Toward practical application of tumor-immune system analysis (Theory of Biomathematics and Its Applications XI)

Tbwardpracticalapplicationof tumorimmunesystem analysis MitsuoTakase

LINFOPSInc.

$1\cdot 21-1-503$OukurayamaKouhoku-kuYokohama222-0037Japan

$E$-mail GZL03154@niftycom

Abstract. The $tumor^{-}\dot{m}lm\iota me$ system interaction seems to be able to be expressed mechanically, mathematically. toa large extent. One of main aims here is to express the state of$tumor^{-}$inmlune

system mechanically, analytically and quantitatively, and another is to make strategies to cure by

making the practical numerical modelandits software. Analytical methods andits software which

givesusconcrete numberstobeableto beappliedtotreatments in the tumor immunesystemhave a

possibilityto giveinformation which

can

not be obtained without them. This analysis is eigenvalue analysis,andthelocalignitionat eigenvalue $\lambda=1$has aspecialmeaningforthe

cure.

The distance tothelocalignition isalsoimportant wheretreatments tomakethedistance smaller or

zero

shouldbe made. Here, the behaviors under thesub ignition stateto cure andthe movement mechanismfrom the sub-ignitionto the localignition withthecontrol mechanism ofthe immune systemare shown. Theeffectof thetendencyfor tumor cells totendto become lessdifferentiatedandmoreindependent

isinspected.

1.Introduction

As $t\iota mlor-$ inmune system interaction has the aspects ofinformationprocessing anddymamic

system. One of the aims here is to express the interaction mechanically, mathematically,

analytically and quantitatively One of main aims here is to express the state of tumorimmune

systemin suchaway, and another isto makestrategies tocurebymaking thepracticalnumerical

model andits software.

An analytical method and its software which gives us concrete numbers to be applied to treatments haveapossibilityto giveinformation which

can

not beobtainedwithoutthem.

Moreover, itsdiscrete numericalanalysis with the numerical model and the useofMonte Carlo

simulation are considered, and its computer software is being developed for the calculation of the model. This situation leads to the numerical realization of the phenomena and gives us the quantitative analysis bya discrete numerical simulation

even

ifthereare notenoughmathematical equations.

Here,

one

ofmainaims isapracticalapplication which

means

thereahzation ofthe numerical total

modelandits concrete numerical useful results.

Thismodelngand their simulationsarewidelyconductedin theareas likepowerplants.

The numerical simulationcanbe realizedby hmitingthesimulation

area

likeExtendedfieldmodel showninsection2 inthe solidtumoralthough theroute tolymph nodesand from them istaken into account.These analyticalconditionsgiveusthe eigenvalue problemsituationandcharacteristics like eigenvalues greaterthan 1

or

not and quantitativevalues liketothe distance to eigenvalue $\lambda=1$

althoughtheeigenvalue problemsituation varies. Herethestate $\lambda=1$is namedlocalignitionwhere

the immunesystemis activatedabruptlyandlocally..Sowe

can

inspectvarious conditions to reach the localignition tocure. One ofcharacteristicsofthisanalysisis thatcontactprobability between a tumor peptideand$T$cellreceptorisconsideredto haveanequaleffect with affinity betweenatumorpeptide

and$T$cellreceptorandkillingprobabihty.Theygivethe

same

effecttoachieve thelocalignition.

Here, The effectof$Toe\grave{1}1s$

are

_{mainly considered, this}is dueto the largenessof the $T$oell effect to

tumor(ref. 2).

with

additional

explanation to

aim

for practical application and the production of the $\infty$mputer

software. Thepractical applicationis akeyaimhere.

Moreover,the following two themes

are

inspected.(1)and(2)

_are

_{showninsection 3 and 4}

respectively.

(1) The behavior and the effects of the immmune system to attack a solid tumor in the state of sub-ignition,which

means

here thestate where the localignitionis notyetachieved $\lambda<1$_,is shown. The behaviors under the sub-ignition state to

cure

and the movement mechanism ffom the

sub.ignitionto thelocalignition withthe$\infty$ntrol mechanism ofthe immmumesystemarealso shown.

(2) _TLmor $\infty \mathbb{I}s$ tend to not only evolve to

overcome

and survive the state of bad $\infty$nditional

environments around them where thereareless oxygen, less nutrition and muchwaste butalso become thestate less and lessdifferentiatedgradually.

