﻿ A Study on Multipeutics

Applied Mathematics
Vol.08 No.05(2017), Article ID:76677,28 pages
10.4236/am.2017.85059

A Study on Multipeutics

Jens Christian Larsen

Vanløse Alle 50 2. mf. tv, 2720 Vanløse, Copenhagen, Denmark    Received: April 3, 2017; Accepted: May 24, 2017; Published: May 27, 2017

ABSTRACT

Keywords:

Cancer, Mass Action Kinetic System, Quadrapeutics, Immune Therapy 1. Introduction

In  we introduced a discrete three dimensional model of cancer growth and a three dimensional ODE model of cancer growth. The variables are C cancer cells, GF growth factors (positive control) and growth inhibitors GI (negative control) in both models. In  we showed that there are cancer ODE models with oscillatory behaviour and in  we found a discrete mathematical model of C, GF, GI with a nontrivial attractor. In the same paper, we introduced mathematical models of the four phases of adoptive T cell therapy. The first phase is the resection of the tumor and lymphodepletion with a chemo therapeutic drug. The second phase is the pre REP phase where the TILs (tumor infiltrating lymphocytes) from the patients’ tumor are expanded with IL-2 treatment. The third phase is the REP phase of expanding and activating the TILs with IL-2 and anti CD3. The fourth phase is the injection of the expanded and activated TIL s back into the patient. We showed that this last phase of the treatment is bistable in the following sense. There is a positive sink and there exist initial conditions such that the cancer burden goes to infinity as $t\to +{t}^{+},$ where $\right]{t}^{-},{t}^{+}\left[\text{\hspace{0.17em}}{t}^{-}<0,{t}^{+}>0$ is the domain of definition of the maximal integral curve $c\left(t\right)$ of the corresponding vector field.

In the present paper, we propose to combine quadrapeutics and immune therapy.

We have written a paper  on fundamental concepts in dynamics intended for researchers with a background in medicine.

There are several important monographs related to the present paper, see  -  .  is a mathematical model of miRNAs.    are papers on quadrapeutics.   are two papers of mine on cancer and mathematics.

2. The Multipeutics System

Consider the mass action kinetic system

$C+N\to 0$ (1)

$C+D\to 0$ (2)

$D+G\to M$ (3)

$M\to N$ (4)

$N\to 0$ (5)

$M\to 0$ (6)

$C+X\to 0$ (7)

$C\to 2C$ (8)

$C⇆0$ (9)

$D⇆0$ (10)

$G⇆0$ (11)

$C+{I}_{i}\to 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\cdots ,n$ (12)

${I}_{i}⇆0$ (13)

$n\in {ℕ}_{0}$ ( $n=0$ means that the reactions (12) and (13) are omitted). The complexes are $C\left(1\right)=C+N,$ $C\left(2\right)=0,$ $C\left(3\right)=C+D,$ $C\left(4\right)=D+G,$ $C\left(5\right)=M,$ $C\left(6\right)=N,$ $C\left(7\right)=C+X,$ $C\left(8\right)=C,$ $C\left(9\right)=2C,$ $C\left(10\right)=D,$ $C\left(11\right)=G,$ $C\left(10+2i\right)=C+{I}_{i},$ $C\left(11+2i\right)={I}_{i},i=1,\cdots ,n$ The variables here are C cancer cells, D lysosomal chemo therapeutic drug, G nanogold particles conjugated to an antibody panitumumab against the epidermal growth factor receptor EGFR, M mixed clusters in cancer cells, N nanobubbles and finally ${I}_{i}$ immune or chemo therapies. X, the x rays, is not a variable but a parameter, a positive real number. (1) says that nanobubbles kill cancer cells and (2) that the chemo therapeutic drug kills cancer cells. (3) means that drug and nano gold particles form mixed clusters in cancer cells and normal cells and the cluster size is the largest in cancer cells. The mixed clusters generate nano bubbles (4), when excited with a laser pulse. The laser threshold pulse decreases with cluster size and since the cluster size is the largest in cancer cells it is mainly in cancer cells that plasmonic nanobubbles kill the host cell. (7) is the killing of cancer cells by x rays. (5) and (6) are decay rates for N and M. (8) says that cancer cells proliferate rapidly. (9), (10) and (11) give birth and decay rates for cancer cells, chemotherapeutic drug and nano gold particles, respectively. (12) and (13) are immune or chemo therapies. The rate constant in (7) is ${\stackrel{˜}{k}}_{27}={k}_{27}X.$ The ODEs are, see 

${C}^{\prime }=-{k}_{21}C\cdot N-{k}_{23}C\cdot D+\left({k}_{98}-{k}_{28}-{k}_{27}X\right)C+{k}_{82}-\underset{i=1}{\overset{n}{\sum }}\text{ }\text{ }{k}_{2,10+2i}C\cdot {I}_{i}$ (14)

${D}^{\prime }=-{k}_{23}C\cdot D-{k}_{54}D\cdot G+{k}_{10,2}-{k}_{2,10}D$ (15)

${G}^{\prime }=-{k}_{54}D\cdot G+{k}_{11,2}-{k}_{2,11}G$ (16)

${M}^{\prime }={k}_{54}D\cdot G-\left({k}_{65}+{k}_{25}\right)M$ (17)

${N}^{\prime }=-{k}_{21}C\cdot N+{k}_{65}M-{k}_{26}N$ (18)

${{I}^{\prime }}_{i}=-{k}_{2,10+2i}C\cdot {I}_{i}+{k}_{11+2i,2}-{k}_{2,11+2i}{I}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\cdots ,n$ (19)

The corresponding vector field is denoted f. We are going to consider subsystems of (14) to (19). First of all the chemo radiation system, the vector field in (14) and (15) with the rate constants in (16) to (19) equal to zero and the rest positive. This vector field is denoted

${f}_{cr}:{ℝ}^{2}\to {ℝ}^{2}$ (20)

The quadrapeutics system (14) to (18) with the rate constants in (19) equal to zero and the rest positive. This vector field is denoted

${f}_{q}:{ℝ}^{5}\to {ℝ}^{5}$ (21)

Finally the multipeutics systems: the system (14) to (19) with the rate constants in (19) equal to zero, when $i=n-j+1,\cdots ,n$ and the rest positive. This vector field is denoted

${f}_{n-j}:{ℝ}^{n-j+5}\to {ℝ}^{n-j+5}$ (22)

$j\in \left\{0,\cdots ,n\right\}.$ Now introduce some important notation. A singular point of ${f}_{cr}$ is denoted ${c}_{cr}=\left({C}_{\ast }^{cr},{D}_{\ast }^{cr}\right).$ A singular point of ${f}_{q}$ is denoted ${c}_{q}=$ $\left({C}_{\ast }^{q},{D}_{\ast }^{q},{G}_{\ast }^{q},{M}_{\ast }^{q},{N}_{\ast }^{q}\right).$ A singular point of ${f}_{n-j}$ is denoted

${c}_{n-j}=\left({C}_{\ast }^{n-j},{D}_{\ast }^{n-j},{G}_{\ast }^{n-j},{M}_{\ast }^{n-j},{N}_{\ast }^{n-j},{I}_{\ast }^{1,n-j},\cdots ,{I}_{\ast }^{n-j,n-j}\right)$ (23)

We are going to compare the cancer burden for the different systems. In fact we shall show that there are stable equilibria of the different systems such that

${C}_{\ast }^{n}<{C}_{\ast }^{n-1}<\cdots <{C}_{\ast }^{1}<{C}_{\ast }^{q}<{C}_{\ast }^{cr}$ (24)

Looking from right to left, this says that the more treatments we apply the lower the cancer burden.

In Figure 1 we have plotted $C\left(t\right)$ versus t for the chemo radiation model and $C\left(t\right)$ versus t for the quadrapeutics model. All rate constants equal to 1, except ${k}_{21}=10,{k}_{65}=10,{k}_{11,2}=20.$ Also $X=1.$ This supports the proof of Theorem 1 that if you make the nanobubble kill rate ${k}_{21}$ sufficiently big, then quadrapeutics outperforms chemo radiation. I have iterated the Euler maps

${H}_{cr}\left(C,D\right)=\left(C,D\right)+h{f}_{cr}\left(C,D\right)$ (25)

${H}_{q}\left(C,D,G,M,N\right)=\left(C,D,G,M,N\right)+h{f}_{q}\left(C,D,G,M,N\right)$ (26)

where h is the step size and equal to 0.02. There are 1000 iterations. The numerical analysis indicated, that $C\left(t\right),D\left(t\right),G\left(t\right),M\left(t\right),N\left(t\right)$ for the quadrapeutics system converged, when t goes to infinity. And similarly for $C\left(t\right),D\left(t\right)$ of the chemo radiation model, see Figure 1 and Figure 2.

Theorem 1. Assume, that $\stackrel{^}{a}\triangleq {k}_{98}-{k}_{28}-{k}_{27}X<0.$ There exist positive values of the rate constants ${k}_{ij}$ , such that (24) holds and the singular points are all positive and stable.

Proof. Let K be defined by

${k}_{1}=\left({k}_{82},{k}_{10,2},{k}_{11,2},{\left\{{k}_{11+2i,2}\right\}}_{i=1,\cdots ,n}\right)$ (27)

${k}_{2}=\left({k}_{21},{k}_{23},{k}_{98},{k}_{28},{\stackrel{˜}{k}}_{27},{k}_{54},{k}_{2,10},{k}_{2,11},{k}_{65},{k}_{25},{k}_{26},{\left\{{k}_{2,10+2i},{k}_{2,11+2i}\right\}}_{i=1,\cdots ,n}\right)$ (28)

$K=\left({k}_{1},{k}_{2}\right)$ (29)

Figure 1. (r, c1) is C(t) versus t for the chemo radiation model and (r, c) is C(t) versus t for the quadrapeutics model.

Figure 2. (r, d1) is D(t) versus t for the chemo radiation system and (r, d) is D(t) versus t for the quadrapeutics system.

Define

${k}_{82}=t{k}_{82}^{0}$ (30)

${k}_{10,2}=t{k}_{10,2}^{0}$ (31)

${k}_{11,2}=t{k}_{11,2}^{0}$ (32)

${k}_{11+2i,2}=t{k}_{11+2i,2}^{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\cdots ,n\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in ℝ$ (33)

when $\ast =n$ . Here

${k}_{82}^{0},{k}_{10,2}^{0},{k}_{11,2}^{0},{k}_{11+2i,2}^{0}\in {ℝ}_{+}$ (34)

$i=1,\cdots ,n.$ Notice that $\left(0,\cdots ,0\right)$ is a singular point of f, when $t=0$ . The rate constants ${k}_{ij}$ in ${k}_{2}$ are positive real numbers. Also note that ${D}_{1}{f}_{*}{}_{0}$ is an isomorphism.

