Thus, singular controls are locally maximizing for this problem. This brings us to the next result

Theorem 2

If (N*,u*) is an optimal controlled trajectory for problem 1, then there does not exist an interval on which the control u* is singular.

We show some examples of extremal bang-bang controls that have been computed using the values in Table 1, which are shown in Figure 2.

Fig. 2

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Fig. 2

Examples of locally optimal controls (left) and their corresponding trajectories (right) for T = 7 (top), T = 21 (middle) and T = 60 (bottom) for the parameter values given in Table 1.

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00029

Table 1

Numerical values for the coefficients and parameters used in numerical computations for the optimal control problem 1, extracted from [41].

CoefficientInterpretationNumerical value
a1Inverse transit time through G0/G1+ S0.197
a2Inverse transit time through G2/M0.356
N10Initial condition for N10.7012
N20Initial condition for N20.2988
umaxMaximum dose rate/concentration effectiveness of the drug0.90
sPenalty/weight for the total dose of cytotoxic agent0.50
q1Penalty/weight in the objective for the average number of cancer cells in G0/G1+ S during therapy0.10
q2Penalty/weight in the objective for the average number of cancer cells in G2/M during therapy0.10
r1Penalty/weight in the objective for the average number of cancer cells in G0/G1+ S at the end of therapy3
r2Penalty/weight in the objective for the average number of cancer cells in G2/M at the end of therapy3
TTherapy horizonT = 7,21,60

2.2 A 3 compartment model with a cytotoxic and recruiting agent

This model was originally formulated by Swierniak et al. [47], where a cytotoxic agent, which can act in S or G2/M phases, is combined with a recruitment agent, which attracts inactive cells in the G0 compartment to re-enter the active cell cycle and thus be attacked by the cytotoxic agent. One of the biggest problems in the chemotherapeutic treatment of some leukemias is a large residue of inactive cells in the G0 phase, which are not sensitive to most of the cytotoxic agents. This problem has also been reported in breast and ovarian cancer. In this model, three compartments are considered: G0, G1 and S +G2/M. The state space is thus the first orthant P in R3. The variables N0, N1 and N2 represent the states. The newborn cells initiate the process of cell division, but they can remain dormant and remain in the quiescent stage G0. Let p0 and p1 be positive numbers, with p0+ p1 =1, representing the probabilities that the daughter cells will pass through the first growth phase without obstacles (p1) or fall into the state of quiescence (p0). A recruiting agent is applied to reduce the average transit time through the G0 compartment, resulting in a higher outflow from G0. If w represents the concentration of the recruitment agent, it is assumed that the outflow is increased by a factor of 1 +w, 0 ≤ w ≤ wmax, where w = 0 represents that no drug is being applied and wmax corresponds to a full dose treatment. A cytotoxic agent u, 0 ≤ u ≤ umax, is applied in the third compartment, where umax corresponds to the maximum dose (see Figure 3 ). The combination of these drugs with the standard characteristics of cell division and cell death (and under the log-kill hypothesis), results in a bilinear control system, of the form

Fig. 3

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Fig. 3

A 3-compartment model with cytotoxic and recruiting agent.

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00029

N˙(t)=A+uB1+wB2N(t),N(0)=N0,

where the matrices A, B1, B2 are

A=a002p0a2a0a12p1a20a1a2,B1=002p0a2002p1a2000,B2=a000a000000.

As in the 2 compartment model considered earlier, the ai are positive coefficients related to the inverse transit times of the cancer cells through the compartments.

Thus, we considere the following optimal control problem:

Problem 2

For a fixed therapy horizon [0,T], minimize the objective

J=rN(T)+0TqN(t)+s1u(t)+s2w(t)dt

over all Lebesgue-measurable (respectively, piecewise continuous) functions (u,w):[0,T][0,umax]×[0,wmax],umax12,subject to equations [47]

N˙(t)=(1+w)a002p0(1u)a2(1+w)a0a12p1(1u)a20a1a2N(t),N(0)=N0.

Using the Pontryagin maximum principle, the adjoint equation is given by

λ˙=λ(A+uB1+wB2)q,λ(T)=r,

which, in terms of its components can be written as,

λ˙0=(1+w)a0(λ0λ1)q0,λ0(T)=r0,

λ˙1=a1(λ1λ2)q1,λ1(T)=r1,

λ˙2=2a2(1u)(p0λ0+p1λ1)+a2λ2q2,λ(T)=r2.

The switching functions for the controls u and w are, respectively

Φ1(t)=s1+λ(t)B1N(t)=s12a2{p0λ0(t)+p1λ1(t)}N2(t),

Φ2(t)=s2+λ(t)B2N(t)=s2a0{λ0(t)λ1(t)}N0(t).

The following theorem shows that the optimal control for the cytotoxic agent u can not be singular:

Theorem 3

Suppose (N*,u*,w*) is an optimal controlled trajectory for problem 2. Then there does not exist an interval on which the control u* is singular, i.e., the cytotoxic agent is not given at partial dose rates/concentrations.

The proof of this theorem can be found in [47].

The following theorem establishes what the behavior of the recruitment agent w should be:

Theorem 4

Let (N*,u*,w*) be an optimal controlled trajectory for problem 2. If the weight s2 in the Lagrangian for the recruiting agent w is zero, s2 = 0 then there does not exist an interval on which the control w* is singular. However, if this coefficient is positive, s2 > 0, then the Legendre-Clebsch condition for optimality of a singular control w is satisfied.

