1. 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
where
is the domain of definition of the maximal integral curve
of the corresponding 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.
There are several important monographs related to the present paper, see [6] - [11] . [12] is a mathematical model of miRNAs. [13] [14] [15] are papers on quadrapeutics. [16] [17] are two papers of mine on cancer and mathematics.
2. The Multipeutics System
Consider the mass action kinetic system
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(
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
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
The ODEs are, see [18]
(14)
(15)
(16)
(17)
(18)
(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
(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
(21)
Finally the multipeutics systems: the system (14) to (19) with the rate constants in (19) equal to zero, when
and the rest positive. This vector field is denoted
(22)
Now introduce some important notation. A singular point of
is denoted
A singular point of
is denoted
A singular point of
is denoted
(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
(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
versus t for the chemo radiation model and
versus t for the quadrapeutics model. All rate constants equal to 1, except
Also
This supports the proof of Theorem 1 that if you make the nanobubble kill rate
sufficiently big, then quadrapeutics outperforms chemo radiation. I have iterated the Euler maps
(25)
(26)
where h is the step size and equal to 0.02. There are 1000 iterations. The numerical analysis indicated, that
for the quadrapeutics system converged, when t goes to infinity. And similarly for
of the chemo radiation model, see Figure 1 and Figure 2.
Theorem 1. Assume, that
There exist positive values of the rate constants
, such that (24) holds and the singular points are all positive and stable.
Proof. Let K be defined by
(27)
(28)
(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
(30)
(31)
(32)
(33)
when
. Here
(34)
Notice that
is a singular point of f, when
. The rate constants
in
are positive real numbers. Also note that
is an isomorphism.
Here
(35)
For instance in f
(36)
is an isomorphism, where
all other
.
So there exist
maps,
(37)
such that
(38)
by the implicit function theorem.
are defined below.
Also
(39)
(40)
(41)
For
define
(42)
and
(43)
For
we define
(44)
and
(45)
Finally for the multipeutics system
(46)
and
(47)
Now let
and
and finally
above is an open subset containing
in
. Define
Then
(48)
Define
(49)
For the chemo radiation model we let
(50)
For the quadrapeutics model we let
And for the multipeutics model we let
(51)
Differentiate (48) with respect to t to get
(52)
Thus
(53)
regardless of the system. We have
#Math_123# (54)
Differentiate this with respect to t to get
(55)
(56)
Also differentiate this equation with respect to t to find
(57)
(58)
(59)
(60)
From this equation it follows that
(61)
anticipating that
(62)
Observe that
(63)
(64)
The third equality holds for
The fourth equality holds for the multipeutics system
By the fundamental theorem of calculus applied twice
(65)
So
(66)
To see the first inequality, observe that there exists a positive constant c, such that
(67)
because a positive function on a compact interval
assumes a minimum
. The other inequalities follow in the same way. And this proves all the inequalities in (24) except
. Henceforth assume
(68)
Finally differentiate (57) to (60) with respect to t to get
(69)
(70)
(71)
when
But
(72)
However when
(73)
To see this differentiate (17) twice with respect to t to find
(74)
If we now differentiate (18) twice with respect to t we find
(75)
which is what we sought to show.
Furthermore
(76)
and
(77)
when
and small. Differentiate (15) twice with respect to t when
to get
(78)
Now
(79)
Also
(80)
(81)
(82)
where we have evaluated in
on the right hand side. So if
(83)
we have
(84)
when
and hence also for small
The fundamental theorem of calculus applied three times gives
(85)
So
(86)
for small
and the theorem follows in the same way as (67). Define the matrices
(87)
and then write the definition of the determinant
(88)
where
is the set of permutations of
and
if
is even and
when
is odd. This formula shows immediately that
(89)
where
is a polynomial in
and hence is smooth in
You can prove this by induction. Now apply the continuous dependence of roots of a polynomial on its coefficients to show that
is stable for
, when t is small, see [19] . □
Remark. If the quadrapeutics chemo rate is
(90)
and the x ray rate is
(91)
where
(92)
while
Here
(93)
Then by taking
big we find
(94)
since
(95)
as
Hence
(96)
for small
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
(97)
replacing (7). The ODE s are the same (15) to (19) except (14), which is
(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
are positive. (19) gives
(99)
when
and (18) gives
(100)
(16) and (17) give
(101)
or
(102)
Also
(103)
Insert these formulas in (14)
(104)
(105)
(106)
. Insert the formula for D in
to get
(107)
We can isolate C in this equation to find
(108)
where
(109)
and
(110)
Finally
(111)
We now get by multiplying with the product of the denominators in (104), (105) and (106)
(112)
(113)
(114)
(115)
(116)
(117)
If we now insert the formulas for
and multiply with
or
we get a polynomial p of degree at most
or
in M, such that a positive singular point
(118)
will have
Let
Then we have
Theorem 2. For
there exist rate constants
such that (24) holds and all equilibria are positive and stable.
Proof. Now let
(119)
and assume that it is negative. Then
(120)
is the same for all treatments. But now the first coordinate of formula (48) becomes
(121)
So
(122)
(123)
(124)
(125)
and when
(126)
(127)
(128)
We can now take
big and argue as in theorem 1. □
3. The Chemo Radiation Model
Consider the mass action kinetic system
(129)
(130)
(131)
(132)
(133)
Here the complexes are
With mass action kinetics the vector field is
(134)
all
This system with
is similar to the reduced system from [17] .
(135)
(136)
(137)
(138)
(139)
(140)
where we have added the last reaction. The complexes are
The vector field is
(141)
We can assume that
is at equilibrium
(142)
The reduced system is then
(143)
(144)
We shall find the singular points of the chemo radiation system.
gives
(145)
when
and then
amounts to
(146)
which is equivalent to
(147)
(148)
when
and when
it is equivalent to
(149)
(150)
The linearization of f at a singular point
is
(151)
when
and when
it is
(152)
If
(153)
and
is a positive singular point, then
(154)
hence this is a stable, positive singular point.
