research article modeling the treatment of glioblastoma...

Research ArticleModeling the Treatment of Glioblastoma Multiforme andCancer Stem Cells with Ordinary Differential Equations

Kristen Abernathy1 and Jeremy Burke2

1Department of Mathematics Winthrop University Rock Hill SC 29733 USA2Department of Mathematics Vassar College Poughkeepsie NY 12604 USA

Correspondence should be addressed to Kristen Abernathy abernathykwinthropedu

Received 3 September 2015 Revised 12 November 2015 Accepted 22 December 2015

Academic Editor Chung-Min Liao

Copyright copy 2016 K Abernathy and J Burke This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Despite improvements in cancer therapy and treatments tumor recurrence is a common event in cancer patients One explanationof recurrence is that cancer therapy focuses on treatment of tumor cells and does not eradicate cancer stem cells (CSCs) CSCsare postulated to behave similar to normal stem cells in that their role is to maintain homeostasis That is when the population oftumor cells is reduced or depleted by treatment CSCs will repopulate the tumor causing recurrence In this paper we study theapplication of the CSC Hypothesis to the treatment of glioblastoma multiforme by immunotherapy We extend the work of Koganet al (2008) to incorporate the dynamics of CSCs prove the existence of a recurrence state and provide an analysis of possiblecancerous states and their dependence on treatment levels

1 Introduction

Dynamical systems continue to play an important role inunderstanding cancer dynamics [1ndash4] A recent developmentin cancer dynamics is the cancer stem cell hypothesis Thecancer stem cell hypothesis states that malignant tumors areinitiated and maintained by a population of tumor cells thatshare similar biologic properties to normal adult stem cells[5] With evidence mounting in support of the cancer stemcell hypothesis recent work has been devoted to the inclusionof cancer stem cells (CSCs) in current cancer models [6ndash9]

Cancer stem cells are a specialized type of cancer cell thatare believed to be responsible for populating tumors CSCshave a very small population in comparison to normal cancercells because of their specialized function While tumor cellsare only able to undergo a limited number of divisions CSCsare able to repopulate a depleted tumor even if there are onlya few CSCs left [10] Once the number of CSCs begins todrop usually due to treatment they cease creating cancer cellsand focus on repopulating themselves The small populationof CSCs is hard to detect and therefore treatment is often

stopped before all CSCs have been eradicated which leads torecurrence of cancer [11] It is clear that for treatment to beeffective wemust focus our efforts on eliminating both tumorcells and CSCs

The cancer stem cell hypothesis has been biologicallyverified for many solid tumors including brain cancer [12]Glioblastoma is a type of brain tumor which forms in thecerebral hemisphere of the brain often in the frontal andtemporal lobe These types of tumors are highly malignantforming from normal brain cells astrocytes or star-shapedglial cells which support nerve cells These cells can growrapidly due to ample amounts of blood available in thebrain Immunotherapy is a cancer treatmentwhich stimulatesthe immune system to work harder to attack cancer cellsThe therapy uses additive components such as man-madeproteins vaccines or white blood cells to further attack can-cer cells Immunotherapy is essential to treating multiformeglioblastomas because of their sensitive location the brainwhich is too delicate to be treated by chemotherapy or surgery[10 13]

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2016 Article ID 1239861 11 pageshttpdxdoiorg10115520161239861

2 Computational and Mathematical Methods in Medicine

2 Presentation of the Model

In this paper we extend the previous work of Kogan et al[13] to include cancer stem cells in modeling the treatmentof glioblastomamultiformewith immunotherapyWe presentan abstract model that can be adapted to fit various biologicalassumptions We analyze the stability of the model both withand without treatment and derive sufficient conditions ontreatment to ensure a globally asymptotically stable cure stateWe conclude with an example illustrating the transition fromcoexistence of cancer cells to eradication of cancer cells withvarious treatment levels Much of this model is based onexperimental results obtained by Kruse et al [14]

The following system models the dynamics of tumorcells (119879) cancer stem cells (119878) alloreactive cytotoxic-T-lymphocytes (119862) TGF-120573 (119865120573) IFN-120574 (119865120574) and major histo-compatibility complex classes I and II (119872I and119872II resp)

1198791015840= 120572 (119879 119878) + 1198771 (119879) 119879

minus 119891119879 (119865120573) 119892119879 (119872I) ℎ119879 (119879) 119862119879

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 119891119878 (119865120573) 119892119878 (119872I) ℎ119878 (119878) 119862119878

1198621015840= 119891119862 ((119879 + 119878) sdot 119872II) 119892119862 (119865120573) minus 120583119862119862 + 119873 (119905)

1198651015840120573 = 119891120573 (119879 + 119878) minus 120583120573119865120573

1198651015840120574 = 119891120574 (119862) minus 120583120574119865120574

1198721015840I = 119891119872I

(119865120574) minus 120583119872I119872I

1198721015840II = 119891119872II

(119865120573) 119892119872II(119865120574) minus 120583119872II

119872II

(1)

To understand the formation of the system above wediscuss the biological interpretations of each equation in themodel

1198791015840= 120572 (119879 119878) + 1198771 (119879) 119879 minus 119891119879 (119865120573) 119892119879 (119872I) ℎ119879 (119879) 119862119879 (2)

The first term on the right hand side (RHS) of theequation represents differentiated tumor cells produced bythe CSCs without immune intervention where 120572(119879 119878) is therate at which CSCs produce tumor cells and 119879 is the numberof nonstem tumor cells (TCs) currently present The secondterm stands for normal tumor growth the cells producedby regular reproduction of nonstem tumor cells Both thefirst and second terms use classical logistic growth (note thatthe carrying capacities for 119878 and 119879 are distinct) The thirdterm on the RHS represents tumor elimination by CTL inproportion to both 119879 and 119862 CTLs are white blood cellsresponsible for attacking tumor cells in this case The thirdterm also introduces the effects of MHC class I receptors(119872I) and TGF-120573 (119865120573) which is assumed to be a majorimmunosuppressive factor for CTL activity [10] Consider

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 119891119878 (119865120573) 119892119878 (119872I) ℎ119878 (119878) 119862119878 (3)

The first term on the RHS stands for the rate of stemcell growth without immune intervention As before this

follows a logistical growthmodel with a carrying capacity themaximal tumor cell burden The second term like the firstof the previous equation stands for differentiated tumor cellsproduced by CSCs The third term is almost identical to thethird term of the previous equation except that the functions119891119878 and 119892119878 represent the interaction of CSCs and CTLs withregard to TGF-120573 and MHC class I (as opposed to TCs in theabove equation) [11] Consider

1198621015840= 119891119862 ((119879 + 119878) sdot 119872II) 119892119862 (119865120573) minus 120583119862119862 + 119873 (119905) (4)

The first summand of the RHS stands for CTL recruit-ment from the blood system The recruitment function ispositively affected by MHC class II (119872II) and the number ofTCs (119879) and CSCs (119878) The cytokine TGF-120573 suppresses theproliferation and activation of T-lymphocytes [2] as well asleukocyte migration across the brain-blood boundary (BBB)[15] these are collectively represented by the function 119892119862We assume a constant death rate for 119862 represented by 120583119862The term119873(119905) describes the rate of infusion of primed CTLsdirectly to the tumor site119873(119905) is set equal to 0 in absence ofimmunotherapy [14 16] Consider

1198651015840120573 = 119891120573 (119879 + 119878) minus 120583120573119865120573

1198651015840120574 = 119891120574 (119862) minus 120583120574119865120574

(5)

The above two equations describe the dynamics of TGF-120573 and IFN-120574 respectively In the first equation the first termrepresents the natural basal level in the CNS (central nervoussystem) This includes TGF-120573 produced by the tumor whichis assumed to be proportional to the tumorrsquos size Thedegradation of TGF-120573 is assumed to be constant with the rate120583120573 and is represented by the second term

In the second equation the first term on the RHS is alinear production of IFN-120574 119865120574 We assume the only sourceof IFN-120574 is CTL The second term is the natural degradationof IFN-120574 with constant rate 120583120574 [15] Consider

1198721015840I = 119891119872I

(119865120574) minus 120583119872I119872I

1198721015840II = 119891119872II

(119865120573) 119892119872II(119865120574) minus 120583119872II

119872II(6)

The above two equations represent the dynamics ofMHCclasses I and II respectively For the first equation the firstterm on the RHS is the basal rate of119872I receptor expressionper tumor cell This includes the stimulation by IFN-120574 of119872Iexpression on the surface of a glioblastoma cell The secondterm is the natural degradation of119872I with constant rate 120583119872I

In the second equation the first term represents the rate

of119872II per tumor cell as a function of IFN-120574 and TGF-120573 [17]The second term is the degradation of119872II with constant rate120583119872II

[15]

3 Preliminary Results

Throughout the paper we will assume that system (1) issubject to nonnegative initial conditions In addition thefunctions 120572 1198771 1198772 119891119879 119891119878 119891119862 119891120573 119891120574 119891119872I

119891119872II 119892119879 119892119878

Computational and Mathematical Methods in Medicine 3

119892119862 119892119872II ℎ119879 and ℎ119878 are all C1 functions with nonnegative

values Here we use C1 to denote the space of continuouslydifferentiable functions These assumptions imply that thenonnegative orthant is invariant under (1) and there exists aunique solution to (1) subject to initial conditions To ensuresolutions to (1) stay bounded over time we need additionalassumptions We make the following mathematical assump-tionsmodified from [13] to account for cancer stem cells (A1)

(1) 1198771(119879) and 1198772(119878) are at most linear(2) 120572(119879 119878) is increasing(3) 119891119879(119865120573) and119891119878(119865120573) are decreasing and bounded below

119892119879(119872I) and 119892119878(119872I) are increasing and boundedabove

(4) ℎ119879 and ℎ119878 are decreasing and bounded below(5) 119891119862((119879 + 119878) sdot 119872II) is increasing and bounded above

119892119862(119865120573) is decreasing and bounded below(6) 119873(119905) is nonnegative and bounded above(7) 119891120573(119879) 119891120573(119878) and 119891120574(119862) are increasing(8) 119891119872I

(119865120574) is increasing and bounded above(9) 119892119872II

(119865120574) is increasing and bounded above 119891119872II(119865120573) is

decreasing and bounded below

We will use the following substitutions to simplify ourequations

119865120573 = 119909

119891120573 = 119891119909

120583120573 = 120583119909

119865120574 = 119910

119891120574 = 119891119910

120583120574 = 120583119910

119872I = 119906

119891119872I= 119891119906

120583119872I= 120583119906

119872II = V

119891119872II= 119891V

120583119872II= 120583V

(7)

These substitutions give us the following system equiva-lent to (1)

1198791015840= 120572 (119879 119878) + 1198771 (119879) 119879 minus 119891119879 (119909) 119892119879 (119906) ℎ119879 (119879) 119862119879

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 119891119878 (119909) 119892119878 (119906) ℎ119878 (119878) 119862119878

1198621015840= 119891119862 ((119879 + 119878) sdot V) 119892119862 (119909) minus 120583119862119862 + 119873 (119905)

1199091015840= 119891119909 (119879 + 119878) minus 120583119909119909

1199101015840= 119891119910 (119862) minus 120583119910119910

1199061015840= 119891119906 (119910) minus 120583119906119906

V1015840 = 119891V (119909) 119892V (119910) minus 120583VV(8)

Also as in [13] and included here for the readerrsquos benefitwe make the following biological assumptions on our system(A2)

(1) 1198771(119879) and 1198772(119878) are decreasing on [0 1198701] [0 1198702]respectively where 1198701 and 1198702 are their respectivecarrying capacities Furthermore 1198771(1198701) = 0 and1198772(1198702) = 0 Also 1198771(0) = 1199031 gt 0 and 1198772(0) = 1199032 gt 0

(2) 120572(119879 119878) = 119903120572(1198781198702)(1198791198701)(1198701 minus 119879) (when the CSCpopulation is small CSCs repopulate themselves butas their population grows they focus on populatingthe TC population) [11]

(3) 119891119879(119909) and 119891119878(119909) are decreasing 119891119879(0) = 119891119878(0) = 1and lim119909rarrinfin119891119879(119909) = 119886119879119909 lim119909rarrinfin119891119878(119909) = 119886119878119909 forsome 119886119879119909 119886119878119909 gt 0 (TGF-120573 decreases the efficacy oftumors killed by CTLs up to some limit)

(4) 119892119879(119906) 119892119878(119906) are increasing 119892119879(0) = 119892119878(0) = 0 andlim119906rarrinfin119892119879(119906) = 119886119879 gt 0 lim119906rarrinfin119891119878(119906) = 119886119878 gt 0

(MHC class I receptors are necessary for the action ofCTLs and increase their efficiency up to some limit)

(5) 119892119878(119906) lt 119892119879(119906) (CTLs are less efficient when attackingCSCs) [11]

(6) ℎ119879(119879) and ℎ119878(119878) are decreasing ℎ119879(0) = ℎ119878(0) = 1and lim119879rarrinfinℎ119879(119879) = 0 lim119878rarrinfinℎ119878(119878) = 0 (largetumormass hampers the access of CTLs to tumor cellsand reduces their kill rate)

(7) 119891119862(119879V+119878V) is increasing from 0 to 119886119862V gt 01198911015840119862(0) gt 0

and lim119879V+119878Vrarrinfin1198911015840119862(119879V+ 119878V) = 0 (the total number of

MHC class II receptors on all tumor cells and stemcells determines the recruitment of CTLs the rate ofCTL recruitment is limited and its growth decreasesto zero)

(8) 119892119862(119909) is decreasing from 1 to some bound greater than0 (TGF-120573 reduces recruitment of CTLs)

(9) 119891119909(119879 + 119878) = 119892119909 + 119886119909119879119879 + 119886119909119878119878 119891119910(119862) = 119886119910119862119862 where119886119909119879 119886119909119878 119886119910119862 gt 0 (TGF-120573 and IFN-120574 are secreted bythe cancerous cells and CTLs resp at constant ratethere is base level secretion of TGF-120573)

(10) 119891119906(0) = 119892119906 gt 0 and lim119910rarrinfin119891119906(119910) = 119892119906+119886119906119910 119886119906119910 gt 0(there is a constant basic production of MHC class IIreceptors at the cell surface while IFN-120574 increases thisproduction up to some level)

(11) 119891V(0) = 1 and 119891V(119909) is decreasing to 0 (TGF-120573decreases MHC class II production to 0)

(12) 119892V(0) = 0 119892V(119910) is increasing to 119886119906119910 gt 0 1198921015840V gt 0and lim119910rarrinfin119892

1015840V(119910) = 0 (IFN-120574 is necessary to induce


production of MHC class II receptors and increasesup to some level with increase declining to zero)

(13) 119873(119905) equiv 119873 (treatment is assumed to be constant withrespect to time)

Theorem 1 Under the (A2) assumptions system (1) is dissipa-tive on [0 1198701] times [0 1198702] times (Rge0)

5

Proof Let 119888max = max119891119862(119879V + 119878V)119892119862(119909) 119891119909 = (119892119879 + 119892119878) +(1198861199091198791198701 + 1198861199091198781198702) ge 119891119909(119879 + 119878) for all values of 119879 and 119878 119891119910 =119886119910119862119862max where119862max = max(119888max +119873)120583119862 119862(0) 119891119906 = 119892119906 +119886119906119910 and119891V = 119886119906119910These values are well defined based on the(A2) assumptions