Here

$\ast =\left(\begin{array}{c}cr\\ q\\ 1\\ ⋮\\ n\end{array}$ (35)

For instance in f

${D}_{1}{f}_{0}={\left\{{b}_{ij}\right\}}_{i,j=1,\cdots ,n}$ (36)

is an isomorphism, where ${b}_{11}=\stackrel{^}{a},$ ${b}_{2,2}=-{k}_{2,10},$ ${b}_{3,3}=-{k}_{2,11},$ ${b}_{4,4}=-\left({k}_{65}+{k}_{25}\right),$ ${b}_{5,5}=-{k}_{26},$ ${b}_{5,4}={k}_{65},$ ${b}_{66}=-{k}_{2,13},\cdots ,{b}_{n+5,n+5}=-{k}_{2,11+2n},$ all other ${b}_{ij}=0$ .

So there exist ${C}^{k}$ maps, $k\ge 5,$

${c}_{*}:{V}_{*}\subset {ℝ}^{{p}_{*}}\to {ℝ}^{{s}_{*}}$ (37)

such that

${f}_{*}\left({c}_{*}\left({K}_{*}\right),{K}_{*}\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{*}\left({\stackrel{^}{K}}_{*}\right)=\left(0,\cdots ,0\right)$ (38)

by the implicit function theorem. ${V}_{*},{K}_{*},{\stackrel{^}{K}}_{*}$ are defined below.

Also

${p}_{cr}=7\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{s}_{cr}=2$ (39)

${p}_{q}=14\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{s}_{q}=5$ (40)

${p}_{n-j}=14+3\left(n-j\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{s}_{n-j}=5+n-j$ (41)

For $\ast =cr$ define

${k}_{1}^{cr}=\left({k}_{82},{k}_{10,2}\right)$ (42)

and

${k}_{2}^{cr}=\left({k}_{23},{k}_{98},{k}_{28},{\stackrel{˜}{k}}_{27},{k}_{2,10}\right)$ (43)

For $\ast =q$ we define

${k}_{1}^{q}=\left({k}_{82},{k}_{10,2},{k}_{11,2}\right)$ (44)

and

${k}_{2}^{q}=\left({k}_{2}^{cr},{k}_{21},{k}_{2,11},{k}_{54},{k}_{65},{k}_{25},{k}_{26}\right)$ (45)

Finally for the multipeutics system

${k}_{1}^{n-j}=\left({k}_{82},{k}_{10,2},{k}_{11,2},{k}_{11+2i,2}{}_{i=1,\cdots ,n-j}\right)$ (46)

and

${k}_{2}^{n-j}=\left({k}_{2}^{q},{\left({k}_{2,10+2i},{k}_{2,11+2i}\right)}_{i=1,\cdots ,n-j}\right)$ (47)

Now let ${\stackrel{^}{K}}_{cr}=\left({k}_{1}^{cr},{k}_{2}^{cr}\right),{k}_{1}^{cr}=0$ and ${\stackrel{^}{K}}_{q}=\left({k}_{1}^{q},{k}_{2}^{q}\right),{k}_{1}^{q}=0,$ and finally ${\stackrel{^}{K}}_{n-j}=\left({k}_{1}^{n-j},{k}_{2}^{n-j}\right),{k}_{1}^{n-j}=0.$ ${V}_{*}$ above is an open subset containing ${\stackrel{^}{K}}_{*}$ in ${ℝ}^{p*}$ . Define ${K}_{*}=\left({k}_{1}^{*},{k}_{2}^{*}\right).$ Then

${f}_{*}\left({c}_{*}\left(t{k}_{1}^{*,0},{k}_{2}^{*}\right),t{k}_{1}^{*,0},{k}_{2}^{*}\right)=0$ (48)

Define

${d}_{*}\left(t\right)={c}_{*}\left(t{k}_{1}^{*,0},{k}_{2}^{*}\right)$ (49)

For the chemo radiation model we let

${d}_{*}^{G}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{d}_{*}^{M}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{d}_{*}^{N}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{d}_{*}^{{I}_{i}}=0$ (50)

$i=1,\cdots ,n.$ For the quadrapeutics model we let ${d}_{*}^{{I}_{i}}=0,i=1,\cdots ,n.$ And for the multipeutics model we let

${d}_{*}^{{I}_{i}}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=n-j+1,\cdots ,n\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}j\in \left\{0,\cdots ,n\right\}$ (51)

Differentiate (48) with respect to t to get

${D}_{1}f\left(\frac{\partial {d}_{*}}{\partial t}\right)+\frac{\partial f}{\partial t}=0$ (52)

Thus

$\frac{\partial {d}_{*}^{C}}{\partial t}\left(0\right)=-\frac{1}{\stackrel{^}{a}}{k}_{82}^{0}$ (53)

regardless of the system. We have

#Math_123# (54)

Differentiate this with respect to t to get

$-{k}_{21}\left(\frac{\partial {d}_{*}^{C}}{\partial t}{d}_{*}^{N}+{d}_{*}^{C}\frac{\partial {d}_{*}^{N}}{\partial t}\right)-{k}_{23}\left(\frac{\partial {d}_{*}^{C}}{\partial t}{d}_{*}^{D}+{d}_{*}^{C}\frac{\partial {d}_{*}^{D}}{\partial t}\right)$ (55)

$+\stackrel{^}{a}\frac{\partial {d}_{*}^{C}}{\partial t}+{k}_{82}^{0}-\underset{i=1}{\overset{n}{\sum }}\text{ }\text{ }{k}_{2,10+2i}\left(\frac{\partial {d}_{*}^{C}}{\partial t}{d}_{*}^{{I}_{i}}+{d}_{*}^{C}\frac{\partial {d}_{*}^{{I}_{i}}}{\partial t}\right)=0$ (56)

Also differentiate this equation with respect to t to find

$-{k}_{21}\left(\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}{d}_{*}^{N}+2\frac{\partial {d}_{*}^{C}}{\partial t}\frac{\partial {d}_{*}^{N}}{\partial t}+{d}_{*}^{C}\frac{{\partial }^{2}{d}_{*}^{N}}{\partial {t}^{2}}\right)$ (57)

$-{k}_{23}\left(\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}{d}_{*}^{D}+2\frac{\partial {d}_{*}^{C}}{\partial t}\frac{\partial {d}_{*}^{D}}{\partial t}+{d}_{*}^{C}\frac{{\partial }^{2}{d}_{*}^{D}}{\partial {t}^{2}}\right)$ (58)

$-\underset{i=1}{\overset{n}{\sum }}\text{ }\text{ }{k}_{2,10+2i}\left(\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}{d}_{*}^{{I}_{i}}+2\frac{\partial {d}_{*}^{C}}{\partial t}\frac{\partial {d}_{*}^{{I}_{i}}}{\partial t}+{d}_{*}^{C}\frac{{\partial }^{2}{d}_{*}^{{I}_{i}}}{\partial {t}^{2}}\right)$ (59)

$+\stackrel{^}{a}\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}=0$ (60)

From this equation it follows that

$\frac{{\partial }^{2}{d}_{cr}^{C}}{\partial {t}^{2}}\left(0\right)=\frac{{\partial }^{2}{d}_{q}^{C}}{\partial {t}^{2}}\left(0\right)>\frac{{\partial }^{2}{d}_{1}^{C}}{\partial {t}^{2}}\left(0\right)>\cdots >\frac{{\partial }^{2}{d}_{n}^{C}}{\partial {t}^{2}}\left(0\right)$ (61)

anticipating that

${d}_{*}^{N}\left(0\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {d}_{*}^{N}}{\partial t}\left(0\right)=0$ (62)

Observe that

$\frac{\partial {d}_{*}^{C}}{\partial t}\left(0\right)=-\frac{{k}_{82}^{0}}{\stackrel{^}{a}}>0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {d}_{*}^{D}}{\partial t}\left(0\right)=\frac{{k}_{10,2}^{0}}{{k}_{2,10}}>0$ (63)

$\frac{\partial {d}_{*}^{G}}{\partial t}\left(0\right)=\frac{{k}_{11,2}^{0}}{{k}_{2,11}}>0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {d}_{*}^{{I}_{i}}}{\partial t}\left(0\right)=\frac{{k}_{11+2i,2}^{0}}{{k}_{2,11+2i}}>0$ (64)

$i=1,\cdots ,n.$ The third equality holds for $\ast =q,1,\cdots ,n.$ The fourth equality holds for the multipeutics system ${f}_{n-j},i=1,\cdots ,n-j,j\in \left\{0,1,\cdots ,n\right\}.$

By the fundamental theorem of calculus applied twice

${d}_{*}^{C}\left(t\right)={\int }_{0}^{t}\left({\int }_{0}^{s}\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}\left(v\right)\text{d}v+\frac{\partial {d}_{*}^{C}}{\partial t}\left(0\right)\right)\text{d}s$ (65)

So

${d}_{q}^{C}\left(t\right)>{d}_{1}^{C}\left(t\right)>\cdots >{d}_{n}^{C}\left(t\right)$ (66)

To see the first inequality, observe that there exists a positive constant c, such that

${d}_{q}^{C}\left(t\right)-{d}_{1}^{C}\left(t\right)={\int }_{0}^{t}{\int }_{0}^{s}\left(\frac{{\partial }^{2}{d}_{q}^{C}}{\partial {t}^{2}}\left(v\right)-\frac{{\partial }^{2}{d}_{1}^{C}}{\partial {t}^{2}}\left(v\right)\right)\text{d}v\text{d}s\ge \frac{1}{2}c{t}^{2}>0$ (67)

because a positive function on a compact interval $\left[0,\delta \right]$ assumes a minimum $c>0$ . The other inequalities follow in the same way. And this proves all the inequalities in (24) except ${C}_{*}^{q}<{C}_{*}^{cr}$ . Henceforth assume

${k}_{2,10+2i}={k}_{11+2i,2}={k}_{2,11+2i}=0$ (68)

$i=1,\cdots ,n.$ Finally differentiate (57) to (60) with respect to t to get

$-3{k}_{21}\left(\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}\frac{\partial {d}_{*}^{N}}{\partial t}+\frac{\partial {d}_{*}^{C}}{\partial t}\frac{{\partial }^{2}{d}_{*}^{N}}{\partial {t}^{2}}\right)$ (69)

$-3{k}_{23}\left(\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}\frac{\partial {d}_{*}^{D}}{\partial t}+\frac{\partial {d}_{*}^{C}}{\partial t}\frac{{\partial }^{2}{d}_{*}^{D}}{\partial {t}^{2}}\right)$ (70)

$+\stackrel{^}{a}\frac{{\partial }^{3}{d}_{*}^{C}}{\partial {t}^{3}}=0$ (71)

when $t=0.$ But

$\frac{\partial {d}_{*}^{M}}{\partial t}\left(0\right)=\frac{\partial {d}_{*}^{N}}{\partial t}\left(0\right)=0$ (72)

However when $\ast =q$

$\frac{{\partial }^{2}{d}_{*}^{N}}{\partial {t}^{2}}\left(0\right)>0$ (73)

To see this differentiate (17) twice with respect to t to find

$\left({k}_{65}+{k}_{25}\right)\frac{{\partial }^{2}{d}_{*}^{M}}{\partial {t}^{2}}\left(0\right)=2{k}_{54}\frac{\partial {d}_{*}^{D}}{\partial t}\left(0\right)\frac{\partial {d}_{*}^{G}}{\partial t}\left(0\right)>0$ (74)

If we now differentiate (18) twice with respect to t we find

${k}_{26}\frac{{\partial }^{2}{d}_{*}^{N}}{\partial {t}^{2}}\left(0\right)={k}_{65}\frac{{\partial }^{2}{d}_{*}^{M}}{\partial {t}^{2}}\left(0\right)>0$ (75)

which is what we sought to show.

