Cancer and immune system interaction model like a neural network model, analysis of cancer mass effect and meaning of vaccine (Theory of Biomathematics and its Applications V)

Cancer and immune systeminteractionmodellikeaneural network model, analysis of

cancer mass

effectandmeaningofvaccine

LINFOPS

有限会社 高瀬 光雄 (Mitsuo Takase)

LINFOPS

(life _information processing systems) Inc.

3013-1-503

Futoochou Kouhoku-ku Yokohama

222-0031

Japan [email protected]

Abstract.A numerical interaction model between a

cancer

mass and the immune system is shown basedon aneural networkpart anddiffusive recurrentparts. Usingthe numericalmodel

as

the basis ofbehavioranalysis, how

canoer

mass

e&ct

weakens the efficacyof immunity, under what$\infty ndition$ theimmunesystemignites andthemeaningsofvaccinetherapyareexplained especiallyfirom$\infty ntact$

frequency betweenlymphocytes and

canoer

oells. 1.Introduction

With

contact ftequencyprobability at

_ffl

$\alpha(\{d)$, affinitybetweenlymphoqtesand

cancer

cells at

$\{x\}\beta(u)$ and killing probability by lymphocytes at $u\gamma(\{x\}),$ $\alpha(u)\cdot\beta(\{x\})\cdot\gamma(\{x\})$ inhibit the

proli&rationofcanoercells at$\{x\}$

.

Then $\alpha(\{x\})$has

an

equaleffict to $\gamma(\{x\})$to inhibitproli&rationrate

$\lambda(u)$ofcancercellsmathematically. $\alpha(u)\cdot\beta$($\{d)$

can

have amaineffect$f$₎$r$thebeginningandthe

responseintensityofthe immunesystem.

Onthe other hand, free

cancer

cells isolated$\Re m$

a

cancer mass

can

hardly survive in

a

healthy

body. Because iffiee

cancer

cellseasilysurvived,

canoer

cells would $\infty ntinue$toincrease in not only

$bl\infty d$ butalso in anywhere inbody. This means a cancer

mass

may _get an _{advantage especially to}

reduce the attackofthe immunesystem. Soitisin&rredthat $\alpha(\{x\})$may give usphysicalbehaviors

tomake such

an

advantageand letusknow treatment methodsbreakingtheadvantage.

For theaim, thefollowings

are

shownhere.

(1) _{To make the simulation model of}

_cancer

_{mass-immune system interaction to support the}

quantitative $\infty mpoehension$ of the behaviors based on a neural network and

a

$sink\cdot sour\infty$

diffusion analysis (ref. 1, 2 and 3). _{Necessary densities of}$T$ cells for the _{$\infty mpleteoe\infty very$}

are

thought to be causedby$T$cellp$\infty$li&rationrate $>1$inthe recurrentdynamical system.

(2)_{Togetignitioncondition ffir the immune systemagainstasmallcanoer}

_mass

(3)_{Toknowtheeffictsand meaningsofvaminetherapy from the analysismodel in}(1).

(4)_{Analysisofcancer}

_mass

_{effictwhichlowers the}

_e&ct

_{oftheimmunesystem.}

The model shown here

can

be applied not only tocancer,but alsoin&ctious

cases.

2.Simulation model

2.1the immunesystemforsimulation

2.1.1.Elements considered oftheimmunitysystem (1)_Elements$\infty nsideoed$oftheimmunity_system

Th oell helper$Toe\mathbb{L}$

Tccell cytotoxic$T$cell.This isactivated byanantigenwithsimultaneousactivation ofTh$oe\mathbb{L}$

IL2 interleukin2

It is assumed thatthereisonlyone

canoer mass

inabody.

There

are

actualexampleswhereTc cells work for theextinctionof

canoer

cells

as

a mainplayer (Ref l).

(2)_{Elements not to be considered inthe immunity system}

.

