A Robust Optimization Approach to Cancer Treatment under Toxicity Uncertainty

Abstract

The design of optimal protocols plays an important role in cancer treatment. However, in clinical applications, the outcomes under the optimal protocols are sensitive to variations of parameter settings such as drug effects and the attributes of age, weight, and health conditions in human subjects. One approach to overcoming this challenge is to formulate the problem of finding an optimal treatment protocol as a robust optimization problem (ROP) that takes parameter uncertainty into account. In this chapter, we describe a method to model toxicity uncertainty. We then apply a mixed integer ROP to derive the optimal protocols that minimize the cumulative tumor size. While our method may be applied to other cancers, in this work we focus on the treatment of chronic myeloid leukemia (CML) with tyrosine kinase inhibitors (TKI). For simplicity, we focus on one particular mode of toxicity arising from TKI therapy, low blood cell counts, in particular low absolute neutrophil count (ANC). We develop optimization methods for locating optimal treatment protocols assuming that the rate of decrease of ANC varies within a given interval. We further investigated the relationship between parameter uncertainty and optimal protocols. Our results suggest that the dosing schedule can significantly reduce tumor size without recurrence in 360 weeks while insuring that toxicity constraints are satisfied for all realizations of uncertain parameters.

Appendices

Appendix 1: Nominal Optimization Problem

The nominal optimization problem can be formulated as a MIOP as below:

$$ \min \sum \limits_{m\in M,\kern0.75em i\in I\backslash \left\{1\right\}}{x}_{i,m} $$

(7a)

$$ \mathrm{s}.\mathrm{t}.\kern1em {\overset{\cdotp }{x}}_1(t)=\sum \limits_{j=0}^3{z}^{m,j}\left({b}_1^j{\psi}_{x_1}-d\right){x}_1,\kern0.5em t\in \left[m\Delta t,\left(m+1\right)\Delta t\right],m\in M\backslash \left\{M\right\} $$

(7b)

$$ {\overset{\cdotp }{x}}_2(t)=\sum \limits_{j=0}^3{z}^{m,j}\left({b}_2^j\left(1-\left(n-2\right)\mu \right){\psi}_{x_2}-d\right){x}_2,t\in \left[m\Delta t,\left(m+1\right)\Delta t\right],m\in M\backslash \left\{M\right\} $$

(7c)

$$ {\overset{\cdotp }{x}}_i(t)=\sum \limits_{j=0}^3{z}^{m,j}\left(\left({b}_i^j{\psi}_{x_2}-d\right){x}_i+\mu {b}_2^j{\psi}_{x_2}{x}_2\right),\kern0.5em t\in \left[m\Delta t,\left(m+1\right)\Delta t\right],m\in M\backslash \left\{M\right\},3\le \mathrm{i}\le n $$

(7d)

$$ \sum \limits_{j\in J}{z}^{m,j}=1,\kern1.75em m\in M\backslash \left\{M\right\}, $$

(7e)

$$ {y}^{m+1}={\widehat{y}}^m-\sum \limits_{j\in J}{d}_{\mathrm{anc},j}{z}^{m,j},\kern1.75em m\in M\backslash \left\{M\right\}, $$

(7f)

$$ {\widehat{y}}^m=\min \left({y}^m,{\mathrm{ANC}}_{\mathrm{normal}}\right),\kern1.75em m\in M, $$

(7g)

$$ {L}_{anc}\le {\widehat{y}}^m,\kern1.75em m\in M, $$

(7h)

$$ {z}^{m,j}\in \left\{0,1\right\},\kern0.5em m\in M\backslash \left\{M\right\},j\in J $$

(7i)

where x(0) , y ⁰ are given. In Eqs. 7b, 7c, and 7d, the dynamics of NSC, WSC, and MSC are described, respectively. Equations 7e, and 7i indicate that during each week, only one type of drug or no drug is allowed. Equations 7f, 7g, and 7h describe the toxicity constraints.