In order to show that our system is dissipative we need toconstruct a119882 such that

nabla119882 sdot 119865 le 119860 minus 120575119882 where 119860 120575 gt 0 (9)

where 119865 is the RHS of system (1) Let119882(119879 119878 119862 119909 119910 119906 V) =119879+119878+119862+119909+119910+119906+VThen since1198771(0) = 0 and1198771(1198701) = 01198771(119879)119879 le 1198861 minus 1198871119879 for some 1198861 1198871 gt 0 on [0 1198701] Similarly1198772(119878)119878 le 1198862 minus 1198872119878 for some 1198862 1198872 gt 0 on [0 1198702]Thus

nabla119882 sdot 119865 = 1198771 (119879) 119879 minus 119891119879 (119909) 119892119879 (119906) 119862119879ℎ119879 (119879)

+ 1198772 (119878) 119878 minus 119891119878 (119909) 119892119878 (119906) 119862119878ℎ119878 (119878)

+ 119891119862 (119879V + 119878V) 119892119862 (119909) minus 120583119862119862 + 119873

+ 119891119909 (119879 + 119878) minus 120583119909119909 + 119891119910 (119862) minus 120583119910119910

+ 119891119906 (119910) minus 120583119906119906 + 119891V (119909) 119892V (119910) minus 120583VV

le 1198861 minus 1198871119879 + 1198862 minus 1198872119878 + 119888max minus 120583119862119862 + 119873 + 119891119909

minus 120583119909119909 + 119891119910 minus 120583119910119910 + 119891119906 minus 120583119906119906 + 119891V minus 120583V

le 119860 minus 120575119882

(10)

where 120575 = min1198871 1198872 120583119862 120583119909 120583119910 120583119906 120583V and 119860 = 1198861 + 1198862 +

119888max + 119873 + 119891119909 + 119891119910 + 119891119906 + 119891V

Since our inequalities1198771(119879)119879 le 1198861minus11988711198791198772(119878)119878 le 1198862minus1198872119878hold for all values of119879 119878 in this space the system is dissipativeeverywhere on [0 1198701] times [0 1198702] times (Rge0)

5 and by a theoremfrom Robinson [18] we get the following corollary

Corollary 2 System (1) has a compact global attractor on[0 1198701] times [0 1198702] times (Rge0)

5

4 Stability Analysis

In the following section we present an analysis of threepotential steady states tumor elimination (where CSC andTC populations are eradicated) recurrence (where the TCpopulation is eradicated but the CSC population persists)and coexistence (where CSC and TC populations persist) Ineach case we discuss sufficiency conditions on the treatmentterm 119873 which will allow for a globally asymptotically stablecure state (tumor elimination)

41 Semitrivial Tumor Elimination We begin our analysiswith tumor elimination Setting 119879 = 0 and 119878 = 0 we findthe equilibrium values

119862lowast=119873

120583119862

119909lowast=119891119909 (0)

120583119909

119910lowast=

119891119910 (119862lowast)

120583119910

119906lowast=119891119906 (119910lowast)

120583119906

Vlowast =119891V (119909lowast) 119892V (119910

lowast)

120583V

(11)

Substituting this equilibrium point into the Jacobian matrix

(((((((((((((((((

(

minus119873119891119878 (119909

lowast) 119892119878 (119906

lowast)

120583119862

+ 1199032 0 0 0 0 0 0

0 minus119873119891119879 (119909

lowast) 119892119879 (119906

lowast)

120583119862

+ 1199031 0 0 0 0 0

119891V (119909lowast) 119892119862 (119909

lowast) 119892V (119909

lowast) 1198911015840119862 (0)

120583V

119891V (119909lowast) 119892119862 (119909

lowast) 119892V (119909

lowast) 1198911015840119862 (0)

120583Vminus120583119862 0 0 0 0

0 0 1198911015840119910 (119862lowast) minus120583119910 0 0 0

1198911015840119909 (0) 119891

1015840119909 (0) 0 0 minus120583119909 0 0

0 0 0 1198911015840119906 (119910lowast) 0 minus120583119906 0

0 0 0 119891V (119909lowast) 1198921015840V (119910lowast) 1198911015840V (119909lowast) 119892V (119910

lowast) 0 minus120583V

)))))))))))))))))

)

(12)


yields eigenvalues 120582 = minus120583119862 minus120583119906 minus120583V minus120583119909 minus120583119910minus119873119891119878(119909

lowast)119892119878(119906lowast)120583119862 + 1199032 and minus119873119891119879(119909

lowast)119892119879(119906

lowast)120583119862 + 1199031 So

as long as we choose119873 large enough so that

119873 gt1199032 sdot 120583119862

119891119878 (119909lowast) 119892119878 (119906

lowast)

119873 gt1199031 sdot 120583119862

119891119879 (119909lowast) 119892119879 (119906

lowast)

(13)

we are guaranteed to have a locally asymptotically stable curestate

We now show that under necessary condition (13) (0 0119862lowast 119909lowast 119910lowast 119906lowast Vlowast) is locally asymptotically stable Let 119873 =

119873lowast where 119873lowast meets condition (13) Then for some initial

conditions there exists 119905120598 such that for 119905 ge 119905120598 119862 gt 119862lowast=

(119873lowast120583119862)(1 minus 120598) for arbitrarily small 120598 If we also let 119910lowast =

(119891119910(119862lowast)120583119910)(1 minus 1205981) and 119906

lowast= (119891119906(119910

lowast)120583119906)(1 minus 1205982) for

1205981 1205982 arbitrarily small we have that 119906 gt 119906lowast from some

starting moment and therefore 119892119879(119906)119862 gt 119892119879(119906lowast)119862lowast By our

assumptions (A1) ℎ119879(119879) ge ℎ119879(1198701) and 119891119879(119909) ge 119891119879(119909max)so if we let 119873lowast be large enough so that 119892119879(119906

lowast)119862lowastgt (1199031 +

119903120572119870214)ℎ119879(1198701)119891119879(119909max) then we have that

1198791015840le minus1198861119879

where 1198861 = 119892119879 (119906lowast) 119862lowastminus

1199031 + 119903120572119870214

ℎ119879 (1198701) 119891119879 (119909max)gt 0

(14)

A parallel argument works to show that for119873lowast large enough

1198781015840le minus1198862119878


1199032

ℎ119878 (1198702) 119891119878 (119909max)gt 0

(15)

Thus by increasing the treatment value119873 we are able to showthat for some set of initial conditions the CSC population119878 and TC population 119879 decay exponentially to 0 leading totumor elimination

42 Persistence of Tumor We now wish to consider steadystates in which some subset of cancer cell populations persistLet our hypothetical equilibrium point be (119879119901 119878119901 119862119901 119909119901119910119901 119906119901 V119901) where 119878119901 gt 0Then

119909119901 =

119891119909 (119879119901 + 119878119901)

120583119909

119910119901 =

119891119910 (119862119901)

120583119910

119906119901 =

119891119906 (119910119901)

120583119906

V119901 =119891V (119909119901) 119892V (119910119901)

120583V

(16)

We wish to show existence of 119879119901 119878119901 gt 0 and 119862119901 gt 0 To doso we must solve the system

1198771 (119879119901) + 120572 (119879119901 119878119901) = 119891119879(

119891119909 (119879119901 + 119878119901)

120583119909

)

sdot 119892119879(

119891119906 (119891119910 (119862119901120583119910))

120583119906

)ℎ119879 (119879119901) 119862119901

1198772 (119878119901) 119878119901 = 120572 (119879119901 119878119901) 119879119901 + 119891119878(

119891119909 (119879119901 + 119878119901)

120583119909

)

sdot 119892119878(

119891119906 (119891119910 (119862119901120583119910))

120583119906

)ℎ119878 (119878119901) 119878119901119862119901

120583119862119862119901

= 119891119862(

119891V (119891119909 (119879119901 + 119878119901) 120583119909) 119892V (119891119910 (119862119901) 120583119910) (119879119901 + 119878119901)

120583V)

sdot 119892119862(

119891119909 (119879119901 + 119878119901)

120583119909

) +119873

(17)

Defining the auxiliary function

119867119879119878 (119862)

= 119891119862(

119891V (119891119909 (119879 + 119878) 120583119909) 119892V (119891119910 (119862) 120583119910) (119879 + 119878)

120583V)

sdot 119892119862 (119891119909 (119879 + 119878)

120583119909

) minus 120583119862119862 + 119873

(18)

we see that for every 119879 119878 ge 0119867119879119878(0) = 119873 gt 0In addition taking the derivative with respect to119862 we get

1198671015840119879119878 (119862)

= 1198911015840119862(

119891V (119891119909 (119879 + 119878) 120583119909) 119892V (119891119910 (119862) 120583119910) (119879 + 119878)

120583V)

sdot119891V (119891119909 (119879 + 119878) 120583119909) (119879 + 119878)

120583V1198921015840V (

119891119910 (119862)

120583119910

)

1198911015840119910 (119862)

120583119910

sdot 119892119862 (119891119909 (119879 + 119878)

120583119909

) minus 120583119862

(19)

From assumptions (A2) we have that 1198671015840119879119878(119862) is decreasingso there is exactly one positive 119862 for which 119867119879119878(119862) = 0 forany given 119879 119878 Thus such 119862119901 gt 0 exists but to further solvesystem (17) we need more information about the arbitraryfunctions present

To move forward in the analysis of system (8) we willneed to make the following simplifying assumptions (A3)

(1) The dynamics of TGF-120573 are much faster than those ofthe other system components

(2) The inflow of CTLs is constant


With these assumptions we can assume 119909 is determined byits steady-state 119909lowast = 119891119909(119879 + 119878)120583119909 and we get the simplifiedsystem

1198791015840= 120572 (119879 119878) + 1198771 (119879) 119879 minus 119891119879 (119879 119878) 119892119879 (119906) ℎ119879 (119879) 119879119862

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 119891119878 (119879 119878) 119892119878 (119906) ℎ119878 (119878) 119878119862

1198621015840= minus120583119862119862 + 119873

1199101015840= 119891119910 (119862) minus 120583119910119910

1199061015840= 119891119906 (119910) minus 120583119906119906

V1015840 = 119891V (119879 119878) 119892V (119910) minus 120583VV

(20)

where

119891119879 (119879 119878) = 119891119879 (119891119909 (119879 + 119878)

120583119909

)

119891119878 (119879 119878) = 119891119878 (119891119909 (119879 + 119878)

120583119909

)

119892119862 (119879 119878) = 119892119862 (119891119909 (119879 + 119878)

120583119909

)

119891V (119879 119878) = 119891V (119891119909 (119879 + 119878)

120583119909

)

(21)

Note that if 119879 = 0 these equations will simply be referred toin terms of 119878 and vice versa

With simplifying assumptions (A3) we are able to studythe possible dynamics of persistence of cancer recurrenceand coexistence in the following subsections

43 Recurrence State Stability Weknow that for system (20)the equilibrium points for 119862 119910 and 119906 must be 119862lowast = 119873120583119862119910lowast= 119891119910(119862

lowast)120583119910 and 119906

lowast= 119891119906(119910

lowast)120583119906 In this section we

wish to study the recurrence steady state so we will set theTC population 119879 = 0 and observe the consequences for theCSC population steady state

1198781015840= 1198772 (119878

lowast) 119878lowastminus 119891119878 (119878

lowast) 119892119878 (119906

lowast) ℎ119878 (119878

lowast) 119878lowast119862lowast= 0 (22)

119878lowast will be a steady state when 119878lowast = 0 or

1198772 (119878lowast) minus 119891119878 (119878

lowast) 119892119878 (119906

lowast) ℎ119878 (119878

lowast) 119862lowast= 0 (23)

For further analysis we denote

119866 (119873) = 119892119878 (119906lowast(119873)) 119862

lowast(119873)

119867 (119878) = 119891119878 (119878) ℎ119878 (119878)

(24)

where 119906lowast(119873) = 119891119906(119873)120583119906 Note that (23) now becomes

1198772 (119878lowast) minus 119866 (119873)119867 (119878

lowast) = 0 (25)

where 119866(119873) is increasing in119873 at least linearly and 1198772(119878) and119867(119878) are both decreasing We recall from assumptions (A2)that 1198772(1198702) = 0 and119867(1198702) = 119891119878(1198702)ℎ119878(1198702) gt 0 We define

119871119873 (119878) = 1198772 (119878) minus 119866 (119873)119867 (119878) (26)

for which we know 119871119873(1198702) lt 0 for any119873 gt 0

We also know that 119871119873(0) = 1198772(0)minus119866(119873)119891119878(119892119909120583119909) Since119866(119873) is an increasing function its inverse exists and we candefine 119873min = 119866

minus1(1198772(0)119891119878(119892119909120583119909)) Notice that for 119873 =

119873min 119871119873(0) = 0 For119873 lt 119873min 119871119873(0) gt 0 and 119871119873(1198702) lt 0Therefore for119873 lt 119873min 119871119873(119878) = 0 has at least one solution119878lowastisin (0 1198702) and in general since the functionmust cross the

119878-axis an odd number of times there are an odd number ofsolutions 119878lowast1 119878

lowast119899

Without loss of generality we can assume 119862 119910 and 119906 areat their respective steady states since these variables will allconverge to their steady-state values exponentially We canalso assume that for large enough 119905 (20) is arbitrarily wellapproximated by

1198781015840= 1198772 (119878) 119878 minus 119867 (119878) 119866 (119873) 119878 = 119878119871119873 (119878) (27)

For119873 lt 119873min the equilibrium point (0 0 119862lowast 119910lowast 119906lowast Vlowast)is unstable since any values of 119878 less than 0 will yield anegative value for 119871119873(119878) and any values of 119878 between 0 and119878lowast1 will yield a positive value for 119871119873(119878) Thus our first stableequilibrium point is at (0 119878lowast1 119862

lowast 119910lowast 119906lowast Vlowast)

In contrast if 119873 is large enough so that 119873 gt 119873minour equilibrium point (0 0 119862lowast 119910lowast 119906lowast Vlowast) is locally stablesince for 119873 gt 119873min 119871119873(0) lt 0 There could however stillexist positive solutions 119878lowast1 119878

lowast119899 where 119899 is even If these

solutions are organized in nondecreasing order 119878lowast119894 is locallystable for even 119894 and unstable for odd 119894 since 119871119873(119878) gt 0

between an odd and even root and 119871119873(119878) lt 0 between aneven and odd root Note that if (0 0 119862lowast 119910lowast 119906lowast Vlowast) is the onlyequilibrium point it is globally asymptotically stable

We now show that if we increase our treatment term119873 we can guarantee the existence of a globally asymp-totically stable cure state Let alefsym be the maximum of1198772(119878)119866(119873min)119867(119878) and choose 119873thr such that 119866(119873thr) =

119866(119873min)alefsym Notice that this is possible since 119866(119873) is increas-ing and continuous Then for 119873 gt 119873thr 119871119873(119878) lt 0 for all0 le 119878 le 1198702 and so (0 0 119862lowast 119910lowast 119906lowast Vlowast) is now a globallyasymptotically stable equilibrium point

