An automatic controlmechanism to ignite theimmune systemlocally against a small
cancer mass
considering reliability
1.Introduction
Tumor-immunesysteminteractionis considered. Then it is known that$T$cellsplaya mainrole[2].
There should be two stepsforagroupof$T$cellsto attack. There isthedetection of tumorpeptides by
each$T$cell causing$T$cell activation, and there canbe theoretically the local ignition of theimmune
system which I already proposed [1]. While the former can be thought to have a threshold with probability oftheactivation,the latter has athresholdbytheeigenvalueequalto 1 which
causes
the local ignition. The realization of thesestepsisexpectedto lead to thecureof the tumor.In this former situation for $T$ cell activation,
an
affimity between a tumor peptide and a $T$ cellreceptorcanbethoughttobeamechanical pattern matching, and thesituation is similar to neural networks[5]although separationbetweenhealthycells and the tumor cellsisimportantinthe tumor-immune interaction. But memory$T$cells
are
thoughttonecessarilymemorizetheantigen pattems.[Problemandpurpose]
(1)$Here$,the attackagainstthe tumor cells isthoughtto be conductedthrough the step of thedetection
oftumor cells. But at the sametime, theprotectionofhealthycellsshould be considered. Especially the local ignition as a mass action isthought to give a serious damage to thehealthytissue ifthe attackis wrong. Onthe otherhand, it maynotbe reahstic to think that each$T$cell hasanenough
reliabilitytoprotect healthycells almostcompletelybecause thereliabilitymustbevery high..
Thethresholdsandtheprobabilities, especiallythe latterone,
are
expectedto besetto giveenough high reliabilitytoprotect healthycellsthinking naturalselection and the evolution.On the other hand,ifwecanmake the thresholdscorrespondto the reliability, wecan grabthe behavior ofthe immune system more analytically and quantitatively thinking about not only the attackeffect, but also the enough reliability for the protection ofhealthy cells, and may lead the situationof tumor-
immune
systeminteractionto the direction of thecuremore
preciselyIf the results oftumor-immune interactionanalysisarecoherentto the necessity from theresultofthereliabihty analysis,we canbemuchmoreconfident to theanalysisresult.It is known that there
are cases
where the $\infty$stimulation of$CD4T$ and $CD8T$ aocompanies the$CD8T$
activation
to lead to its development to a R. Here forthe simplicity of the discussion, weanalyse onlythe$T$cellbehaviorswhich include the
cases
of thecostimulasion.While the consideration ofthe$\infty$stimulation will beexpectedtoheightenthe precision to detect
an
antigencorrectly,it willlower thecontactprobability shownin[1].Thereliabihtydiscussed here
is
verysimilarto thetest of$g\infty$dness of fit instatistics.
There is necessity forhealthy cells to be protected from thelocal ignitionwith
an
enough high reliabihty.Fromthesebackgrounds,thefollowings
are
shown.(1) qkegcell$\infty$ntributes to themechanismto make clearer the separationabilitybetweenbodycells and atumor peptide likein neural networks. in section2.1.
(2) Themechanismof thecell-wise detectionby$T$cellsandthelocalignition
as
themass
action oftheimmunesysteminalocalarea nearthe tumormassis shown insection2.2.
(3) Themodeltocalculatethereliabihtyisshownin section 3.
(4) The calculation example ofthe value which leads to the reliabihtyis conducted and shown in section4
Common
notation
usedisshown here.[Common notation]
Th helperTcell
$?b$ cytotoxic Tcell
Tact
.
activated$T$cell$lkgL2$ regulatory Tcellor suppressorTcell
interleukin 2
2.Possibilityof local ignitionincytotoxicimmunesystem Thefollowingtwo stepsarenecessaryto
cause
local ignition.(1) $T$cellactivation
(2) Necessityof thelocal ignition in
mass
actionof multiple$T$cells2.1 Activation of$T$cells
.
Itis known thata$T$oell becomes a?$kg$cell in itsdevelopmentinthymuswhenthe$T$cellhasabigaffimitywithapeptideofhealthy bodycells.