The proof of this theorem can be found again in [47].

For small positive weights s2, and this is the important case for this model, bang-bang controls are optimal. Figure 4 shows the graphs of such controls u and w and their corresponding trajectories when the weight s2 in the Lagrangian is chosen as s2 = 0 and s2 = 0:1 for parameter values specified in Table 2.

Fig. 4

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Fig. 4

Examples of locally optimal controls u (citotoxic agent, top), w (recruiting agent, middle) and their corresponding trajectories (bottom) for weights s2 = 0 (left) and s2 = 0:1 (right). The parameter values are given in Table 2.

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00029

Table 2

Numerical values for the coefficients and parameters used in numerical computations for the optimal control problem 2 with cytotoxic and recruitment agents, extracted from [41].

CoefficientInterpretationNumerical value
a0Inverse transit time through G00.05
a1Inverse transit time through G10.5
a2Inverse transit time through S +G2/M1
p0Probability that cells enter G00.9
p1 = 1– p0Probability that cells enter G10.1
N0(0)Initial condition for N00.8589
N1(0)Initial condition for N10.0954
N2(0)Initial condition for N20.0456
umaxMaximum dose rate/concentration effectiveness0.95
wmaxMaximum dose rate/concentration effectiveness6
s1Penalty/weight at the cytotoxic agent1
s2Penalty/weight at the recruiting agent0, 0.1
q0Penalty/weight in the objective for the average number of cancer cells in G0 during therapy3
q1Penalty/weight in the objective for the average number of cancer cells in G1 during therapy1
q2Penalty/weight in the objective for the average number of cancer cells in S +G2/M during therapy1
r0Penalty/weight in the objective for the average number of cancer cells in G0 at the end of therapy3
r1Penalty/weight in the objective for the average number of cancer cells in G1 at the end of therapy1
r2Penalty/weight in the objective for the average number of cancer cells in S +G2/M at the end of therapy1
TTherapy horizonT = 21

2.3 A 3 compartment model with a cytotoxic and blocking agent

In this model the combination of two drugs is considered: a cytotoxic agent u, which is responsible for killing tumor cells and acts in the G2/M phase, and a blocking agent v (cytostatic), which slows the development of tumor cells in the S phase, preventing them from passing into the cell division phase [48]. Therefore, three compartments are considered: the first formed by the phases G0 and G1, the second by the S phase and the third is formed by the phases G 2 and M. The space of states is the first orthant P in ℝ3, being Ni(t), i = 1,2,3, the number of cancer cells in the i-th compartment at time t.

As a result, the flow of tumor cells from the second compartment to the third is reduced by a factor given by v(t), from a value of a2N2(t) to (1-v(t))a2N2(t), where 0 ≤ v(t) ≤ vmax < 1. The value v(t) = 0 represents that the treatment is not being applied, while vmax represents a complete dose. The control u represents the dose of cytotoxic agent administered with u = 0 indicating the absence of treatment and u = 1 the maximum dose (see Figure 5). Under all assumed conditions for the previous compartmental models (homogenous tumor population, constituted by cells sensitive to therapy and under the log-kill hypothesis), we get back to a bilinear system again of the form (18), but now the matrices involved have the form

Fig. 5

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Fig. 5

A 3-compartment model with cytostatic agent v and cytotoxic agent u.

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00029

A=a102a3a1a200a2a3,B1=002a3000000,B2=0000a200a20.

We thus have the following optimal control problem

Problem 3

For a fixed therapy horizon [0,T], minimize the objective

J=rN(T)+0TqN(t)+s1u(t)+s2v(t)dtmin

over all Lebesgue-measurable (respectively piecewise continuous) functions (u,v) : [0,T][0,umax]×[0,vmax] subject to the dynamics

N˙(t)=a102(1u)a3a1(1v)a200(1v)a2a3N(t),N(0)=N0.

We again write out the fundamental necessary condition for optimality for this system. In terms of components, the adjoint equations read as

λ˙1=a1(λ1λ2)q1,λ1(T)=r1,λ˙2=(1v)a2(λ2λ3)q2,λ2(T)=r2,λ˙3=a3(λ32(1u)λ1)q3,λ3(T)=r3,

and the switching functions for the controls u and v are given by

Φ1(t)=s12a3λ1(t)N3(t),

and

Φ2(t)=s2+a2{λ2(t)λ3(t)}N2(t),

respectively. Once again, we analyze the optimality properties of singular controls, starting with the cytostatic agent [47].

Theorem 5

If (N*,u*,v*) is an optimal controlled trajectory for problem 3, then there does not exist an interval on which the control v* is singular.

For this model we can no longer assert in general that the cytotoxic agent u can not be singular, but need to impose an inequality relation on the inverse transit times in the cell cycle and the maximum reduction vmax that the cytostatic agent can achieve.

Theorem 6

Suppose that

a1+a2(1vmax)2a30.

If (N*,u*,v*) is an optimal controlled trajectory for problem 3, there also does not exist an interval on which the control u* is singular.

We close this section with some numerical experiments of locally optimal controls for problem 3. The parameter values are given in Table 3 and condition (25) is satisfied. Figure 6 shows two different scenarios.