44 Coexistence State Stability Still working under simplify-ing assumptions (A3) we conclude our stability analysis withconsideration of a coexistence steady state As above for largevalues of 119905 we can expect119862 119910 and 119906 to be at steady state andso we can reduce our system to the two equations

1198791015840= 1198771 (119879) 119879 + 120572 (119879 119878) minus 1198671 (119879 119878) 1198661 (119873) 119879

= 119879(1198711 (119879 119878) + 119903120572

119878

11987021198701

(1198701 minus 119879))

equiv 1198791198721 (119879 119878)

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 1198672 (119879 119878) 1198662 (119873) 119878

= 119878(1198712 (119879 119878) minus 119903120572

119879

11987021198701

(1198701 minus 119879)) equiv 1198781198722 (119879 119878)

(28)

where 1198661(119873) = 119892119879(119906lowast(119873))119862

lowast(119873) 1198671(119879 119878) = 119891119879(119879

119878)ℎ119879(119879) 1198662(119873) = 119892119878(119906lowast(119873))119862

lowast(119873) and 1198672(119879 119878) =

119891119878(119879 119878)ℎ119878(119878) Define 119873min = min119873min119879 119873min119878 where


119873min119879 = 119866minus11 (1198771(0)119891119879(119892119909120583119909)) and 119873min119878 = 119866

minus12 ((1198772(0) minus

119903120572119870141198702)119891119878(119892119909120583119909))

Proposition 3 For119873 lt 119873min system (28) has a locally stablecoexistence steady state

Proof We begin by showing the existence of such a steadystateWithout loss of generality suppose119873min = 119873min119878Thenfor values of 119873 lt 119873min 1198722(119879 0) gt 0 and 1198722(1198791198702) lt 0

for all values of 119879 isin (0 1198701) Therefore there exists a value119878lowastisin (0 1198702) such that 1198722(119879 119878

lowast) = 0 for any 119879 isin (0 1198701)

In general there exist an odd number of solutions 119878lowast1 119878lowast119899

such that1198722(119879 119878lowast119894 ) = 0 for any 119879 isin (0 1198701) 119894 = 1 119899

Likewise by our choice of 119873min 1198721(0 119878) gt 0 and1198721(1198701 119878) lt 0 for all values of 119878 isin (0 1198702) Therefore for119873 lt 119873min there exists a value 119879lowast isin (0 1198701) such that1198721(119879

lowast 119878) = 0 for all 119878 isin (0 1198702) and in general there exist an

odd number of solutions 119879lowast1 119879lowast119898 such that1198721(119879

lowast119895 119878) = 0

for any 119878 isin (0 1198702) 119895 = 1 119898Thus we are able to deduce the existence of an equi-

librium point (119879lowast 119878lowast) in (0 1198701) times (0 1198702) for system (28)Moreover notice that while (0 0) is an equilibrium point ofsystem (28) it is unstable since1198721(119879 119878) gt 0 for 0 le 119879 lt 119879

lowast1

119878 isin (0 1198702) and1198722(119879 119878) gt 0 for 0 le 119878 lt 119878lowast1 119879 isin (0 1198701)

In fact if we denote 1198790 = 0 and 1198780 = 0 equilibrium points(119879lowast119895 119878lowast119894 ) 119895 = 0 119898 119894 = 0 119899 are locally unstable when

119894 or 119895 is even and locally stable when 119894 and 119895 are odd

Remark 4 Note that in the absence of treatment 119873 = 0 lt

119873minTherefore in the case where the tumor is left untreatedcancer persists

We now show that by increasing the treatment term 119873we will achieve a globally asymptotically stable cure steadystate For119873 ge 119873min let us consider the system

1198791015840= 119879 (1198711 (119879 119878) + 1199031205721198701)

1198781015840= 1198781198712 (119879 119878)

(29)

Define alefsym119878 as the maximum of 1198772(119878)1198662(119873min119878)1198672(1198701 119878)and choose 119873thr119878 such that 1198662(119873thr119878) = 1198662(119873min119878)alefsym119878Similarly let alefsym119879 be the maximum of (1198771(119879) + 1199031205721198701)

1198661(119873min119879)1198671(1198791198702) and choose 119873thr119879 such that1198661(119873thr119879) = 1198661(119873min119879)alefsym119879 Let119873cure = max119873thr119879 119873thr119878

Proposition 5 For 119873 gt 119873119888119906119903119890 (0 0) is a globally asymptoti-cally stable equilibrium point of system (29)

Proof For119873 gt 119873thr119879

1198711 (119879 119878) + 1199031205721198701 lt 1198771 (119879) minus 1198661 (119873thr119879)1198671 (119879 119878)

+ 1199031205721198701

= 1198771 (119879) minus alefsym1198791198661 (119873min119879)1198671 (119879 119878)

+ 1199031205721198701

= 1198771 (119879) + 1199031205721198701

minus1198671 (119879 119878)

1198671 (1198791198701)(1198771 (119879) + 1199031205721198701) le 0

(30)

for 0 le 119879 le 1198701 since1198671 is decreasing based on assumptions(A2)Thus 1198711(119879 119878)+1199031205721198701 lt 0 for all (119879 119878) isin [0 1198701]times[0 1198702]Similarly for 119873 gt 119873thr119878 1198712(119879 119878) lt 0 for all (119879 119878) isin

[0 1198701] times [0 1198702] Therefore (0 0) is the only equilibriumpoint for system (29) and (0 0) is globally asymptoticallystable

We conclude our analysis by noting that

1198791015840= 119879(1198711 (119879 119878) + 119903120572

119878

11987021198701

(1198701 minus 119879))

le 119879 (1198711 (119879 119878) + 1199031205721198701)

1198781015840= 119878(1198712 (119879 119878) minus 119903120572

119879

1198702

(1198701 minus 119879)) le 1198781198712 (119879 119878)

(31)

for all (119879 119878) isin [0 1198701] times [0 1198702]

Corollary 6 For119873 gt 119873119888119906119903119890 (0 0) is a globally asymptoticallystable equilibrium solution to system (28)

5 Example

In this section we present an example to illustrate thetheory presented above This model is a modification ofthe biologically verified system presented in Kronik et al[10] to address the CSC Hypothesis as presented in thispaper Modifications include the incorporation of CSCs andamending the CTL equation to satisfy assumptions (A3)Consider the following system

119889119879

119889119905

= 1199031119879(1 minus119879

1198701

) + 119903120572

119879

1198701

119878

1198702

(1198701 minus 119879)

minus 119886119879

119872I119872I + 119890119879

(119886119879120573 +

119890119879120573 (1 minus 119886119879120573)

119865120573 + 119890119879120573

)119862119879

ℎ119879 + 119879

119889119878

119889119905

= 1199032119878 (1 minus119878

1198702

) minus 119903120572

119879

1198701

119878

1198702

(1198701 minus 119879)

minus 119886119878

119872I119872I + 119890119878

(119886119878120573 +

119890119878120573 (1 minus 119886119878120573)

119865120573 + 119890119878120573

)119862119878

ℎ119878 + 119878

119889119862

119889119905= minus120583119862119862 + 119873

119889119865120573

119889119905= 119892120573 + 119886120573119879119879 + 119886120573119878119878 minus 120583120573119865120573


Table 1 Parameter values

Parameter Value Units Reference1199031 001 hminus1 Based on data from Burger et al [19]1198701 10

8 Cell Turner et al [20]119886119879 12 hminus1 Based on data from Arciero et al [21] and Wick et al [22]119890119879 50 recsdotcellminus1 Based on data from Kageyama et al [23]119886119879120573 69 None Thomas and Massague [2]119890119879120573 10

4 pg Based on data from Peterson et al [24]ℎ119879 5 lowast 10

8 Cell Based on data from Kruse et al [14]1199032 1 hminus1 Vainstein et al [8]1198702 10

7 Cell Turner et al [20]119903120572 006 hminus1 Vainstein et al [8]119886119878 1 lowast 119886119879 hminus1 Estimated based on data from Prince et al [11]119890119878 119890119879 recsdotcellminus1 Estimated119886119878120573 119886119879120573 None Estimated119890119878120573 119890119879120573 pg Estimatedℎ119878 ℎ119879 Cell Estimated120583119862 007 hminus1 Taylor et al [25]119892120573 63945 lowast 10

4 pgsdothminus1 Peterson et al [24]119886120573119879 575 lowast 10

minus6 pgsdotcellminus1sdothminus1 Peterson et al [24]119886120573119878 119886120573119879 pgsdotcellminus1sdothminus1 Estimated120583120573 7 hminus1 Coffey Jr et al [26]119892119872I

144 recsdotcellminus1sdothminus1 Based on data from Kageyama et al [23]119886119872I 120574

288 recsdotcellminus1sdothminus1 Based on data from Yang et al [27]119890119872I 120574

338 lowast 105 pg Based on data from Yang et al [27]

120583119872I0144 hminus1 Milner et al [28]

119886119872II 1205748660 recsdotcellminus1sdothminus1 Based on data from Phillips et al [29] and Bosshart and Jarrett [30]

119890119872II 1205741420 pg Based on data from Phillips et al [29] and Bosshart and Jarrett [30]

119886119872II 120573012 None Based on data from Suzumura et al [17]

119890119872II 120573105 pg Based on data from Suzumura et al [17]

120583119872II0144 hminus1 Based on data from Lazarski et al [31]

119886120574119862 102 lowast 10minus4 pgsdotcellminus1sdothminus1 Kim et al [32]

120583120574 102 hminus1 Turner et al [33]

119889119865120574

119889119905= 119886120574119862119862 minus 120583120574119865120574

119889119872I119889119905

= 119892119872I+

119886119872I 120574119865120574

119865120574 + 119890119872I 120574

minus 120583119872I119872I

119889119872II119889119905

=

119886119872II 120574119865120574

119865120574 + 119890119872II 120574

(

119890119872II 120573(1 minus 119886119872II 120573

)

119865120573 + 119890119872II 120573

+ 119886119872II 120573)

minus 120583119872II119872II

(32)

subject to the initial conditions 119879(0) = 70 119878(0) = 30 119862(0) =250 119865120573(0) = 50 119865120574(0) = 50119872I(0) = 50 and119872II(0) = 50Parameter values are given by Table 1 calculations for theseparameter values can be found in [8 10 20]

The equilibrium values for 119862 119865120573 119865120574119872I and119872II are

119862lowast=119873

120583119862

119865lowast120573 =

119892120573 + 119886120573119878119878 + 119886120573119879119879

120583120573

119865lowast120574 =

119886120574119862119862lowast

120583120574

119872lowastI =

119890119872I 120574119892119872I

+ (119886119872I 120574+ 119892119872I

) 119865lowast120574

120583119872I(119890119872I 120574

+ 119865lowast120574 )

119872lowastII =

119886119872II 120574(119890119872II 120573

+ 119886119872II 120573119865lowast120573 ) 119865lowast120574

120583119872II(119890119872II 120573

+ 119865lowast120573) (119890119872II 120574

+ 119865lowast120574 )

(33)


Tumor cells times 108

CSCs times 107

CTLs times 100

05

10

15

20

25

2000500 1000 30002500 35001500

Figure 1119873 = 0

Following calculations from Section 44 we get 1198661(119873) =(119886119879(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

))(119892119872I+ 119886119872I 120574

119865lowast120574 (119865lowast120574 +

119890119872I 120574) + 119890119879))(119873120583119862) and 1198662(119873) = (119886119878(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 +

119890119872I 120574))(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

) + 119890119878))(119873120583119862) This gives

119873min119879

= 119866minus11 (

1199031

119886119879120573 + 119890119879120573 (1 minus 119886119879120573) (119892120573120583120573 + 119890119879120573)

)

= 119866minus11 (00117) = 00245

119873min119878

= 119866minus12 (

1199032 minus 119903120572119870141198702

119886119878120573 + 119890119878120573 (1 minus 119886119878120573) (119892120573120583120573 + 119890119878120573)

)

= 119866minus12 (09976) = 207889

(34)

Thus in the case of no treatment cancer will persist (seeFigure 1)

When we increase treatment past 119873min we are able tofind initial conditions that will allow a locally stable cureor recurrence state This can be seen by adding a treatmentvalue of119873 = 1 and increasing our initial amount of CTLs to119862(0) = 25 lowast 10

10 (see Figure 2)When wemaximize (1198771(119879)+1199031205721198701)1198661(119873min119879)1198671(1198791198702)

on the interval [0 1198701] we calculate alefsym119879 = 360322 lowast 1017

Similarly when we maximize 1198772(119878)1198662(119873min119878)1198672(119879 119878) onthe interval [0 1198702] we calculate alefsym119878 = 14866 lowast 10

15 Since

1198661(119873min119879) = 00245 and 1198662(119873min119878) = 207889 we have

119873thr119879 = 119866minus11 (00245 lowast 360322 lowast 10

17)

= 310199 lowast 1014


15)

= 108783 lowast 1015

(35)


CSCs times 107

CTLs times 1010

1000 1500 2000 2500 3000 3500500

05

10

15

20

25

Figure 2 A locally stable recurrence state


CSCs times 105

1

2

3

4

5

6

7

004 006 008 010002

Figure 3 Even with large initial populations of tumor cells 119879(0) =7lowast10

5 and CSCs 119878(0) = 3lowast105 a cure state is rapidly achieved when119873 = 108783 lowast 10

15

Taking the maximum of these two values we find119873cure =

108783 lowast 1015 (see Figure 3)

6 Conclusion

In this paper we extend a previous model for the treatment ofglioblastomamultiformewith immunotherapy by accountingfor the existence of cancer stem cells that can lead to therecurrence of cancer when not treated to completion Weprove existence of a coexistence steady state (one where bothtumor and cancer stem cells survive treatment) a recurrencesteady state (one where cancer stem cells survive treatmentbut tumor cells do not hence upon discontinuation oftreatment the tumor would be repopulated by the survivingcancer stemcells) and a cure state (onewhere both tumor andcancer stem cells are eradicated by treatment) Furthermorewe categorize the stability of the previously mentioned steadystates depending on the amount of treatment administeredFinally in each case we establish sufficiency conditions onthe treatment term for the existence of a globally asymptot-ically stable cure state It should be noted that the amount


of treatment necessary to eliminate all cancer stem cells aswell as tumor cells is higher than the amount of treatmentnecessary to merely eliminate tumor cells based on the lowervalue of 119892119878 (see (A2)) These results are an improvementon previous models that do not account for the existence ofcancer stem cells and therefore yield an artificially low valueof treatment necessary for a cure state

However it is important to note that the values oftreatment for which we achieve a globally stable cure stateare sufficient but not necessary We make the assumptionthat the production of cytotoxic-T-lymphocytes is constant inorder to simplify our analysis (see (A2) and (A3)) leading toa potentially higher-than-necessary value of treatment (sincethe treatment would ordinarily be helped along by the bodyrsquosnatural production of CTLs not relying on the treatment119873alone) A natural extension of this work would account forthe bodyrsquos natural production of CTLs in the analysis of therecurrence and coexistence steady states as well as allowingthe treatment term119873 to vary over time

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This material is based upon work supported by the NationalScience Foundation under Grant no DMS-1358534