_The activationofThcells and Tc cells by affinitywith the specialpeptide of

canoer

oells inlymph

nodes is not$\infty nsideoed$ because it is assumed here that thepeptide flows outof

canoer

oells isvery

little.

otherinterleukinsandcytokinesexoeptIL2

are

not$\infty nsideoed$

.

(3)_{Summarizedfunctionsin theassumed}$\infty nditions$ofthe immune system

O

The activationoflymphoqtes through lymphnodeshaidly

occurs.

Then ThcellsandTc oellsdirectly$oe\infty gnize$the

canoer

masse

notthrough lymphnodes.

If there

are

multiple canoer

masses

and the activation of the immune system is supported

through lymph nodes, each

canoer mass

causes the attack by the immune system against all the

canoer masses

forminganetwork.

\copyright Thoellsand Tccells have main roles.

O

Antibodies donotwork.

O

Activated Tc and Th oellspi$\mathfrak{v}$li&rate throughIL2which is$p\infty duoed$bytheactivated Tc oells and

activatedTh oells.

O

A

more

precise afiinitytoaspecial

canoer

peptideisalways being$1\infty ked$brthroughthesupport

ofTh oells. This

causes

alsothe beginningoftheimmuneactivationagainstthe

canoer

mass.

2.1.2 The relationshipof $\alpha(\{d),$ $\beta(u)$and $\gamma(u)$in $\alpha(\{d)\cdot\beta(\{x\})\cdot\gamma(\{x\})$

(1)_{Relationshipwith}$PI\mathfrak{v}Rration$rate A in thecanoer

mass

Here $\alpha(u),$ $\beta(\{xI)$and $\gamma(\{d)$

a&ct

equaUytopmli&ration rate $\lambda$ ofcanoer oells.

$\alpha(\{A)$ average$\infty ntact$frequencybetweenactivatedTccells andcanoeroells perunitvolume

at{Xi.Thisdependsonboth$1C(\{x\})]$ and$1Tc(ffl)]$

.

$\beta$(be) affinityof$\infty ntact$vectors between the activatedTcoells and thecanoeroellsat$\{d$

$0\leqq\beta(\{x\})\leqq 1$

.

This ismathematicallythe innerproductofthe two vectors. $\gamma(u)$ probability ffir the Tc to hllthecanoeroell $0\leqq\gamma(\{x\})\leqq 1$

$fA$ apositionvector in thebody especiallyinthe

canoer

massand around it.

$[C(\{x\})]$ densityofcancer oellsat$\{x\}$in the

cancer mass

[Th$(u)$] _{densityofhelper}$T$oells at$td$in the

canoer

_{$mass_{o}$} $[Tc(\{x\})]$ density ofcytotoxic$T$oellsat$\{d$in thecanoer_{$mass_{0}$}

A$(\{x\})$ proli&rationrateofcanoer cellsat$\{x\}$

.

$\chi(\{x\})=\{-\alpha(\{x\})\cdot\beta(\{x\})\cdot\gamma(u)+\lambda^{+}\cdot[C(h\})]\}$

$/[C(\{x\})]$

$\lambda^{+}$ proliferation rate ofcanoeroellswithoutattackby theimmunesystem

$\lambda v$ averagedpi$\mathfrak{v}$li&ration rate ofcanoeroellsinV $\lambda v=J^{\cdot}\lambda(\{x\})dvN$

.

$v$

When an activated Tcoellworks toextinguuisha canceroeU,$\mathfrak{b}uoWulg$steps

are

neoessary. $O1$ (about $\alpha$)The Tc cell$en\infty unter$withthecanoercell

\copyright (about $\beta$)Theaffinitybetween the reoeptoroftheTc cell and thespecial peptideofthe canoeroell

isenoughhighto$oe\infty gniae$the$spe\dot{Q}ah\mathfrak{h}^{r}$ofthe

canoer

cellpeptide.

\copyright (about $\gamma$)The Tc oellworkstoextinguishthe

canoer

oelllikebycausingapoptosis.