Furthermore

$\left({k}_{65}+{k}_{25}\right){d}_{*}^{M}\left(t\right)={k}_{54}{d}_{*}^{D}\cdot {d}_{*}^{G}\left(t\right)>0$ (76)

and

${d}_{*}^{N}\left(t\right)=\frac{{k}_{65}{d}_{*}^{M}\left(t\right)}{{k}_{21}{d}_{*}^{C}\left(t\right)+{k}_{26}}>0$ (77)

when $t>0$ and small. Differentiate (15) twice with respect to t when $\ast =cr,q$ to get

$-{k}_{2,10}\frac{{\partial }^{2}{d}_{*}^{D}}{\partial {t}^{2}}=2{k}_{23}\frac{\partial {d}_{*}^{C}}{\partial t}\frac{\partial {d}_{*}^{D}}{\partial t}+2{k}_{54}\frac{\partial {d}_{*}^{D}}{\partial t}\frac{\partial {d}_{*}^{G}}{\partial t}$ (78)

Now

$\stackrel{^}{a}\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}=2{k}_{23}\frac{\partial {d}_{*}^{C}}{\partial t}\frac{\partial {d}_{*}^{D}}{\partial t}$ (79)

Also

$\stackrel{^}{a}\frac{{\partial }^{3}{d}_{*}^{C}}{\partial {t}^{3}}\left(0\right)=3{k}_{21}\left(\frac{\partial {d}_{*}^{C}}{\partial t}\frac{{k}_{65}{k}_{54}2}{{k}_{26}\left({k}_{65}+{k}_{25}\right)}\frac{\partial {d}_{*}^{D}}{\partial t}\frac{\partial {d}_{*}^{G}}{\partial t}\right)$ (80)

$+3{k}_{23}\frac{1}{\stackrel{^}{a}}\left(2{k}_{23}\frac{\partial {d}_{*}^{C}}{\partial t}\frac{\partial {d}_{*}^{D}}{\partial t}\right)\frac{\partial {d}_{*}^{D}}{\partial t}$ (81)

$-3{k}_{23}\frac{1}{{k}_{2,10}}\frac{\partial {d}_{*}^{C}}{\partial t}\left(2{k}_{23}\frac{\partial {d}_{*}^{C}}{\partial t}\frac{\partial {d}_{*}^{D}}{\partial t}+2{k}_{54}\frac{\partial {d}_{*}^{D}}{\partial t}\frac{\partial {d}_{*}^{G}}{\partial t}\right)$ (82)

where we have evaluated in $t=0$ on the right hand side. So if

$3\frac{{k}_{21}{k}_{65}2}{{k}_{26}\left({k}_{65}+{k}_{25}\right)}-\frac{3{k}_{32}2}{{k}_{2,10}}>0$ (83)

we have

$\frac{{\partial }^{3}{d}_{cr}^{C}}{\partial {t}^{3}}\left(t\right)>\frac{{\partial }^{3}{d}_{q}^{C}}{\partial {t}^{3}}\left(t\right)$ (84)

when $t=0$ and hence also for small $t>0.$ The fundamental theorem of calculus applied three times gives

${d}_{*}^{C}\left(t\right)={\int }_{0}^{t}\left({\int }_{0}^{s}\left({\int }_{0}^{v}\frac{{\partial }^{3}{d}_{*}^{C}}{\partial {t}^{3}}\left(u\right)\text{d}u+\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}\left(0\right)\right)\text{d}v+\frac{\partial {d}_{*}^{C}}{\partial t}\left(0\right)\right)\text{d}s$ (85)

So

${d}_{cr}^{C}\left(t\right)>{d}_{q}^{C}\left(t\right)$ (86)

for small $t>0$ and the theorem follows in the same way as (67). Define the matrices

$B=D{f}_{*}{}_{{c}_{*}}-\lambda \text{\hspace{0.17em}}\text{id}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}A=D{f}_{*}{}_{{c}_{*}}$ (87)

and then write the definition of the determinant

$\mathrm{det}B=\underset{\sigma \in {S}^{p}}{\sum }{\left(-1\right)}^{I\left(\sigma \right)}{B}_{1,\sigma \left(1\right)}\cdots {B}_{p,\sigma \left(p\right)}$ (88)

where ${S}^{p}$ is the set of permutations of $\left\{1,\cdots ,p\right\}$ and $I\left(\sigma \right)=0$ if $\sigma$ is even and $I\left(\sigma \right)=1,$ when $\sigma$ is odd. This formula shows immediately that

$\mathrm{det}B={\left(-1\right)}^{p}{\lambda }^{p}+\underset{i=0}{\overset{p-1}{\sum }}\text{ }\text{ }{a}_{i}\left({A}_{11},\cdots ,{A}_{pp}\right){\lambda }^{i}$ (89)

where ${a}_{i}$ is a polynomial in ${A}_{11},\cdots ,{A}_{pp},$ and hence is smooth in ${A}_{11},\cdots ,{A}_{pp}.$ You can prove this by induction. Now apply the continuous dependence of roots of a polynomial on its coefficients to show that ${d}_{*}\left(t\right)$ is stable for ${f}_{*}$ , when t is small, see  . □

Remark. If the quadrapeutics chemo rate is

${k}_{10,2}^{q}={k}_{10,2}^{cr}\stackrel{˜}{\alpha }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{˜}{\alpha }\in \right]0,1\left[$ (90)

and the x ray rate is

${k}_{27}^{q}X={k}_{27}^{cr}X\stackrel{˜}{\beta }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{˜}{\beta }\in \right]0,1\left[$ (91)

where

${k}_{10,2}^{q}=t{k}_{10,2}^{q,0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{k}_{10,2}^{cr}=t{k}_{10,2}^{cr,0}$ (92)

while $\stackrel{^}{a}={k}_{98}-{k}_{28}-{k}_{27}X<0.$ Here

${k}_{10,2}^{q,0},{k}_{10,2}^{cr,0}\in {ℝ}_{+}$ (93)

Then by taking ${k}_{21}$ big we find

$\frac{{\partial }^{3}{d}_{q}^{C}}{\partial {t}^{3}}\left(0\right)<\frac{{\partial }^{3}{d}_{cr}^{C}}{\partial {t}^{3}}\left(0\right)$ (94)

since

$\stackrel{^}{a}\frac{{\partial }^{3}{d}_{q}^{C}}{\partial {t}^{3}}\left(0\right)\to +\infty$ (95)

as ${k}_{21}\to +\infty .$ Hence

${d}_{q}^{C}\left(t\right)<{d}_{cr}^{C}\left(t\right)$ (96)

for small $t>0,$ when we impose (90) and (91), arguing as in the proof of theorem 1.

Experimentally this is what you see, that quadrapeutics can outperform chemo radiation even if we take the chemo radiation doses smaller, see     .

Consider also the system (1) to (13) with

$2C+X\to 0$ (97)

replacing (7). The ODE s are the same (15) to (19) except (14), which is

${C}^{\prime }=-{k}_{21}C\cdot N-{k}_{23}C\cdot D+\left({k}_{98}-{k}_{28}\right)C+{k}_{82}-2{k}_{27}X\cdot {C}^{2}-\underset{i=1}{\overset{n}{\sum }}\text{ }\text{ }{k}_{2,10+2i}C\cdot {I}_{i}$ (98)

We are now going to find candidates of the positive singular points of the different systems. We start with f and assume that all ${k}_{ij}$ are positive. (19) gives

${I}_{i}=\frac{{k}_{11+2i,2}}{{k}_{2,10+2i}C+{k}_{2,11+2i}}$ (99)

when $C>0$ and (18) gives

$N=\frac{{k}_{65}M}{{k}_{21}C+{k}_{26}}$ (100)

(16) and (17) give

${k}_{11,2}-{k}_{2,11}G-\left({k}_{65}+{k}_{25}\right)M=0$ (101)

or

$G=\frac{1}{{k}_{2,11}}\left({k}_{11,2}-\left({k}_{65}+{k}_{25}\right)M\right)=\alpha M+\beta$ (102)

Also

$D=\frac{{k}_{10,2}}{{k}_{23}C+{k}_{54}G+{k}_{2,10}}$ (103)

Insert these formulas in (14)

$-\text{\hspace{0.17em}}{k}_{21}C\frac{{k}_{65}M}{{k}_{21}C+{k}_{26}}-{k}_{23}C\frac{{k}_{10,2}}{{k}_{23}C+{k}_{54}G+{k}_{2,10}}$ (104)

$+\left({k}_{98}-{k}_{28}-{k}_{27}X\left(1+\delta \left(2C-1\right)\right)\right)C$ (105)

$-\text{\hspace{0.17em}}\underset{i=1}{\overset{n}{\sum }}\text{ }\text{ }{k}_{2,10+2i}C\frac{{k}_{11+2i,2}}{{k}_{2,10+2i}C+{k}_{2,11+2i}}+{k}_{82}=0$ (106)

$\delta =0,1$ . Insert the formula for D in ${M}^{\prime }=0$ to get

${k}_{54}G\frac{{k}_{10,2}}{{k}_{23}C+{k}_{54}G+{k}_{2,10}}-\left({k}_{65}+{k}_{26}\right)M=0$ (107)

We can isolate C in this equation to find

$C=\frac{a{M}^{2}+bM+c}{M}$ (108)

where

$a=\left({k}_{65}+{k}_{25}\right)\frac{{k}_{54}}{{k}_{2,11}{k}_{23}}$ (109)

and

$b=-\frac{{k}_{54}{k}_{10,2}}{{k}_{23}{k}_{2,11}}-\left(\frac{{k}_{54}}{{k}_{2,11}{k}_{23}}{k}_{11,2}+\frac{{k}_{2,10}}{{k}_{23}}\right)$ (110)

Finally

$c=\frac{{k}_{54}{k}_{10,2}{k}_{11,2}}{{k}_{23}{k}_{2,11}\left({k}_{65}+{k}_{25}\right)}$ (111)

We now get by multiplying with the product of the denominators in (104), (105) and (106)

$-\text{\hspace{0.17em}}{k}_{21}C{k}_{65}M\left({k}_{23}C+{k}_{54}G+{k}_{2,10}\right)\underset{i=1}{\overset{n}{\prod }}\left({k}_{2,10+2i}C+{k}_{2,11+2i}\right)$ (112)

$-\text{\hspace{0.17em}}{k}_{23}C{k}_{10,2}\left({k}_{21}C+{k}_{26}\right)\underset{i=1}{\overset{n}{\prod }}\left({k}_{2,10+2i}C+{k}_{2,11+2i}\right)$ (113)