Inmanycases,especiallyin tumortherapyreportsforvaccinetherapy, ‘llregcellstendto work to weakenthe therapyeffect [3, 4],butatthe
same
tmehereit isexpectedthat they notonlyprotect healthycells but alsotheycan
make clearer the differencebetween atumorpeptideandpeptidesof healthycells lowerin$g$the$\infty mmon$partlike showninFig.2.$T$cellshaveaspectstoactivate based onprobabilityand have its distribution function.
Sohereweassumethat theactivationprobabilityofa$T$cellis determinedbytheaffinitybetween
thereceptorsof the$T$celland the antigen which isatumorpeptide.
Wedo not have the data to express the$T$cellactivation
curve
of probabilityvs.affinityThe examples of the
curves
withexpected charactersare
shown inFig, $i.$$Fhe$possibihtyof theassumption]
matchin$g$of thereceptor and the tumor peptide
$2$ Itis knownthat there
are
cases where (a) thereceptorsof$T$cells have anaffimity toanautoantigen which is not
zero
and is not enough strong (b) the antigen is not enough abundant,butan autoimmuneis not caused[2].$3$ There isanexampleof
an
autoimmunebyinsulin wherean autoimmuneis causedbytheabundanceof insuhn[2].
In $2$ and $3$ iftheprobabilityof$T$cellactivationaccordingtoaffimityisnot considered, the$T$cells
haveonlythe threshold ofthe activation to work with the thresholdbytheeigenvalue of the local ignition. Then there will be much less difference between a furious immune response and a mild immuneresponseaslong
as
thereisnodifference among$T$cellactivationlevels.Theseresponses oftheimmunesystemareexpectedtobe caused by not only$T$cells but alsothe
attack by antibodies, etc., sothe upper discussion may be notenoughto prove the existence of the probabilisticaspectof$T$cell activation, buttbeexistence of theprobabilistic aspectisexpected.
2.2$T$cells
mass
actionandlocalignitionWhen there is a positive feedback in a system behavior, it canbe expressed as an eigen value problem with eigen values and its eigen vectors. The feedback mechanism is shown in Fig 3. The exampleof this mechanismisshown in detailin[1, 5]whereTh,Tc andIL2areconsidered..
It
is considered that
each$T$cellworks through the probability expressed by $\alpha\cdot\beta\cdot\gamma$ in[1, 5]. $\alpha$.
contactprobabilityofapathogenwitha$T$cellreceptor$\beta$ recognition probability of the pathogenbythe$T$cell
$\gamma$ deletionprobability ofthepathogen bytheTc cell with the contact ofatumor cell
For the simplicity of thediscussion, $\alpha$ isassumedtobeconstant inthis article.
3. Model tocalculatereliabilityforseparationbetweenatumorpeptideand apeptideofhealthycells
[Background]
Qualitatively speakin$g$, it is expected that the total number ofTact cells has a main role to
cause the local ignition thinking from the mechanism. In otherwords, this means that local
$igniti6n$does not dependonthelocations ofTact cellsaroundthemassbutthe totalnumber of
Tact cellsaround themass.
We consider$X=$ $(x1^{++}X2 +xN)/N.X$
means
the average ofthesamples$X1,$ $X2,$ $\cdot$,$XN,$
where $x=1$when $T$cell$i$is activated,$Xi=0$when$T$cell $i$is not activated and
xihas activation
probabihty$p$and statisticalindependence.
The total number$N$varies,and the distribution ofxl,$X2,$ $\cdots,$$XN$in themassalsovaries.These
behaviors
are
expressedas an eigenvalue problem, and Xvariationdependsonthe eigenvalue,anditsabruptincreasedependsontheeigenvalue$\lambda>1$whichmeansthe localignition, andthe
distributiondependsonthe eigen vector[1].
So$X1+x2+\cdot$ $+xN$ is much concemedwiththe expression of the behavior of the local ignition
as a massaction.