References

[1] Y Louzoun C Xue G B Lesinski and A Friedman ldquoA math-ematical model for pancreatic cancer growth and treatmentsrdquoJournal of Theoretical Biology vol 351 pp 74ndash82 2014

[2] D A Thomas and J Massague ldquoTGF-120573 directly targetscytotoxic T cell functions during tumor evasion of immunesurveillancerdquo Cancer Cell vol 8 no 5 pp 369ndash380 2005

[3] J Mason A stochastic Markov chain model to describe cancermetastasis [PhD thesis] University of Southern California2013

[4] N L Komarova ldquoSpatial stochastic models of cancer fitnessmigration invasionrdquoMathematical Biosciences and Engineeringvol 10 no 3 pp 761ndash775 2013

[5] B T Tan C Y Park L E Ailles and I L Weissman ldquoThecancer stem cell hypothesis a work in progressrdquo LaboratoryInvestigation vol 86 no 12 pp 1203ndash1207 2006

[6] E Beretta V Capasso and N Morozova ldquoMathematical mod-elling of cancer stem cells population behaviorrdquo MathematicalModelling of Natural Phenomena vol 7 no 1 pp 279ndash305 2012

[7] G Hochman and Z Agur ldquoDeciphering fate decision in normaland cancer stem cells mathematical models and their exper-imental verificationrdquo in Mathematical Methods and Models inBiomedicine Lecture Notes on Mathematical Modelling in theLife Sciences pp 203ndash232 Springer New York NY USA 2013

[8] V Vainstein O U Kirnasovsky Y Kogan and Z AgurldquoStrategies for cancer stem cell elimination insights frommathematicalmodelingrdquo Journal ofTheoretical Biology vol 298pp 32ndash41 2012

[9] H YoussefpourMathematical modeling of cancer stem cells andtherapeutic intervention methods [PhD thesis] University ofCalifornia Irvine Calif USA 2013

[10] N Kronik Y Kogan V Vainstein and Z Agur ldquoImprovingalloreactive CTL immunotherapy for malignant gliomas usinga simulation model of their interactive dynamicsrdquo CancerImmunology Immunotherapy vol 57 no 3 pp 425ndash439 2008

[11] M E Prince R Sivanandan A Kaczorowski et al ldquoIdentifica-tion of a subpopulation of cells with cancer stem cell propertiesin head and neck squamous cell carcinomardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 104 no 3 pp 973ndash978 2007

[12] R Ganguly and I K Puri ldquoMathematical model for the cancerstem cell hypothesisrdquo Cell Proliferation vol 39 no 1 pp 3ndash142006

[13] Y Kogan U Forys and N Kronik ldquoAnalysis of theimmunotherapy model for glioblastoma multiform braintumourrdquo Tech Rep 178 Institute of Applied Mathematics andMechanics UW 2008

[14] C A Kruse L Cepeda B Owens S D Johnson J Stears andKO Lillehei ldquoTreatment of recurrent gliomawith intracavitaryalloreactive cytotoxic T lymphocytes and interleukin-2rdquoCancerImmunology Immunotherapy vol 45 no 2 pp 77ndash87 1997

[15] W F Hickey ldquoBasic principles of immunological surveillanceof the normal central nervous systemrdquo Glia vol 36 no 2 pp118ndash124 2001

[16] D Kirschner and J C Panetta ldquoModeling immunotherapyof the tumormdashimmune interactionrdquo Journal of MathematicalBiology vol 37 no 3 pp 235ndash252 1998

[17] A Suzumura M Sawada H Yamamoto and T MarunouchildquoTransforming growth factor-120573 suppresses activation and pro-liferation of microglia in vitrordquoThe Journal of Immunology vol151 no 4 pp 2150ndash2158 1993

[18] J C Robinson Infinite-Dimensional Dynamical Systems Cam-bridge University Press New York NY USA 2001

[19] P C Burger F S Vogel S B Green and T A StrikeldquoGlioblastoma multiforme and anaplastic astrocytoma patho-logic criteria and prognostic implicationsrdquo Cancer vol 56 no5 pp 1106ndash1111 1985

[20] C Turner A R Stinchcombe M Kohandel S Singh and SSivaloganathan ldquoCharacterization of brain cancer stem cells amathematical approachrdquo Cell Proliferation vol 42 no 4 pp529ndash540 2009

[21] J C Arciero T L Jackson andD E Kirschner ldquoAmathematicalmodel of tumor-immune evasion and siRNA treatmentrdquo Dis-crete and Continuous Dynamical Systems Series B vol 4 no 1pp 39ndash58 2004

[22] W D Wick O O Yang L Corey and S G Self ldquoHow manyhuman immunodeficiency virus type 1-infected target cells cana cytotoxic T-lymphocyte killrdquo Journal of Virology vol 79 no21 pp 13579ndash13586 2005

[23] S Kageyama T J Tsomides Y Sykulev and H N EisenldquoVariations in the number of peptide-MHC class I complexesrequired to activate cytotoxic T cell responsesrdquo The Journal ofImmunology vol 154 no 2 pp 567ndash576 1995

[24] P K Peterson C C Chao S Hu KThielen and E G ShaskanldquoGlioblastoma transforming growth factor-120573 and Candidameningitis a potential linkrdquoThe American Journal of Medicinevol 92 no 3 pp 262ndash264 1992

[25] G P Taylor S E Hall S Navarrete et al ldquoEffect of lamivu-dine on human T-cell leukemia virus type 1 (HTLV-1) DNA


copy number T-cell phenotype and anti-tax cytotoxic T-cellfrequency in patients with HTLV-1-associated myelopathyrdquoJournal of Virology vol 73 no 12 pp 10289ndash10295 1999

[26] R J Coffey Jr L J Kost R M Lyons H L Moses and NF LaRusso ldquoHepatic processing of transforming growth factor120573 in the rat uptake metabolism and biliary excretionrdquo TheJournal of Clinical Investigation vol 80 no 3 pp 750ndash757 1987

[27] I Yang T J Kremen A J Giovannone et al ldquoModulationof major histocompatibility complex Class I molecules andmajor histocompatibility complexmdashbound immunogenic pep-tides induced by interferon-120572 and interferon-120574 treatment ofhuman glioblastoma multiformerdquo Journal of Neurosurgery vol100 no 2 pp 310ndash319 2004

[28] E Milner E Barnea I Beer and A Admon ldquoThe turnoverkinetics of MHC peptides of human cancer cellsrdquoMolecular ampCellular Proteomics vol 5 pp 366ndash378 2006

[29] L M Phillips P J Simon and L A Lampson ldquoSite-specificimmune regulation in the brain differential modulation ofmajor histocompatibility complex (MHC) proteins in brain-stem vs hippocampusrdquo Journal of Comparative Neurology vol405 no 3 pp 322ndash333 1999

[30] H Bosshart and R F Jarrett ldquoDeficient major histocompatibil-ity complex class II antigen presentation in a subset ofHodgkinrsquosdisease tumor cellsrdquo Blood vol 92 no 7 pp 2252ndash2259 1998

[31] C A Lazarski F A Chaves S A Jenks et al ldquoThe kineticstability of MHC class II peptide complexes is a key parameterthat dictates immunodominancerdquo Immunity vol 23 no 1 pp29ndash40 2005

[32] J J Kim L K Nottingham J I Sin et al ldquoCD8 positive Tcells influence antigen-specific immune responses through theexpression of chemokinesrdquoThe Journal of Clinical Investigationvol 102 no 6 pp 1112ndash1124 1998

[33] P K Turner J A Houghton I Petak et al ldquoInterferon-gammapharmacokinetics and pharmacodynamics in patients withcolorectal cancerrdquo Cancer Chemotherapy and Pharmacologyvol 53 no 3 pp 253ndash260 2004

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


2 Presentation of the Model

In this paper we extend the previous work of Kogan et al[13] to include cancer stem cells in modeling the treatmentof glioblastomamultiformewith immunotherapyWe presentan abstract model that can be adapted to fit various biologicalassumptions We analyze the stability of the model both withand without treatment and derive sufficient conditions ontreatment to ensure a globally asymptotically stable cure stateWe conclude with an example illustrating the transition fromcoexistence of cancer cells to eradication of cancer cells withvarious treatment levels Much of this model is based onexperimental results obtained by Kruse et al [14]

The following system models the dynamics of tumorcells (119879) cancer stem cells (119878) alloreactive cytotoxic-T-lymphocytes (119862) TGF-120573 (119865120573) IFN-120574 (119865120574) and major histo-compatibility complex classes I and II (119872I and119872II resp)

1198791015840= 120572 (119879 119878) + 1198771 (119879) 119879

minus 119891119879 (119865120573) 119892119879 (119872I) ℎ119879 (119879) 119862119879

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 119891119878 (119865120573) 119892119878 (119872I) ℎ119878 (119878) 119862119878

1198621015840= 119891119862 ((119879 + 119878) sdot 119872II) 119892119862 (119865120573) minus 120583119862119862 + 119873 (119905)

1198651015840120573 = 119891120573 (119879 + 119878) minus 120583120573119865120573

1198651015840120574 = 119891120574 (119862) minus 120583120574119865120574

1198721015840I = 119891119872I

(119865120574) minus 120583119872I119872I

1198721015840II = 119891119872II

(119865120573) 119892119872II(119865120574) minus 120583119872II

119872II

(1)

To understand the formation of the system above wediscuss the biological interpretations of each equation in themodel

1198791015840= 120572 (119879 119878) + 1198771 (119879) 119879 minus 119891119879 (119865120573) 119892119879 (119872I) ℎ119879 (119879) 119862119879 (2)

The first term on the right hand side (RHS) of theequation represents differentiated tumor cells produced bythe CSCs without immune intervention where 120572(119879 119878) is therate at which CSCs produce tumor cells and 119879 is the numberof nonstem tumor cells (TCs) currently present The secondterm stands for normal tumor growth the cells producedby regular reproduction of nonstem tumor cells Both thefirst and second terms use classical logistic growth (note thatthe carrying capacities for 119878 and 119879 are distinct) The thirdterm on the RHS represents tumor elimination by CTL inproportion to both 119879 and 119862 CTLs are white blood cellsresponsible for attacking tumor cells in this case The thirdterm also introduces the effects of MHC class I receptors(119872I) and TGF-120573 (119865120573) which is assumed to be a majorimmunosuppressive factor for CTL activity [10] Consider

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 119891119878 (119865120573) 119892119878 (119872I) ℎ119878 (119878) 119862119878 (3)

The first term on the RHS stands for the rate of stemcell growth without immune intervention As before this

follows a logistical growthmodel with a carrying capacity themaximal tumor cell burden The second term like the firstof the previous equation stands for differentiated tumor cellsproduced by CSCs The third term is almost identical to thethird term of the previous equation except that the functions119891119878 and 119892119878 represent the interaction of CSCs and CTLs withregard to TGF-120573 and MHC class I (as opposed to TCs in theabove equation) [11] Consider

1198621015840= 119891119862 ((119879 + 119878) sdot 119872II) 119892119862 (119865120573) minus 120583119862119862 + 119873 (119905) (4)

The first summand of the RHS stands for CTL recruit-ment from the blood system The recruitment function ispositively affected by MHC class II (119872II) and the number ofTCs (119879) and CSCs (119878) The cytokine TGF-120573 suppresses theproliferation and activation of T-lymphocytes [2] as well asleukocyte migration across the brain-blood boundary (BBB)[15] these are collectively represented by the function 119892119862We assume a constant death rate for 119862 represented by 120583119862The term119873(119905) describes the rate of infusion of primed CTLsdirectly to the tumor site119873(119905) is set equal to 0 in absence ofimmunotherapy [14 16] Consider

1198651015840120573 = 119891120573 (119879 + 119878) minus 120583120573119865120573

1198651015840120574 = 119891120574 (119862) minus 120583120574119865120574

(5)

The above two equations describe the dynamics of TGF-120573 and IFN-120574 respectively In the first equation the first termrepresents the natural basal level in the CNS (central nervoussystem) This includes TGF-120573 produced by the tumor whichis assumed to be proportional to the tumorrsquos size Thedegradation of TGF-120573 is assumed to be constant with the rate120583120573 and is represented by the second term

In the second equation the first term on the RHS is alinear production of IFN-120574 119865120574 We assume the only sourceof IFN-120574 is CTL The second term is the natural degradationof IFN-120574 with constant rate 120583120574 [15] Consider

1198721015840I = 119891119872I

(119865120574) minus 120583119872I119872I

1198721015840II = 119891119872II

(119865120573) 119892119872II(119865120574) minus 120583119872II

119872II(6)

The above two equations represent the dynamics ofMHCclasses I and II respectively For the first equation the firstterm on the RHS is the basal rate of119872I receptor expressionper tumor cell This includes the stimulation by IFN-120574 of119872Iexpression on the surface of a glioblastoma cell The secondterm is the natural degradation of119872I with constant rate 120583119872I

In the second equation the first term represents the rate

of119872II per tumor cell as a function of IFN-120574 and TGF-120573 [17]The second term is the degradation of119872II with constant rate120583119872II

[15]

3 Preliminary Results

Throughout the paper we will assume that system (1) issubject to nonnegative initial conditions In addition thefunctions 120572 1198771 1198772 119891119879 119891119878 119891119862 119891120573 119891120574 119891119872I

119891119872II 119892119879 119892119878












119865120573 = 119909

119891120573 = 119891119909

120583120573 = 120583119909

119865120574 = 119910

119891120574 = 119891119910

120583120574 = 120583119910

119872I = 119906

119891119872I= 119891119906

120583119872I= 120583119906

119872II = V

119891119872II= 119891V

120583119872II= 120583V

(7)


1198791015840= 120572 (119879 119878) + 1198771 (119879) 119879 minus 119891119879 (119909) 119892119879 (119906) ℎ119879 (119879) 119862119879

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 119891119878 (119909) 119892119878 (119906) ℎ119878 (119878) 119862119878

1198621015840= 119891119862 ((119879 + 119878) sdot V) 119892119862 (119909) minus 120583119862119862 + 119873 (119905)

1199091015840= 119891119909 (119879 + 119878) minus 120583119909119909

1199101015840= 119891119910 (119862) minus 120583119910119910

1199061015840= 119891119906 (119910) minus 120583119906119906

V1015840 = 119891V (119909) 119892V (119910) minus 120583VV(8)






















5






+ 1198772 (119878) 119878 minus 119891119878 (119909) 119892119878 (119906) 119862119878ℎ119878 (119878)

+ 119891119862 (119879V + 119878V) 119892119862 (119909) minus 120583119862119862 + 119873

+ 119891119909 (119879 + 119878) minus 120583119909119909 + 119891119910 (119862) minus 120583119910119910

+ 119891119906 (119910) minus 120583119906119906 + 119891V (119909) 119892V (119910) minus 120583VV



le 119860 minus 120575119882

(10)

where 120575 = min1198871 1198872 120583119862 120583119909 120583119910 120583119906 120583V and 119860 = 1198861 + 1198862 +

119888max + 119873 + 119891119909 + 119891119910 + 119891119906 + 119891V




5




119862lowast=119873

120583119862

119909lowast=119891119909 (0)

