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# Optimal control for cancer chemotherapy under tumor heterogeneity with Michealis-Menten pharmacodynamics

• * Corresponding author: Heinz Schättler
• We analyze mathematical models for cancer chemotherapy under tumor heterogeneity as optimal control problems. Tumor heterogeneity is incorporated into the model through a potentially large number of states which represent sub-populations of tumor cells with different chemotherapeutic sensitivities. In this paper, a Michaelis-Menten type functional form is used to model the pharmacodynamic effects of the drug concentrations. In the objective, a weighted average of the tumor volume and the total amounts of drugs administered (taken as an indirect measurement for the side effects) is minimized. Mathematically, incorporating a Michaelis-Menten form creates a nonlinear structure in the controls with partial convexity properties in the Hamiltonian function for the optimal control problem. As a result, optimal controls are continuous and this fact can be utilized to set up an efficient numerical computation of extremals. Second order Jacobi type necessary and sufficient conditions for optimality are formulated that allow to verify the strong local optimality of numerically computed extremal controlled trajectories. Examples which illustrate the cases of both strong locally optimal and non-optimal extremals are given.

Mathematics Subject Classification: Primary: 92C50, 49K15; Secondary: 93C95.

 Citation: • • Figure 1.  A numerically computed extremal control (top row, left) for the effectiveness function $\varphi(x) = \frac{2}{1+x^2}$ (top row, right), weights $\alpha = (1, \ldots, 1)$, $\beta = 500\alpha$ and $\gamma = 5000$, and therapy horizon $T = 10$. The corresponding evolution of the individual cell populations is shown in the middle row (left) along with the evolution of the average of the populations (right). A comparison of the initial density $n(0, x)\equiv \frac{200}{21}$ and the terminal density $n(10, x)$ shown as red curve is given in the bottom row on the left. This control is a strong local minimum as the entries of the solution $S$ to the Riccati differential equation (13) along the extremal controlled trajectory (shown in the bottom row, right) remain finite over the full interval $[0, T]$

Figure 2.  A numerically computed extremal control (top row, left) for the effectiveness function $\varphi_1$ and $\varphi_2$ given in (22), (top row, right), $\alpha = (1, \ldots, 1)$, $\beta = 50 \alpha$, $\gamma = (600, 1400)$ and time horizon $T = 10$. The corresponding evolution of the individual cell populations is shown in the middle row (left) along with the evolution of the average of the populations (right). A comparison of the initial density $n(0, x)\equiv \frac{200}{21}$ and the terminal density $n(10, x)$ shown as red curve is given in the bottom row on the left. This control is a strong local minimum as the entries of the solution $S$ to the Riccati differential equation (13) along the extremal controlled trajectory (shown in the bottom row, right) remain finite over the full interval $[0, T]$

Figure 3.  A numerically computed extremal control (top row, left) for the effectiveness function $\varphi_1$ and $\varphi_2$ given in (23), (top row, right), $\alpha = (1, \ldots, 1)$, $\beta = 50 \alpha$, $\gamma = (600, 1400)$ and time horizon $T = 10$. The corresponding evolution of the individual cell populations is shown in the middle row (left) along with the evolution of the average of the populations (right). A comparison of the initial density $n(0, x)\equiv \frac{200}{21}$ and the terminal density $n(10, x)$ shown as red curve is given in the bottom row on the left. This control is not a strong local minimum as the entries of the solution $S$ to the Riccati differential equation (13) along the extremal controlled trajectory (shown in the bottom row, right) blow up in the interval

Figure 4.  Examples of the system response, for the same parameters as in Fig. 3, for the constant controls $u_1 = 1$ and $u_2 = 2.5$

Figure 5.  Illustration of the functions $\Upsilon$ (top, left), $\eta$ (top, right) and the full Hamiltonian $H$ (bottom) as function of the controls. These graphs depict the actual values for the solution shown in Fig. 6 below at the time $t = 19$