Theless differentiated and

more

independent tumor cells

are

inspected. In the beginning of the

tumor, the state of tumor cells

can

be thought to be differentiated like the healthy cells, then the

behavior of$T$oelkinthe immmumesystemappears to beclear,butin thestate less differentiated and

moreindependent,the behaviorandtheanalysisseemto become much

more

$\infty$mplicated.

2.Thebasic modeltoreahze aims 2.1Explanationofthe model

The basicmodel torealizethese aims is shownin Fig. 1.

$v^{I}$

$v_{s}$

Afield modelas a $s$

$s$

part of the whole

cancer mass blood flow

$\prime$

’

$s$

The area where the analysis should be conducted is shown in the following

as

Extended field model.

Extended field model.

.

.

an

area

where adjacent field models

are

included and $T$ cells

concentration,etc.

are

statistically much lessvariablewith nearly

zero

fluctuationbyinput

andoutput$T$cells

Field model

.

.

.

an

areawith the

common

environmentuseful for

an

efficientcalculation

as

shownin Fig. 1.

$\lceil$

Input$data\rfloor$

Mean freepathof$T$cell

Averaged frequencyof$T$oell movement

The concentration of$T$cells which have higher affinities _{to the tumor peptide around}

the solid

tumor

.

[T]

Initial distribution oftumorcells etc.

$\lceil$

Output d$ata\rfloor$

Maximumeigenvalue.

.

.

This

means

Tact cell (activated$T$cell) proliferation rate,etc.

Eigenvector

. . .

proliferation speedspatialdistributionofTact cellconcentration[Tact], etc.. The distributionshapeand scale graduallychange.

The basic model is

a

$\infty$mparablysmall

area

which determines the environment andthe

eigenvalue problemfromit wherewe$m$get characteristicallyandquantitativelyimportantdatanot

to be obtained without the model.

$\lambda{\rm Im}$

1

$\epsilon$

Fig.2 $\lambda_{1m}v.s.x$and$T_{act}v.s.x$graphs.Here$x$is total effect to increase $\lambda_{m}$like[L2]

$\lambda_{m}$ $x$

. .

Total effecttoincrease $\lambda_{m}$like[IL2]

.

.

.

$[T_{a}d$

$\epsilon$ Entering Th cellsand Tc cellswith

a

high

affinityintothe field modelconstantly

toacancerpeptide

$\lambda_{m}$ The$T$cellsproliferationrate which is the_{eigenvalueoftheeigenvalue problem}

shown

inequation(1)_{insection 2.1.}

[Tact] _The$\infty$ncentrationofactivated$T$cellswhichis,forexample,thenumberofT cellsper unit

[L2] _The$\infty$ncentration of

interleukin

2

[Characteristics_{of the}model]

(1) Availabilityofinputdata

Theinputdata should be knownbyinspectionlike bloodinspectionfor thepracticalapplication

At least theinputdata ofthe model

are

$\infty$ntainedin blood.

(2) $T$cell activation and its works aseffector$T$cells

are

caused through themultiplication$of\alpha_{\backslash }$ $\beta$ and $\gamma$

$\alpha x\beta x\gamma$

$\alpha$

.

$\infty$ntactprobabilitybetween$T$cell and

an

antigen(tumorpeptide)

$\beta$

.

affinity ofT cellreoeptorsto the tumorpeptide

$\gamma$ probabilityto kill tumor cell

a&r

the$\infty$ntact and theaffinity

are

achieved

From this mechanism$T$oell movements andthe

enviromnent

where$T$oelk

move

easily

havealmost the

same

effect with theaffinity,$T$oellconcentration,etc. to$\ovalbox{\tt\small REJECT}$the effect ofT oelk

which

are

increased by vaccine therapyandto achievethelocalignition.

(3) _{Eigenvalue calculation}$\cdots$As theeigenvector changesgradually theequationoftheeigenvalue

problem graduallychanges.

Characteristicmeaning

The achievementofeigenvalue $\lambda=1$(Fig.2)is named localignitionhere.

Quantitativemeaning

Thedistance to $\lambda=1$ to

cause

thelocalignitionhas

an

importantvalue to

cure

(Fig.2

Movementofeigenvector

It isthoughtthatthese

are

due tomainlyL2 which is

a

$\infty$

mmon

element toproliferate$T$oellslike

in Fig.3 and secretedconrnonly byTact(activated$T$oelk)oelk.

