The Bistability Theorem in a Model of Metastatic Cancer

The main theorem of the present paper is the bistability theorem for a four dimensional cancer model, in the variables representing primary cancer C, metastatic cancer , growth factor GF and growth inhibitor GI, respectively. It says that for some values of the para- meters this system is bistable, in the sense that there are exactly two positive singular points of this vector field. And one is stable and the other unstable. We also find an expression for for the discrete model T of the introduction, with variables , where C is cancer, are growth factors and growth inhibitors respectively. We find an affine vector field Y whose time one map is T2 and then compute , where is an integral curve of Y through . We also find a formula for the first escape time for the vector field associated to T, see section four.

A Study on Multipeutics

Multipeutics is the simultaneous application of m ≥ 4 cancer treatments. m = 4 is quadrapeutics, which was invented by researchers at Rice University, Northeastern University, MD Anderson Cancer Centre and China Medical University, see [1]. Multipeutics is our idea. From section 6 Summary, it follows that multipeutics can be more potent than quadrapeutics by comparing these two mathematical models. The first two treatments in quadrapeutics are systemically administered nano gold particles G and lysosomal chemo therapeutic drug D. They form mixed clusters M primarily in cancer cells and can be excited by a laser pulse, the third treatment, to form plasmonic nanobubbles N. These nanobubbles can kill the cancer cells by mechanical impact. If they do not the chemo therapeutic drug can be released into the cytoplasm, which might be lethal to the cancer cell. The fourth treatment is x rays X and the cancer cells have been sensitized to x rays by the treatment. We present an ODE (ordinary differential equations) model of quadrapeutics and of multipeutics, which is quadrapeutics and n ≥ 1 immune or chemo therapies. In the present paper we have found a polynomial p of degree at most 2(n + 3), such that a singular point (C, D, G, M, N, I1, …, In) will have p(M) = 0 Here I1, …, In are immune or chemo therapies. So this gives us candidates for singular points. Quadrapeutics is treated extensively. We find in theorem 3 a polynomium s of degree at most six in M such that a positive singular point (C, D, G, M, N) of the quadrapeutics system will have s(M) = 0. The main theorem of the present paper is the multipeutics theorem, saying that the more treatments we apply the lower the cancer burden, even if we take the doses of each treatment smaller. From the proof of this theorem, we can say, that quadrapeutics can outperform chemo radiation if the nanobubble kill rate k21 is sufficiently big. See also Figure 1 and Figure 2 and the text explaining them.

Optimal Control of Cancer Growth

The purpose of the present paper is to apply the Pontryagin Minimum Principle to mathematical models of cancer growth. In [1], I presented a discrete affine model T of cancer growth in the variables C for cancer, GF for growth factors and GI for growth inhibitors. One can sometimes find an affine vector field X on whose time one map is T. It is to this vector field we apply the Pontryagin Minimum Principle. We also apply the Discrete Pontryagin Minimum Principle to the model T. So we prove that maximal chemo therapy can be optimal and also that it might not depending on the spectral properties of the matrix A, (see below). In section five we determine an optimal strategy for chemo or immune therapy.

Models of Cancer Growth Revisited

In the present paper we study models of cancer growth, initiated in Jens Chr. Larsen: Models of cancer growth [1]. We consider a cancer model in variables C cancer cells, growth factors GFi?,i= 1,,p, (oncogene, tumor suppressor gene or carcinogen) and growth inhibitor GFi?,i= 1,,p, (cells of the immune system or chemo or immune therapy). For q =1 this says, that cancer grows if (1) below holds and is eliminated if the reverse inequality holds. We shall prove formulas analogous to (1) below for arbitrary p, q∈N, p ≥ q . In the present paper, we propose to apply personalized treatment using the simple model presented in the introduction.

Bistability theorem in a cancer model

In this paper, I continue the study of the mathematical models presented in [J. C. Larsen, Models of cancer growth, J. Appl. Math. Comput. 53（1-2） （2015） 613-645] and [J. C. Larsen, The bistability theorem in a model of metastatic cancer, to appear in Appl. Math.]. I shall prove the bistability theorem for the ODE model from [Larsen, 2015]. It is a mass action kinetic system in the variables C cancer, GF growth factor and GI growth inhibitor. This theorem says that for some values of the parameters, there exist two positive singular points c＊＋ = （C＊＋, GF＊., GI＊＋）, c＊2- = （C＊-, GF＊, GI＊-） of the vector field. Here C＊- 〈 C＊＋ and e. is stable and c＊＋ is unstable, see Sec. 2. There is also a discrete model in [Larsen, 2015], it is a linear map （T） on three-dimensional Euclidean vector space with variables （C, GF, GI）, where these variables have the same meaning as in the ODE model above. In [Larsen, 2015], I showed that one can sometimes find attine vector fields on three-dimensional Euclidean vector space whose time one map is T. I shall also show this in the present paper in a more general setting than in [Larsen, 2015]. This enables me to find an expression for the rate of change of cancer growth on the coordinate hyperplane C = 0 in Euclidean vector space. I also present an ODE model of cancer metastasis with variables C, CM, CF,GI, where C is primary cancer and CM is metastatic cancer and GF, GI are growth factors and growth inhibitors, respectively.