Tbe Amctionsof $\beta(\{x\})$ and $\gamma(\{x\})$

are

usuaUy taken into_{$ac\infty unt$}

as

the

e&ct

of Tccells,but the

effict $\alpha$

seems

to be not usually considered in medical discussion E&ct of $\alpha(k\})$ has a meaning

equal to $\beta(u)$

or

$\gamma(\{x\})$ in

a

mathematicalequation and has a hidden efficacyto the extinction of

canoerceiblike $\alpha(u)$

.

(2)_{Functions of} $\alpha(u)\cdot\beta(u)$in$\alpha(\{d)\cdot\beta(u)\cdot\gamma(u)$

O

Beginning of the immune system _activation by $\alpha(\{x\})\cdot\beta(\{x\})$ which has the function to detect

canoeroells.

\copyright _{Refinement of the receptor affinity of Tc by both Th and Tc by} $\alpha(\{x\})\cdot\beta(u)$

.

There $\beta(\{x\})$

increases

O

Memorization ofapeptideofthe

cancer

oellby memory Thand memoryTcthrough $\alpha(\{x\})\cdot\beta(R)$

Memorization strength degtee is assumed to be determined by spatiotemporal strength or the number of memory$T$oells.

O

Secretion$ofIL2$_{byactivatedThand Tc. Thand Tc}

are

_activatedby $\alpha(\{x\})\cdot\beta(\Re)$

.

These

cause

the inmaaes ofthe proliSbration rate of Tc and Thoells, $[Tc(\{x\})]$and $\alpha(\{x\})$

.

2.2Neural network model

$\dot{\mathfrak{R}}=K^{t}\{\dot{\eta}\}$

.

{Xi}$)$ (2.1)

$\Delta\{d=Cf\dot{l}$

{Xi}

(2.2) $c$is$\infty nstant$

.

$\{wj\}_{t}=\Delta${wj} $+\{wj\}_{t-\Delta^{t}}$ (2.3)

$ is the$input- ou\Phi ut$monotonic linearfunction withsaturation and

a

threshold.

{Xi}

input vector$i$hke avisual image. This_{$\infty raesponds$}to the vector givenbya

canoer

peptide

shown mainly with MHCI ofthe

canoer

oell

[wjl a vector fOrmed by the electrical $\infty nductivities$ at all the synapses to

neuron

$j$

.

This

$\infty msponds$to the vectorgiven by

a

reoeptorofTcor Th in this immune model.

yj excitation and output levelofneuronj causedby the input

{xi}.

This corresponds tothe activation leveloftheThcellorthe Tccell.

$\Delta$[wj] changeof[wj]by the input{Xi}.

This

means

thememorizationofvector{Xi}.This

$\infty msponds$to the increment of the number and memorization_strengthof memory$T$ oells toan

antigeninthisimmunemodel.

The purposes ofaneuralnetworkmodel

are

similar to thoseofthe immunesystem.

(1) ${\rm Re}\infty gnition$ofinputpatternsbycorrelation. This_{$\infty msponds$}to affinityin the immunesystem. (2) _{Memorizationofnew inputpattems.This corresponds to memory}$T$oells intheimmunesystem.

(3) _{Remembranoeacoording to theimportanoeof each inputpattem}

This

can

be donebymemory$T$oell in the immunesystem.

(4) _{Search of memorizedpatternsbytheproductionofchaoticpattemsrelatedtoaninput pattem}

This can$\infty mspond$ to a oertain extent to theproductionof random patternsofreoeptorsof Th

and Tc So to

use a

neuralnetwork model

as

thetemplateto express the immunesystem has an advantage to express andcomprehendthe immunesystem.

$\alpha(u)\infty roesponds$totheinput prooesstoa

neuron

in theneural network model

$\beta(\Re)\infty msponds$to$\infty rrelation$intheneuralnetworkmodel

$\gamma(u)\infty msponds$to the selection ofanaction for the bodyprotectiondetermined inneural networks.

2.2.1Neoessityofdiffusioncalculation to know

{Tl

and

{Tact]

distributions.