As discussed in the previous work [26], the ODEs can be approximated by linear functions:

$$ \min \sum \limits_{m\in M,\kern0.75em i\in I\backslash \left\{1\right\}}{x}_{i,m} $$

$$ \mathrm{s}.\mathrm{t}.\kern1em {x}_{i,m+1}=\sum \limits_{j=0}^3{z}^{m,j}\left({C}_{i,0}^j+\sum \limits_{k=1}^n{C}_{i,k}^j{x}_{k,m}\right),\kern0.5em t\in \left[m\Delta t,\left(m+1\right)\Delta t\right],m\in M\backslash \left\{M\right\} $$

$$ \sum \limits_{j\in J}{z}^{m,j}=1,\kern1.75em m\in M\backslash \left\{M\right\} $$

$$ {y}^{m+1}={\widehat{y}}^m-\sum \limits_{j\in J}{d}_{\mathrm{anc},j}{z}^{m,j},\kern1.75em m\in M\backslash \left\{M\right\} $$

$$ {\widehat{y}}^m=\min \left({y}^m,{\mathrm{ANC}}_{\mathrm{normal}}\right),\kern1.75em m\in M $$

$$ {L}_{\mathrm{anc}}\le {\widehat{y}}^m,\kern1.75em m\in M $$

$$ {z}^{m,j}\in \left\{0,1\right\},\kern0.5em m\in M\backslash \left\{M\right\},j\in J $$

where x(0) , y ⁰ are given.

There are two types of nonlinear terms here: z ^m , j x _i , m and ${\widehat{y}}^m=\min \left({y}^m,{\mathrm{ANC}}_{\mathrm{normal}}\right)$.

To linearize z ^m , j x _i , m , we introduce a new variable

$$ 0\le {v}_i^{m,j}\le {U}_i{z}^{m,j} $$

$$ -{U}_i\left(1-{z}^{m,j}\right)\le {v}_i^{m,j}-{x}_{i,m}\le {U}_i\left(1-{z}^{m,j}\right) $$

To linearize ${\widehat{y}}^m=\min \left({y}^m,{\mathrm{ANC}}_{\mathrm{normal}}\right)$, we introduce a binary variable p ^m

$$ {\widehat{y}}^m\ge {\mathrm{ANC}}_{\mathrm{normal}}-{U}_y\left(1-{p}^m\right), $$

$$ {\widehat{y}}^m\ge {y}^m-{U}_y{p}^m, $$

$$ {\widehat{y}}^m\le {y}^m, $$

$$ {\widehat{y}}^m\le {\mathrm{ANC}}_{\mathrm{normal}}, $$

$$ {p}^m\in \left\{0,1\right\}. $$

The nominal problem can be transformed into a MILP as

$$ \min \sum \limits_{m\in M,\kern0.75em i\in I\backslash \left\{1\right\}}{x}_{i,m} $$

(8a)

$$ \mathrm{s}.\mathrm{t}.\kern0.5em {x}_{i,m+1}=\sum \limits_{j=0}^3\left({z}^{m,j}{C}_{i,0}^j+\sum \limits_{k=1}^n{C}_{i,k}^j{v}_k^{m,j}\right),\kern0.5em t\in \left[m\Delta t,\left(m+1\right)\Delta t\right],m\in M\backslash \left\{M\right\}, $$

(8b)

$$ 0\le {v}_i^{m,j}\le {U}_i{z}^{m,j}, $$

(8c)

$$ -{U}_i\left(1-{z}^{m,j}\right)\le {v}_i^{m,j}-{x}_{i,m}, $$

(8d)

$$ {v}_i^{m,j}-{x}_{i,m}\le {U}_i\left(1-{z}^{m,j}\right), $$

(8e)

$$ \sum \limits_{j\in J}{z}^{m,j}=1,\kern1em m\in M\backslash \left\{M\right\}, $$

(8f)

$$ {y}^{m+1}={\widehat{y}}^m-\sum \limits_{j\in J}{d}_{\mathrm{anc},j}{z}^{m,j},\kern1em m\in M\backslash \left\{M\right\}, $$

(8g)

$$ {\widehat{y}}^m\ge {\mathrm{ANC}}_{\mathrm{normal}}-{U}_y\left(1-{p}^m\right), $$

(8h)

$$ {\widehat{y}}^m\ge {y}^m-{U}_y{p}^m, $$

(8i)

$$ {\widehat{y}}^m\le {y}^m, $$

(8j)

$$ {\widehat{y}}^m\le {\mathrm{ANC}}_{\mathrm{normal}}, $$

(8k)

$$ {p}^m\in \left\{0,1\right\}, $$

(8l)

$$ {L}_{\mathrm{anc}}\le {\widehat{y}}^m,\kern1em m\in M, $$

(8m)

$$ {z}^{m,j}\in \left\{0,1\right\},\kern1em m\in M\backslash \left\{M\right\},j\in J, $$

(8n)

where x(0) , y ⁰ are given.

