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.


Introduction
In [2] 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 [3] we showed that there are cancer ODE models with oscillatory behaviour and in [4] 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 t + → + where , 0, 0 t t t t is the domain of definition of the maximal integral curve ( ) c t of the corres- ponding vector field.
In the present paper, we propose to combine quadrapeutics and immune therapy.
We have written a paper [5] on fundamental concepts in dynamics intended for researchers with a background in medicine.

The Multipeutics System
Consider the mass action kinetic system J. C. Larsen 0 n ∈  ( 0 n = means that the reactions (12) and ( 13) are omitted).The complexes are ( ) 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) The ODEs are, see [18] ( ) ( ) 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 The quadrapeutics system ( 14) to (18) with the rate constants in (19) equal to zero and the rest positive.This vector field is denoted : Finally the multipeutics systems: the system ( 14) to (19) with the rate constants in (19) equal to zero, when 1, , i n j n = − +  and the rest positive.This vector field is denoted , , , , .
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 Looking from right to left, this says that the more treatments we apply the lower the cancer burden.
In Figure 1 we have plotted ( ) where h is the step size and equal to 0.02.There are 1000 iterations.The numerical analysis indicated, that There exist positive values of the rate constants ij k , such that (24) holds and the singular points are all positive and stable.
Proof.Let K be defined by { } ( ) , when n * = .Here , , Notice that ( ) 0, , 0  is a singular point of f, when 0 t = .The rate constants ij k in 2 k are positive real numbers.Also note that 1 *0 D f is an isomorphism.
For the chemo radiation model we let For the quadrapeutics model we let * 0, 1, , .
Differentiate (48) with respect to t to get regardless of the system.We have Differentiate this with respect to t to get Also differentiate this equation with respect to t to find From this equation it follows that anticipating that ( ) ( ) Observe that ( ) ( ) The third equality holds for ,1, , .
By the fundamental theorem of calculus applied twice To see the first inequality, observe that there exists a positive constant c, such that because a positive function on a compact interval [ ] 0,δ assumes a minimum 0 c > .The other inequalities follow in the same way.And this proves all the in- equalities in (24) except * * q cr C C < . Henceforth assume Finally differentiate (57) to (60) with respect to t to get However when To see this differentiate (17) twice with respect to t to find If we now differentiate (18) twice with respect to t we find ( ) ( ) which is what we sought to show.Furthermore ( ) ( ) ( ) and when 0 t > and small.Differentiate (15) twice with respect to t when where we have evaluated in 0 t = on the right hand side.So if ( ) we have ( ) ( ) when 0 t = and hence also for small 0.

t >
The fundamental theorem of calculus applied three times gives for small 0 t > and the theorem follows in the same way as (67).Define the matrices and then write the definition of the determinant where p S is the set of permutations of { } 1, , p  and ( ) 0 I σ = when σ is odd.This formula shows immediately that ( ) ( ) where i a is a polynomial in Then by taking 21 k big we find ( ) ( ) for small 0, t > 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 [1] [13] [14] [15].Consider also the system (1) to ( 13) with replacing (7).The ODE s are the same (15) to ( 19) except (14), which is ( ) We are now going to find candidates of the positive singular points of the different systems.We start with f and assume that all ij k are positive.(19) gives We can isolate C in this equation to find )  there exist rate constants 0, ij k > such that (24) holds and all equilibria are positive and stable.
We can now take 21 k big and argue as in theorem 1. □

The Chemo Radiation Model
Consider the mass action kinetic system Here the complexes are ( ) This system with 0 δ = is similar to the reduced system from [17].
where we have added the last reaction.The complexes are ( ) ) , , We can assume that GF is at equilibrium The reduced system is then We shall find the singular points of the chemo radiation system.
and when and ( ) hence this is a stable, positive singular point.
Consider the chemo radiation system where 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 q f in the singular points.I have plotted a phase portrait for these values of the parameters in Figure 3.

The Quadrapeutics Equilibrium Equation
We shall find the quadrapeutics equilibrium equation too.Theorem 3. Suppose 0.

δ =
There exists a polynomial of degree atmost six ( ) 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 ) ) where we have multiplied with ) We shall now insert  , , and this is ( ) and this is ( ) which is ( ) Then by the previous computation ( ) and by the computation above Then ( ) And ( ) We also find ( ) The last two ds give ( ) Then we find from the above computations ( )

J. C. Larsen
We can also find    The quadrapeutics polynomial is ( ) ( ) This polynomium has three real roots

The Extended Quadrapeutics Model
This is the system (1) to (11) and the reactions taking into account, that plasmonic nanobubbles destroy the liposomes D and thus the chemo therapeutic drug D  is injected into the cytoplasm.But some of the liposomes decay, producing chemo therapeutic drug.The complexes here are ( ) ( ) ( ) ( ) The vector field is denoted q g .Denote by cr g 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.holds and the singular points are both positive and stable.Proof.Define for q * = ( ) , , , , , , , ( ) Use the implicit function theorem to find a mapping : : and define ( ) ( ) , q cr V V are open subsets of 18 10  ,   respectively.For  2 We also have the last equality when q * = .We now get 2 where we have used that and taking 21 k big, this implies the theorem, arguing as in the proof of theo- rem 1.
and insert this in Let a  be the coefficient to and Proof.We have We also can compute ( ) ( ) Now we get

□
If we assume there exists 0 and ( )

Summary
Consider the extended quadrapeutics system and the additional reactions Denote this vector field q g and let cr g 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 : Here , is the same as in section 5. Also , cr q V V are open subsets of 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 [20].So we can ask: can cancer cells repair nanobubble injury.

)Figure 1 .Figure 2 .
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.
by the implicit function theorem.* * * , , V K K are defined below.

= 1 ,
we now insert the formulas for , G C and multiply with 3 we get a polynomial p of degree at most ( ) in M, such that a positive singular point ( ) , , , , , , n C D G M N I I 

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

2 N
and let ( ) b M  be the coefficient to N and ( ) 65 .

18 
the implicit function theorem.Here U is an open neighbourhood of 0 k in .So the roots of s are nearly zeroes of * p − , when are small and (333) and (334) holds.The example in the end of section 4 applies here.
see[4]. a L are activated and expanded tumor infiltrating lympho- cytes.They are injected back into the patient.Then the differential equations are )