$+\left({k}_{82}+\left({k}_{98}-{k}_{28}-{k}_{27}X\left(1+\delta \left(2C-1\right)\right)\right)C\right)\left({k}_{21}C+{k}_{26}\right)$ (114)

$\cdot \left({k}_{23}C+{k}_{54}G+{k}_{2,10}\right)\underset{i=1}{\overset{n}{\prod }}\left({k}_{2,10+2i}C+{k}_{2,11+2i}\right)$ (115)

$-\text{\hspace{0.17em}}\underset{i=1}{\overset{n}{\sum }}\text{ }\text{ }{k}_{2,10+2i}C{k}_{11+2i,2}\left(\underset{j=1,j\ne i}{\overset{n}{\prod }}\left({k}_{2,10+2j}C+{k}_{2,11+2j}\right)\right)$ (116)

$\cdot \left({k}_{23}C+{k}_{54}G+{k}_{2,10}\right)\left({k}_{21}C+{k}_{26}\right)=0$ (117)

If we now insert the formulas for $G,C$ and multiply with ${M}^{n+3},\delta =0$ or ${M}^{n+4},\delta =1,$ we get a polynomial p of degree at most $2\left(n+3\right)$ $\left(\delta =0\right)$ or $2\left(n+4\right)$ $\left(\delta =1\right)$ in M, such that a positive singular point

$\left(C,D,G,M,N,{I}_{1},\cdots ,{I}_{n}\right)$ (118)

will have $p\left(M\right)=0.$

Let $\delta =1.$ Then we have

Theorem 2. For $\delta =1,$ there exist rate constants ${k}_{ij}>0,$ such that (24) holds and all equilibria are positive and stable.

Proof. Now let

$\stackrel{^}{a}={k}_{98}-{k}_{28}$ (119)

and assume that it is negative. Then

$\frac{\partial {d}_{*}^{C}}{\partial t}\left(0\right)=-\frac{{k}_{82}^{0}}{\stackrel{^}{a}}$ (120)

is the same for all treatments. But now the first coordinate of formula (48) becomes

$-{k}_{21}{d}_{*}^{C}{d}_{*}^{N}-{k}_{23}{d}_{*}^{C}{d}_{*}^{D}+\stackrel{^}{a}{d}_{*}^{C}-2{k}_{27}X{\left({d}_{*}^{C}\right)}^{2}+{k}_{82}-\underset{i=1}{\overset{n}{\sum }}\text{ }\text{ }{k}_{2,10+2i}{d}_{*}^{C}{d}_{*}^{{I}_{i}}=0$ (121)

So

$\stackrel{^}{a}\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}\left(0\right)={k}_{21}\left(\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}{d}_{*}^{N}+2\frac{\partial {d}_{*}^{C}}{\partial t}\frac{\partial {d}_{*}^{N}}{\partial t}+{d}_{*}^{C}\frac{{\partial }^{2}{d}_{*}^{N}}{\partial t}\right)$ (122)

$+\text{\hspace{0.17em}}{k}_{23}\left(\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}{d}_{*}^{D}+2\frac{\partial {d}_{*}^{C}}{\partial t}\frac{\partial {d}_{*}^{D}}{\partial t}+{d}_{*}^{C}\frac{{\partial }^{2}{d}_{*}^{D}}{\partial t}\right)$ (123)

$+\underset{i=1}{\overset{n}{\sum }}\text{ }\text{ }{k}_{2,10+2i}\left(\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}{d}_{*}^{{I}_{i}}+2\frac{\partial {d}_{*}^{C}}{\partial t}\frac{\partial {d}_{*}^{{I}_{i}}}{\partial t}+{d}_{*}^{C}\frac{{\partial }^{2}{d}_{*}^{{I}_{i}}}{\partial t}\right)$ (124)

$+\text{\hspace{0.17em}}4{k}_{27}X{\left(\frac{\partial {d}_{*}^{C}}{\partial t}\right)}^{2}$ (125)

and when ${k}_{2,10+2i}={k}_{2,11+2i}={k}_{11+2i,2}=0,i=1,\cdots ,n.$

$\stackrel{^}{a}\frac{{\partial }^{3}{d}_{*}^{C}}{\partial {t}^{3}}\left(0\right)=3{k}_{21}\left(\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}\frac{\partial {d}_{*}^{N}}{\partial t}+\frac{\partial {d}_{*}^{C}}{\partial t}\frac{{\partial }^{2}{d}_{*}^{N}}{\partial {t}^{2}}\right)$ (126)

$+\text{\hspace{0.17em}}3{k}_{23}\left(\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}\frac{\partial {d}_{*}^{D}}{\partial t}+\frac{\partial {d}_{*}^{C}}{\partial t}\frac{{\partial }^{2}{d}_{*}^{D}}{\partial {t}^{2}}\right)$ (127)

$+12{k}_{27}X\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}\frac{\partial {d}_{*}^{C}}{\partial t}$ (128)

We can now take ${k}_{21}$ big and argue as in theorem 1. □

Consider the mass action kinetic system

$C+D\to 0$ (129)

$\left(1+\delta \right)C+X\to 0,\delta =0,1$ (130)

$C⇆2C$ (131)

$C⇆0$ (132)

$D⇆0$ (133)

Here the complexes are $C\left(2\right)=0,$ $C\left(3\right)=C+D,$ $C\left(7\right)=\left(1+\delta \right)C+X,$ $C\left(8\right)=C,$ $C\left(9\right)=2C,$ $C\left(10\right)=D.$ With mass action kinetics the vector field is

$f\left(C,D\right)=\left(\begin{array}{c}{k}_{82}-{k}_{23}C\cdot D+\left({k}_{98}-{k}_{28}-{k}_{27}X\left(1+\delta \left(2C-1\right)\right)\right)C\\ -{k}_{23}C\cdot D+{k}_{10,2}-{k}_{2,10}D\end{array}\right)$ (134)

all ${k}_{ij}>0.$

This system with $\delta =0$ is similar to the reduced system from  .

$GF\to C$ (135)

$C+GI\to 0$ (136)

$C\to 2C$ (137)

$GF⇆0$ (138)

$GI⇆0$ (139)

$C⇆0$ (140)

where we have added the last reaction. The complexes are $C\left(1\right)=GF,$ $C\left(2\right)=C,$ $C\left(3\right)=C+GI,$ $C\left(4\right)=0,$ $C\left(5\right)=2C,$ $C\left(6\right)=GI.$ The vector field is

$g\left(C,GF,GI\right)=\left(\begin{array}{c}{k}_{21}GF-{k}_{43}C\cdot GI+\left({k}_{52}-{k}_{42}\right)C+{k}_{24}\\ {k}_{14}-\left({k}_{21}+{k}_{41}\right)GF\\ -{k}_{43}C\cdot GI+{k}_{64}-{k}_{46}GI\end{array}\right)$ (141)

We can assume that $GF$ is at equilibrium

$GF=\frac{{k}_{14}}{{k}_{21}+{k}_{41}}$ (142)

The reduced system is then

$\stackrel{^}{g}\left(C,GI\right)=\left(\begin{array}{c}k-{k}_{43}C\cdot GI+\left({k}_{52}-{k}_{42}\right)C\\ -{k}_{43}C\cdot GI+{k}_{64}-{k}_{46}GI\end{array}\right)$ (143)

$k=\frac{{k}_{21}{k}_{14}}{{k}_{21}+{k}_{41}}+{k}_{24}$ (144)

We shall find the singular points of the chemo radiation system. ${D}^{\prime }=0$ gives

$D=\frac{{k}_{10,2}}{{k}_{23}C+{k}_{2,10}}$ (145)

when $C>0$ and then ${C}^{\prime }=0$ amounts to

${k}_{82}-{k}_{23}C\frac{{k}_{10,2}}{{k}_{23}C+{k}_{2,10}}+\left({k}_{98}-{k}_{28}-{k}_{27}X\left(1+\delta \left(2C-1\right)\right)\right)C=0$ (146)

which is equivalent to

${k}_{23}\left({k}_{98}-{k}_{28}-{k}_{27}X\right){C}^{2}$ (147)

$+\left({k}_{2,10}\left({k}_{98}-{k}_{28}-{k}_{27}X\right)+{k}_{23}{k}_{82}-{k}_{23}{k}_{10,2}\right)C+{k}_{82}{k}_{2,10}=0$ (148)

when $\delta =0,C>0$ and when $\delta =1,C>0$ it is equivalent to

$-2{k}_{23}{k}_{27}X{C}^{3}+\left(\left({k}_{98}-{k}_{28}\right){k}_{23}-2{k}_{27}X{k}_{2,10}\right){C}^{2}$ (149)

$+\left({k}_{2,10}\left({k}_{98}-{k}_{28}\right)+{k}_{82}{k}_{23}-{k}_{23}{k}_{10,2}\right)C+{k}_{82}{k}_{2,10}=0$ (150)

The linearization of f at a singular point ${c}_{*}=\left(C,D\right)$ is

$A=D{f}_{{c}_{*}}=\left(\begin{array}{cc}-{k}_{23}D+\left({k}_{98}-{k}_{28}-{k}_{27}X\right)& -{k}_{23}C\\ -{k}_{23}D& -{k}_{2,10}-{k}_{23}C\end{array}\right)$ (151)

when $\delta =0$ and when $\delta =1$ it is

$\stackrel{˜}{A}=D{f}_{{c}_{*}}=\left(\begin{array}{cc}-{k}_{23}D+\left({k}_{98}-{k}_{28}-4{k}_{27}XC\right)& -{k}_{23}C\\ -{k}_{23}D& -{k}_{2,10}-{k}_{23}C\end{array}\right)$ (152)

If $\delta =0$

$\stackrel{^}{a}={k}_{98}-{k}_{28}-{k}_{27}X<0$ (153)

and $\left(C,D\right)$ is a positive singular point, then

$\sigma =\text{trace}\text{ }A<0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{det}A=-\stackrel{^}{a}{k}_{23}C+{k}_{2,10}\left({k}_{23}D-\stackrel{^}{a}\right)>0$ (154)

hence this is a stable, positive singular point.

Consider the chemo radiation system where

${k}_{10,2}=23\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{k}_{98}=13\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}X=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\delta =1$ (155)

and all other ${k}_{ij}=1.$ Then the cubic polynomial giving candidates of singular points is

$-2{C}^{3}+10{C}^{2}-10C+1=0$ (156)

and this gives three singular points

$\left(C,D\right)=\left(\begin{array}{l}\left(0.112,20.68\right)\hfill \\ \left(1.21,10.41\right)\hfill \\ \left(3.677,4.918\right)\hfill \end{array}$ (157)

where we have used that

$D=\frac{{k}_{10,2}}{{k}_{23}C+{k}_{2,10}}$ (158)

It is a simple matter to check that the first and last of these are stable equilibria and the middle is an unstable saddle by computing the trace and determinant of the linearization of ${f}_{q}$ in the singular points. I have plotted a phase portrait for these values of the parameters in Figure 3.