$\lceil$

Additionalexplanationofthe localignition$\rfloor$

(1) _Thelogical$\infty$nnection haspositivefeedback loops.

(2) AbruptproliferationofTact cells is causedbythe achievement of $\lambda=1.$

(3) _{The localignition hasasimilarity withassociationin the brain}

(4) _There

_seem

_oftenpositivefeedbacks in inmune cells which

cause

the localignition

like

$T$oelk

Fig.3 Positive feedbackrelationship 2.$2An$exampleofexpected control_mechanism

Anexampleofexpected$\infty$ntrol mechanism is showninFig.4.

$\lceil The$effects ofthemodel andtheanalysis$\rfloor$

(1) $T$ cell movement (mean free path and movement frequency) affects the $\infty$ntact probabihty

betweentumor cellpeptidesand$T$cellreoeptors,and the$\infty$ntactprobabilityis$\infty$nsideredto havean

equal effectwith affinity betweena tumor peptide and$T$oellreceptor. Theygive the

same

effectto

achievethelocalignitionlikeshowninsection 2.1.

(2)_{On the otherhand,atherapylikevaccinetherapy}(ref. 3)_{is thoughttoincrease theaffimtyand}$T$

cell $\infty$ncentration [T]. L2 therapy (ref. 4) increases the $\infty$noentration $nL2$] and enhance the

proliferation rate ofT cellsasshownin Fig.3. If

some

therapieswhich affect the$T$cellmovementsand

heightenthem,they$\infty$ntnbutetoheighten$T$cellactivityandtoachievethelocalignition.

ignition,it

can

haveabigmeaningforthecure. Themultipletherapies

can

be anytherapiesinduding alternativetherapiesiftheyhave the effects.

(3)By knowingthe distance to the local ignition, $a$$\infty$ncreteaimwith a quantitative numbercanbe

made, andstrategiestoapproachthelocal ignition

can

be planned. Ifso,this may lead to

psychologicallybettereffect. Constant input

Aboutwhy eigenvalueproblem

can

be applied Fig.$4An$example_{of mechanism to}cause_thelocalignition

Thact

.

activated CD4$T$oell

Tcact activatedCD8$T$cell

Thm memory CD4$T$cell

Tcm memory CD8$T$cell

3. Behavior inthe stateofsub-ignition, and

move

tothe state ofthelocalignitionandits automatic control

3.1 Behaviorin thestate ofsub-ignition

the behavior and the effects of the immune system to attack the solid tumor in the state of

discussed. Iftheproliferationrate of tumor cells isenough low,the state of thesub-ignition mayhave

an

enough ability toehminate the solid tumor. This state

can

beexpressed bythefollowing equations. Thebehaviors

can

be inferredfrom theequationtoa certain extent.

Equation(1)_{isexpressed by eigenvalue problem expression.}

$\lambda lm$

means

eigenvaluewhich is the growthofeigenvector.

$\lambda_{un}$ $\{\begin{array}{l}\{Tact\}\{T\rangle\{Tm\}\end{array}\}$ $=[A1Bm2B2$ $A2B1[0]Bm1A3[0]$ $\{\begin{array}{l}\{Tact\}\{T\}\{Tm\rangle\end{array}\}$

.

.

.

(1) Equation$(\emptyset$

is expressedbyrecurrentformmeaningthe

same

phenomena ofequation(1)

$\{T\}$_,

{Tact}

and

{Tm}

in equation (1) and

{Tbffe}

inequation (2)

can

include both

cases

of_$CD4T$ and

$CD8T$

.

Here the expressions

are

simplifiedinequation(1)and(2).

$T$

.

. .

$T$oell

Tact activated$T$cell

Tm memory$T$cell

Tbffe effector$T$cell

Each element Tiof$\{T\}$

means

the$\infty$noentration of$T$cells atposition $i$which

means

the

number ofTcellsina unit volumeat position$\dot{\iota}$

Here forexample Blsubmatrixdependson tumor cells and

can

beexpressed bylike

Bl$=b_{1}\cdot E\cdot\{C\}$ where blis$\infty$nstant anddependson the$T$cell movementand the affimty between

$T$cellreceptorandtumor cellpeptides

$\{C\}$ tumor cells$\infty$noentrationdistribution like(T)

$E$ unitmatffix

$\{\begin{array}{l}\{Tact\rangle\{B\{Tm\}\end{array}\}i+1$ $=\{\begin{array}{lll}A1 B1 Bm1B2 A2 [0]Bm2 [0] A3\end{array}\}$ $\{\begin{array}{l}\{Tact\}\{l?\{Tm\rangle\end{array}\}i$

. .