120583119909

119910lowast=

119891119910 (119862lowast)

120583119910

119906lowast=119891119906 (119910lowast)

120583119906


lowast)

120583V

(11)


(((((((((((((((((

(

minus119873119891119878 (119909

lowast) 119892119878 (119906

lowast)

120583119862

+ 1199032 0 0 0 0 0 0

0 minus119873119891119879 (119909

lowast) 119892119879 (119906

lowast)

120583119862

+ 1199031 0 0 0 0 0

119891V (119909lowast) 119892119862 (119909

lowast) 119892V (119909

lowast) 1198911015840119862 (0)

120583V

119891V (119909lowast) 119892119862 (119909

lowast) 119892V (119909

lowast) 1198911015840119862 (0)

120583Vminus120583119862 0 0 0 0

0 0 1198911015840119910 (119862lowast) minus120583119910 0 0 0

1198911015840119909 (0) 119891

1015840119909 (0) 0 0 minus120583119909 0 0

0 0 0 1198911015840119906 (119910lowast) 0 minus120583119906 0



)))))))))))))))))

)

(12)




lowast)119892119879(119906

lowast)120583119862 + 1199031 So


119873 gt1199032 sdot 120583119862

119891119878 (119909lowast) 119892119878 (119906

lowast)

119873 gt1199031 sdot 120583119862

119891119879 (119909lowast) 119892119879 (119906

lowast)

(13)






(119891119910(119862lowast)120583119910)(1 minus 1205981) and 119906

lowast= (119891119906(119910

lowast)120583119906)(1 minus 1205982) for






1198791015840le minus1198861119879


1199031 + 119903120572119870214

ℎ119879 (1198701) 119891119879 (119909max)gt 0

(14)


1198781015840le minus1198862119878


1199032

ℎ119878 (1198702) 119891119878 (119909max)gt 0

(15)



119909119901 =

119891119909 (119879119901 + 119878119901)

120583119909

119910119901 =

119891119910 (119862119901)

120583119910

119906119901 =

119891119906 (119910119901)

120583119906

V119901 =119891V (119909119901) 119892V (119910119901)

120583V

(16)


1198771 (119879119901) + 120572 (119879119901 119878119901) = 119891119879(

119891119909 (119879119901 + 119878119901)

120583119909

)

sdot 119892119879(

119891119906 (119891119910 (119862119901120583119910))

120583119906

)ℎ119879 (119879119901) 119862119901

1198772 (119878119901) 119878119901 = 120572 (119879119901 119878119901) 119879119901 + 119891119878(

119891119909 (119879119901 + 119878119901)

120583119909

)

sdot 119892119878(

119891119906 (119891119910 (119862119901120583119910))

120583119906

)ℎ119878 (119878119901) 119878119901119862119901

120583119862119862119901

= 119891119862(

119891V (119891119909 (119879119901 + 119878119901) 120583119909) 119892V (119891119910 (119862119901) 120583119910) (119879119901 + 119878119901)

120583V)

sdot 119892119862(

119891119909 (119879119901 + 119878119901)

120583119909

) +119873

(17)


119867119879119878 (119862)

= 119891119862(

119891V (119891119909 (119879 + 119878) 120583119909) 119892V (119891119910 (119862) 120583119910) (119879 + 119878)

120583V)

sdot 119892119862 (119891119909 (119879 + 119878)

120583119909

) minus 120583119862119862 + 119873

(18)


1198671015840119879119878 (119862)

= 1198911015840119862(

119891V (119891119909 (119879 + 119878) 120583119909) 119892V (119891119910 (119862) 120583119910) (119879 + 119878)

120583V)

sdot119891V (119891119909 (119879 + 119878) 120583119909) (119879 + 119878)

120583V1198921015840V (

119891119910 (119862)

120583119910

)

1198911015840119910 (119862)

120583119910

sdot 119892119862 (119891119909 (119879 + 119878)

120583119909

) minus 120583119862

(19)







1198791015840= 120572 (119879 119878) + 1198771 (119879) 119879 minus 119891119879 (119879 119878) 119892119879 (119906) ℎ119879 (119879) 119879119862

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 119891119878 (119879 119878) 119892119878 (119906) ℎ119878 (119878) 119878119862

1198621015840= minus120583119862119862 + 119873

1199101015840= 119891119910 (119862) minus 120583119910119910

1199061015840= 119891119906 (119910) minus 120583119906119906

V1015840 = 119891V (119879 119878) 119892V (119910) minus 120583VV

(20)

where

119891119879 (119879 119878) = 119891119879 (119891119909 (119879 + 119878)

120583119909

)

119891119878 (119879 119878) = 119891119878 (119891119909 (119879 + 119878)

120583119909

)

119892119862 (119879 119878) = 119892119862 (119891119909 (119879 + 119878)

120583119909

)

119891V (119879 119878) = 119891V (119891119909 (119879 + 119878)

120583119909

)

(21)




lowast)120583119910 and 119906

lowast= 119891119906(119910



1198781015840= 1198772 (119878


lowast) 119892119878 (119906

lowast) ℎ119878 (119878



1198772 (119878lowast) minus 119891119878 (119878

lowast) 119892119878 (119906

lowast) ℎ119878 (119878



119866 (119873) = 119892119878 (119906lowast(119873)) 119862

lowast(119873)

119867 (119878) = 119891119878 (119878) ℎ119878 (119878)

(24)


1198772 (119878lowast) minus 119866 (119873)119867 (119878

lowast) = 0 (25)


119871119873 (119878) = 1198772 (119878) minus 119866 (119873)119867 (119878) (26)






lowast119899


1198781015840= 1198772 (119878) 119878 minus 119867 (119878) 119866 (119873) 119878 = 119878119871119873 (119878) (27)










1198791015840= 1198771 (119879) 119879 + 120572 (119879 119878) minus 1198671 (119879 119878) 1198661 (119873) 119879

= 119879(1198711 (119879 119878) + 119903120572

119878

11987021198701

(1198701 minus 119879))

equiv 1198791198721 (119879 119878)

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 1198672 (119879 119878) 1198662 (119873) 119878

= 119878(1198712 (119879 119878) minus 119903120572

119879

11987021198701

(1198701 minus 119879)) equiv 1198781198722 (119879 119878)

(28)

where 1198661(119873) = 119892119879(119906lowast(119873))119862

lowast(119873) 1198671(119879 119878) = 119891119879(119879

119878)ℎ119879(119879) 1198662(119873) = 119892119878(119906lowast(119873))119862

lowast(119873) and 1198672(119879 119878) =



119873min119879 = 119866minus11 (1198771(0)119891119879(119892119909120583119909)) and 119873min119878 = 119866

minus12 ((1198772(0) minus

119903120572119870141198702)119891119878(119892119909120583119909))










lowast119895 119878) = 0



lowast1







1198791015840= 119879 (1198711 (119879 119878) + 1199031205721198701)

1198781015840= 1198781198712 (119879 119878)

(29)




Proof For119873 gt 119873thr119879

1198711 (119879 119878) + 1199031205721198701 lt 1198771 (119879) minus 1198661 (119873thr119879)1198671 (119879 119878)

+ 1199031205721198701


+ 1199031205721198701

= 1198771 (119879) + 1199031205721198701

minus1198671 (119879 119878)

1198671 (1198791198701)(1198771 (119879) + 1199031205721198701) le 0

(30)




1198791015840= 119879(1198711 (119879 119878) + 119903120572

119878

11987021198701

(1198701 minus 119879))

le 119879 (1198711 (119879 119878) + 1199031205721198701)

1198781015840= 119878(1198712 (119879 119878) minus 119903120572

119879

1198702

(1198701 minus 119879)) le 1198781198712 (119879 119878)

(31)



5 Example


119889119879

119889119905

= 1199031119879(1 minus119879

1198701

) + 119903120572

119879

1198701

119878

1198702

(1198701 minus 119879)

minus 119886119879

119872I119872I + 119890119879

(119886119879120573 +

119890119879120573 (1 minus 119886119879120573)

119865120573 + 119890119879120573

)119862119879

ℎ119879 + 119879

119889119878

119889119905

= 1199032119878 (1 minus119878

1198702

) minus 119903120572

119879

1198701

119878

1198702

(1198701 minus 119879)

minus 119886119878

119872I119872I + 119890119878

(119886119878120573 +

119890119878120573 (1 minus 119886119878120573)

119865120573 + 119890119878120573

)119862119878

ℎ119878 + 119878

119889119862

119889119905= minus120583119862119862 + 119873

119889119865120573

119889119905= 119892120573 + 119886120573119879119879 + 119886120573119878119878 minus 120583120573119865120573





















119889119865120574

119889119905= 119886120574119862119862 minus 120583120574119865120574

119889119872I119889119905

= 119892119872I+

119886119872I 120574119865120574

119865120574 + 119890119872I 120574

minus 120583119872I119872I

119889119872II119889119905

=

119886119872II 120574119865120574

119865120574 + 119890119872II 120574

(

119890119872II 120573(1 minus 119886119872II 120573

)

119865120573 + 119890119872II 120573

+ 119886119872II 120573)

minus 120583119872II119872II

(32)



119862lowast=119873

120583119862

119865lowast120573 =

119892120573 + 119886120573119878119878 + 119886120573119879119879

120583120573

119865lowast120574 =

119886120574119862119862lowast

120583120574

119872lowastI =

119890119872I 120574119892119872I

+ (119886119872I 120574+ 119892119872I

) 119865lowast120574

120583119872I(119890119872I 120574

+ 119865lowast120574 )

119872lowastII =

119886119872II 120574(119890119872II 120573

+ 119886119872II 120573119865lowast120573 ) 119865lowast120574

120583119872II(119890119872II 120573

+ 119865lowast120573) (119890119872II 120574

+ 119865lowast120574 )

(33)



CSCs times 107

CTLs times 100

05

10

15

20

25

2000500 1000 30002500 35001500

Figure 1119873 = 0


+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

))(119892119872I+ 119886119872I 120574

119865lowast120574 (119865lowast120574 +

119890119872I 120574) + 119890119879))(119873120583119862) and 1198662(119873) = (119886119878(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 +

119890119872I 120574))(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

) + 119890119878))(119873120583119862) This gives

119873min119879

= 119866minus11 (

1199031

119886119879120573 + 119890119879120573 (1 minus 119886119879120573) (119892120573120583120573 + 119890119879120573)

)

= 119866minus11 (00117) = 00245

119873min119878

= 119866minus12 (

1199032 minus 119903120572119870141198702

119886119878120573 + 119890119878120573 (1 minus 119886119878120573) (119892120573120583120573 + 119890119878120573)

)

= 119866minus12 (09976) = 207889

(34)






15 Since



17)

= 310199 lowast 1014


15)

= 108783 lowast 1015

(35)


CSCs times 107

CTLs times 1010

1000 1500 2000 2500 3000 3500500

05

10

15

20

25



CSCs times 105

1

2

3

4

5

6

7

004 006 008 010002



15



6 Conclusion







Acknowledgment


References









































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



























119865120573 = 119909

119891120573 = 119891119909

120583120573 = 120583119909

119865120574 = 119910

119891120574 = 119891119910

120583120574 = 120583119910

119872I = 119906

119891119872I= 119891119906

120583119872I= 120583119906

119872II = V

119891119872II= 119891V

120583119872II= 120583V

(7)


1198791015840= 120572 (119879 119878) + 1198771 (119879) 119879 minus 119891119879 (119909) 119892119879 (119906) ℎ119879 (119879) 119862119879

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 119891119878 (119909) 119892119878 (119906) ℎ119878 (119878) 119862119878

1198621015840= 119891119862 ((119879 + 119878) sdot V) 119892119862 (119909) minus 120583119862119862 + 119873 (119905)

1199091015840= 119891119909 (119879 + 119878) minus 120583119909119909

1199101015840= 119891119910 (119862) minus 120583119910119910

1199061015840= 119891119906 (119910) minus 120583119906119906

V1015840 = 119891V (119909) 119892V (119910) minus 120583VV(8)






















5






+ 1198772 (119878) 119878 minus 119891119878 (119909) 119892119878 (119906) 119862119878ℎ119878 (119878)

+ 119891119862 (119879V + 119878V) 119892119862 (119909) minus 120583119862119862 + 119873

+ 119891119909 (119879 + 119878) minus 120583119909119909 + 119891119910 (119862) minus 120583119910119910

+ 119891119906 (119910) minus 120583119906119906 + 119891V (119909) 119892V (119910) minus 120583VV



le 119860 minus 120575119882

(10)

where 120575 = min1198871 1198872 120583119862 120583119909 120583119910 120583119906 120583V and 119860 = 1198861 + 1198862 +

119888max + 119873 + 119891119909 + 119891119910 + 119891119906 + 119891V




5




119862lowast=119873

120583119862

119909lowast=119891119909 (0)

120583119909

119910lowast=

119891119910 (119862lowast)

120583119910

119906lowast=119891119906 (119910lowast)

120583119906


lowast)

120583V

(11)


(((((((((((((((((

(

minus119873119891119878 (119909

lowast) 119892119878 (119906

lowast)

120583119862

+ 1199032 0 0 0 0 0 0

0 minus119873119891119879 (119909

lowast) 119892119879 (119906

lowast)

120583119862

+ 1199031 0 0 0 0 0

119891V (119909lowast) 119892119862 (119909

lowast) 119892V (119909

lowast) 1198911015840119862 (0)

120583V

119891V (119909lowast) 119892119862 (119909

lowast) 119892V (119909

lowast) 1198911015840119862 (0)

120583Vminus120583119862 0 0 0 0

0 0 1198911015840119910 (119862lowast) minus120583119910 0 0 0

1198911015840119909 (0) 119891

1015840119909 (0) 0 0 minus120583119909 0 0

0 0 0 1198911015840119906 (119910lowast) 0 minus120583119906 0



)))))))))))))))))

)

(12)




lowast)119892119879(119906

lowast)120583119862 + 1199031 So


119873 gt1199032 sdot 120583119862

119891119878 (119909lowast) 119892119878 (119906

lowast)

119873 gt1199031 sdot 120583119862

119891119879 (119909lowast) 119892119879 (119906

lowast)

(13)






(119891119910(119862lowast)120583119910)(1 minus 1205981) and 119906

lowast= (119891119906(119910

lowast)120583119906)(1 minus 1205982) for






1198791015840le minus1198861119879


1199031 + 119903120572119870214

ℎ119879 (1198701) 119891119879 (119909max)gt 0

(14)


1198781015840le minus1198862119878


1199032

ℎ119878 (1198702) 119891119878 (119909max)gt 0

(15)



119909119901 =

119891119909 (119879119901 + 119878119901)

120583119909

119910119901 =

119891119910 (119862119901)

120583119910

119906119901 =

119891119906 (119910119901)

120583119906

V119901 =119891V (119909119901) 119892V (119910119901)

120583V

(16)


1198771 (119879119901) + 120572 (119879119901 119878119901) = 119891119879(

119891119909 (119879119901 + 119878119901)

120583119909

)

sdot 119892119879(

119891119906 (119891119910 (119862119901120583119910))

120583119906

)ℎ119879 (119879119901) 119862119901

1198772 (119878119901) 119878119901 = 120572 (119879119901 119878119901) 119879119901 + 119891119878(