Herewe useequation (1)for X. Because the equation(1)is similar tothenormalizedeigenvector in
an
eigen valueproblem,andwe think the responses of$T$cellmassaction.betweenhealthycells and
a
tumorpeptideis determinedby theaverage
and the diversion. Thenearer
to 1 the averageis,thehigherthe reliabihtyis.The smaller thedivergenceis,thehigherthe reliabihtyis.
.
$X=$ $(X1+X2+\cdot \cdot +m)/N$.
(1)Theaverage$ofX$ $E(X)=p$
Thediversion$ofX\sigma^{2}(X)=pq/N=p(1-p)/$Nq$=1-p$
$N$
.
‘Ibtal number of$T$cells ina unitvolumeinthemass
of tumor$P.$ $T$cellactivationprobabihtywhenthereceptors$\infty$ntactto
a
kin$d$of tumor peptides..Here, $\sigma^{2}(X)$becomes smaller when$N$becomeslarger.
4,Evaluation ofreliabihty by comparisonbetweentwocases
Atpresentwecannotcalculatedirectly
an
actualvalueofreliabihty mainlybecauseof the lack of necessary data, so here we inspect the effects ofthe total number ofparticipated$T$ cellsnear
thetumormass,the affimity and the number ofTactcellsby$\infty$mparisonbetweentwo
cases
oneofwhichhas
an
enough high reliabihty tocause
thelocal ignition without anydamage tohealthy cells, the othercase
whichcan
notcause
thelocalignition.Here, intheformercase, theaffinitybetween the$T$cellreceptorandthe tumorpeptideisenough
highto
cause
thelocal ignition, inthe othercase,theaffimityislesshigh andcan
notcause
thelocal ignition.We call the formercase
2 and the lattercase1.This relationshipisshowninFig.4
The probability$p$of the$T$cellactivationand the total number$N$of$T$cellswhich
means
the totalnumberof$T$cells with
an
affinity of the receptors toa
tumorpeptideare
givenaboutcase
1andcase
2respectivelyasfollows. Case1: $p_{1}$and$N_{1}$
$E_{1}$ $\cdots$its average
$\sigma\iota^{2}$ its$\cdot$
diversion
$Z1$ $\ldots$itsaffinitybetween theraeeptorandthe tumorpeptide Case2: p2andN2
$E_{2}$ $\cdots$its average
$\sigma 2^{2}$ its diversion
z2 itsaffinitybetween thereceptorandthe tumorpeptide
Here, weconduct the following modifications to make the reliabilityofcase 1 same with that of
case
2.(1) Modificationofdiversionbychange of$T$cell number inthelocal
area
aroundthetumormass
The diversion$ofX$isgiven byequation(2).
$\sigma^{2}\{X\}=p(1-p)/N \cdots\cdot\cdot(2)$
Where$0\leqq X\leqq 1.$
When diversionissmaller,the separationbetween healthy cells and tumor cell isclearer. Because the reliabihtybecomeshigher.
Usually $\sigma 2^{2}$ $<\sigma 1^{2}$becausep2is
nearer
to 1than$p_{1}.$
We
can
make $\sigma 1^{2}$ nearlysame
with $\sigma^{2_{2}}$byincreasing$N_{1}.$(2)Modificationof averageby changeof$T$cell numberinthe local
area
around the tumormassThe average ofXisgiven by equation(3).
$E\alpha)=p$ $\cdots\cdot\cdot(3)$
When the averageislarger,the separation betweenhealthycells and tumor cell isclearer,andthe reliabilitybecomeshigher.
Usually $E_{1}<$ E2 becausep2is
nearer
to 1 than$p_{1}.$Wecanmake$E_{1}$ nearlysamewithE2byincreasing$N_{1}.$
(1) Modificationto make diversionsequalusing the total number of$T$cells $\sigma 1^{2_{=}}\sigma^{2}\otimes 1)=p_{1}(1-p_{1})/N_{1}$
$\sigma 2^{2_{=}}\sigma^{2}\alpha_{2})=p_{2}(1-p_{2})/N_{2}$
From $\sigma 1^{2_{=}}\sigma 2^{2}$
$p_{1}(1-p_{1})/N_{la}=p_{2}(1-p_{2})/N_{2}$
$N_{la}=N_{2}\cdot p_{1}(1-p_{1})/(p_{2}(1-p_{2}))$ $=N_{2}\cdot(p_{1}/p_{2})\{(1-p_{1})/(1-p_{2})\}$
(2) Modificationusingboth averageanddiversion
We conduct modificationby$E_{1^{2}},/\sigma 1^{2_{=E_{2^{2\int}\sigma 2^{2}}}}$where the dimensionsaremadeequal,because both
average and diversion affect thereliabihtyof the separation and have different dimensions..