.

(2)

$\{\begin{array}{ll} \{Tact\} \{T\} \{Tm\rangle- \{bff e\}\end{array}\}$ $=[A1Bm2B3B2[0]A2[0]B1$

$Bm1A3[0][0]$ $A4[0][0][0]$

$\{\{Tm\rangle--\{^{r}I\‘{e} ffe\rangle\{T\}\{Tact\}$ $\}_{i}$

.

. .

(3) $\{C\}_{i+1}=\lambda c\cdot E\cdot\{C\}_{i}$ –$\alpha\cdot(E\cdot\{T\infty ffe\})\cdot\{C\}_{i}$

$=$ $(\lambda c\cdot E-\alpha$

.

($E$

.

{Reffe})

$)\cdot\{C\}_{i}$ (4)

$i$

Tceffe effector$T$oell of$CD8T$_oellwhich

means

cytotoxic$T$oelloftenexpressed byCTL.

This

means

Teffein equation(3).

$\lambda c$ The proliferationrate oftumorcellswithoutany suppressionbytheimmunesystem.

Equation(4)_{shows thedynamicbehavior with thesuppression by}$T$oells,and

for $\lambda c$to beenough$sma\mathbb{I}$in$\infty$mparisonwith $\parallel\{T\infty ffe\}\int\int$ can

cause

thedeletionoftumor cellsby

$T$cellsunder thesub ignition.

$\chi_{c2}\{C\}=$ $(\lambda c\cdot E-\alpha$

.

($E$

.

{Toeffe})

$)\cdot\{C\}$ (5)

$\lambda_{c2}$ the proliferationrate oftumorcells withthesuppression byTceffe cells. Equation(5)

_means

_{eigenvalue problem expressionof}equation(4). $\lambda_{c2}<1$which

means

the

3.2Move tothestateof thelocalignition anditsautomatic$\infty$ntrol

If theproliferationrateof tumor cells isenoughlowfor$T$cells todelete the tumorcells completely

underthesub.ignition,thetumorcells disappearand the diseasewill be$\infty$mpletely cured.

When the tumorcells proliferate

more

and

more

although$T$cellsattackthetumorcellsunder the

sub ignitionevenif $\lambda_{m}$becomesbiggerandbiggerunder the sub ignition, it is necessary for

$\lambda_{m}$to achieve thelocalignition.

When the local ignition isachieved,the$\infty$ncentration of the activated$T$cells $[T_{act}]$becomeslarger

and larger without hnmt theoretically beyond the point where the proliferationrate $\lambda_{c2}$oftumor cells become less than 1 forthe tumorto become smalleruntil the tumor cells

are

completelydeletedor

nearlycompletelydeletedtheoretically.

At that time, as shown in Fig.4, when there is no tumor cell for the positive feedback to be

maintained,andthepositivefeedbackdisappears.

Here, these show the example ofthe automatic $\infty$ntrolmechanism of the immune system by$T$

cellsaccordingto the existence of tumor cells..

At theprocessfiromthe sub-ignitiontothe localignition, the clearrecognitionof the existenoe of tumor cellsby$T$cells is necessary.The clearrecognition may notbeeasy The

some

_{mechanism from}

reliabilityaboutthis isshown inref.6.

4. Astudyfor why tumors are not easily cured without the achievement ofthe state of the local

ignitionbythinking mechanically

Here, we thinkaboutone ofthe

reasons

whyinmany

cases

of solid tumorsthey

can

notbe cured

easily.

(1) _{Tumor cells tend to become the state less and less differentiated and more independent}

gradually.

(2) _{Tumorcells evolvegradually togetabihties like angiogenesis and metastasissurvivingthe} state of bad $\infty$nditions where there

can

be less oxygen, less nutrition and much waste

materials.

In the

case

(1),theanalysisandthe model

seem

to becomecomplicated abruptlynottocatch the

dynamicbehavioreasilyandclearlyin$\infty$mparisonwith the state of differentiatedtumor cells which

means

acomparably earlystateoftumorasshowninthe followings.