Fig. 6

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Fig. 6

Examples of locally optimal controls u (cytotoxic agent) (top), v (cytostatic agent) (middle) and their corresponding trajectories (bottom) for different weights r = (1,1,1) and q = (1,1,1) on the left and r = (8:25,8:25,8.25) and q = (0.1,0.1,0.1) on the right. The parameter values are given in Table 3.

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00029

Table 3

Numerical values for the coefficients and parameters used in computations for the optimal control problem 3 with cytostatic and cytotoxic agents, extracted from [41].

CoefficientInterpretationNumerical value
a1Inverse transit time through G1/G00.197
a2Inverse transit time through S0.395
a3Inverse transit time through G2/M0.107
N1(0)Initial condition for N10.3866
N2(0)Initial condition for N20.1722
N3(0)Initial condition for N30.4412
umaxMaximum dosage/concentration/effectiveness of cytotoxic agent0.95
vmaxMaximum blocking effect of cytostatic agent0.30
s1Penalty/weight at the cytotoxic agent1
s2Penalty/weight at the cytostatic agent0.01
q1Penalty/weight in the objective for the average number of cancer cells in G1/G0 during therapy1, resp. 0.1
q2Penalty/weight in the objective for the average number of cancer cells in S during therapy1, resp. 0.1
q3Penalty/weight in the objective for the average number of cancer cells in G2/M during therapy1, resp. 0.1
r1Penalty/weight in the objective for the average number of cancer cells in G1/G0 at the end of therapy1, resp. 8.25
r2Penalty/weight in the objective for the average number of cancer cells in S at the end of therapy1, resp. 8.25
r3Penalty/weight in the objective for the average number of cancer cells in G2/M at the end of therapy1, resp. 8.25
TTherapy horizonT = 21

3 Low grade glioma and chemotherapy

In this section, we consider a low grade glioma (LGG), which is a term denoting a set of primary progressive brain tumor of astrocytic or oligodendroglial origin characterized by slow and continuous growth preceding the anaplastic transformation [30]. Such model was considered by Ribba and colaborators [37] and assumes that a LGG is composed of proliferative P and nonproliferative quiescent tissue Q. The chemotherapy treatment u(t) eliminates proliferative cells and the quiescent cells are also affected by the treatment and become damaged quiescent cells Qp. Damaged quiescent cells, when re-entering the cell cycle, be able to repair their DNA and become proliferative once again or can die because of unrepaired damages. For simplicity, here it is assumed that the dosage, the concentration and the effect of the drug are equal. Thus, it is assumed that the drug kills a proportion of u of proliferating cells, γuP, and damages a proportion of u of quiescent cells, γuQ. Therefore, the system of differential equations that describes the proliferative, non-proliferative and damaged cells and the effect of chemotherapy on these populations is given by [40]

dPdt=cpP1NK+cQPQpcPQPγPu,P(0)=P0,

dQdt=cPQPγQu,Q(0)=Q0,

dQpdt=γQucQPQpδQPQp,Qp(0)=0,

where N = P+Q+Qp is the total number of cells and P,Q,Qp 0. Consider now the following functional objective

Jw(u)=(1w)(P(T)+Q(T))+w0T(P(t)+Q(t))dt,w[0,1],

where w is a weight parameter. We obtain two different functionals with biological relevance for w = 0 and w = 1. Thus, for w = 0 we obtain a functional in the form of Mayer that focuses on the tumor cells at the end of the treatment,

J0(u)=P(T)+Q(T),

while for w =1 we obtain a functional in the form of Lagrange that takes into account the tumor cells throughout the dynamic process:

J1(u)=0TP(t)+Q(t)dt.

In addition, we introduce a restriction for additional control by imposing a limit on the total dose of the drug:

0Tu(t)dtM.

Thus, we have the following optimal control problem:

Problem 4

For a fixed time T > 0, minimize Jw(u) subject to the system of equations (26) for all measurable functions Lebesgue (respectively, piecewise continuous) u : [0,T][0,umax] and satisfying the constraint (30).

First of all, existence of optimal controls are proved:

Theorem 7

Suppose that the initial conditions of the control system (26) satisfy P(0) = P0 > 0, Q(0) = Q0 > 0, Qp(0)=0 and N (0)=P0+ Q0+Qp0 < K. Then there is an optimal control for the function Jw(u), 0 ≤w ≤1, in (29).

The proof of this theorem can be found in [40]. On the other hand, system (26) can be written in a matrix form as

x˙=f(x)+g(x)u,

where x = (P,Q,Qp,z)t and

f(x)=cpP1NK+cQPQpcPQPcPQPcQPQpδQPQp0,g(x)=γPγQγQ1.

Defining the attached variable λ = (λ1234), the Hamiltonian is given by

H(x,λ,u)=w(P+Q)+λ1cpP1NK+cQPQpcPQPγPu+λ2[cPQPγQu]+λ3[γQucQPQpδQPQp]+λ4u.

The adjoint equation becomes

λ˙=aλ(Df(x)+uDg(x)),λ(T)=m,

where a = (w,w,0,0)t , m = (1-w,1-w,0)t and Df , resp., Dg represent the matrices of partial derivatives of the vectors f , resp., g.

The switching function Φ for this problem is

Φ(t)=λ4γ(λ1P+λ2Qλ3Q).

Then, the optimal controls are given by

u(t)=uMifΦ(t)<0,0ifΦ(t)>0.