We shall find the quadrapeutics equilibrium equation too.

Theorem 3. Suppose $\delta =0.$ There exists a polynomial of degree atmost six

$s\left(M\right)=\underset{i=0}{\overset{6}{\sum }}\text{ }\text{ }{q}_{i}{M}^{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{i}\in ℝ$ (159)

such that if $\left(C,D,G,M,N\right)$ is a positive singular point of ${f}_{q}$ then

$s\left(M\right)=0$ (160)

Figure 3. A tristable chemo radiation system. There are three singular points marked with a circle.

Proof. Isolate D, G, M, N from (15), (16), (17), (18) and insert it in ${f}_{q}{}_{1}\left(C,D,G,M,N\right)=0$ to find

$p\left(C,G,M\right)=-{k}_{21}C{k}_{65}M\left({k}_{23}C+{k}_{54}G+{k}_{2,10}\right)$ (161)

$-{k}_{23}C{k}_{10,2}\left({k}_{21}C+{k}_{26}\right)$ (162)

$+\left({k}_{82}+\left({k}_{98}-{k}_{28}-{k}_{27}X\left(1+\delta \left(2C-1\right)\right)\right)C\right)$ (163)

$\cdot \left({k}_{21}C+{k}_{26}\right)\left({k}_{23}C+{k}_{54}G+{k}_{2,10}\right)=0$ (164)

where we have multiplied with

$\left({k}_{21}C+{k}_{26}\right)\left({k}_{23}C+{k}_{54}G+{k}_{2,10}\right)$ (165)

We shall now insert

$C=\frac{a{M}^{2}+bM+c}{M}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}G=\alpha M+\beta$ (166)

where

$\alpha =-\frac{{k}_{65}+{k}_{25}}{{k}_{2,11}}$ (167)

$\beta =\frac{{k}_{11,2}}{{k}_{2,11}}$ (168)

in the polynomial p. Compute

${M}^{3}{C}^{3}=\underset{i=0}{\overset{6}{\sum }}\text{ }\text{ }{\stackrel{˜}{a}}_{i}{M}^{i}$ (169)

where

${\stackrel{˜}{a}}_{6}={a}^{3}$ (170)

${\stackrel{˜}{a}}_{5}=3{a}^{2}b$ (171)

${\stackrel{˜}{a}}_{4}=3\left(a{b}^{2}+{a}^{2}c\right)$ (172)

${\stackrel{˜}{a}}_{3}=6abc+{b}^{3}$ (173)

${\stackrel{˜}{a}}_{2}=3\left(a{c}^{2}+{b}^{2}c\right)$ (174)

${\stackrel{˜}{a}}_{1}=3b{c}^{2}$ (175)

${\stackrel{˜}{a}}_{0}={c}^{3}$ (176)

Now define

${p}_{1}\left(C,G,M\right)={M}^{3}\left(-{k}_{21}C{k}_{65}M\left({k}_{23}C+{k}_{54}G+{k}_{2,10}\right)\right)$ (177)

and this is

${s}_{1}\left(M\right)={p}_{1}\left(\frac{a{M}^{2}+bM+c}{M},\alpha M+\beta ,M\right)$ (178)

Also put

${p}_{2}\left(C,G,M\right)={M}^{3}\left(-{k}_{23}C{k}_{10,2}\left({k}_{21}C+{k}_{26}\right)\right)$ (179)

and this is

${s}_{2}\left(M\right)={p}_{2}\left(\frac{a{M}^{2}+bM+c}{M},\alpha M+\beta ,M\right)$ (180)

Finally set

${p}_{3}\left(C,G,M\right)={M}^{3}\left({k}_{82}+\stackrel{^}{a}C\right)\left({k}_{21}C+{k}_{26}\right)\left({k}_{23}C+{k}_{54}G+{k}_{2,10}\right)$ (181)

which is

${s}_{3}\left(M\right)={p}_{3}\left(\frac{a{M}^{2}+bM+c}{M},\alpha M+\beta ,M\right)$ (182)

So

$s\left(M\right)={s}_{1}\left(M\right)+{s}_{2}\left(M\right)+{s}_{3}\left(M\right)$ (183)

Now compute

#Math_332# (184)

$-\text{\hspace{0.17em}}{k}_{21}\left(a{M}^{2}+bM+c\right){k}_{65}{M}^{3}\left({k}_{54}\left(\alpha M+\beta \right)\right)+{k}_{2,10}\right)$ (185)

$=-{k}_{21}{k}_{65}{k}_{23}\left({a}^{2}{M}^{6}+{b}^{2}{M}^{4}+{c}^{2}{M}^{2}+2ab{M}^{5}+2ac{M}^{4}+2bc{M}^{3}\right)$ (186)

$\begin{array}{l}-\text{\hspace{0.17em}}{k}_{21}{k}_{65}\left(a{M}^{2}+bM+c\right){M}^{3}{k}_{54}\alpha M\\ -{k}_{21}{k}_{65}\left(a{M}^{2}+bM+c\right){M}^{3}\left({k}_{54}\beta +{k}_{2,10}\right)\end{array}$ (187)

Write

${s}_{1}\left(M\right)=\underset{i=0}{\overset{6}{\sum }}\text{ }\text{ }{a}_{i}{M}^{i}$ (188)

Then by the previous computation

${a}_{6}=-{k}_{21}a{k}_{65}\left({k}_{23}a+{k}_{54}\alpha \right)$ (189)

${a}_{5}=-{k}_{21}b{k}_{65}\left({k}_{23}a+{k}_{54}\alpha \right)-{k}_{21}a{k}_{65}\left({k}_{54}\beta +{k}_{23}b+{k}_{2,10}\right)$ (190)

${a}_{4}=-{k}_{21}c{k}_{65}\left({k}_{23}a+{k}_{54}\alpha \right)-{k}_{21}b{k}_{65}\left({k}_{54}\beta +{k}_{23}b+{k}_{2,10}\right)-{k}_{21}{k}_{65}{k}_{23}ac$ (191)

${a}_{3}=-{k}_{21}c{k}_{65}\left({k}_{54}\beta +{k}_{23}b+{k}_{2,10}\right)-{k}_{21}{k}_{65}{k}_{23}bc$ (192)

${a}_{2}=-{k}_{21}{c}^{2}{k}_{65}{k}_{23}$ (193)

${a}_{1}=0$ (194)

${a}_{0}=0$ (195)

Now compute

${s}_{2}\left(M\right)={M}^{3}\left(-{k}_{23}C{k}_{10,2}\left({k}_{21}C+{k}_{26}\right)\right)$ (196)

$\begin{array}{l}=-{M}^{3}{k}_{23}{k}_{10,2}\left({k}_{21}\frac{{a}^{2}{M}^{4}+{b}^{2}{M}^{2}+{c}^{2}+2ab{M}^{3}+2ac{M}^{2}+2bcM}{{M}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{k}_{26}\frac{a{M}^{2}+bM+c}{M}\right)\end{array}$ (197)

$=-{k}_{23}{k}_{10,2}{k}_{21}\left({a}^{2}{M}^{5}+{b}^{2}{M}^{3}+{c}^{2}M+2ab{M}^{4}+2ac{M}^{3}+2bc{M}^{2}\right)$ (198)

$-\text{\hspace{0.17em}}{k}_{23}{k}_{10,2}{k}_{26}\left(a{M}^{4}+b{M}^{3}+c{M}^{2}\right)$ (199)

Write

${s}_{2}\left(M\right)=\underset{i=0}{\overset{6}{\sum }}\text{ }\text{ }{b}_{i}{M}^{i}$ (200)

and by the computation above

${b}_{6}=0$ (201)

${b}_{5}=-{k}_{23}{a}^{2}{k}_{10,2}{k}_{21}$ (202)

${b}_{4}=-{k}_{23}b{k}_{10,2}{k}_{21}a-{k}_{23}a{k}_{10,2}\left({k}_{21}b+{k}_{26}\right)$ (203)

${b}_{3}=-2{k}_{23}{k}_{10,2}{k}_{21}ac-{k}_{23}{k}_{10,2}b\left({k}_{21}b+{k}_{26}\right)$ (204)

${b}_{2}=-{k}_{23}c{k}_{10,2}\left({k}_{21}b+{k}_{26}\right)-{k}_{23}b{k}_{10,2}{k}_{21}c$ (205)

${b}_{1}=-{k}_{23}{c}^{2}{k}_{10,2}{k}_{21}$ (206)

${b}_{0}=0$ (207)

Now consider

${p}_{3}\left(C,G,M\right)={k}_{21}{k}_{23}\stackrel{^}{a}{C}^{3}+{k}_{21}{k}_{54}\stackrel{^}{a}{C}^{2}G$ (208)

$+\text{\hspace{0.17em}}{k}_{21}\stackrel{^}{a}{C}^{2}{k}_{2,10}+{k}_{26}{k}_{23}\stackrel{^}{a}{C}^{2}$ (209)

$+\text{\hspace{0.17em}}{k}_{26}{k}_{54}G\stackrel{^}{a}C+{k}_{26}{k}_{2,10}\stackrel{^}{a}C$ (210)

$+\text{\hspace{0.17em}}{k}_{82}{k}_{21}{k}_{23}{C}^{2}+{k}_{82}{k}_{21}{k}_{54}CG$ (211)

$+\text{\hspace{0.17em}}{k}_{82}{k}_{21}{k}_{2,10}C+{k}_{82}{k}_{26}{k}_{23}C$ (212)

$+\text{\hspace{0.17em}}{k}_{82}{k}_{26}{k}_{54}G+{k}_{82}{k}_{26}{k}_{2,10}$ (213)

Name these 12 summands ${d}_{1},\cdots ,{d}_{12}.$ Now

${M}^{3}{d}_{1}={k}_{21}{k}_{23}\stackrel{^}{a}\underset{i=0}{\overset{6}{\sum }}\text{ }\text{ }{\stackrel{˜}{a}}_{i}{M}^{i}$ (214)

and

${M}^{3}{d}_{2}={k}_{21}{k}_{54}\stackrel{^}{a}\left(\alpha \left({a}^{2}{M}^{6}+{b}^{2}{M}^{4}+{c}^{2}{M}^{2}+2ab{M}^{5}+2ac{M}^{4}+2bc{M}^{3}\right)$ (215)

$+\beta \left({a}^{2}{M}^{5}+{b}^{2}{M}^{3}+{c}^{2}M+2ab{M}^{4}+2ac{M}^{3}+2bc{M}^{2}\right)\right)$ (216)

Also

#Math_366# (217)

Similarly

${M}^{3}{d}_{4}={k}_{26}\stackrel{^}{a}{k}_{23}\left({a}^{2}{M}^{5}+{b}^{2}{M}^{3}+{c}^{2}M+2ab{M}^{4}+2ac{M}^{3}+2bc{M}^{2}\right)$ (218)

Then

#Math_368# (219)

And

${M}^{3}{d}_{6}={k}_{26}{k}_{2,10}\stackrel{^}{a}\left(a{M}^{4}+b{M}^{3}+c{M}^{2}\right)$ (220)