Appendix 2: ROP Model

In this section, we describe the mathematical details of the robust problem. We introduce parameters Γ^m that take values in the bounded intervals [0,|V ^m|], where V ^m is the index sets of parameters with uncertainty. Γ^m is not necessarily an integer. The role of parameters Γ^m is to adjust the robustness of the proposed model against the conservation level of solution, thus it is called protection level. The motivation of Γ^m is that it is unlikely that all the parameters with uncertainty vary at the same time and reach the maximal uncertainty. In other words, the model assumes that there exists only a subset of the parameter drift that influence the solution. More specifically, it assumes that there are up to ⌊Γ^m⌋ of uncertainty parameters which are allowed to deviate from their nominal values, and the toxicity decreasing rate ${d}_{\mathrm{anc},j}^m$ changes by at most $\left({\Gamma}^m-\left\lfloor {\Gamma}^m\right\rfloor \right){\widehat{C}}^j$, where ⌊Γ^m⌋ is the greatest integer ≤Γ^m. Note that, if we choose Γ^m = 0, we completely ignore the influence of parameter uncertainty and are using the nominal values of the uncertain parameters, and if we choose Γ^m = V ^m, then all the uncertain parameters are subjected to deviate from their nominal values. In this project, the maximum value of Γ^m is 3 since there are three parameters with uncertainty. Note however that only one drug is chosen for each period, parameter uncertainty of the other two drugs will not affect the robust optimal solution. Thus, the robust optimal solution under Γ^m = 1 is exactly the same as the ones under Γ^m > 1. The proposed robust counterpart of problem (Eq. 6) is as follows:

$$ \min \sum \limits_{m\in M,\kern0.75em i\in I\backslash \left\{1\right\}}{x}_{i,m} $$

(9a)

$$ \mathrm{s}.\mathrm{t}.\kern1em {y}^{m+1}-{\widehat{y}}^m+\sum \limits_{j\in J}{L}^j{z}^{m,j}+\underset{C_m^{\mathrm{RO}}}{\max}\left\{\sum \limits_{j\in {S}^m}{\widehat{C}}^j{z}^{m,j}+\left({\Gamma}^m-\left\lfloor {\Gamma}^m\right\rfloor \right){\widehat{C}}^{t^m}{z}^{m,{t}^m}\right\}, $$

(9b)

$$ {\displaystyle \begin{array}{l}(8b),(8c),(8d),(8e),(8f),(8h),\\ {}(8i),(8j),(8k),(8l),(8m),(8n)\end{array}} $$

(9c)

where ${C}_m^{\mathrm{RO}}=\left\{{S}^m\cup \left\{{t}^m\right\}|{S}^m\subseteq {V}^m,\left|{S}^m\right|\le \left\lfloor {\Gamma}^m\right\rfloor, {t}^m\in {V}^m\backslash {S}^m\right\}$, S ^m is the index sets of uncertain parameters which are allowed to deviate from their nominal values. According to the method developed in [20], the maximization problem in Eq. 9b is equivalent to the following auxiliary problem:

$$ \max \kern0.75em \sum \limits_{j\in J}{\widehat{C}}^j{\eta}^{m,j}{z}^{m,j} $$

(10a)

$$ \mathrm{s}.\mathrm{t}.\kern1em 0\le {\eta}^{m,j}\le 1,\kern1em j\in J, $$

(10b)

$$ \sum \limits_{j\in J}{\eta}^{m,j}\le {\Gamma}^m, $$

(10c)

Equation 10c indicates that the total variation of the parameters cannot exceed some threshold Γ^m. Notice that problem (Eq. 11) is bounded. It is clear that η ^m , j = 0 is a feasible solution of (Eq. 11). By strong duality, the optimal objective value of problem (Eq. 11) is the same as the optimal objective value of its dual problem. It is easy to check that the dual problem can be written as

$$ \max\ {q}^m{\Gamma}^m+\sum \limits_{j\in J}{p}^{m,j} $$

(11a)

$$ \mathrm{s}.\mathrm{t}.\kern1em -{\widehat{C}}^j{z}^{m,j}+{\mathrm{q}}^{\mathrm{m}}+{p}^{m,j}\ge 0, $$

(11b)

$$ {q}^m\ge 0, $$

(11c)

$$ {p}^{m,j}\ge 0, $$

(11d)

