Antigen receptor therapy in bone metastasis via optimal control for different human life stages

Abstract

In this work we propose a bone metastasis model using power law growth functions in order to describe the biochemical interactions between bone cells and cancer cells. Experimental studies indicate that bone remodeling cycles are different for human life stages: childhood, young adulthood, and adulthood. In order to include such differences in our study, we estimate the model parameter values for each human life stage via bifurcation analysis. Results reveal an intrinsic relationship between the active period of remodeling cycles and the proliferation of cancer cells. Subsequently, using optimal control theory we analyze a possible antigen receptor therapy as a new treatment for bone metastasis. Theoretical results such as existence of optimal solutions are proved. Numerical simulations for late stages of bone metastasis are presented and a discussion of our results is carried out.

Appendices

Bifurcation analysis

Based on the oscillations frequency, we carried out a bifurcation analysis with the numerical software XPPAUT/AUTO to give parameter values in the childhood, young adults and adulthood. Recalling that the solution of system (1) exhibits limit cycles or stable/unstable oscillations when equation (2) is satisfied. For this analysis, we study the effect of varying the parameters $g_{11}$ and $g_{22}$ on osteoclast steady states (c) in the bone remodeling model.

Fig. 11

a Stability diagram of the osteoclast, c equilibrium point as a function of $g_{11}$. Red line is stable, and black line is unstable. Hopf bifurcation appears at $g_{11}=1.1$ and $g_{22}=0.001$. b Bifurcation diagram of the osteoclast, c equilibrium for system (1). Green dots represent stable limit cycle and blue circles represent unstable limit cycles (colour figure online)

First, the bifurcation parameter was $g_{11}$ and the other parameters were taken as $g_{22}=0.0001$, $g_{21} =1$ and $g_{12}=-0.3$. Figure 11 shows the stability diagram of the equilibrium point and the Hopf bifurcation varying $g_{11}$. The structure of the bifurcation diagram for model (1) is illustrated in Fig. 11b. Using a two parameter variation we obtained that limit cycles occur when $g_{11}\in [1.099\;\; 1.161048]$ and $g_{22}\in [-0.6104841\;\; 0)$. Recall that negative real-valued for kinetic orders describe inhibition activity. For the second case, the bifurcation parameter was $g_{22}$. The other parameters were: $g_{11}=0.90$, $g_{21} =1$ and $g_{12}=-0.3$ (the smallest/largest allowable values for $g_{21}$ and $g_{12}$ are 0.2 and $-0.1$ respectively, after that the system does not present oscillations).

Fig. 12

Hopf bifurcation diagrams of the osteoclast (c) equilibrium point of model (1) as a function of $g_{22}$. Red line is stable, and black line is unstable. Green dots represent stable limit cycle and blue circles represent unstable limit cycles. Hopf bifurcation occurs at values listed (colour figure online)

Figure 12 shows the diagram of the variation of the equilibrium point and the Hopf bifurcation. Notice that for values of $g_{11}$ less than 0.90 the Hopf bifurcation was not found. Fig. 12a, b show some of the bifurcation dynamics: when parameter $g_ {22}$ decreases then the stability region also decreases, and the instability region increases (see red and black lines). In such analysis, we obtained that limit cycles occur when $g_{11}\in [0.90\;\; 1.161]$ and $g_{22}\in [-0.61\;\; 2.0]$, with $g_{12}\in [-1.0\;\; -0.1]$ and $g_{21}\in [0.2\;\; 1.0]$. We also analyzed the oscillations frequency for parameter values close to the bifurcation, which allowed us to classify them in three stage: (a) If $g_{12}\in [-0.3\;\; -0.1]$ the BMU is less active, that is BMU has long periods of inactivity (adulthood stage); (b) if $g_{12}\in [-0.7\; \; -0.3)$ the BMU is more active, that is, the period of activity is approximately of 6 months (young adult stage), (c) finally if $g_{12}\in [-1.0\; -0.7)$ the BMU is more active and does not have inactive periods, that is, BMU is constantly remodeling (childhood stage). Therefore, with all the previous information we were able to classify the BMU behavior for three different stages in Table 1.

Proof of Theorem 3

The existence of optimal solutions for control problem (9)–(12) is obtained if the sufficient conditions of the Filippov-Cesari theorem (Fleming and Rishel 2012) are fulfilled:

(i):

The right-hand side of the model (10) is composed of continuous functions, and for each one of these functions $f_{i}$ there exist positive constant ${\mathcal {C}}_{1}$, ${\mathcal {C}}_{2}$ such that $\mid f_{i}(t,x,u)\mid \le {\mathcal {C}}_{1}(1+\mid x\mid +\mid u\mid )$ and $\mid f_{i}(t,{\overline{x}},u)-f_{i}(t,x,u)\mid \le {\mathcal {C}}_{2}\mid {\overline{x}}-x\mid (1+\mid u\mid )$ for all $0\le t\le t_{f}$ and $i=1,2,3$ and $x(t)=(C(t),B(t),T(t))$.