We also find

${M}^{3}{d}_{7}={k}_{82}{k}_{21}{k}_{23}\left({a}^{2}{M}^{4}+{b}^{2}{M}^{2}+{c}^{2}+2ab{M}^{3}+2ac{M}^{2}+2bcM\right)M$ (221)

#Math_371# (222)

Combine

${M}^{3}\left({d}_{9}+{d}_{10}\right)=\left({k}_{82}{k}_{21}{k}_{2,10}+{k}_{82}{k}_{26}{k}_{23}\right)\left(a{M}^{4}+b{M}^{3}+c{M}^{2}\right)$ (223)

The last two ds give

${M}^{3}{d}_{11}={k}_{82}{k}_{26}{k}_{54}\left(\alpha {M}^{4}+\beta {M}^{3}\right)$ (224)

${M}^{3}{d}_{12}={k}_{82}{k}_{26}{k}_{2,10}{M}^{3}$ (225)

Write

${s}_{3}\left(M\right)=\underset{i=0}{\overset{6}{\sum }}\text{ }\text{ }{c}_{i}{M}^{i}$ (226)

Then we find from the above computations

${c}_{6}=\stackrel{^}{a}{a}^{2}{k}_{21}\left({k}_{23}a+{k}_{54}\alpha \right)$ (227)

and

${c}_{5}={k}_{21}{k}_{23}\stackrel{^}{a}3{a}^{2}b+{k}_{21}{k}_{54}\stackrel{^}{a}\left(\alpha 2ab+\beta {a}^{2}\right)+{k}_{21}\stackrel{^}{a}{k}_{2,10}{a}^{2}$ (228)

$+\text{\hspace{0.17em}}{k}_{26}{k}_{23}\stackrel{^}{a}{a}^{2}+{k}_{82}{k}_{21}{k}_{23}{a}^{2}+{k}_{82}{k}_{21}{k}_{54}a\alpha +{k}_{26}{k}_{54}\stackrel{^}{a}\alpha a$ (229)

Now collect terms

${c}_{4}={k}_{21}{k}_{23}\stackrel{^}{a}3\left(a{b}^{2}+{a}^{2}c\right)+{k}_{21}{k}_{54}\stackrel{^}{a}\alpha \left({b}^{2}+2ac\right)$ (230)

$+\text{\hspace{0.17em}}{k}_{21}{k}_{54}\stackrel{^}{a}\beta 2ab+{k}_{21}\stackrel{^}{a}{k}_{2,10}2ab+{k}_{26}{k}_{23}\stackrel{^}{a}2ab$ (231)

$+\text{\hspace{0.17em}}{k}_{26}{k}_{54}\stackrel{^}{a}\left(\alpha b+\beta a\right)+{k}_{26}{k}_{2,10}\stackrel{^}{a}a$ (232)

$+\text{\hspace{0.17em}}{k}_{82}{k}_{21}{k}_{23}2ab+{k}_{82}{k}_{21}{k}_{54}\left(\alpha b+\beta a\right)$ (233)

$+\left({k}_{82}{k}_{21}{k}_{2,10}+{k}_{82}{k}_{26}{k}_{23}\right)a+{k}_{82}{k}_{26}{k}_{54}\alpha$ (234)

We can also find

${c}_{3}={k}_{21}{k}_{23}\stackrel{^}{a}\left(6abc+{b}^{3}\right)+{k}_{21}{k}_{54}\stackrel{^}{a}\left(2bc\alpha +\beta \left({b}^{2}+2ac\right)\right)$ (235)

$+\text{\hspace{0.17em}}{k}_{21}\stackrel{^}{a}{k}_{2,10}\left({b}^{2}+2ac\right)+{k}_{26}{k}_{23}\stackrel{^}{a}\left({b}^{2}+2ac\right)$ (236)

$+\text{\hspace{0.17em}}{k}_{26}{k}_{54}\stackrel{^}{a}\left(c\alpha +\beta b\right)+{k}_{26}{k}_{2,10}\stackrel{^}{a}b$ (237)

$+\text{\hspace{0.17em}}{k}_{82}{k}_{21}{k}_{23}\left({b}^{2}+2ac\right)+{k}_{82}{k}_{21}{k}_{54}\left(\alpha c+\beta b\right)$ (238)

$+\left({k}_{82}{k}_{21}{k}_{2,10}+{k}_{82}{k}_{26}{k}_{23}\right)b+{k}_{82}{k}_{26}{k}_{54}\beta +{k}_{82}{k}_{26}{k}_{2,10}$ (239)

Now

${c}_{2}={k}_{21}{k}_{23}\stackrel{^}{a}3\left(a{c}^{2}+{b}^{2}c\right)+{k}_{21}{k}_{54}\stackrel{^}{a}\left(\alpha {c}^{2}+\beta 2bc\right)$ (240)

$+\left({k}_{21}\stackrel{^}{a}{k}_{2,10}+{k}_{26}{k}_{23}\stackrel{^}{a}\right)2bc+{k}_{26}{k}_{54}\stackrel{^}{a}\beta c$ (241)

$+\text{\hspace{0.17em}}{k}_{26}{k}_{2,10}\stackrel{^}{a}c+{k}_{82}{k}_{21}{k}_{23}2bc$ (242)

$+\text{\hspace{0.17em}}{k}_{82}{k}_{21}{k}_{54}\beta c+\left({k}_{82}{k}_{21}{k}_{2,10}+{k}_{82}{k}_{26}{k}_{23}\right)c$ (243)

Also

${c}_{1}={k}_{21}{k}_{23}\stackrel{^}{a}3b{c}^{2}+{k}_{21}{k}_{54}\stackrel{^}{a}{c}^{2}\beta +{k}_{21}\stackrel{^}{a}{k}_{2,10}{c}^{2}$ (244)

$+\text{\hspace{0.17em}}{k}_{82}{k}_{21}{k}_{23}{c}^{2}+{k}_{23}{c}^{2}\stackrel{^}{a}{k}_{26}$ (245)

Finally

${c}_{0}={k}_{21}{k}_{23}\stackrel{^}{a}{c}^{3}$ (246)

$s\left(M\right)=\underset{i=0}{\overset{6}{\sum }}\left({a}_{i}+{b}_{i}+{c}_{i}\right){M}^{i}$ (247)

If all rate constants ${k}_{ij}=1,X=1$ then

$s\left(M\right)=2{M}^{5}-8{M}^{4}+10{M}^{3}-\frac{23}{4}{M}^{2}+\frac{3}{2}M-\frac{1}{8}=0$ (248)

This polynomium has three real roots

$M=0.148638,\text{\hspace{0.17em}}M=\frac{1}{2},\text{\hspace{0.17em}}M=2.32234$ (249)

and two imaginary roots

$0.514511±0.312089i$ (250)

There are thus two candidates of singular points ( $M=\frac{1}{2}$ does not give a singular point).

$\left(C,D,G,M,N\right)=\left(\begin{array}{l}\left(0.661153,0.423634,0.702724,0.148638,0.089479\right)\hfill \\ \left(1.85998,-1.27437,-3.64468,2.32234,0.812013\right)\hfill \end{array}$ (251)

and they are both singular points (numerical evidence). We have used, that

$G=\alpha M+\beta$ (252)

$C=\frac{a{M}^{2}+bM+c}{M}$ (253)

$D=\frac{{k}_{10,2}}{{k}_{23}C+{k}_{54}G+{k}_{2,10}}$ (254)

$N=\frac{{k}_{65}M}{{k}_{21}C+{k}_{26}}$ (255)

This is the system (1) to (11) and the reactions

$\stackrel{˜}{D}+C\to 0$ (256)

$N+D\to \stackrel{˜}{D}$ (257)

$\stackrel{˜}{D}\to 0$ (258)

$D\to \stackrel{˜}{D}$ (259)

taking into account, that plasmonic nanobubbles destroy the liposomes $D$ and thus the chemo therapeutic drug $\stackrel{˜}{D}$ is injected into the cytoplasm. But some of the liposomes decay, producing chemo therapeutic drug. The complexes here are $C\left(12\right)=N+D,C\left(13\right)=\stackrel{˜}{D},C\left(14\right)=\stackrel{˜}{D}+C.$ The ODEs are

${C}^{\prime }=-{k}_{21}C\cdot N-{k}_{23}C\cdot D+\left({k}_{98}-{k}_{28}-{k}_{27}X\right)C+{k}_{82}-{k}_{2,14}\stackrel{˜}{D}\cdot C$ (260)

${\stackrel{˜}{D}}^{\prime }=-{k}_{2,14}\stackrel{˜}{D}\cdot C+{k}_{13,12}N\cdot D-{k}_{2,13}\stackrel{˜}{D}+{k}_{13,10}D$ (261)

${D}^{\prime }=-{k}_{23}C\cdot D-{k}_{54}D\cdot G+{k}_{10,2}-{k}_{2,10}D-{k}_{13,12}N\cdot D-{k}_{13,10}D$ (262)

${G}^{\prime }=-{k}_{54}D\cdot G+{k}_{11,2}-{k}_{2,11}G$ (263)

${M}^{\prime }={k}_{54}D\cdot G-\left({k}_{65}+{k}_{25}\right)M$ (264)

${N}^{\prime }=-{k}_{21}C\cdot N+{k}_{65}M-{k}_{26}N-{k}_{13,12}N\cdot D$ (265)

The vector field is denoted ${g}_{q}$ . Denote by ${g}_{cr}$ the vector field in (260), (261) and (262) with the rate constants in (263), (264) and (265) all equal to zero and the rest positive.

Theorem 4. There exist positive values of the rate constants such that

${C}_{*}^{q}<{C}_{*}^{cr}$ (266)

holds and the singular points are both positive and stable.