To

cause

$t\{wj\}$

.

{Xi}

inequation(2.1),the$\infty ntact$ofa

cancer

oell anda$T$cellis necessary,

so

thecalculation of$1C(ffl)]$ _{distributionand thedistributions of[Th(U)l and} $[Tc(\{x\})]$ includingthose of activated Tc and Th oells are neoessary. These distnbutionequations ofdiscrete expression are shown by (2.4), (2.5)_and (2.6). _These equations

are

shownbythe recurrent brm althoughthetine steps

are

notshown

$\{\{Tact\}\{\Gamma\}\}$ $=$ $[A1B2$ $A2B1]\{\{Tact\}\{\Gamma\}\}$ (2.4)

affinitytoa

canoer

peptide is very high Hereprecisely speaking, $f\Gamma$} should be divided into$f\Gamma h$} and

{Tc}, but the$\infty mmon$expressionisused. Generally each elementof{$T]$ at{Ahas adistribution inthe multidimensional$\infty ntinuous$region$ac\infty rd\dot{m}g$tovaniousaffinity between

a canoer

peptideand$T$oells.

{Tact]

is the density vector ofactivated$T$oellsin spaoewhoseaffinityto a

canoer

peptideis very high

Aland A2arethediffusion submatrioes of

{Tact}

and$\{\Gamma\}$with the extinction ofTact cellsand$T$oells. Submatrix Bl givestheadditional production ofTact oellsthroughthe$\infty ntact$with canoer cells.

Submatrix B2 gives the additional production ofT oells withthe same reoeptorvector andits huigh

affinity through IL2 distribution produoed by Tact oells. Here IL2 gives mutual excitatory

pmli&ration stimulus like inaneural networkwith mutual$\infty nnections$

.

$\{\{Tm\}\{Tact\}\}$ $=$ $[A1Bm1Bm2A2]\{\{Tm\}\{Tact\}\}$ (2.5)

ftrm]_{istbe densityvectorofmemory}$T$cells inspace.

Equation(2.5)_{is similar to}equation(2.4). _{Tact],Al and A2

are

$\infty mmon$withequation(2.4).

Buttheelement values of submatrix Bml

are

larger thanBl,because memory$T$oells

are more

easily

excitedthroughthe$\infty ntact$with

canoer

cells than$T$oelk.

$\{C\}=E]\{C\}$ (2.6)

$\{C\}$istheexistenoe vectorofcanoeroells in spaoe.

[E]_{matrixgives the growth and theextinctionofcancer oells.}

Equation$(2.1)\sim(2.6)$givethetotalanalysis equationsofcanoerimmune interactionmodelcausing

behaviors likein neuralnetworks.Theseequationscanbeexpressedlike inFig. 1.

Fig. 1 (A),, (B)_and(C)_modelpartsafficteach other simultaneously in the prooessofthestimulation. Equation (2.4) _and (2.6) _{and their behaviors}

_are

_{sinilar to those in nudear} analysis for neutron distribution.

2.3 The neoessity of $\lambda_{I}^{l}\iota>1$ kept br

a

while $br\infty mpletere\infty veiy$ ffim canoer disease and the

$\infty fflitions$toignitethe immunesystem. $\lambda$

qbis the$pm\mathbb{R}ration$rateofTc cells inthe part(B)ofFig.1.

(1)_Activated$T$cellsproduoe IL2, and IL2makes activated$T$oells proliferate and produoe$T$oellswith

the samehigh affimity reoeptors. So IL2brmsmutually excitatorynetwork like neural networks with

mutual connections. IL2therapyexists(Ref 4). _Butit

_seems

_{to benoteasy tokeepIL2density enough}

highbrtheignihonagainstdiffusion especiallywhenthe

canoer mass

issmall.

(2)_{The neoessity of} $\lambda_{Tt}>1$keptforawhuile forthe$\infty mplete$

re

$\infty very$from

canoer.

And

{Tact}

and$\{\Gamma\}$

with enough big

norms are

neoessary to extinguish all the

canoer mass

$\infty mpletely$

.