Thus, the optimal solution of our robust problem can be obtained by solving the MILP:

$$ \min \sum \limits_{m\in M,\kern0.75em i\in I\backslash \left\{1\right\}}{x}_{i,m} $$

(12a)

$$ s.t.\kern1em {x}_{i,m+1}=\sum \limits_{j=0}^3\left({z}^{m,j}{C}_{i,0}^j+\sum \limits_{k=1}^n{C}_{i,k}^j{v}_k^{m,j}\right),\kern0.5em t\in \left[m\Delta t,\left(m+1\right)\Delta t\right],m\in M\backslash \left\{M\right\}, $$

(12b)

$$ 0\le {v}_i^{m,j}\le {U}_i{z}^{m,j}, $$

(12c)

$$ -{U}_i\left(1-{z}^{m,j}\right)\le {v}_i^{m,j}-{x}_{i,m}, $$

(12d)

$$ {v}_i^{m,j}-{x}_{i,m}\le {U}_i\left(1-{z}^{m,j}\right), $$

(12e)

$$ \sum \limits_{j\in J}{z}^{m,j}=1,\kern1em m\in M\backslash \left\{M\right\}, $$

(12f)

$$ {y}^{m+1}\le {\widehat{y}}^m-\sum \limits_{j\in J}{L}^j{z}^{m,j}-{q}^m{\Gamma}^m-\sum \limits_{j\in J}{p}^{m,j},\kern1em m\in M\backslash \left\{M\right\}, $$

(12g)

$$ {\widehat{y}}^m\ge {\mathrm{ANC}}_{\mathrm{normal}}-{U}_y\left(1-{p}^m\right), $$

(12h)

$$ {\widehat{y}}^m\ge {y}^m-{U}_y{p}^m, $$

(12i)

$$ {\widehat{y}}^m\le {y}^m, $$

(12j)

$$ {\widehat{y}}^m\le {\mathrm{ANC}}_{\mathrm{normal}}, $$

(12k)

$$ {p}^m\in \left\{0,1\right\}, $$

(12l)

$$ {L}_{\mathrm{anc}}\le {\widehat{y}}^m,\kern1em m\in M, $$

(12m)

$$ {z}^{m,j}\in \left\{0,1\right\},\kern1em m\in M\backslash \left\{M\right\},j\in J, $$

(12n)

$$ -{\widehat{C}}^j{z}^{m,j}+{q}^m+{p}^{m,j}\ge 0, $$

(12o)

$$ {q}^m\ge 0, $$

(12p)

$$ {p}^{m,j}\ge 0, $$

(12q)

$$ x(0),{y}^1,{\widehat{y}}^1\ \mathrm{are}\ \mathrm{given},\kern0.5em {p}^1=0 $$

(12r)

Appendix 3: Robust Optimal Solutions for 30 Weeks

In this section, we summarize all the robust optimal solutions discussed in Subheading. 4.4 (see Figs. 13, 14, 15, 16, and 17).

Fig. 13

Robust optimal solutions under ${\widehat{C}}^j=0.2\times {L}^j$ for 30 weeks. The initial conditions for NSC, WSC, Y 253F, and F317L are 9E + 06, 9E + 05, 1E + 04, and 1E + 04, respectively

Fig. 14

Robust optimal solutions under ${\widehat{C}}^j=0.2\times {L}^j$ for 30 weeks. The initial conditions for NSC, WSC, Y 253F, and F317L are 9E + 06, 5E + 05, 3E + 05, and 3E + 05, respectively

Fig. 15

Robust optimal solutions under ${\widehat{C}}^j=0.3\times {L}^j$ for 30 weeks. The initial conditions for NSC, WSC, Y 253F, and F317L are 9E + 06, 9E + 05, 1E + 04, and 1E + 04, respectively

Fig. 16

Robust optimal solutions under ${\widehat{C}}^j=0.3\times {L}^j$ for 30 weeks. The initial conditions for NSC, WSC, Y 253F, and F317L are 9E + 06, 9E + 05, 1E + 05, and 1E + 05, respectively

Fig. 17

Robust optimal solutions under ${\widehat{C}}^j=0.3\times {L}^j$ for 30 weeks. The initial conditions for NSC, WSC, Y 253F, and F317L are 9E + 06, 5E + 05, 3E + 05, and 3E + 05, respectively