The control is not determined a priori by the minimum condition when Φ(t) = 0. Thus, we consider two cases:

  • If the switching function Φ(t) has only a finite number of zeros in an interval Ib [0,T], the control u* is called bang-bang over Ib. In this case we have u*(t) ϵ {0,uM} for all t ϵ Ib.

  • If Φ(t) 0 is maintained over a time interval Is = [t1,t2], 0 ≤t1 < t2 ≤ T, we say that u* is singular about Is.

For the optimal control problems that are linear in the control, it is typical that the optimal solutions are concatenations of bang-bang and singular controls, but this does not derive directly from the necessary conditions. In principle, since the controls are measurable Lebesgue, more degenerate situations can not be excluded a priori.

As in the previous section, we can use the strict Legendre-Clebsch condition as a necessary condition for the existence of singular controls. Thus, if we assume that the strict Legendre-Clebsch condition is satisfied

ud2dt2ϕ(t)=N(x(t),λ(t))>0,tIs,

where

N(x(t),λ(t))=γ2λ1cpKP2cQP(Q+Qp)γ2λ3L4γ2w(P+Q),

with

L4=cPQP+(cQP+δQP)Q,

the relation d2dt2ϕ(t)=0gives rise to the following formula for singular control in terms of the state variable x and the adjoint variable λ:

using(x,λ)=M(x,λ)N(x,λ),

where

M(x,λ)=γλ1L52cpKPL1+L2L3cQPcpKPL4+γwL1+L3+cPQP+γλ2cPQL3+γλ3cPQL1+(cQP+δQP)(cPQP+L4),

and

L1=cpP1NK+cQPQpcPQP,L2=cp1NKcpKPcPQ,L3=cpKP2+cQPQp+cQPQ,L4=cPQP+(cQP+δQP)Q,L5=cQP(cQP+δQP)QpcQPcPQP.

Thus, the strict Legendre-Clebsch condition (35) requires checking if the inequality

λ1cpKP2cQP(Q+Qp)+λ3cPQP+(cQP+δQP)Q+w(P+Q)>0

is satisfied on the singular interval Is.

3.0.1 Numerical solutions

We solve the optimal control problem 4 using AMPL software [19], which is linked to the IPOPT Interior Point optimization code. We use a high number N 2 [3000,10000] of grid points and implement the trapezoidal integration scheme [5].

The objective of this section is to calculate a standard chemotherapy protocol for a LGG, where the total amount of TMZ dose is M = 5 and the treatment time is T = 1 (30 days). We use the parameters of Table 4 and the initial conditions

Table 4

Values of the biological parameters in the model of LGG evolution.

VariableValue (Units)
P00.924 mm
Q042.3 mm
Qp00 mm
K100 mm
cp0.114 mo -1
cPQ0.0226 mo-1
cQP0.0045 mo-1
δQP0.0214 mo-1
γ0.842

P(0)=0.924,Q(0)=42.3,Qp(0)=0.

  1. w = 0. The state variables, the optimal control and the change function are shown in Figure 7. The optimal treatment consists of DMT therapy. Initially, the dose level is zero and then jump to the maximum tolerated dose (DMT) for 5 days. Initially, the proliferative and quiescent cells are growing, but after applying the chemotherapy, both types of cells are drastically decreasing until the end of the treatment period. We obtain the objective value and terminal states

Fig. 7

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Fig. 7

Optimal solution for functional Jw(u) in (29) with w = 0 and total dose M = 5, the time horizon is 30 days. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ.

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00029

J0(u)=37.0623,P(T)=0.8396,Q(T)=36.7801,Qp(T)=5.5286.

  1. w =1. Figure 8 shows the optimal control and the state variables for the weight w =1 in the objective (29). The optimal treatment is based on DMT therapy, but now, in contrast to the solution for w = 0, the initial dose is the maximum dose until the total dose M = 5 is reached. Subsequently, no dose is administered. We get the following final value for the objective and the state variables

    Fig. 8

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    Fig. 8

    Optimal solution for functional Jw(u) in (29) with w = 1 and total dose M = 5, the time horizon is 30 days. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ.

    Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00029

    J1(u)=38.0623,P(T)=0.8607,Q(T)=36.7804,Qp(T)=5.4084.

  2. w = ws = 0:004067. For this value, the optimal control is given by

0w<ws:u(t)=0for0t251for25<t30

ws<w1:u(t)=1for0t50for5<t30

Therefore, for wws all controls are bang-bang with a single switching that occurs at the time when the total amount of dose, M =5, has been administered. But the structure of the control "switch" in the weight ws = 0:004067 being, for this value, a totally singular optimal control.

Figure 9 show this fact.

Fig. 9

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Fig. 9

Optimal solution for functional Jw(u) in (29) with w = ws = 0:004067 and total dose M = 5, the time horizon is 30 days. (left) optimal control u(t) which is totally singular on [0,1], (right) proliferative cells P.

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00029

Now, we solve the optimal control problem (29) for T = 30 months.

  1. w = 0. In this case, we get control with the bang-singular-bang structure:

    u(t)=0for0t<t1,using(x(t),λ(t))fort1tt2,1fort2<t30.