119891119909 (119879119901 + 119878119901)

120583119909

)

sdot 119892119878(

119891119906 (119891119910 (119862119901120583119910))

120583119906

)ℎ119878 (119878119901) 119878119901119862119901

120583119862119862119901

= 119891119862(

119891V (119891119909 (119879119901 + 119878119901) 120583119909) 119892V (119891119910 (119862119901) 120583119910) (119879119901 + 119878119901)

120583V)

sdot 119892119862(

119891119909 (119879119901 + 119878119901)

120583119909

) +119873

(17)


119867119879119878 (119862)

= 119891119862(

119891V (119891119909 (119879 + 119878) 120583119909) 119892V (119891119910 (119862) 120583119910) (119879 + 119878)

120583V)

sdot 119892119862 (119891119909 (119879 + 119878)

120583119909

) minus 120583119862119862 + 119873

(18)


1198671015840119879119878 (119862)

= 1198911015840119862(

119891V (119891119909 (119879 + 119878) 120583119909) 119892V (119891119910 (119862) 120583119910) (119879 + 119878)

120583V)

sdot119891V (119891119909 (119879 + 119878) 120583119909) (119879 + 119878)

120583V1198921015840V (

119891119910 (119862)

120583119910

)

1198911015840119910 (119862)

120583119910

sdot 119892119862 (119891119909 (119879 + 119878)

120583119909

) minus 120583119862

(19)







1198791015840= 120572 (119879 119878) + 1198771 (119879) 119879 minus 119891119879 (119879 119878) 119892119879 (119906) ℎ119879 (119879) 119879119862

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 119891119878 (119879 119878) 119892119878 (119906) ℎ119878 (119878) 119878119862

1198621015840= minus120583119862119862 + 119873

1199101015840= 119891119910 (119862) minus 120583119910119910

1199061015840= 119891119906 (119910) minus 120583119906119906

V1015840 = 119891V (119879 119878) 119892V (119910) minus 120583VV

(20)

where

119891119879 (119879 119878) = 119891119879 (119891119909 (119879 + 119878)

120583119909

)

119891119878 (119879 119878) = 119891119878 (119891119909 (119879 + 119878)

120583119909

)

119892119862 (119879 119878) = 119892119862 (119891119909 (119879 + 119878)

120583119909

)

119891V (119879 119878) = 119891V (119891119909 (119879 + 119878)

120583119909

)

(21)




lowast)120583119910 and 119906

lowast= 119891119906(119910



1198781015840= 1198772 (119878


lowast) 119892119878 (119906

lowast) ℎ119878 (119878



1198772 (119878lowast) minus 119891119878 (119878

lowast) 119892119878 (119906

lowast) ℎ119878 (119878



119866 (119873) = 119892119878 (119906lowast(119873)) 119862

lowast(119873)

119867 (119878) = 119891119878 (119878) ℎ119878 (119878)

(24)


1198772 (119878lowast) minus 119866 (119873)119867 (119878

lowast) = 0 (25)


119871119873 (119878) = 1198772 (119878) minus 119866 (119873)119867 (119878) (26)






lowast119899


1198781015840= 1198772 (119878) 119878 minus 119867 (119878) 119866 (119873) 119878 = 119878119871119873 (119878) (27)










1198791015840= 1198771 (119879) 119879 + 120572 (119879 119878) minus 1198671 (119879 119878) 1198661 (119873) 119879

= 119879(1198711 (119879 119878) + 119903120572

119878

11987021198701

(1198701 minus 119879))

equiv 1198791198721 (119879 119878)

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 1198672 (119879 119878) 1198662 (119873) 119878

= 119878(1198712 (119879 119878) minus 119903120572

119879

11987021198701

(1198701 minus 119879)) equiv 1198781198722 (119879 119878)

(28)

where 1198661(119873) = 119892119879(119906lowast(119873))119862

lowast(119873) 1198671(119879 119878) = 119891119879(119879

119878)ℎ119879(119879) 1198662(119873) = 119892119878(119906lowast(119873))119862

lowast(119873) and 1198672(119879 119878) =



119873min119879 = 119866minus11 (1198771(0)119891119879(119892119909120583119909)) and 119873min119878 = 119866

minus12 ((1198772(0) minus

119903120572119870141198702)119891119878(119892119909120583119909))










lowast119895 119878) = 0



lowast1







1198791015840= 119879 (1198711 (119879 119878) + 1199031205721198701)

1198781015840= 1198781198712 (119879 119878)

(29)




Proof For119873 gt 119873thr119879

1198711 (119879 119878) + 1199031205721198701 lt 1198771 (119879) minus 1198661 (119873thr119879)1198671 (119879 119878)

+ 1199031205721198701


+ 1199031205721198701

= 1198771 (119879) + 1199031205721198701

minus1198671 (119879 119878)

1198671 (1198791198701)(1198771 (119879) + 1199031205721198701) le 0

(30)




1198791015840= 119879(1198711 (119879 119878) + 119903120572

119878

11987021198701

(1198701 minus 119879))

le 119879 (1198711 (119879 119878) + 1199031205721198701)

1198781015840= 119878(1198712 (119879 119878) minus 119903120572

119879

1198702

(1198701 minus 119879)) le 1198781198712 (119879 119878)

(31)



5 Example


119889119879

119889119905

= 1199031119879(1 minus119879

1198701

) + 119903120572

119879

1198701

119878

1198702

(1198701 minus 119879)

minus 119886119879

119872I119872I + 119890119879

(119886119879120573 +

119890119879120573 (1 minus 119886119879120573)

119865120573 + 119890119879120573

)119862119879

ℎ119879 + 119879

119889119878

119889119905

= 1199032119878 (1 minus119878

1198702

) minus 119903120572

119879

1198701

119878

1198702

(1198701 minus 119879)

minus 119886119878

119872I119872I + 119890119878

(119886119878120573 +

119890119878120573 (1 minus 119886119878120573)

119865120573 + 119890119878120573

)119862119878

ℎ119878 + 119878

119889119862

119889119905= minus120583119862119862 + 119873

119889119865120573

119889119905= 119892120573 + 119886120573119879119879 + 119886120573119878119878 minus 120583120573119865120573





















119889119865120574

119889119905= 119886120574119862119862 minus 120583120574119865120574

119889119872I119889119905

= 119892119872I+

119886119872I 120574119865120574

119865120574 + 119890119872I 120574

minus 120583119872I119872I

119889119872II119889119905

=

119886119872II 120574119865120574

119865120574 + 119890119872II 120574

(

119890119872II 120573(1 minus 119886119872II 120573

)

119865120573 + 119890119872II 120573

+ 119886119872II 120573)

minus 120583119872II119872II

(32)



119862lowast=119873

120583119862

119865lowast120573 =

119892120573 + 119886120573119878119878 + 119886120573119879119879

120583120573

119865lowast120574 =

119886120574119862119862lowast

120583120574

119872lowastI =

119890119872I 120574119892119872I

+ (119886119872I 120574+ 119892119872I

) 119865lowast120574

120583119872I(119890119872I 120574

+ 119865lowast120574 )

119872lowastII =

119886119872II 120574(119890119872II 120573

+ 119886119872II 120573119865lowast120573 ) 119865lowast120574

120583119872II(119890119872II 120573

+ 119865lowast120573) (119890119872II 120574

+ 119865lowast120574 )

(33)



CSCs times 107

CTLs times 100

05

10

15

20

25

2000500 1000 30002500 35001500

Figure 1119873 = 0


+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

))(119892119872I+ 119886119872I 120574

119865lowast120574 (119865lowast120574 +

119890119872I 120574) + 119890119879))(119873120583119862) and 1198662(119873) = (119886119878(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 +

119890119872I 120574))(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

) + 119890119878))(119873120583119862) This gives

119873min119879

= 119866minus11 (

1199031

119886119879120573 + 119890119879120573 (1 minus 119886119879120573) (119892120573120583120573 + 119890119879120573)

)

= 119866minus11 (00117) = 00245

119873min119878

= 119866minus12 (

1199032 minus 119903120572119870141198702

119886119878120573 + 119890119878120573 (1 minus 119886119878120573) (119892120573120583120573 + 119890119878120573)

)

= 119866minus12 (09976) = 207889

(34)






15 Since



17)

= 310199 lowast 1014


15)

= 108783 lowast 1015

(35)


CSCs times 107

CTLs times 1010

1000 1500 2000 2500 3000 3500500

05

10

15

20

25



CSCs times 105

1

2

3

4

5

6

7

004 006 008 010002



15



6 Conclusion







Acknowledgment


References









































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of




















5






+ 1198772 (119878) 119878 minus 119891119878 (119909) 119892119878 (119906) 119862119878ℎ119878 (119878)

+ 119891119862 (119879V + 119878V) 119892119862 (119909) minus 120583119862119862 + 119873

+ 119891119909 (119879 + 119878) minus 120583119909119909 + 119891119910 (119862) minus 120583119910119910

+ 119891119906 (119910) minus 120583119906119906 + 119891V (119909) 119892V (119910) minus 120583VV



le 119860 minus 120575119882

(10)

where 120575 = min1198871 1198872 120583119862 120583119909 120583119910 120583119906 120583V and 119860 = 1198861 + 1198862 +

119888max + 119873 + 119891119909 + 119891119910 + 119891119906 + 119891V




5




119862lowast=119873

120583119862

119909lowast=119891119909 (0)

120583119909

119910lowast=

119891119910 (119862lowast)

120583119910

119906lowast=119891119906 (119910lowast)

120583119906


lowast)

120583V

(11)


(((((((((((((((((

(

minus119873119891119878 (119909

lowast) 119892119878 (119906

lowast)

120583119862

+ 1199032 0 0 0 0 0 0

0 minus119873119891119879 (119909

lowast) 119892119879 (119906

lowast)

120583119862

+ 1199031 0 0 0 0 0

119891V (119909lowast) 119892119862 (119909

lowast) 119892V (119909

lowast) 1198911015840119862 (0)

120583V

119891V (119909lowast) 119892119862 (119909

lowast) 119892V (119909

lowast) 1198911015840119862 (0)

120583Vminus120583119862 0 0 0 0

0 0 1198911015840119910 (119862lowast) minus120583119910 0 0 0

1198911015840119909 (0) 119891

1015840119909 (0) 0 0 minus120583119909 0 0

0 0 0 1198911015840119906 (119910lowast) 0 minus120583119906 0



)))))))))))))))))

)

(12)




lowast)119892119879(119906

lowast)120583119862 + 1199031 So


119873 gt1199032 sdot 120583119862

119891119878 (119909lowast) 119892119878 (119906

lowast)

119873 gt1199031 sdot 120583119862

119891119879 (119909lowast) 119892119879 (119906

lowast)

(13)






(119891119910(119862lowast)120583119910)(1 minus 1205981) and 119906

lowast= (119891119906(119910

lowast)120583119906)(1 minus 1205982) for






1198791015840le minus1198861119879


1199031 + 119903120572119870214

ℎ119879 (1198701) 119891119879 (119909max)gt 0

(14)


1198781015840le minus1198862119878


1199032

ℎ119878 (1198702) 119891119878 (119909max)gt 0

(15)



119909119901 =

119891119909 (119879119901 + 119878119901)

120583119909

119910119901 =

119891119910 (119862119901)

120583119910

119906119901 =

119891119906 (119910119901)

120583119906

V119901 =119891V (119909119901) 119892V (119910119901)

120583V

(16)


1198771 (119879119901) + 120572 (119879119901 119878119901) = 119891119879(

119891119909 (119879119901 + 119878119901)

120583119909

)

sdot 119892119879(

119891119906 (119891119910 (119862119901120583119910))

120583119906

)ℎ119879 (119879119901) 119862119901

1198772 (119878119901) 119878119901 = 120572 (119879119901 119878119901) 119879119901 + 119891119878(

119891119909 (119879119901 + 119878119901)

120583119909

)

sdot 119892119878(

119891119906 (119891119910 (119862119901120583119910))

120583119906

)ℎ119878 (119878119901) 119878119901119862119901

120583119862119862119901

= 119891119862(

119891V (119891119909 (119879119901 + 119878119901) 120583119909) 119892V (119891119910 (119862119901) 120583119910) (119879119901 + 119878119901)

120583V)

sdot 119892119862(

119891119909 (119879119901 + 119878119901)

120583119909

) +119873

(17)


119867119879119878 (119862)

= 119891119862(

119891V (119891119909 (119879 + 119878) 120583119909) 119892V (119891119910 (119862) 120583119910) (119879 + 119878)

120583V)

sdot 119892119862 (119891119909 (119879 + 119878)

120583119909

) minus 120583119862119862 + 119873

(18)


1198671015840119879119878 (119862)

= 1198911015840119862(

119891V (119891119909 (119879 + 119878) 120583119909) 119892V (119891119910 (119862) 120583119910) (119879 + 119878)

120583V)

sdot119891V (119891119909 (119879 + 119878) 120583119909) (119879 + 119878)

120583V1198921015840V (

119891119910 (119862)

120583119910

)

1198911015840119910 (119862)

120583119910

sdot 119892119862 (119891119909 (119879 + 119878)

120583119909

) minus 120583119862

(19)







1198791015840= 120572 (119879 119878) + 1198771 (119879) 119879 minus 119891119879 (119879 119878) 119892119879 (119906) ℎ119879 (119879) 119879119862

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 119891119878 (119879 119878) 119892119878 (119906) ℎ119878 (119878) 119878119862

1198621015840= minus120583119862119862 + 119873

1199101015840= 119891119910 (119862) minus 120583119910119910

1199061015840= 119891119906 (119910) minus 120583119906119906

V1015840 = 119891V (119879 119878) 119892V (119910) minus 120583VV

(20)

where

119891119879 (119879 119878) = 119891119879 (119891119909 (119879 + 119878)

120583119909

)

119891119878 (119879 119878) = 119891119878 (119891119909 (119879 + 119878)

120583119909

)

119892119862 (119879 119878) = 119892119862 (119891119909 (119879 + 119878)

120583119909

)

119891V (119879 119878) = 119891V (119891119909 (119879 + 119878)

120583119909

)

(21)




lowast)120583119910 and 119906

lowast= 119891119906(119910



1198781015840= 1198772 (119878


lowast) 119892119878 (119906

lowast) ℎ119878 (119878



1198772 (119878lowast) minus 119891119878 (119878

lowast) 119892119878 (119906

lowast) ℎ119878 (119878



119866 (119873) = 119892119878 (119906lowast(119873)) 119862

lowast(119873)

119867 (119878) = 119891119878 (119878) ℎ119878 (119878)

(24)


1198772 (119878lowast) minus 119866 (119873)119867 (119878

lowast) = 0 (25)


119871119873 (119878) = 1198772 (119878) minus 119866 (119873)119867 (119878) (26)






lowast119899


1198781015840= 1198772 (119878) 119878 minus 119867 (119878) 119866 (119873) 119878 = 119878119871119873 (119878) (27)










1198791015840= 1198771 (119879) 119879 + 120572 (119879 119878) minus 1198671 (119879 119878) 1198661 (119873) 119879

= 119879(1198711 (119879 119878) + 119903120572

119878

11987021198701

(1198701 minus 119879))

equiv 1198791198721 (119879 119878)