Proof. Define for $\ast =q$

${k}_{1}=\left({k}_{82},{k}_{10,2},{k}_{11,2}\right)$ (267)

#Math_425# (268)

And for $\ast =cr$

${k}_{1}^{cr}=\left({k}_{82},{k}_{10,2}\right)$ (269)

${k}_{2}^{cr}=\left({k}_{23},{k}_{98},{k}_{28},{\stackrel{˜}{k}}_{27},{k}_{2,14},{k}_{2,13},{k}_{2,10},{k}_{13,10}\right)$ (270)

Define

${K}_{cr}=\left({k}_{1}^{cr},{k}_{2}^{cr}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{K}_{q}=\left({k}_{1}^{q},{k}_{2}^{q}\right)$ (271)

and

${\stackrel{^}{K}}_{*}=\left({k}_{1}^{*},{k}_{2}^{*}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{k}_{1}^{*}=0$ (272)

Use the implicit function theorem to find a mapping

${c}_{q}:{V}_{q}\subset {ℝ}^{18}\to {ℝ}^{6}$ (273)

and

${c}_{cr}:{V}_{cr}\subset {ℝ}^{10}\to {ℝ}^{3}$ (274)

such that

${g}_{*}\left({c}_{*}\left({K}_{*}\right),{K}_{*}\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{*}\left({\stackrel{^}{K}}_{*}\right)=0$ (275)

and define

${d}_{*}\left(t\right)={c}_{*}\left(t{k}_{1}^{*,0},{k}_{2}^{*}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\ast =\left(\begin{array}{l}cr\hfill \\ q\hfill \end{array}$ (276)

${V}_{q},{V}_{cr}$ are open subsets of ${ℝ}^{18},{ℝ}^{10}$ respectively. For $\ast =cr$ let ${d}_{*}^{G}={d}_{*}^{M}$ Math_439# By the implicit function theorem

${d}_{*}^{\stackrel{˜}{D}}\left(0\right)=0$ (277)

Differentiate (262) and (261) with respect to t to get

$\frac{\partial {d}_{*}^{D}}{\partial t}=\frac{{k}_{10,2}^{0}}{{k}_{2,10}+{k}_{13,10}}$ (278)

and

$\frac{\partial {d}_{*}^{\stackrel{˜}{D}}}{\partial t}=\frac{{k}_{10,2}^{0}{k}_{13,10}}{{k}_{2,13}\left({k}_{2,10}+{k}_{13,10}\right)}$ (279)

Now differentiate (261) twice with respect to t to find

${k}_{2,13}\frac{{\partial }^{2}{d}_{*}^{\stackrel{˜}{D}}}{\partial {t}^{2}}={k}_{13,10}\frac{{\partial }^{2}{d}_{*}^{D}}{\partial {t}^{2}}-2{k}_{2,14}\frac{\partial {d}_{*}^{\stackrel{˜}{D}}}{\partial t}\frac{\partial {d}_{*}^{C}}{\partial t}$ (280)

and differentiate (262) with respect to t to get

$\frac{{\partial }^{2}{d}_{*}^{D}}{\partial {t}^{2}}=-\frac{2}{{k}_{2,10}+{k}_{13,10}}\left({k}_{23}\frac{\partial {d}_{*}^{C}}{\partial t}\frac{\partial {d}_{*}^{D}}{\partial t}+{k}_{54}\frac{\partial {d}_{*}^{D}}{\partial t}\frac{\partial {d}_{*}^{G}}{\partial t}\right)$ (281)

We also have

$\frac{\partial {d}_{*}^{C}}{\partial t}=-\frac{{k}_{82}^{0}}{\stackrel{^}{a}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {d}_{*}^{G}}{\partial t}=\frac{{k}_{11,2}^{0}}{{k}_{2,11}}$ (282)

the last equality when $\ast =q$ . We now get

$\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}=\frac{1}{\stackrel{^}{a}}\left(2{k}_{23}\frac{\partial {d}_{*}^{C}}{\partial t}\frac{\partial {d}_{*}^{D}}{\partial t}+2{k}_{2,14}\frac{\partial {d}_{*}^{\stackrel{˜}{D}}}{\partial t}\frac{\partial {d}_{*}^{C}}{\partial t}+{k}_{21}\frac{\partial {d}_{*}^{C}}{\partial t}\frac{\partial {d}_{*}^{N}}{\partial t}\right)$ (283)

so

$\frac{{\partial }^{2}{d}_{cr}^{C}}{\partial {t}^{2}}\left(0\right)=\frac{{\partial }^{2}{d}_{q}^{C}}{\partial {t}^{2}}\left(0\right)$ (284)

and

$\frac{{\partial }^{2}{d}_{*}^{N}}{\partial {t}^{2}}=\frac{{k}_{65}2{k}_{54}}{{k}_{26}\left({k}_{65}+{k}_{25}\right)}\frac{\partial {d}_{*}^{D}}{\partial t}\frac{\partial {d}_{*}^{G}}{\partial t}$ (285)

where we have used that

$\frac{\partial {d}_{*}^{M}}{\partial t}\left(0\right)=0$ (286)

hence

$\frac{\partial {d}_{*}^{N}}{\partial t}\left(0\right)=0$ (287)

Finally

$\stackrel{^}{a}\frac{{\partial }^{3}{d}_{*}^{C}}{\partial {t}^{3}}=3{k}_{23}\left(\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}\frac{\partial {d}_{*}^{D}}{\partial t}+\frac{\partial {d}_{*}^{C}}{\partial t}\frac{{\partial }^{2}{d}_{*}^{D}}{\partial {t}^{2}}\right)$ (288)

$+\text{\hspace{0.17em}}3{k}_{2,14}\left(\frac{{\partial }^{2}{d}_{*}^{\stackrel{˜}{D}}}{\partial {t}^{2}}\frac{\partial {d}_{*}^{C}}{\partial t}+\frac{\partial {d}_{*}^{\stackrel{˜}{D}}}{\partial t}\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}\right)$ (289)

$+\text{\hspace{0.17em}}3{k}_{21}\frac{\partial {d}_{*}^{C}}{\partial t}\frac{{\partial }^{2}{d}_{*}^{N}}{\partial {t}^{2}}$ (290)

and taking ${k}_{21}$ big, this implies the theorem, arguing as in the proof of theorem 1. □

From ${D}^{\prime }=0$ isolate

$C=\frac{1}{{k}_{23}D}\left(-{k}_{54}D\cdot G+{k}_{10,2}-\left({k}_{2,10}+{k}_{13,10}\right)D-{k}_{13,12}N\cdot D\right)$ (291)

and insert this in ${N}^{\prime }=0$ to get using

$G=\alpha M+\beta \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha <0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\beta >0$ (292)

and

$D=\frac{-\alpha {k}_{2,11}M}{{k}_{54}\left(\alpha M+\beta \right)}$ (293)

(this equation follows from (263)), that

$\frac{{k}_{13,12}{k}_{21}}{{k}_{23}}{N}^{2}+N\left({k}_{54}\frac{{k}_{21}}{{k}_{23}}\left(\alpha M+\beta \right)+\frac{{k}_{21}{k}_{10,2}{k}_{54}\left(\alpha M+\beta \right)}{{k}_{23}{k}_{2,11}\alpha M}$ (294)

$+{k}_{21}\frac{{k}_{2,10}+{k}_{13,10}}{{k}_{23}}-{k}_{26}+{k}_{13,12}\frac{{k}_{2,11}\alpha M}{{k}_{54}\left(\alpha M+\beta \right)}\right)+{k}_{65}M=0$ (295)

$M>0,M\ne -\frac{\beta }{\alpha }.$ Let $\stackrel{˜}{a}$ be the coefficient to ${N}^{2}$ and let $\stackrel{˜}{b}\left(M\right)$ be the coefficient to N and $\stackrel{˜}{c}\left(M\right)={k}_{65}M.$ Then we get when

$\Delta =\stackrel{˜}{b}{\left(M\right)}^{2}-4\stackrel{˜}{a}\stackrel{˜}{c}\left(M\right)>0$ (296)

${N}^{±}\left(M\right)=\frac{-\stackrel{˜}{b}\left(M\right)±\sqrt{\Delta }}{2\stackrel{˜}{a}}$ (297)

$\stackrel{˜}{a}>0.$ Thus ${D}^{\prime }=0$ gives

${C}^{±}=\frac{1}{{k}_{23}D\left(M\right)}\left(-{k}_{54}D\cdot G+{k}_{10,2}-\left({k}_{2,10}+{k}_{13,10}\right)D-{k}_{13,12}{N}^{±}\cdot D\right)$ (298)

and

${\stackrel{˜}{D}}^{±}=\frac{{k}_{13,12}{N}^{±}D+{k}_{13,10}D}{{k}_{2,14}{C}^{±}+{k}_{2,13}}$ (299)

$D=\frac{-{k}_{2,11}\alpha M}{{k}_{54}\left(\alpha M+\beta \right)}$ (300)

So let

${p}_{*}\left(M\right)={p}_{*}^{±}\left(M\right)=-{k}_{21}{C}^{±}{N}^{±}-{k}_{23}{C}^{±}D+\stackrel{^}{a}{C}^{±}+{k}_{82}-{k}_{2,14}{\stackrel{˜}{D}}^{±}{C}^{±}$ (301)

If $\left(C,\stackrel{˜}{D},D,G,M,N\right)$ is a positive singular point of ${g}_{q}$ , then

${p}_{*}^{±}\left(M\right)=0$ (302)

Define h by

$\sqrt{1+x}-1={\int }_{0}^{1}\frac{\partial }{\partial s}\sqrt{1+xs}\text{d}s$ (303)

$=x{\int }_{0}^{1}\frac{1}{2\sqrt{1+xs}}\text{d}s=xh\left(x\right)$ (304)

where h is smooth and $x\in \right]-1,+\infty \left[.$

Proposition 5. The function

${N}^{-}\left(M\right)\triangleq -2\frac{\stackrel{˜}{c}\left(M\right)}{\stackrel{˜}{b}\left(M\right)}h\left(-\frac{4\stackrel{˜}{a}\stackrel{˜}{c}}{{\stackrel{˜}{b}}^{2}}\right)$ (305)

satisfies (294) and (295), when

$-\frac{4\stackrel{˜}{a}\stackrel{˜}{c}}{{\stackrel{˜}{b}}^{2}}\in \right]-1,+\infty \left[,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{˜}{b}\left(M\right)<0$ (306)

Proof. If $\stackrel{˜}{b}\left(M\right)<0$ then

${N}^{-}\left(M\right)=-\frac{2\stackrel{˜}{c}\left(M\right)}{\stackrel{˜}{b}\left(M\right)}h\left(-\frac{4\stackrel{˜}{a}\stackrel{˜}{c}\left(M\right)}{\stackrel{˜}{b}{\left(M\right)}^{2}}\right)$ (307)

is smooth and

${N}^{+}\left(M\right)=\frac{-\stackrel{˜}{b}\left(M\right)+\sqrt{\Delta \left(M\right)}}{2\stackrel{˜}{a}}$ (308)

when $\stackrel{˜}{a}>0.$ If $\stackrel{˜}{b}\left(M\right)>0$ then

${N}^{+}\left(M\right)=-\frac{2\stackrel{˜}{c}\left(M\right)}{\stackrel{˜}{b}\left(M\right)}h\left(-\frac{4\stackrel{˜}{a}\stackrel{˜}{c}\left(M\right)}{\stackrel{˜}{b}{\left(M\right)}^{2}}\right)$ (309)

is smooth and

${N}^{-}\left(M\right)=\frac{-\stackrel{˜}{b}\left(M\right)-\sqrt{\Delta \left(M\right)}}{2\stackrel{˜}{a}}$ (310)

when $\stackrel{˜}{a}>0.$ These are both negative, so $\stackrel{˜}{b}\left(M\right)>0$ is not interesting. □

Proposition 6. If ${k}_{13,12}={k}_{2,13}={k}_{2,14}={k}_{13,10}=0$ and (306) holds, then

$s\left(M\right)=q\left(M\right){p}_{*}^{-}\left(M\right)$ (311)

where

$q\left(M\right)={M}^{3}\left({k}_{21}C+{k}_{26}\right)\left({k}_{23}C+{k}_{54}G+{k}_{2,10}\right)$ (312)

and

$C=\frac{a{M}^{2}+bM+c}{M}$ (313)