The singular control using is obtained in (36). On the other hand, we can obtain the change times t1 = 20:1 and t2 = 25:2. The control and state variables are shown in Figure 10. We get the following final value for the objective and the state variables

Fig. 10

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Fig. 10

Optimal solution for functional Jw(u) in (29) with w = 0 and total dose M = 5, the time horizon is T = 30 months. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ, the switching function ϕ(t) is zero on the singular arc [t1,t2].

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00029

J0(u)=0.9059,P(T)=0.2545,Q(T)=0.6514,Qp(T)=38.2335.

Here, the treatment starts with zero doses, then jumps to a single dose with fairly small values followed by a total dose at the end of the treatment.

2. w = 1. Here, we consider the functional

J1(u)=0T(P(t)+Q(t))dt,

and we find again the control structure bang-singular-bang

u(t)=1for0t<t1,using(x(t),λ(t))fort1tt2,0fort2<t30.

The switching times for are t1 =3:00 and t2 =18:4 showing a fairly long singular arc. We get the following final value for the objective and the state variables

J1(u)=139.31,P(T)=4.7982,Q(T)=1.6042,Qp(T)=20.0689.

Figure 11 shows the state variable and the optimal control.

Fig. 11

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Fig. 11

Optimal solution for functional Jw(u) in (29) with w = 1 and total dose M = 5, the time horizon is T = 30 months. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ, the switching function ϕ(t) is zero on the singular arc [t1, t2].

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00029

4 Cancer chemotherapy and drug resistance

Resistance to drugs is one of the most important problems in the treatment of cancer, causing a high number of deaths. There are several probabilistic models for the development of drug resistance [8, 53]. The amplification of a gene is an increase in the number of copies of that gene present in the cell after cell division, the depletion corresponds to a decrease in its number of copies. The hypothesis of amplification of a gene to a copy establishes that in cell division at least one of the two daughter cells will be an exact copy of the mother cell while the second, with some positive probability, will undergo genetic amplifications. Next, we present a model studied by Ledzewicz et al. in [26].

4.1 A 2 compartment model with sensitive and resistant subpopulations

Here, we consider two compartments consisting of drug sensitive and resistant cells, and we denote the numbers of cells in the sensitive and resistant compartments by S and R, respectively. Within the one copy forward gene amplification model, once a sensitive cell undergoes cell division, the mother cell dies and one of the daughters will remain sensitive. The other daughter, with probability q, 0 < q < 1, changes into a resistant cell. Similarly, if a resistant cell undergoes cell division, then the mother cell dies, and one of the daughters remains resistant. However, for cancer cells it is possible that a resistant cell may mutate back into a sensitive cell. This phenomenon is well-documented in the medical literature where experiments have shown that the resistant cell population decreases in a drug free medium [44]. We therefore include a probability r ≥ 0 that one of the daughters of a resistant cell may become sensitive. Generally r will be small.

Thus, the controlled dynamics can be represented as

S˙=aS+(1u)(2q)aS+rcR,

R˙=cR+(2r)cR+(1u)qaS,

and the initial condition is given by positive numbers S0 and R0 at time t = 0. Setting N = (S,R)T , the dynamics is described again by a bilinear system

N˙=(A+uB)N,

where

A=(1q)arcqa(1r)c,B=aq20q0.

The problem that we want to solve is to delay the onset of the time when drugs are no longer effective. Mathematically, we consider the problem to minimize an objective of the form

J=kN(T)+0T(lN(t)+u(t))dtmin

where k and l are row-vectors of weights with the components of k positive and those of l non-negative. The penalty term kN(T) represents a weighted average of the total number of cancer cells at the end of an assumed fixed therapy interval [0,T] and the term lN(t) models cumulative effects during the therapy. The control term u(t) in the Lagrangian models the negative side effects of the drugs, measured in the L1 norm. The parameters k and l can also be used to put a stronger emphasis on the number of cancer cells since it can be argued that a duplication of the cancer or the drug dose is more hazardous than a duplication of the objective would represent.

Thus, the optimal control problem has the form

Problem 5

For a fixed therapy interval [0,T], minimize the objective J (46), over all Lebesgue measurable functions (respectively, piecewise continuous) u : [0,T][0,umax] subject to the equations (42) and (43).

Using the Pontryagin Maximum Principle, there exists a continuous function λ satisfying the adjoint equation

λ˙=λ(A+uB)l,λ(T)=k,

such that the optimal control u* minimizes the Hamiltonian H,

H=lN+u+λ(A+uB)N

over the control set [0,umax] along (λ(t),N*(t)). Since the Hamiltonian is linear in u and the control set is an interval, defining the switching function Φ by

Φ(t)=1+λ(t)BN(t)

and, therefore, we have

u(t)=0ifΦ(t)>0,using(x(t),λ(t))ifΦ(t)0,umaxifΦ(t)<0.

Thus, the singular control can be computed and to get

using=λ(t)[A,[A,B]]N(t)l[A,B]N(t)lBAN(t)λ(t)[B,[A,B]]N(t)lB2N(t).

Whether the singular control will be admissible, i.e., whether it will obey the control bounds 0 ≤ u ≤ umax, will depend on the parameter values, but also on the region in the state space where the state (S,R)T lies and cannot be answered in general. In this case the singular control is called of order 1 and it is again where we have to use the generalized Legendre-Clebsch condition as before we did:

ud2dt2Hu<0.

Thus, we get

ud2dt2Hu=2a2rc(2q)λ0+qλ1(qS(2q)R).

