Fractional kinetics under external forcing

Abstract

Fractional tumor development is considered in the framework of one-dimensional continuous time random walks (CTRW) in the presence of chemotherapy. The chemotherapy influence on the CTRW is studied by observations of both stationary solutions due to proliferation and fractional evolution in time.

Appendices

Appendix A: Fractional integro-differentiation

A basic introduction to fractional calculus can be found, e.g., in Ref. [15–17]. Fractional integration of the order of $\alpha$ is defined by the operator

$$\begin{aligned} {}_aI_t^{\alpha }f(t)=\frac{1}{\varGamma (\alpha )} \int \limits _a^tf(\tau )(t-\tau )^{\alpha -1}d\tau , ~~(\alpha >0)\, .\nonumber \\ \end{aligned}$$

(41)

There is no constraint on the limit $a$. In our consideration, $a=0$ since this is a natural limit for the time. A fractional derivative is defined as an inverse operator to ${}_aI_t^{\alpha }\equiv I_t^{\alpha }$ as $\frac{d^{\alpha }}{dt^{\alpha }}=I_t^{-\alpha }=D_t^{\alpha }$, correspondingly $I_t^{\alpha }=\frac{d^{-\alpha }}{dt^{-\alpha }} =D_t^{-\alpha }$. It’s explicit form is convolution

$$\begin{aligned} D_t^{\alpha }=\frac{1}{\varGamma (-\alpha )}\int \limits _0^t \frac{f(\tau )}{(t-\tau )^{\alpha +1}}d\tau \, . \end{aligned}$$

(42)

For arbitrary $\alpha >0$ this integral is, in general, divergent. As a regularization of the divergent integral, the following two alternative definitions for $D_t^{\alpha }$ exist [27]

$$\begin{aligned}&{}_{RL}D_{(0,t)}^{\alpha }f(t)\equiv D_{RL}^{\alpha }f(t)= D^nI^{n-\alpha }f(t) \nonumber \\&\quad =\frac{1}{\varGamma (n-\alpha )}\frac{d^n}{dt^n}\int \limits _0^t \frac{f(\tau )d\tau }{(t-\tau )^{\alpha +1-n}} \, ,\nonumber \\\end{aligned}$$

(43)

$$\begin{aligned}&D_C^{\alpha }f(t)= I^{n-\alpha }D^nf(t) \nonumber \\&\quad =\frac{1}{\varGamma (n-\alpha )}\int \limits _0^t \frac{f^{(n)d\tau }(\tau )}{(t-\tau )^{\alpha +1-n}} \, , \end{aligned}$$

(44)

where $n-1<\alpha <n,~~n=1,2,\dots$. Eq. (43) is the Riemann–Liouville derivative, while Eq. (44) is the fractional derivative in the Caputo form [15, 27]. Performing integration by part in Eq. (43) and then applying Leibniz’s rule for the derivative of an integral and repeating this procedure $n$ times, we obtain

$$\begin{aligned} D_{RL}^{\alpha }f(t)=D_C^{\alpha }f(t)+\sum _{k=0}^{n-1}f^{(k)}(0^+) \frac{t^{k-\alpha }}{\varGamma (k-\alpha +1)} \, .\nonumber \\ \end{aligned}$$

(45)

The Laplace transform can be obtained for Eq. (44). If $\hat{L}f(t)=\tilde{f}(s)$, then

$$\begin{aligned} \hat{L}[D_C^{\alpha }f(t)]=s^{\alpha }\tilde{f}(s)- \sum _{k=0}^{n-1}f^{(k)}(0^+)s^{\alpha -1-k}\, . \end{aligned}$$

(46)

The following fractional derivatives are helpful for the present analysis

$$\begin{aligned} D_{RL}^{\alpha }[1]=\frac{t^{-\alpha }}{\varGamma (1-\alpha )}\, , ~~ D_C^{\alpha }[1]=0\, . \end{aligned}$$

(47)

We also note that

$$\begin{aligned} D_{RL}^{\alpha }t^{\beta }=\frac{t^{\beta -\alpha }\varGamma (\beta +1)}{\varGamma (\beta +1-\alpha )}\, , \end{aligned}$$

(48)

where $\beta >-1$ and $\alpha >0$. The fractional derivative from an exponential function can be simply calculated as well by virtue of the Mittag–Leffler function (see e.g., [15]):

$$\begin{aligned} E_{\gamma ,\delta }(z)=\sum _{k=0}^{\infty } \frac{z^k}{\varGamma (\gamma k+\delta )} \, . \end{aligned}$$

(49)

Therefore, we have the following expression

$$\begin{aligned} D_{RL}^{\alpha }e^{\lambda t}=t^{\alpha }E_{1,1-\alpha }(\lambda t)\, . \end{aligned}$$

(50)

Appendix B: Chemotherapy

One should recognize that Eq. (5) generalizes a standard chemotherapy scheme, based on experimental data and suggested in the framework of the linear reaction–diffusion equation [21, 22]

$$\begin{aligned} \text{ rate } \text{ of } \text{ change } \text{ of } \text{ cells } =\left\{ \begin{array}{l} \text{ motility } \text{ of } \text{ cells } \\ +\, \text{ net } \text{ proliferation }\\ -\, \text{ chemotherapy } \end{array}\right\} \, . \end{aligned}$$

(51)

These scheme follows the seminal result obtained by analyzing a recurrent anaplastic astrocytoma, treated by chemotherapy [21]. New chemotherapeutic strategies are represented by the combination of multi-targeted drugs with cytotoxic chemotherapy and radiotherapy in order to overcome tumor resistance [28]. One accounts also heterogeneous drug delivery due to complicated vascular structure (in particular, in brain [29]) that leads to the chemotherapy term to be a complicated function of time, space, and the cancer cell concentration $G(P)=G(t,x,P)$, see also [30].