(ii):

A solution of the dynamical system (10) is well defined and is unique for an admissible $u\in \varGamma$.

(iii):

The set of solutions to system (10) is non-empty and bounded for admissible control functions $u\in \varGamma$.

(iv):

The control set $\varGamma$ is closed, bounded and convex in $\mathrm{I\!R}$.

(v):

The right-hand side of the dynamical system (10) is linear in control u.

(vi):

The integrand of function (9) is convex with respect to u and satisfies

$$\begin{aligned} T(t)+\frac{B}{2}u^{2}(t)\ge {\mathcal {K}}_{1}\mid u\mid ^{\theta }-{\mathcal {K}}_{2}, \end{aligned}$$

with constants ${\mathcal {K}}_{1}>0$, $\theta >1$ and ${\mathcal {K}}_{2}$.

Item (i) holds since the model functions are of class $C^2$ in $\varOmega$. Item (ii) holds since the admissible control set $\varGamma$ contains continuous bounded functions and the right-hand side of (10) is Lipschitz continuous with respect to the three variables C, B and T for admissible $u(t)\in \varGamma$. Using Picard-Lindelöf theorem, there exists a unique solution $(C(t), B(T), T(t))\in \varOmega$ corresponding to the admissible control $u(t):\mathrm{I\!R}^{+}\mapsto [0,u_{c}]$. When the control variable takes its minimum value, that is, $u(t)=0$ in (10), we recover the initial model without control (4) whose orbits C(t), B(t) and T(t) of $(C(0),B(0), T(0))\in \varOmega$ are bounded for all $t\ge 0$. Therefore, solutions of the system (10) with $u(t)=0$ can be regarded as its super-solution. On the other hand, when the control variable is at the upper bound, that is, $u(t)=u_{c}$, the underlying solution of the system (10) can be regarded as its sub-solutions. Moreover, applying the Carathéodory’s existence theorem guarantees the existence of solutions for Cauchy problems. The latter implies that item (iii) holds. Additionally, by the definition of the admissible control set $\varGamma$ in (12) is clear that this set control is closed, bounded and convex in $\mathrm{I\!R}$. Thus, (iv) holds. To proof the item (v), let f(t, x, u) be the vector function defined by right-hand side of (10), we will find suitable bounds for the states:

$$\begin{aligned} \frac{dC(t)}{dt}= & {} \alpha _{1}C(t)B(t)^{g_{12}} -\beta _{1}C(t)+ (\sigma _{1}-a_{1}u(t))C(t)T(t)\nonumber \\\le & {} C(t)(\alpha _{1}m_{1} + \sigma _{1}C_{max})\; \; (g_{12}<0\;\;and\;\; m_{1}:=(B_{min})^{g_{12}}). \end{aligned}$$

(18)

On the other hand,

$$\begin{aligned} \frac{dB(t)}{dt}= & {} \alpha _{2}C(t)^{g_{21}}B(t)-\beta _{2}B(t)+(\sigma _{2}-a_{2}u(t))B(t)T(t),\nonumber \\\le & {} \left\{ \begin{array}{lcc} B(t)(\alpha _{2}m_{2} + \sigma _{2}C_{max})&{}&{} if\;\; \sigma _{2}\ge 0, \\ B(t)\alpha _{2}m_{2} &{}&{}if\;\;\sigma _{2}<0, \end{array} \right. \end{aligned}$$

(19)

where $m_{2}:=(C_{max})^{g_{21}}$. Finally,

$$\begin{aligned} \frac{dT(t)}{dt}= & {} \alpha _{3}C(t)^{g_{31}}T(t)\left( 1-\frac{T(t)}{K}\right) -(\beta _{3}+u(t))T(t)\nonumber \\\le & {} T(t)(\alpha _{3}m_{2} -\beta _{3}). \end{aligned}$$

(20)

Thus, from (18)–(20) we have that our model is bounded by a linear system. Item vi), the integrand of (9) is quadratic in u, and therefore convex. Condition $T(t)+\frac{B}{2}u^{2}(t)\ge {\mathcal {K}}_{1}\mid u\mid ^{\theta }-{\mathcal {K}}_{2}$ is fulfilled with ${\mathcal {K}}_{2}=0$, $\theta =2>1$ and ${\mathcal {K}}_{1}=\frac{B}{2}>0$.