$\lceil Anexample\rfloor$ Thereis acasewhere tumor cells tend to_presentlessMHC_{IandMHC IIgradually}

(ref 2). _Then$T$cells with $\alpha/\beta$ receptorcan not recognizeeasilyeven ifthey have$\infty$ntactwith the tumor cell.

Thissituation issimilar to alowerstateofT cell concentration[T]_{with highaffinities.}

NK cellsbeginto work tokillthetumor cells insteadof CTLwhich isaTbffe cell ofa CD8$T$cell(ref.

2)._{NK cells notonlyworkto kill the tumor cells without bEfCI,but also effectively kin thembythe}

supportofantibodieswhich

are

producedby$B$cells. The situation is namedADCC.

Here

the existenceofthelocalignition is importantto

cause

astrongresponse oftheimmunesystem,

butbothNK cells and$B$oellsdonotseemto havesostrong positivefeedbackas$T$cells,and

IL2

causes

the proliferationofboth NKcellsand$B$oells,but thelocalignitionbythe setofTcells

andIL2willnotbe madeeasilyfrom less MHCI and less MHCII althoughthere

are

$T$oells with

$\gamma/\delta$ receptorwhichrecognize directly antigenwithoutMHCIand MHC II (ref.2).These situations

areshownin Fig.5.

Ifwe think aboutonlythis situation in Fig. 5mechanica]ly thelocalignitionby$T$cells and IL2 will

becomemoreandmoredifficult to beachieved,because thepositivefeedback causedby$T$cellsand

This situationmay

mean an

actualhurdleto

cure

even

ifthere is a$1\infty p$like

NK$oe\mathbb{I}arrow FN-\gammaarrow T_{h1}$(type1helper$Toe11$) $arrow$ Tact_{$arrow IL2arrow NK$}cellproliferation.

So from these mechanicaldiscussions,various methods should be $\infty$nsideredenoughtoheighten

theactivity.

Fig.5 Asimple proliferation andstimulationrelationship 5. Discussion

The model is a framemodel whichwill beadded

more

detailedexpressions accordingtoneoessity

although the present model will $\infty ver$

an

effective and wide behaviors in the tumorimmune

interaction.Moreover, when thedetailedexpressions willbeadded, themodel isexpectedtopresent

results nearertothe actual results.

The matrixexpressionismainlyused here. Theexpression

can

have

more

directrelationshipwith Monte Carlo simulation method andmakingthe software,

_mooeover

it

seems

to have the abilityto expressthetumorimmuneinteraction situation and behaviorfreely.

In ref. 7, 4 $\infty$nditions to keep the parts of the body normal are proposed. These seem to be

neoessary to avoid tumor. Referenoes

1.Takase,M.(2010)_Inductionandapplicationof

an

equationtoanalyzea localignitionofthe immume

systemfora $\infty$mpletedeletion ofa

cancer

mass

Theory of Biomathematicsand its applications VI.

RISM1704,53-60Kyoto University.

2. CharlesA. Janeway Jr.et.al.,Immunobiologytheinmmunesystemin healthanddisease,Garland. 3. Yue Zhang, $Y$ et al. (2006) Thl cell adjuvant therapy $\infty$mbined with tumor vaccination.

InternationalImmunology 19, $151\cdot 161$

4. Ewend, M. G. et al. (2000) _Intracranial pmmne interleukin 2 therapy stimulates prolonged

antitumor immunity that extends outsidethecentralnervoussystem.Journal ofimmunology

5.Takase,M.(2009)_{Cancer and immunesysteminteraction model like a neural network}

model, analysis ofcancer

mass

effect and meaning of vaccine Theory of Biomathematics and its

applications V.RISM1663,35-40KyotoUniversity.

6.Takase, M. (2013)_Aautomatic$\infty$ntrol mechanism toignite the immunesystemlocally against

a

small

canoer mass

$\infty$nsidering reliabihtyTheory ofBiomathematics and its applications IX. RISM

1853, 185-193Kyoto University.

7.Takase,M.(2012)_{Definitionoftumor}bythelossofstabihty andfunctionalanalytic approachwith

scalechangesfor tumorbehaviorsbasedongenes.TheoryofBiomathematicsanditsapplications