So $\lambda q\iota>1$ is

neoessary.

(3)_The$\infty ndihon$for theignitionoftheimmunesystemand vaccinetherapy

Vaccine therapy can $\infty ntribute$ to the fOllowing $($

!

$)$, \copyright, $\lambda\tau b>1$ and the increase of $\alpha(u)$

through the$\infty ntact$with

canoer

cells inlymphnodes and all thebodyespeciallywhenthe

canoer

mass

is small.These

cause

the ignitionand$\lambda\pi>1$

.

Ol

Enough high density$ofIL2$

.

so

$\mathfrak{b}r$ the enough density to be kept, enough IL2 must be produoed fipm activated $T$ oells to

compensatethe loss.

\copyright Neoessity ofenough high density of activated $T$ cells in the canoer mass. Enough number of

activated$T$cellsproduoed by the vaccine inall the body and lymphnodescanbe gatheredtoproduoe

thestate of$O1$ into the

canoer mass

throughadhesionmolecules(Ref. 1).

2.4Additionalelements to the model

2.4.1 Possibility of $\sigma/n^{1/2}$ andincreasedprotectionbrhealthyoellsagainstTc cellattack.

As shown in section2.3, IL2

causes

mutual$pro\mathbb{R}ration$stimulation amongactivated$T$ oelklike neural networks with mutual$\infty nnections$.

(1) _{It is} imagined that there

can

be mechanicaUy variational matchings from mutual locational

combinations of

a

canoer

peptide and

a

$T$oell reoeptor. Thenthere

can

be statistical distribution

around

a

maximumaffinitywhich the$T$oellhaswith the

canoer

oell.

(2) _Then$\sigma/n^{1/2}$isthe neoessary standard deviation for _$nT$oells to activate simultaneously by IL2 where $\sigma$ is the standard deviation about the actual effect ofaffimityofeach$T$oellreoeptor.

$n$isthe number ofactivated$T$cellsmutuallystimulatedbyIL2.This

means

that$brnT$cellsto be activated simultaneously,higheraffimtyis neoessary.

2.4.2 Filter effect. If

canoer

oells produoe a lot of fiber proteins in the

canoer

mass, then the fiber proteins

can

work tolowerdiffusion$\infty effi\dot{\mathfrak{a}}ent$oflymphocytes inthe

mass.

2.4.3Thoells and Tb cells whichrespondtobodyoells

are

extinguished.At the sametime,there must

be Th oells and Tc cells whose reoeptor vectors distribute densely near the vectors which kdy oell

peptidesexpress, becausethen the oellscan$oe\infty gnize$mutated cells.

3.Canoer

mass

e&ct

3.1Theanalysisof

mass

effect

Mass

e&ct,

its relationship with $\alpha(\{x\})$and the levelof

mass

e&ct

The situation of

a

canoer mass

which

causes

the

mass

e&ct

[Assumption]

(1)_{It isassumedthat the}

_{canoer mass}

_{isaspherewith radius} $r$

.

Theunitofr isthe scale ofone

canoer

oellwhen

canoer

cells

are

denseinthe

mass.

(2)_{There isno}$bl\infty d$vesselsinthe

canoer mass.

Sothe massissmal. If thereare bloodvesselsin the

mass,the

mass

effecttends tosaturateaocordingtothe

mass

growth.

(3)_{Thereexistonly}

_{one cancer}

_mass.

[Two

_cases

_{ofthe entranoe}prooessesofaThoellor

a

Tccell into the

canoer

mass] (1)_The

_case

_{ofalow affinitybetweentheTccellandthe}peptideofthe

canoer

cells The

case

whichis hardly expectedto

cause

mass

effict

Ol

A Tc cell is attached toapointonthesurfioe of

a

canoer mass

byadhesion proteins.

\copyright _{The Tc cellenters into the}canoer mass.