$G=\alpha M+\beta$ (314)

Proof. We have

${k}_{23}C+{k}_{54}G+{k}_{2,10}$ (315)

$={k}_{23}\frac{a{M}^{2}+bM+c}{M}+{k}_{54}\left(\alpha M+\beta \right)+{k}_{2,10}$ (316)

and here the terms with $a,\alpha$ cancel out to give

${k}_{23}b+{k}_{54}\beta +{k}_{2,10}+{k}_{23}\frac{c}{M}$ (317)

$=-\frac{{k}_{54}{k}_{10,2}}{{k}_{2,11}}+{k}_{23}\frac{c}{M}$ (318)

So

$\frac{{k}_{10,2}}{{k}_{23}C+{k}_{54}G+{k}_{2,10}}$ (319)

$=\frac{{k}_{10,2}}{-\frac{{k}_{54}{k}_{10,2}}{{k}_{2,11}}+{k}_{23}\frac{c}{M}}$ (320)

which becomes

$\frac{\left({k}_{65}+{k}_{25}\right)M}{\frac{{k}_{54}{k}_{11,2}}{{k}_{2,11}}-{k}_{54}\frac{{k}_{65}+{k}_{25}}{{k}_{2,11}}M}$ (321)

$=\frac{-{k}_{2,11}\alpha M}{{k}_{54}\left(\alpha M+\beta \right)}$ (322)

We also can compute

$\stackrel{˜}{b}\left(M\right)={k}_{54}\frac{{k}_{21}}{{k}_{23}}\left(\alpha M+\beta \right)+\frac{{k}_{21}{k}_{10,2}{k}_{54}}{{k}_{23}{k}_{2,11}}\frac{\alpha M+\beta }{\alpha M}$ (323)

$+\text{\hspace{0.17em}}{k}_{21}\frac{{k}_{2,10}}{{k}_{23}}-{k}_{26}$ (324)

$={k}_{54}\frac{{k}_{21}}{{k}_{23}}\left(-\frac{{k}_{65}+{k}_{25}}{{k}_{2,11}}M+\frac{{k}_{11,2}}{{k}_{2,11}}\right)$ (325)

$+\text{\hspace{0.17em}}{k}_{21}\frac{{k}_{2,10}}{{k}_{23}}+\frac{{k}_{21}{k}_{10,2}{k}_{54}}{{k}_{23}{k}_{2,11}}-\frac{{k}_{21}{k}_{10,2}{k}_{54}}{M{k}_{23}{k}_{2,11}}\frac{{k}_{11,2}}{{k}_{65}+{k}_{25}}-{k}_{26}$ (326)

$=-{k}_{21}\frac{a{M}^{2}+bM+c}{M}-{k}_{26}=-{k}_{21}C-{k}_{26}$ (327)

Notice that

${N}^{-}\left(M\right)=-\frac{\stackrel{˜}{c}\left(M\right)}{\stackrel{˜}{b}\left(M\right)}=\frac{\stackrel{˜}{c}\left(M\right)}{{k}_{21}C+{k}_{26}}$ (328)

and

$D=\frac{{k}_{10,2}}{{k}_{23}C+{k}_{54}G+{k}_{2,10}}$ (329)

Now we get

${p}_{*}^{-}\left(M\right)q\left(M\right)=\left(-{k}_{21}C\cdot {N}^{-}-{k}_{23}C\cdot D+\left({k}_{82}+\stackrel{^}{a}C\right)\right)q\left(M\right)$ (330)

$=-{k}_{65}M\cdot C\left({k}_{23}C+{k}_{54}G+{k}_{2,10}\right){M}^{3}$ (331)

#Math_516# (332)

If we assume there exists ${M}_{i}>0$ such that

$s\left({M}_{i}\right)=0$ (333)

$\frac{\partial s}{\partial M}\left({M}_{i}\right)\ne 0$ (334)

and $q\left({M}_{i}\right)>0$ (this is so if $G\left({M}_{i}\right),C\left({M}_{i}\right)>0$ ). Then

${p}_{*}^{-}\left({M}_{i},{k}_{0}\right)=0$ (335)

$\frac{\partial {p}_{*}}{\partial M}\left({M}_{i},{k}_{0}\right)\ne 0$ (336)

where

${k}_{0}={K}_{q}{}_{|{k}_{13,12}={k}_{13,10}={k}_{2,13}={k}_{2,14}={k}_{13,10}=0}$ (337)

Then there exists a ${C}^{1}$ function

${M}_{*}^{i}:U\to ℝ$ (338)

such that

${p}_{*}^{-}\left({M}_{*}^{i}\left(k\right),k\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k\in U$ (339)

${M}_{*}^{i}\left({k}_{0}\right)={M}_{i}$ (340)

by the implicit function theorem. Here U is an open neighbourhood of ${k}_{0}$ in ${ℝ}^{18}$ . So the roots of s are nearly zeroes of ${p}_{*}^{-}$ , when

${k}_{2,14},{k}_{13,12},{k}_{2,13},{k}_{13,10}$ (341)

are small and (333) and (334) holds. The example in the end of section 4 applies here.

6. Summary

${L}_{a}+C\to 0$ (342)

${L}_{a}⇆0$ (343)

Here the complexes are $C\left(15\right)={L}_{a}+C,C\left(16\right)={L}_{a},$ modelling adoptive T cell therapy, see  . ${L}_{a}$ are activated and expanded tumor infiltrating lymphocytes. They are injected back into the patient. Then the differential equations are

$\frac{\text{d}C}{\text{d}t}=-{k}_{21}C\cdot N-{k}_{23}C\cdot D+\stackrel{^}{a}C+{k}_{82}-{k}_{2,14}\stackrel{˜}{D}\cdot C-{k}_{2,15}{L}_{a}\cdot C$ (344)

${\stackrel{˜}{D}}^{\prime }=-{k}_{2,14}\stackrel{˜}{D}\cdot C+{k}_{13,12}N\cdot D-{k}_{2,13}\stackrel{˜}{D}+{k}_{13,10}D$ (345)

${D}^{\prime }=-{k}_{23}C\cdot D-{k}_{54}D\cdot G+{k}_{10,2}-{k}_{2,10}D-{k}_{13,12}N\cdot D-{k}_{13,10}D$ (346)

${G}^{\prime }=-{k}_{54}D\cdot G+{k}_{11,2}-{k}_{2,11}G$ (347)

${M}^{\prime }={k}_{54}D\cdot G-\left({k}_{65}+{k}_{25}\right)M$ (348)

${N}^{\prime }=-{k}_{21}C\cdot N+{k}_{65}M-{k}_{26}N-{k}_{13,12}N\cdot D$ (349)

$\frac{\text{d}{L}_{a}}{\text{d}t}=-{k}_{2,15}{L}_{a}\cdot C+{k}_{16,2}-{k}_{2,16}{L}_{a}$ (350)

Denote this vector field ${g}_{q}$ and let ${g}_{cr}$ denote the vector field in (344), (345), (346) where the rate constants in (347), (348), (349), (350) are all zero and the rest positive. Use the implicit function theorem to find maps

${c}_{cr}:{V}_{cr}\subset {ℝ}^{10}\to {ℝ}^{3}$ (351)

and

${c}_{q}:{V}_{q}\subset {ℝ}^{21}\to {ℝ}^{7}$ (352)

such that

${g}_{*}\left({c}_{*}\left({K}_{*}\right),{K}_{*}\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{*}\left({\stackrel{^}{K}}_{*}\right)=0$ (353)

Here

${K}_{cr}=\left({k}_{1}^{cr},{k}_{2}^{cr}\right)$ (354)

is the same as in section 5. Also ${V}_{cr},{V}_{q}$ are open subsets of ${ℝ}^{10}$ and ${ℝ}^{21}$ respectively. And

${k}_{1}^{q}=\left({k}_{82},{k}_{10,2},{k}_{11,2},{k}_{16,2}\right)$ (355)

Also

${k}_{2}^{q}=\left({k}_{21},{k}_{23},{k}_{98},{k}_{28},{\stackrel{˜}{k}}_{27},{k}_{2,14},{k}_{13,12},{k}_{2,13},{k}_{54},{k}_{2,10},{k}_{2,11},{k}_{65},{k}_{25},$ (356)

${k}_{26},{k}_{2,15},{k}_{2,16},{k}_{13,10}\right)$ (357)

Finally set

${K}_{*}=\left({k}_{1}^{*},{k}_{2}^{*}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{^}{K}}_{*}=\left({k}_{1}^{*},{k}_{2}^{*}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{k}_{1}^{*}=0$ (358)

Define

${d}_{*}\left(t\right)={c}_{*}\left(t{k}_{1}^{*,0},{k}_{2}^{*}\right)$ (359)

Let

$\stackrel{^}{a}={k}_{98}-{k}_{28}-{k}_{27}X$ (360)

and assume that it is negative. Then

$\stackrel{^}{a}\frac{{\partial }^{2}{d}_{*}^{C}}{\partial {t}^{2}}=2{k}_{23}\frac{\partial {d}_{*}^{C}}{\partial t}\frac{\partial {d}_{*}^{D}}{\partial t}+2{k}_{2,15}\frac{\partial {d}_{*}^{{L}_{a}}}{\partial t}\frac{\partial {d}_{*}^{C}}{\partial t}+2{k}_{2,14}\frac{\partial {d}_{*}^{\stackrel{˜}{D}}}{\partial t}\frac{\partial {d}_{*}^{C}}{\partial t}$ (361)

But

$\frac{\partial {d}_{*}^{C}}{\partial t}=-\frac{{k}_{82}^{0}}{\stackrel{^}{a}}$ (362)

$\frac{\partial {d}_{*}^{D}}{\partial t}=\frac{{k}_{10,2}^{0}}{{k}_{2,10}}$ (363)

$\frac{\partial {d}_{*}^{{L}_{a}}}{\partial t}=\frac{{k}_{16,2}^{0}}{{k}_{2,16}}$ (364)

$\frac{\partial {d}_{*}^{\stackrel{˜}{D}}}{\partial t}=\frac{{k}_{10,2}^{0}{k}_{13,10}}{{k}_{2,13}\left({k}_{2,10}+{k}_{13,10}\right)}$ (365)

suggesting, that in this model multipeutics can be more potent than quadrapeutics.

In the present paper, we considered quadrapeutics and multipeutics cancer therapies. We proved the multipeutics theorem, stating that the more treatments we apply the smaller the cancer burden. We also found a polynomial of degree at most 6, giving candidates of singular points for the quadrapeutics system.

The cells of the immune system have a plasma membrane repair system and it turns out that this system is much more efficient in cancer cells than in normal cells. The two proteins S100A11 and Annexin A2 are involved in this plasma membrane repair system. They are commonly upregulated in cancer cells, see  . So we can ask: can cancer cells repair nanobubble injury.

Cite this paper

Larsen, J.C. (2017) A Study on Multipeutics. Applied Mathematics, 8, 746-773. https://doi.org/10.4236/am.2017.85059

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