Figure 6.  A numerically computed locally optimal control (top row, left) for $\alpha = (1, \ldots, 1)$, $\beta = 15 \alpha$, $\gamma = (90, 30)$ with therapy horizon $T = 20$ and effectiveness functions $\varphi_1 = \frac{2}{1+x^2}$ and $\varphi_2 = \frac{2}{1+3x^6}$ (top row, right). The corresponding evolution of the individual cell populations is shown in the middle row (left) along with the evolution of the average of the populations (right). A comparison of the initial density $n(0, x)\equiv \frac{200}{21}$ and the terminal density $n(20, x)$ shown as red curve is given in the bottom row on the left. This control is a strong local minimum as the entries of the solution $S$ to the Riccati differential equation (30) along the extremal controlled trajectory (shown in the bottom row, right) remain finite in the full interval $[0, 20]$

Figure 7.  A numerically computed locally optimal control (top row, left) for α = (1, ...., 1), β = 15α, γ = (90, 30) with therapy horizon T = 20 and effectiveness functions ${\varphi _1}{\rm{ = }}\frac{2}{{1 + 5{x^2}}}$ and ${\varphi _2}{\rm{ = }}\frac{2}{{1 + 60{{\left( {1 - x} \right)}^6}}}$ (top row, right). The corresponding evolution of the individual cell populations is shown in the middle row (left) along with the evolution of the average of the populations (right). A comparison of the initial density n(0, x) ≡ $\frac{{200}}{{21}}$ and the terminal density n(20, x) shown as red curve is given in the bottom row on the left. This control is a strong local minimum as the entries of the solution S to the Riccati differential equation (30) along the extremal controlled trajectory (shown in the bottom row, right) blow up in the interval.

Figure 8.  A numerically computed extremal control (top row, left) for α = (1, ..., 1), β = 15α, γ = (90, 30) with shortened therapy horizon T = 5 and effectiveness function ${\varphi _1}{\rm{ = }}\frac{2}{{1 + 5{x^2}}}$ and ${\varphi _2}{\rm{ = }}\frac{2}{{1 + 60{{\left( {1 - x} \right)}^6}}}$ (top row, right). The corresponding evolution of the individual cell populations is shown in the middle row (left) along with the evolution of the average of the populations (right). A comparison of the initial density n(0, x) ≡ $\frac{{200}}{{21}}$ and the terminal density n(5, x) shown as red curve is given in the bottom row on the left. This control still is a strong local minimum as the entries of the solution S to the Riccati differential equation (30) along the extremal controlled trajectory (shown in the bottom row, right) remain finite in the full interval [0, T].

Table Algorithm 1.  Shooting Method

 Require: $N(t)>0$, $\forall t \in [0, T]$ and pick a threshold $\epsilon >0$. Ensure: optimal control   start with an initial guess $\lambda(0)$ and the stepsize $s=0.01$. while $\|\lambda(T)-\alpha\|>\epsilon$ do  solve $N, \; \lambda$.   $\lambda(0)\Leftarrow \lambda(0)-s\Big( \frac{\delta \lambda(T)}{\delta \lambda(0)}\Big) ^{-1} (\lambda(T)-\alpha)$ end while

Table Algorithm 2.  Iterative Method

 Require: $N(t)>0$, $\forall t \in [0, T]$ and pick a threshold $\epsilon >0$. Ensure: initial guess   Start with an initial trajectory $N^{(0)}$ and $k=1$. while $\|N^{(k)}-N^{(k-1)}\|>\epsilon$ do   $A^{(k-1)} \Leftarrow$ function of $N^{(k-1)}$   $O^{(k-1)} \Leftarrow$ function of $N^{(k-1)}$   Solve $S^{(k)}, \; \nu^{(k)}$ backward in time by assuming $\lambda^{(k)} = S^{(k)}N^{(k)}+\nu^{(k)}$.  Update $N^{(k)}$ forward in time:  $\qquad \dot{N}^{(k)} =[A^{(k-1)}+O^{(k-1)}S^{(k)}]N^{(k)}+O^{(k-1)}\nu^{(k)}$)  $u^{(k)} \Leftarrow$ function of $N^{(k)}$, $\lambda^{(k)}$  $k+1 \Leftarrow k$ end while
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