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 1198672 (119879 119878) 1198662 (119873) 119878

= 119878(1198712 (119879 119878) minus 119903120572

119879

11987021198701

(1198701 minus 119879)) equiv 1198781198722 (119879 119878)

(28)

where 1198661(119873) = 119892119879(119906lowast(119873))119862

lowast(119873) 1198671(119879 119878) = 119891119879(119879

119878)ℎ119879(119879) 1198662(119873) = 119892119878(119906lowast(119873))119862

lowast(119873) and 1198672(119879 119878) =



119873min119879 = 119866minus11 (1198771(0)119891119879(119892119909120583119909)) and 119873min119878 = 119866

minus12 ((1198772(0) minus

119903120572119870141198702)119891119878(119892119909120583119909))










lowast119895 119878) = 0



lowast1







1198791015840= 119879 (1198711 (119879 119878) + 1199031205721198701)

1198781015840= 1198781198712 (119879 119878)

(29)




Proof For119873 gt 119873thr119879

1198711 (119879 119878) + 1199031205721198701 lt 1198771 (119879) minus 1198661 (119873thr119879)1198671 (119879 119878)

+ 1199031205721198701


+ 1199031205721198701

= 1198771 (119879) + 1199031205721198701

minus1198671 (119879 119878)

1198671 (1198791198701)(1198771 (119879) + 1199031205721198701) le 0

(30)




1198791015840= 119879(1198711 (119879 119878) + 119903120572

119878

11987021198701

(1198701 minus 119879))

le 119879 (1198711 (119879 119878) + 1199031205721198701)

1198781015840= 119878(1198712 (119879 119878) minus 119903120572

119879

1198702

(1198701 minus 119879)) le 1198781198712 (119879 119878)

(31)



5 Example


119889119879

119889119905

= 1199031119879(1 minus119879

1198701

) + 119903120572

119879

1198701

119878

1198702

(1198701 minus 119879)

minus 119886119879

119872I119872I + 119890119879

(119886119879120573 +

119890119879120573 (1 minus 119886119879120573)

119865120573 + 119890119879120573

)119862119879

ℎ119879 + 119879

119889119878

119889119905

= 1199032119878 (1 minus119878

1198702

) minus 119903120572

119879

1198701

119878

1198702

(1198701 minus 119879)

minus 119886119878

119872I119872I + 119890119878

(119886119878120573 +

119890119878120573 (1 minus 119886119878120573)

119865120573 + 119890119878120573

)119862119878

ℎ119878 + 119878

119889119862

119889119905= minus120583119862119862 + 119873

119889119865120573

119889119905= 119892120573 + 119886120573119879119879 + 119886120573119878119878 minus 120583120573119865120573





















119889119865120574

119889119905= 119886120574119862119862 minus 120583120574119865120574

119889119872I119889119905

= 119892119872I+

119886119872I 120574119865120574

119865120574 + 119890119872I 120574

minus 120583119872I119872I

119889119872II119889119905

=

119886119872II 120574119865120574

119865120574 + 119890119872II 120574

(

119890119872II 120573(1 minus 119886119872II 120573

)

119865120573 + 119890119872II 120573

+ 119886119872II 120573)

minus 120583119872II119872II

(32)



119862lowast=119873

120583119862

119865lowast120573 =

119892120573 + 119886120573119878119878 + 119886120573119879119879

120583120573

119865lowast120574 =

119886120574119862119862lowast

120583120574

119872lowastI =

119890119872I 120574119892119872I

+ (119886119872I 120574+ 119892119872I

) 119865lowast120574

120583119872I(119890119872I 120574

+ 119865lowast120574 )

119872lowastII =

119886119872II 120574(119890119872II 120573

+ 119886119872II 120573119865lowast120573 ) 119865lowast120574

120583119872II(119890119872II 120573

+ 119865lowast120573) (119890119872II 120574

+ 119865lowast120574 )

(33)



CSCs times 107

CTLs times 100

05

10

15

20

25

2000500 1000 30002500 35001500

Figure 1119873 = 0


+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

))(119892119872I+ 119886119872I 120574

119865lowast120574 (119865lowast120574 +

119890119872I 120574) + 119890119879))(119873120583119862) and 1198662(119873) = (119886119878(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 +

119890119872I 120574))(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

) + 119890119878))(119873120583119862) This gives

119873min119879

= 119866minus11 (

1199031

119886119879120573 + 119890119879120573 (1 minus 119886119879120573) (119892120573120583120573 + 119890119879120573)

)

= 119866minus11 (00117) = 00245

119873min119878

= 119866minus12 (

1199032 minus 119903120572119870141198702

119886119878120573 + 119890119878120573 (1 minus 119886119878120573) (119892120573120583120573 + 119890119878120573)

)

= 119866minus12 (09976) = 207889

(34)






15 Since



17)

= 310199 lowast 1014


15)

= 108783 lowast 1015

(35)


CSCs times 107

CTLs times 1010

1000 1500 2000 2500 3000 3500500

05

10

15

20

25



CSCs times 105

1

2

3

4

5

6

7

004 006 008 010002



15



6 Conclusion







Acknowledgment


References









































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















lowast)119892119879(119906

lowast)120583119862 + 1199031 So


119873 gt1199032 sdot 120583119862

119891119878 (119909lowast) 119892119878 (119906

lowast)

119873 gt1199031 sdot 120583119862

119891119879 (119909lowast) 119892119879 (119906

lowast)

(13)






(119891119910(119862lowast)120583119910)(1 minus 1205981) and 119906

lowast= (119891119906(119910

lowast)120583119906)(1 minus 1205982) for






1198791015840le minus1198861119879


1199031 + 119903120572119870214

ℎ119879 (1198701) 119891119879 (119909max)gt 0

(14)


1198781015840le minus1198862119878


1199032

ℎ119878 (1198702) 119891119878 (119909max)gt 0

(15)



119909119901 =

119891119909 (119879119901 + 119878119901)

120583119909

119910119901 =

119891119910 (119862119901)

120583119910

119906119901 =

119891119906 (119910119901)

120583119906

V119901 =119891V (119909119901) 119892V (119910119901)

120583V

(16)


1198771 (119879119901) + 120572 (119879119901 119878119901) = 119891119879(

119891119909 (119879119901 + 119878119901)

120583119909

)

sdot 119892119879(

119891119906 (119891119910 (119862119901120583119910))

120583119906

)ℎ119879 (119879119901) 119862119901

1198772 (119878119901) 119878119901 = 120572 (119879119901 119878119901) 119879119901 + 119891119878(

119891119909 (119879119901 + 119878119901)

120583119909

)

sdot 119892119878(

119891119906 (119891119910 (119862119901120583119910))

120583119906

)ℎ119878 (119878119901) 119878119901119862119901

120583119862119862119901

= 119891119862(

119891V (119891119909 (119879119901 + 119878119901) 120583119909) 119892V (119891119910 (119862119901) 120583119910) (119879119901 + 119878119901)

120583V)

sdot 119892119862(

119891119909 (119879119901 + 119878119901)

120583119909

) +119873

(17)


119867119879119878 (119862)

= 119891119862(

119891V (119891119909 (119879 + 119878) 120583119909) 119892V (119891119910 (119862) 120583119910) (119879 + 119878)

120583V)

sdot 119892119862 (119891119909 (119879 + 119878)

120583119909

) minus 120583119862119862 + 119873

(18)


1198671015840119879119878 (119862)

= 1198911015840119862(

119891V (119891119909 (119879 + 119878) 120583119909) 119892V (119891119910 (119862) 120583119910) (119879 + 119878)

120583V)

sdot119891V (119891119909 (119879 + 119878) 120583119909) (119879 + 119878)

120583V1198921015840V (

119891119910 (119862)

120583119910

)

1198911015840119910 (119862)

120583119910

sdot 119892119862 (119891119909 (119879 + 119878)

120583119909

) minus 120583119862

(19)







1198791015840= 120572 (119879 119878) + 1198771 (119879) 119879 minus 119891119879 (119879 119878) 119892119879 (119906) ℎ119879 (119879) 119879119862

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 119891119878 (119879 119878) 119892119878 (119906) ℎ119878 (119878) 119878119862

1198621015840= minus120583119862119862 + 119873

1199101015840= 119891119910 (119862) minus 120583119910119910

1199061015840= 119891119906 (119910) minus 120583119906119906

V1015840 = 119891V (119879 119878) 119892V (119910) minus 120583VV

(20)

where

119891119879 (119879 119878) = 119891119879 (119891119909 (119879 + 119878)

120583119909

)

119891119878 (119879 119878) = 119891119878 (119891119909 (119879 + 119878)

120583119909

)

119892119862 (119879 119878) = 119892119862 (119891119909 (119879 + 119878)

120583119909

)

119891V (119879 119878) = 119891V (119891119909 (119879 + 119878)

120583119909

)

(21)




lowast)120583119910 and 119906

lowast= 119891119906(119910



1198781015840= 1198772 (119878


lowast) 119892119878 (119906

lowast) ℎ119878 (119878



1198772 (119878lowast) minus 119891119878 (119878

lowast) 119892119878 (119906

lowast) ℎ119878 (119878



119866 (119873) = 119892119878 (119906lowast(119873)) 119862

lowast(119873)

119867 (119878) = 119891119878 (119878) ℎ119878 (119878)

(24)


1198772 (119878lowast) minus 119866 (119873)119867 (119878

lowast) = 0 (25)


119871119873 (119878) = 1198772 (119878) minus 119866 (119873)119867 (119878) (26)






lowast119899


1198781015840= 1198772 (119878) 119878 minus 119867 (119878) 119866 (119873) 119878 = 119878119871119873 (119878) (27)










1198791015840= 1198771 (119879) 119879 + 120572 (119879 119878) minus 1198671 (119879 119878) 1198661 (119873) 119879

= 119879(1198711 (119879 119878) + 119903120572

119878

11987021198701

(1198701 minus 119879))

equiv 1198791198721 (119879 119878)

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 1198672 (119879 119878) 1198662 (119873) 119878

= 119878(1198712 (119879 119878) minus 119903120572

119879

11987021198701

(1198701 minus 119879)) equiv 1198781198722 (119879 119878)

(28)

where 1198661(119873) = 119892119879(119906lowast(119873))119862

lowast(119873) 1198671(119879 119878) = 119891119879(119879

119878)ℎ119879(119879) 1198662(119873) = 119892119878(119906lowast(119873))119862

lowast(119873) and 1198672(119879 119878) =



119873min119879 = 119866minus11 (1198771(0)119891119879(119892119909120583119909)) and 119873min119878 = 119866

minus12 ((1198772(0) minus

119903120572119870141198702)119891119878(119892119909120583119909))










lowast119895 119878) = 0



lowast1







1198791015840= 119879 (1198711 (119879 119878) + 1199031205721198701)

1198781015840= 1198781198712 (119879 119878)

(29)




Proof For119873 gt 119873thr119879

1198711 (119879 119878) + 1199031205721198701 lt 1198771 (119879) minus 1198661 (119873thr119879)1198671 (119879 119878)

+ 1199031205721198701


+ 1199031205721198701

= 1198771 (119879) + 1199031205721198701

minus1198671 (119879 119878)

1198671 (1198791198701)(1198771 (119879) + 1199031205721198701) le 0

(30)




1198791015840= 119879(1198711 (119879 119878) + 119903120572

119878

11987021198701

(1198701 minus 119879))

le 119879 (1198711 (119879 119878) + 1199031205721198701)

1198781015840= 119878(1198712 (119879 119878) minus 119903120572

119879

1198702

(1198701 minus 119879)) le 1198781198712 (119879 119878)

(31)



5 Example


119889119879

119889119905

= 1199031119879(1 minus119879

1198701

) + 119903120572

119879

1198701

119878

1198702

(1198701 minus 119879)

minus 119886119879

119872I119872I + 119890119879

(119886119879120573 +

119890119879120573 (1 minus 119886119879120573)

119865120573 + 119890119879120573

)119862119879

ℎ119879 + 119879

119889119878

119889119905

= 1199032119878 (1 minus119878

1198702

) minus 119903120572

119879

1198701

119878

1198702

(1198701 minus 119879)

minus 119886119878

119872I119872I + 119890119878

(119886119878120573 +

119890119878120573 (1 minus 119886119878120573)

119865120573 + 119890119878120573

)119862119878

ℎ119878 + 119878

119889119862

119889119905= minus120583119862119862 + 119873

119889119865120573

119889119905= 119892120573 + 119886120573119879119879 + 119886120573119878119878 minus 120583120573119865120573





















119889119865120574

119889119905= 119886120574119862119862 minus 120583120574119865120574

119889119872I119889119905

= 119892119872I+

119886119872I 120574119865120574

119865120574 + 119890119872I 120574

minus 120583119872I119872I

119889119872II119889119905

=

119886119872II 120574119865120574

119865120574 + 119890119872II 120574

(

119890119872II 120573(1 minus 119886119872II 120573

)

119865120573 + 119890119872II 120573

+ 119886119872II 120573)

minus 120583119872II119872II

(32)



119862lowast=119873

120583119862

119865lowast120573 =

119892120573 + 119886120573119878119878 + 119886120573119879119879

120583120573

119865lowast120574 =

119886120574119862119862lowast

120583120574

119872lowastI =

119890119872I 120574119892119872I

+ (119886119872I 120574+ 119892119872I

) 119865lowast120574

120583119872I(119890119872I 120574

+ 119865lowast120574 )

119872lowastII =

119886119872II 120574(119890119872II 120573

+ 119886119872II 120573119865lowast120573 ) 119865lowast120574

120583119872II(119890119872II 120573

+ 119865lowast120573) (119890119872II 120574

+ 119865lowast120574 )

(33)



CSCs times 107

CTLs times 100

05

10

15

20

25

2000500 1000 30002500 35001500

Figure 1119873 = 0


+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

))(119892119872I+ 119886119872I 120574

119865lowast120574 (119865lowast120574 +

119890119872I 120574) + 119890119879))(119873120583119862) and 1198662(119873) = (119886119878(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 +

119890119872I 120574))(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

) + 119890119878))(119873120583119862) This gives

119873min119879

= 119866minus11 (

1199031

119886119879120573 + 119890119879120573 (1 minus 119886119879120573) (119892120573120583120573 + 119890119879120573)

)

= 119866minus11 (00117) = 00245

119873min119878

= 119866minus12 (

1199032 minus 119903120572119870141198702

119886119878120573 + 119890119878120573 (1 minus 119886119878120573) (119892120573120583120573 + 119890119878120573)

)

= 119866minus12 (09976) = 207889

(34)






15 Since



17)

= 310199 lowast 1014


15)

= 108783 lowast 1015

(35)


CSCs times 107

CTLs times 1010

1000 1500 2000 2500 3000 3500500

05

10

15

20

25



CSCs times 105

1

2

3

4

5

6

7

004 006 008 010002



15



6 Conclusion







Acknowledgment


References









































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















1198791015840= 120572 (119879 119878) + 1198771 (119879) 119879 minus 119891119879 (119879 119878) 119892119879 (119906) ℎ119879 (119879) 119879119862

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 119891119878 (119879 119878) 119892119878 (119906) ℎ119878 (119878) 119878119862