Since (2-q)λ0+ qλ1 > 0 the generalized Legendre-Clebsch condition therefore implies:

Proposition 8

Singular controls are not optimal in regions of the state space where qS > (2-q)R.

This holds as long as the resistant population is very small and then optimal controls will be bang-bang, i.e., correspond to sessions of full dose chemotherapy with rest periods interlaced. However, as the portion of resistant cells R increases, the Legendre-Clebsch condition will be satisfied and if admissible, there are mathematical reasons to suspect that it indeed will be the optimal control in this case. It is also quite intuitive that a full dose may do more harm than good once resistance builds up and thus the optimal strategy may switch to give partial doses, i.e., use singular controls.

5 Cancer chemotherapy and antiangiogenic treatment

Cancer is fundamentally a genetic disease: it arises as a consequence of pathological changes in the information contained in the DNA. That is, the uncontrolled growth of the cells is the result of alterations or mutations in the genetic material. These mutations lead to a process of altering the cell division and the inability to undergo programmed cell death (or apoptosis). The growth of a single cancer (or primary cancer) does not usually lead to the death of the organism. However, some cancer cells can acquire the ability to enter the bloodstream, travel to another part of the body and begin to grow in a different organ. This process is known as metastasis. Generally, metastatic growth is what kills the body. When a cancer cell is created, it can multiply and lead to a tumor of small size without the need for blood supply. At this stage, the growth of an avascular tumor stops. Subsequently, to continue growing, since the tumor needs oxygen and nutrients to grow, provided by the blood, the tumor sends out chemical signals called promoters. This induces the blood vessels to grow towards the tumor and leads to the complete vascularization of the tumor, which allows it to increase in size. The formation of new blood vessels is called angiogenesis. The understanding of the role of angiogenesis in the development of cancer has advanced significantly due to the studies conducted by Folkman [17]. We now know that without angiogenesis, the size of most primary tumors does not exceed 3 mm. One type of very important promoters are the so-called Vascular Endothelial Growth Factors (VEGF). These factors stimulate cellular responses joining the tyrosine kinase receptors (a type of enzyme) on the surface of the endothelial cells and stimulate their proliferation and migration to the tumor. Other pro-angiogenic factors are PDGF, PIGF or FGF. Therefore, anti-angiogenic therapies are important in the treatment of certain cancers. But, by itself, the antiangiogenic treatment only prevents the tumor from developing its support of blood vessels and does not directly destroy the cancer cells. In fact, the benefits of using anti-angiogenic therapies alone are, at best, transient and are followed by the restoration of tumor growth and progression [4]. As a consequence, it is common not to limit the treatment to a body, but to combine different treatments in the hope of achieving a synergistic effect. In this way, antiangiogenic treatment should be combined with other treatments, such as radiotherapy or chemotherapy. In this section, we combine antiangiogenic therapy with chemotherapy.

The mathematical model studied in [39] is

dpdt(t)=ξp(t)logp(t)s(t)φv(t)p(t),

dsdt(t)=χq(t)μs(t),

dqdt(t)=χq(t)+bp(t)dp(t)2/3q(t)γu(t)q(t),

where p(t) is the size of the tumor, s(t) is the number of stable vessels, q(t) is the number of unstable vessels and u(t) and v(t) are the antiangiogenic and cytotoxic agents respectively. Table 5 shows the values of the parameters of the model.

Table 5

Values of the biological parameters in the model.

VariableDescriptionValue (Units)
ξTumor growth parameter0.084 day-1
bTumor-induced stimulation parameter5.85 day-1
dTumor-induced inhibition parameter0.00873 mm-2 day-1
μLoss of vascular support0.02 day-1
χMaduration of unstable vessels parameter0.025 day-1
γAnti-angiogenic elimination parameter0.15 kg/mg day-1
φCytotoxic killing parameter for the tumor0.1 kg/mg day-1

Thus, the problem that we want to solve is to minimize the size of the tumor p(t):

J(u)=p(T)

for a free final time T and subject to the constraints

0Tu(t)dtM,0Tv(t)dtN,

where the set of admissible controls is given by

uU={u(t)|0u(t)uM,t[0,T]},vV={v(t)|0v(t)vM,t[0,T]}.

Thus, we have the following problem:

Problem 6

For a free final time T, minimize the objective J, equation (55), over all Lebesgue measurable functions (respectively, piecewise continuous) u : [0,T][0,uM] and v : [0,T][0,vM] subject to the equations (52), (53) and (54) and the constraints (56).

As was done in previous problems, the switching functions can be calculated to get

ϕ1(t)=λ4γλ3q,

ϕ2(t)=λ5φλ1p,

and then, the optimal controls satisfy

u(t)=0,ifϕ1(t)>0,using,ifϕ1(t)0,uM,ifϕ1(t)<0,

and

v(t)=0,ifϕ2(t)>0,vsing,ifϕ2(t)0,vM,ifϕ2(t)<0,

and the adjoint equations are

λ˙1=λ1ξlog(ps)+ξ+φvλ3b23dp1/3q,λ1(T)=1,λ˙2=λ1ξps+μλ2,λ2(T)=0,λ˙3=χλ2+λ3(χ+dp2/3+γu),λ3(T)=0,λ˙4=0,λ˙5=0.