Consider the chemo radiation system where
(155)
and all other
Then the cubic polynomial giving candidates of singular points is
(156)
and this gives three singular points
(157)
where we have used that
(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
in the singular points. I have plotted a phase portrait for these values of the parameters in Figure 3.
4. The Quadrapeutics Equilibrium Equation
We shall find the quadrapeutics equilibrium equation too.
Theorem 3. Suppose
There exists a polynomial of degree atmost six
(159)
such that if
is a positive singular point of
then
(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
to find
(161)
(162)
(163)
(164)
where we have multiplied with
(165)
We shall now insert
(166)
where
(167)
(168)
in the polynomial p. Compute
(169)
where
(170)
(171)
(172)
(173)
(174)
(175)
(176)
Now define
(177)
and this is
(178)
Also put
(179)
and this is
(180)
Finally set
(181)
which is
(182)
So
(183)
Now compute
#Math_332# (184)
(185)
(186)
(187)
Write
(188)
Then by the previous computation
(189)
(190)
(191)
(192)
(193)
(194)
(195)
Now compute
(196)
(197)
(198)
(199)
Write
(200)
and by the computation above
(201)
(202)
(203)
(204)
(205)
(206)
(207)
Now consider
(208)
(209)
(210)
(211)
(212)
(213)
Name these 12 summands
Now
(214)
and
(215)
(216)
Also
#Math_366# (217)
Similarly
(218)
Then
#Math_368# (219)
And
(220)
We also find
(221)
#Math_371# (222)
Combine
(223)
The last two ds give
(224)
(225)
Write
(226)
Then we find from the above computations
(227)
and
(228)
(229)
Now collect terms
(230)
(231)
(232)
(233)
(234)
We can also find
(235)
(236)
(237)
(238)
(239)
Now
(240)
(241)
(242)
(243)
Also
(244)
(245)
Finally
(246)
The quadrapeutics polynomial is
(247)
□
If all rate constants
then
(248)
This polynomium has three real roots
(249)
and two imaginary roots
(250)
There are thus two candidates of singular points (
does not give a singular point).
(251)
and they are both singular points (numerical evidence). We have used, that
(252)
(253)
(254)
(255)
5. The Extended Quadrapeutics Model
This is the system (1) to (11) and the reactions
(256)
(257)
(258)
(259)
taking into account, that plasmonic nanobubbles destroy the liposomes
and thus the chemo therapeutic drug
is injected into the cytoplasm. But some of the liposomes decay, producing chemo therapeutic drug. The complexes here are
The ODEs are
(260)
(261)
(262)
(263)
(264)
(265)
The vector field is denoted
. Denote by
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
(266)
holds and the singular points are both positive and stable.
Proof. Define for
(267)
#Math_425# (268)
And for
(269)
(270)
Define
(271)
and
(272)
Use the implicit function theorem to find a mapping
(273)
and
(274)
such that
(275)
and define
(276)
are open subsets of
respectively. For
let
Math_439# By the implicit function theorem
(277)
Differentiate (262) and (261) with respect to t to get
(278)
and
(279)
Now differentiate (261) twice with respect to t to find
(280)
and differentiate (262) with respect to t to get
(281)
We also have
(282)
the last equality when
. We now get
(283)
so
(284)
and
(285)
where we have used that
(286)
hence
(287)
Finally
(288)
(289)
(290)
and taking
big, this implies the theorem, arguing as in the proof of theorem 1. □
From
isolate
(291)
and insert this in
to get using
(292)
and
(293)
(this equation follows from (263)), that
(294)
(295)
Let
be the coefficient to
and let
be the coefficient to N and
Then we get when
(296)
(297)
Thus
gives
(298)
and
(299)
(300)
So let
(301)
If
is a positive singular point of
, then
(302)
Define h by
(303)
(304)
where h is smooth and
Proposition 5. The function
(305)
satisfies (294) and (295), when
(306)
Proof. If
then
(307)
is smooth and
(308)
when
If
then
(309)
is smooth and
(310)
when
These are both negative, so
is not interesting. □
Proposition 6. If
and (306) holds, then
(311)
where
(312)
and
(313)
(314)
Proof. We have
(315)
(316)
and here the terms with
cancel out to give
(317)
(318)
So
(319)
(320)
which becomes
(321)
(322)
We also can compute
(323)
(324)
(325)
(326)
(327)
Notice that
(328)
and
(329)
Now we get
(330)
(331)
#Math_516# (332)
□
If we assume there exists
such that
(333)
(334)
and
(this is so if
). Then
(335)
(336)
where
(337)
Then there exists a
function
(338)
such that
(339)
(340)
by the implicit function theorem. Here U is an open neighbourhood of
in
. So the roots of s are nearly zeroes of
, when
(341)
are small and (333) and (334) holds. The example in the end of section 4 applies here.
6. Summary
Consider the extended quadrapeutics system and the additional reactions
(342)
(343)
Here the complexes are
modelling adoptive T cell therapy, see [4] .
are activated and expanded tumor infiltrating lymphocytes. They are injected back into the patient. Then the differential equations are
(344)
(345)
(346)
(347)
(348)
(349)
(350)
Denote this vector field
and let
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
(351)
and
(352)
such that
(353)
Here
(354)
is the same as in section 5. Also
are open subsets of
and
respectively. And
(355)
Also
(356)
(357)
Finally set
(358)
Define
(359)
Let
(360)
and assume that it is negative. Then
(361)
But
(362)
(363)
(364)
(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 [20] . So we can ask: can cancer cells repair nanobubble injury.