1198621015840= minus120583119862119862 + 119873

1199101015840= 119891119910 (119862) minus 120583119910119910

1199061015840= 119891119906 (119910) minus 120583119906119906

V1015840 = 119891V (119879 119878) 119892V (119910) minus 120583VV

(20)

where

119891119879 (119879 119878) = 119891119879 (119891119909 (119879 + 119878)

120583119909

)

119891119878 (119879 119878) = 119891119878 (119891119909 (119879 + 119878)

120583119909

)

119892119862 (119879 119878) = 119892119862 (119891119909 (119879 + 119878)

120583119909

)

119891V (119879 119878) = 119891V (119891119909 (119879 + 119878)

120583119909

)

(21)




lowast)120583119910 and 119906

lowast= 119891119906(119910



1198781015840= 1198772 (119878


lowast) 119892119878 (119906

lowast) ℎ119878 (119878



1198772 (119878lowast) minus 119891119878 (119878

lowast) 119892119878 (119906

lowast) ℎ119878 (119878



119866 (119873) = 119892119878 (119906lowast(119873)) 119862

lowast(119873)

119867 (119878) = 119891119878 (119878) ℎ119878 (119878)

(24)


1198772 (119878lowast) minus 119866 (119873)119867 (119878

lowast) = 0 (25)


119871119873 (119878) = 1198772 (119878) minus 119866 (119873)119867 (119878) (26)






lowast119899


1198781015840= 1198772 (119878) 119878 minus 119867 (119878) 119866 (119873) 119878 = 119878119871119873 (119878) (27)










1198791015840= 1198771 (119879) 119879 + 120572 (119879 119878) minus 1198671 (119879 119878) 1198661 (119873) 119879

= 119879(1198711 (119879 119878) + 119903120572

119878

11987021198701

(1198701 minus 119879))

equiv 1198791198721 (119879 119878)

1198781015840= 1198772 (119878) 119878 minus 120572 (119879 119878) minus 1198672 (119879 119878) 1198662 (119873) 119878

= 119878(1198712 (119879 119878) minus 119903120572

119879

11987021198701

(1198701 minus 119879)) equiv 1198781198722 (119879 119878)

(28)

where 1198661(119873) = 119892119879(119906lowast(119873))119862

lowast(119873) 1198671(119879 119878) = 119891119879(119879

119878)ℎ119879(119879) 1198662(119873) = 119892119878(119906lowast(119873))119862

lowast(119873) and 1198672(119879 119878) =



119873min119879 = 119866minus11 (1198771(0)119891119879(119892119909120583119909)) and 119873min119878 = 119866

minus12 ((1198772(0) minus

119903120572119870141198702)119891119878(119892119909120583119909))










lowast119895 119878) = 0



lowast1







1198791015840= 119879 (1198711 (119879 119878) + 1199031205721198701)

1198781015840= 1198781198712 (119879 119878)

(29)




Proof For119873 gt 119873thr119879

1198711 (119879 119878) + 1199031205721198701 lt 1198771 (119879) minus 1198661 (119873thr119879)1198671 (119879 119878)

+ 1199031205721198701


+ 1199031205721198701

= 1198771 (119879) + 1199031205721198701

minus1198671 (119879 119878)

1198671 (1198791198701)(1198771 (119879) + 1199031205721198701) le 0

(30)




1198791015840= 119879(1198711 (119879 119878) + 119903120572

119878

11987021198701

(1198701 minus 119879))

le 119879 (1198711 (119879 119878) + 1199031205721198701)

1198781015840= 119878(1198712 (119879 119878) minus 119903120572

119879

1198702

(1198701 minus 119879)) le 1198781198712 (119879 119878)

(31)



5 Example


119889119879

119889119905

= 1199031119879(1 minus119879

1198701

) + 119903120572

119879

1198701

119878

1198702

(1198701 minus 119879)

minus 119886119879

119872I119872I + 119890119879

(119886119879120573 +

119890119879120573 (1 minus 119886119879120573)

119865120573 + 119890119879120573

)119862119879

ℎ119879 + 119879

119889119878

119889119905

= 1199032119878 (1 minus119878

1198702

) minus 119903120572

119879

1198701

119878

1198702

(1198701 minus 119879)

minus 119886119878

119872I119872I + 119890119878

(119886119878120573 +

119890119878120573 (1 minus 119886119878120573)

119865120573 + 119890119878120573

)119862119878

ℎ119878 + 119878

119889119862

119889119905= minus120583119862119862 + 119873

119889119865120573

119889119905= 119892120573 + 119886120573119879119879 + 119886120573119878119878 minus 120583120573119865120573





















119889119865120574

119889119905= 119886120574119862119862 minus 120583120574119865120574

119889119872I119889119905

= 119892119872I+

119886119872I 120574119865120574

119865120574 + 119890119872I 120574

minus 120583119872I119872I

119889119872II119889119905

=

119886119872II 120574119865120574

119865120574 + 119890119872II 120574

(

119890119872II 120573(1 minus 119886119872II 120573

)

119865120573 + 119890119872II 120573

+ 119886119872II 120573)

minus 120583119872II119872II

(32)



119862lowast=119873

120583119862

119865lowast120573 =

119892120573 + 119886120573119878119878 + 119886120573119879119879

120583120573

119865lowast120574 =

119886120574119862119862lowast

120583120574

119872lowastI =

119890119872I 120574119892119872I

+ (119886119872I 120574+ 119892119872I

) 119865lowast120574

120583119872I(119890119872I 120574

+ 119865lowast120574 )

119872lowastII =

119886119872II 120574(119890119872II 120573

+ 119886119872II 120573119865lowast120573 ) 119865lowast120574

120583119872II(119890119872II 120573

+ 119865lowast120573) (119890119872II 120574

+ 119865lowast120574 )

(33)



CSCs times 107

CTLs times 100

05

10

15

20

25

2000500 1000 30002500 35001500

Figure 1119873 = 0


+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

))(119892119872I+ 119886119872I 120574

119865lowast120574 (119865lowast120574 +

119890119872I 120574) + 119890119879))(119873120583119862) and 1198662(119873) = (119886119878(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 +

119890119872I 120574))(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

) + 119890119878))(119873120583119862) This gives

119873min119879

= 119866minus11 (

1199031

119886119879120573 + 119890119879120573 (1 minus 119886119879120573) (119892120573120583120573 + 119890119879120573)

)

= 119866minus11 (00117) = 00245

119873min119878

= 119866minus12 (

1199032 minus 119903120572119870141198702

119886119878120573 + 119890119878120573 (1 minus 119886119878120573) (119892120573120583120573 + 119890119878120573)

)

= 119866minus12 (09976) = 207889

(34)






15 Since



17)

= 310199 lowast 1014


15)

= 108783 lowast 1015

(35)


CSCs times 107

CTLs times 1010

1000 1500 2000 2500 3000 3500500

05

10

15

20

25



CSCs times 105

1

2

3

4

5

6

7

004 006 008 010002



15



6 Conclusion







Acknowledgment


References









































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















119873min119879 = 119866minus11 (1198771(0)119891119879(119892119909120583119909)) and 119873min119878 = 119866

minus12 ((1198772(0) minus

119903120572119870141198702)119891119878(119892119909120583119909))










lowast119895 119878) = 0



lowast1







1198791015840= 119879 (1198711 (119879 119878) + 1199031205721198701)

1198781015840= 1198781198712 (119879 119878)

(29)




Proof For119873 gt 119873thr119879

1198711 (119879 119878) + 1199031205721198701 lt 1198771 (119879) minus 1198661 (119873thr119879)1198671 (119879 119878)

+ 1199031205721198701


+ 1199031205721198701

= 1198771 (119879) + 1199031205721198701

minus1198671 (119879 119878)

1198671 (1198791198701)(1198771 (119879) + 1199031205721198701) le 0

(30)




1198791015840= 119879(1198711 (119879 119878) + 119903120572

119878

11987021198701

(1198701 minus 119879))

le 119879 (1198711 (119879 119878) + 1199031205721198701)

1198781015840= 119878(1198712 (119879 119878) minus 119903120572

119879

1198702

(1198701 minus 119879)) le 1198781198712 (119879 119878)

(31)



5 Example


119889119879

119889119905

= 1199031119879(1 minus119879

1198701

) + 119903120572

119879

1198701

119878

1198702

(1198701 minus 119879)

minus 119886119879

119872I119872I + 119890119879

(119886119879120573 +

119890119879120573 (1 minus 119886119879120573)

119865120573 + 119890119879120573

)119862119879

ℎ119879 + 119879

119889119878

119889119905

= 1199032119878 (1 minus119878

1198702

) minus 119903120572

119879

1198701

119878

1198702

(1198701 minus 119879)

minus 119886119878

119872I119872I + 119890119878

(119886119878120573 +

119890119878120573 (1 minus 119886119878120573)

119865120573 + 119890119878120573

)119862119878

ℎ119878 + 119878

119889119862

119889119905= minus120583119862119862 + 119873

119889119865120573

119889119905= 119892120573 + 119886120573119879119879 + 119886120573119878119878 minus 120583120573119865120573





















119889119865120574

119889119905= 119886120574119862119862 minus 120583120574119865120574

119889119872I119889119905

= 119892119872I+

119886119872I 120574119865120574

119865120574 + 119890119872I 120574

minus 120583119872I119872I

119889119872II119889119905

=

119886119872II 120574119865120574

119865120574 + 119890119872II 120574

(

119890119872II 120573(1 minus 119886119872II 120573

)

119865120573 + 119890119872II 120573

+ 119886119872II 120573)

minus 120583119872II119872II

(32)



119862lowast=119873

120583119862

119865lowast120573 =

119892120573 + 119886120573119878119878 + 119886120573119879119879

120583120573

119865lowast120574 =

119886120574119862119862lowast

120583120574

119872lowastI =

119890119872I 120574119892119872I

+ (119886119872I 120574+ 119892119872I

) 119865lowast120574

120583119872I(119890119872I 120574

+ 119865lowast120574 )

119872lowastII =

119886119872II 120574(119890119872II 120573

+ 119886119872II 120573119865lowast120573 ) 119865lowast120574

120583119872II(119890119872II 120573

+ 119865lowast120573) (119890119872II 120574

+ 119865lowast120574 )

(33)



CSCs times 107

CTLs times 100

05

10

15

20

25

2000500 1000 30002500 35001500

Figure 1119873 = 0


+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

))(119892119872I+ 119886119872I 120574

119865lowast120574 (119865lowast120574 +

119890119872I 120574) + 119890119879))(119873120583119862) and 1198662(119873) = (119886119878(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 +

119890119872I 120574))(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

) + 119890119878))(119873120583119862) This gives

119873min119879

= 119866minus11 (

1199031

119886119879120573 + 119890119879120573 (1 minus 119886119879120573) (119892120573120583120573 + 119890119879120573)

)

= 119866minus11 (00117) = 00245

119873min119878

= 119866minus12 (

1199032 minus 119903120572119870141198702

119886119878120573 + 119890119878120573 (1 minus 119886119878120573) (119892120573120583120573 + 119890119878120573)

)

= 119866minus12 (09976) = 207889

(34)






15 Since



17)

= 310199 lowast 1014


15)

= 108783 lowast 1015

(35)


CSCs times 107

CTLs times 1010

1000 1500 2000 2500 3000 3500500

05

10

15

20

25



CSCs times 105

1

2

3

4

5

6

7

004 006 008 010002



15



6 Conclusion







Acknowledgment


References









































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of




































119889119865120574

119889119905= 119886120574119862119862 minus 120583120574119865120574

119889119872I119889119905

= 119892119872I+

119886119872I 120574119865120574

119865120574 + 119890119872I 120574

minus 120583119872I119872I

119889119872II119889119905

=

119886119872II 120574119865120574

119865120574 + 119890119872II 120574

(

119890119872II 120573(1 minus 119886119872II 120573

)

119865120573 + 119890119872II 120573

+ 119886119872II 120573)

minus 120583119872II119872II

(32)



119862lowast=119873

120583119862

119865lowast120573 =

119892120573 + 119886120573119878119878 + 119886120573119879119879

120583120573

119865lowast120574 =

119886120574119862119862lowast

120583120574

119872lowastI =

119890119872I 120574119892119872I

+ (119886119872I 120574+ 119892119872I

) 119865lowast120574

120583119872I(119890119872I 120574

+ 119865lowast120574 )

119872lowastII =

119886119872II 120574(119890119872II 120573

+ 119886119872II 120573119865lowast120573 ) 119865lowast120574

120583119872II(119890119872II 120573

+ 119865lowast120573) (119890119872II 120574

+ 119865lowast120574 )

(33)



CSCs times 107

CTLs times 100

05

10

15

20

25

2000500 1000 30002500 35001500

Figure 1119873 = 0


+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

))(119892119872I+ 119886119872I 120574

119865lowast120574 (119865lowast120574 +

119890119872I 120574) + 119890119879))(119873120583119862) and 1198662(119873) = (119886119878(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 +

119890119872I 120574))(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

) + 119890119878))(119873120583119862) This gives

119873min119879

= 119866minus11 (

1199031

119886119879120573 + 119890119879120573 (1 minus 119886119879120573) (119892120573120583120573 + 119890119879120573)

)

= 119866minus11 (00117) = 00245

119873min119878

= 119866minus12 (

1199032 minus 119903120572119870141198702

119886119878120573 + 119890119878120573 (1 minus 119886119878120573) (119892120573120583120573 + 119890119878120573)

)

= 119866minus12 (09976) = 207889

(34)






15 Since



17)

= 310199 lowast 1014


15)

= 108783 lowast 1015

(35)


CSCs times 107

CTLs times 1010

1000 1500 2000 2500 3000 3500500

05

10

15

20

25



CSCs times 105

1

2

3

4

5

6

7

004 006 008 010002



15



6 Conclusion







Acknowledgment


References









































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















CSCs times 107

CTLs times 100

05

10

15

20

25

2000500 1000 30002500 35001500

Figure 1119873 = 0


+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

))(119892119872I+ 119886119872I 120574

119865lowast120574 (119865lowast120574 +

119890119872I 120574) + 119890119879))(119873120583119862) and 1198662(119873) = (119886119878(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 +

119890119872I 120574))(119892119872I

+ 119886119872I 120574119865lowast120574 (119865lowast120574 + 119890119872I 120574

) + 119890119878))(119873120583119862) This gives

119873min119879

= 119866minus11 (

1199031

119886119879120573 + 119890119879120573 (1 minus 119886119879120573) (119892120573120583120573 + 119890119879120573)

)

= 119866minus11 (00117) = 00245

119873min119878

= 119866minus12 (

1199032 minus 119903120572119870141198702

119886119878120573 + 119890119878120573 (1 minus 119886119878120573) (119892120573120583120573 + 119890119878120573)

)

= 119866minus12 (09976) = 207889

(34)






15 Since



17)

= 310199 lowast 1014


15)

= 108783 lowast 1015

(35)


CSCs times 107

CTLs times 1010

1000 1500 2000 2500 3000 3500500

05

10

15

20

25



CSCs times 105

1

2

3

4

5

6

7

004 006 008 010002



15



6 Conclusion







Acknowledgment


References









































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















Acknowledgment


References









































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















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