The generalized Legendre-Clebsch condition

(1)qud2qdt2q0,

for our problem becomes

ud2dt2ϕ1(t)=2bγ2λ3(t)p(t).

Therefore, the generalized Legendre-Clebsch condition is satisfied if λ3(t) > 0, ∀t 2 I. We have then the following proposition:

Proposition 9

If the optimal control u* is singular over an interval I =(a,b), then λ2345 >0. In addition, the control u* ends in an interval (t1,T) where u* = 0.

Then, we suppose the control u is singular and let’s calculate that control. After some tedious calculations, we get the following singular optimal control

using=λ1ψ1+λ2ψ2+λ3(φbv+ψ3)2bγλ3,

where

ψ1(p,q,s)=χξqs,ψ2(p,q,s)=2χb+χqp(μχ)dp1/3qχ,ψ3(p,q,s)=bξlogpsχdp2/3.

Regarding to the control v, analytically it is complicated to characterize it for what the numerical methods are used. Thus, in all the numerical simulations, ϕ < 0 and thus the control v is of the bang-bang type.

Some numerical solutions are shown in Figure 12 and Figure 13.

Fig. 12

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Fig. 12

[Color Online]. a) Evolution of the solutions to Eqs. (52), (53) and (54), p (solid line), s (dotted line), and q (solid-dot line) with the initial conditions (p0,s0,q0) = (12000,5000,14000) b) Optimal control u(t) (solid line) and switching function ϕ1 (dotted line) for M = 300 doses and uM = 75 c) Optimal control v(t) (solid line) and switching function ϕ2 (dotted line) for N = 5 doses and vM = 1.

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00029

Fig. 13

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Fig. 13

[Color Online]. a) Evolution of the solutions to Eqs. (52), (53) and (54), p (solid line), s (dotted line), and q (solid-dot line) with the initial conditions (p0,s0,q0) = (12000,5000,5000) b) Optimal control u(t) (solid line) and switching function ϕ1 (dotted line) for M = 300 doses and uM = 75 c) Optimal control v(t) (solid line) and switching function ϕ2 (dotted line) for N = 5 doses and vM = 1.

Citation: Applied Mathematics and Nonlinear Sciences 3, 2; 10.21042/AMNS.2018.2.00029

6 Conclusions

In this survey, a series of problems of optimal control theory applied to cancer treatments have been addressed. First of all, a class of cell cycle specific compartmental models for cancer chemotherapy was analyzed. By considering the phases of the cell cycle separately, it is possible to appropriately model the different actions of various drugs involved. A common characteristic of the models analyzed in the second section is that it is implicitly assumed that the cancer population is homogeneous and consists of cells that are sensitive to the chemotherapy applied. If we then minimize a weighted average of the tumor population and the total dose of chemotherapy given over a fixed therapy interval, it will be seen that in this scenario it indeed is optimal to give chemotherapy in one full dose session upfront at the beginning of the therapy interval. These results are fully consistent with and confirm as optimal the classical MTD (maximum tolerated dose) regimen. Essentially, the underlying scenario in these models is one where cancer is growing rapidly at a critical level, but sensitive to chemotherapeutic agents. The intuitive optimal solution then is to hit it as hard as possible, as soon as possible.

In the third section, low grade glioma, a type of brain tumor, together with the action of a cytotoxic agent, as the temozolomide, is studied [39]. For this problem, the optimal control solutions depend on the treatment time. If the treatment time is long enough (T = 30 months), the optimal control will be unique. On the other hand, if the treatment time is T = 30 days, the control will be of the bang-bang type.

Many tumors consist of heterogeneous agglomerations of subpopulations of cells that show widely varying sensitivities toward the actions of a particular chemotherapeutic agent [15, 29]. Coupled with the fact that growing tumors also exhibit considerable evolutionary ability to enhance cell survival in an environment that is becoming hostile, this leads to multi-drug resistance of some strains of the cells. Naturally, it makes sense to combine drugs with different activation mechanisms to reach a larger population of the tumor cells-and this is what is being done-but the sad fact remains that some cells develop multi-drug resistance to a wide variety of even structurally unrelated drugs. There may even exist subpopulations of cells that are not sensitive to the treatment from the beginning (ab initio, intrinsic resistance). For certain types of cancer cells, there are simply no effective agents known. We have explored this aspect in section four, studying a 2 compartment model with sensitive and resistant populations.

Finally, we have considered tumor’s microenvironment. One of the most important component of a tumor’s microenvironment is its vasculature. In order to grow beyond a small size, a growing tumor needs to develop its own network of blood vessels and capillaries that will provide it with nutrients and oxygen. This process is called angiogenesis and was already pointed out as a therapeutic target by J. Folkman in the early 1970s [23,24]. Antiangiogenenic treatments aim at depriving the tumor of this needed vasculature by either disrupting the signaling process that the tumor uses to recruit surrounding, mature, host blood vessels or by directly inhibiting the growth of endothelial cells that form the lining of the newly developing blood vessels and capillaries. We have studied this kind of treatment together with the chemotherapy [39] which is based on this model by Hahnfeldt et al. [21].

We hope that this survey will be useful in understanding the optimal treatment protocols in tumors. Mathematical models need to be developed to aid in our understanding of how to implement this work and hopefully to show how to develop new optimal treatment strategies.