O

The Tc oell diffUses with amoeboid movement into the oenter of the

mass

with a diffMsion coefficientmechanically attachingto

canoer

cellswithoutre$\infty gnition$

.

(2)_The

_case

_of

_a

_{highaffimitybetween the Tccelland thepeptide pattemofthe}

_canoer

_cells

The

case

whichisexpectedto

cause

mass

effect

Ol

A Tccell is attached to apointon the$suraoe$_ofa

canoer

mass

byadhesion molecules. \copyright _{The Tc oell enters into the}

_canoer

_mass.

O

The Tc isattachedtoa

canoer

cellbyamechanical$\infty i\dot{\eta}unction$withadhesion molecules.

O

The Tcdoesaset ofactions$\mathfrak{b}r$the prooess toextinguish thecanoercellthroughlike theinjectionof

perfOrine andapoptosis.Thisprooessdelaysthe diffusion.Thediffusioncoefficient is made smaller.

This prooess

can

tendto protect inner

cancer

oellsincomparisonwithcanoeroellsnearthe

suraoe.

The proRration ofinner oells in the

mass

alsopreventsthe Tc cell fromentering.

V the volume ofthemass. The unitis the spaoe occupied by each

canoer

oell. $\lambda$

.

1

canoer

cell pmRrationrate loweoedbyimmumity.

$\lambda=[(S+c\cdot V)\cdot\lambda 0+(1-c)\cdot V\cdot\lambda^{+}]/(S+V)$ $c$

a

$\infty nstant$with$0\leqq c<1$

$c$isexpectedto bealmost

zero.

$\lambda^{+}\cdots$

canoer

oell proRration rate without inhibitionbyimmunity.

$\lambda 0\cdots$proffirationrate under

a

situation with aplateformedby

canoer

cellslike

canoer

mass$suraoe$ andTc oelldensity[Tc]_{outside it.}$\lambda 0$ $=\lambda o([Tc], \beta, \lambda^{+})$ $\beta$ is theunifOrmvalue of $\beta(\{x\})$

.

$\lambda=\lambda 0^{\cdot}SN+\lambda^{+}$ $=$ $\lambda 0^{\cdot}[(y4)/r]+\lambda^{+}$ This

means

that when the

mass

becomessmaller, $\lambda k\infty mes$smffler and the

mass can

be extinguuishedmoreeasily.

3.2 Mathematicalmeaningof

mass

effect

When the shape of

{Tc}

witha$\infty nstant$

norm

is the

same

with that of$\{C\}$with

a

$\infty nstant$norm, the innerproductbetweenthe two vectors is maximizedmathematically. This

means

the most

e&ctive

state of the immune system. The

canoer

mass

effect can be one of the elements whichpoevent the

inmune system from attachngcanoer

masses

$espe\dot{\mathfrak{a}}aIly$thuough $\alpha(\{x\})$in$\alpha(u)\cdot\beta(\{d)\cdot\gamma(\{x\})$ by

destroyinng the mathematical $\infty ndibon$ofinner product. Cytokine TNF $\beta$ is known to work to kill

canoer

cells. There

can

be possibly cytokines like TNF $\beta$ which work to kill

canoer

oells through

increasing$\alpha(u)$and theparameterslike

a

diffusion coefficient.

Re&oenoes

1. Charles A. Janeway

_Jr.

et. al. , Inmuno biology the immune system inhealth anddisease, Garland.

2.Takase,M.(2000) _Creativebindingandachievement ofhigh intelligenoe by top down

&edback.

6th InternationalCon&oenoe

on

Soft Computing $IIZUKA20\alpha$)

$-$

3. Takase, M. (1998) $Effic\mathfrak{t}$ of

&edback

$\infty nnections$ to memory $\infty mpoession$ and formation of

knowledge structure based on Hebb rule and inlubitory cells. Proc. Int. Conf Neural InfOrmation

Prooessing,$981\cdot 9\Re$

4. Ewend, M. G. et $d$

.

(2000) Intracranial paracrine $interleuhn\cdot 2$ therapy stimulates prolonged