Communicated by Elbaz I. Abouelmagd

Acknowledgements

This work has been supported by Ministerio de Economía y Competitividad/FEDER, Spain [grant numbers: MTM2012-31073 and MTM2015-71200-R].

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Ledzewicz, U., Maurer, H. and Schättler, H. (2011), Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Mathematical Biosciences and Engineering 8(2), pp. 307-323, doi 10.3934/mbe.2011.8.307.

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Rojas, C., Belmonte-Beitia, J., Pérez-García, V.M. and Maurer, H. (2016), Dynamics and optimal control of chemotherapy for low grade gliomas: Insights from a mathematical model, Discrete and Continuous Dynamical Systems - Series B 21(6), pp. 1895-1915, doi 10.3934/dcdsb.2016028.

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Świerniak, A., Ledzewicz, U. and Schättler, H. (2003), Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. Appl. Math. Comput. Sci 13(3), pp. 257-368.

[48]

Świerniak, A. and Polanski, A. and Kimmel, M. (1996), Optimal control problems arising in cell-cycle-specific cancer chemotherapy, Cell Proliferation 29(3), pp. 117–139, doi 10.1046/j.1365-2184.1996.00995.x.

[49]

Świerniak, A. (1995), Cell cycle as an object of control, Journal of Biological Systems 3(1), pp. 41-54, doi 10.1142/S0218339095000058.

[50]

Świerniak, A., Polanski, A. and Kimmel, M. (1996), Optimal control problems arising in cell cycle specific cancer chemotherapy, Cell Proliferation 29(3), pp. 117-139.

[51]

Świerniak, A., Duda, Z. (1995), Bilinear models of cancer chemoterapy-singularity of optimal solutions, Mathematical population dynamics 2, pp. 347-358.

[52]

Vincent, T. L., Brown, J.S. (2007), Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics, Cambridge University Press, UK

[53]

Westman, J. J. and Fabijonas, B. R. and Hanson, F. B. (2002), Cancer Treatment Using Multiple Chemotheraputic Agents Subject to Drug Resistance.

[54]

Wooley, T. E., Belmonte-Beitia, J., Calvo, G. F., Hopewell, J. W., Gaffney, E. A. and Jones, B. (2018), Changes in the retreatment radiation tolerance of the spinal cord with time after the initial treatment, International Journal of Radiation Biology 94(6), pp. 515-531, doi 10.1080/09553002.2018.1430911.

Journal Information

Figures

  • View in gallery

    A 2-compartment model with G2/M-specific cytotoxic agent.

  • View in gallery

    Examples of locally optimal controls (left) and their corresponding trajectories (right) for T = 7 (top), T = 21 (middle) and T = 60 (bottom) for the parameter values given in Table 1.

  • View in gallery

    A 3-compartment model with cytotoxic and recruiting agent.

  • View in gallery

    Examples of locally optimal controls u (citotoxic agent, top), w (recruiting agent, middle) and their corresponding trajectories (bottom) for weights s2 = 0 (left) and s2 = 0:1 (right). The parameter values are given in Table 2.

  • View in gallery

    A 3-compartment model with cytostatic agent v and cytotoxic agent u.

  • View in gallery

    Examples of locally optimal controls u (cytotoxic agent) (top), v (cytostatic agent) (middle) and their corresponding trajectories (bottom) for different weights r = (1,1,1) and q = (1,1,1) on the left and r = (8:25,8:25,8.25) and q = (0.1,0.1,0.1) on the right. The parameter values are given in Table 3.

  • View in gallery

    Optimal solution for functional Jw(u) in (29) with w = 0 and total dose M = 5, the time horizon is 30 days. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ.

  • View in gallery

    Optimal solution for functional Jw(u) in (29) with w = 1 and total dose M = 5, the time horizon is 30 days. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ.

  • View in gallery

    Optimal solution for functional Jw(u) in (29) with w = ws = 0:004067 and total dose M = 5, the time horizon is 30 days. (left) optimal control u(t) which is totally singular on [0,1], (right) proliferative cells P.

  • View in gallery

    Optimal solution for functional Jw(u) in (29) with w = 0 and total dose M = 5, the time horizon is T = 30 months. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ, the switching function ϕ(t) is zero on the singular arc [t1,t2].

  • View in gallery

    Optimal solution for functional Jw(u) in (29) with w = 1 and total dose M = 5, the time horizon is T = 30 months. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ, the switching function ϕ(t) is zero on the singular arc [t1, t2].

  • View in gallery

    [Color Online]. a) Evolution of the solutions to Eqs. (52), (53) and (54), p (solid line), s (dotted line), and q (solid-dot line) with the initial conditions (p0,s0,q0) = (12000,5000,14000) b) Optimal control u(t) (solid line) and switching function ϕ1 (dotted line) for M = 300 doses and uM = 75 c) Optimal control v(t) (solid line) and switching function ϕ2 (dotted line) for N = 5 doses and vM = 1.

  • View in gallery

    [Color Online]. a) Evolution of the solutions to Eqs. (52), (53) and (54), p (solid line), s (dotted line), and q (solid-dot line) with the initial conditions (p0,s0,q0) = (12000,5000,5000) b) Optimal control u(t) (solid line) and switching function ϕ1 (dotted line) for M = 300 doses and uM = 75 c) Optimal control v(t) (solid line) and switching function ϕ2 (dotted line) for N = 5 doses and vM = 1.

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