$p_{1^{2}}/\sigma^{2}(X_{1})=p_{2^{2}}/\sigma^{2}(X_{2})$
$p_{1^{2}}/(p_{1}(1-p_{1})/N_{lb})=p_{2^{2}}/(p_{2}(1-p_{2})/N_{2})$
$N_{lb}$ $p_{1^{2}}/(p_{1}(1-p_{1}))=N_{2}\cdot p_{2^{2}}/(p_{2}(1-p_{2}))$
$N_{lb}$ $=N_{2}\cdot(p_{2^{2}}/p_{1^{2}})\{(p_{1}(1-p_{1}))\nearrow(p_{2}(1-p_{2}))\}$
$=N_{2}\cdot(p_{2}/p_{1})(1rightarrow p_{1})/(1-p_{2})$
Weadopt$N_{lb}$becauseof$N_{lb}>N_{la}$fromp2 $>p_{1}$
$\lceil The$meanmgs oftheresults$\rfloor$
(1)When$N_{1}$is increased to$N_{lb}$, the reliabilityachievedbyboth$N_{lb}$ andthe affimity of$Z1$ becomes
nearlysamewith thereliabilityachievedbybothN2 andtheaffinityof$Z2.$
$N_{lb} = N_{2}\cdot(p_{2}/p_{1})\{(1-p_{1})/(1-p_{2})\}$
(2)When$N_{1}$of$T$cells with theaffinityof
$Z1$isincreased to$N_{lb}$,the number ofTact
can
be calculated inequation(4).$N_{lb}\cdot p_{1}=(N_{2}\cdot p_{2})\cdot\{(1-p_{1})/(1-p_{2})\} \cdots\cdot(4)$
Equation (4) means that the number ofTact cells with the affimity $Z1$ is nearly same with the
number ofTact with theaffimity$Z2$although$\{(1-p_{1})/(1-p_{2})\}$ismultiplied.
$\lceil the$modified number of Tact withaffimity$Z1\rfloor=\lceil$(thenumberofTact incase2)$\cdot$$\{(1-p_{1})/(1-p_{2})\}\rfloor$
This
means
that the numbers of Tact inthetwocasesare
almost equal each other to a certainextent after theequahzationofthereliabilities.
The meanin$g$of$(1-p_{1})/(1-p_{2})$isthat when$p_{2}=1,$ $\sigma 2^{2_{=}}p2(1-p_{2})/N_{2}=0$There isnoerrorin the
ditection,so$N_{lb}$must be $\infty.$
$\{(1-p_{1})/(1-p_{2})\}arrow\infty$ when$p_{2}arrow 1$because $\sigma 2^{2_{=}}p2(1-p_{2})/Narrow 0$ when$p_{2}arrow 1.$
5.Conclusion
The results of theanalysisconducted herecanbe arrangedasfollows.
(1) The recogmitionof a tumor peptide by $T$ cells is considered as a separation problem between
healthy cells andtumorcells through the tumor peptide
(2) lkeg has a function to expand and clarify a difference between healthy cells and tumor cells
throughthetumorpeptide.
(3) The reliability of the separation between healthy cells and tumor cells by probability
can
be expressedusingbinomial distributionalthoughitisdivided by thetotalsample number$N.$(4) Even if theaffimity of$T$cellreceptorstoatumor peptideis weakor$T$cells
are
repressedby qkegincreasesgetting
an
enough reliabihtyto be able to achieve the local ignition from the pointview
of reliabilitysafely(5)Whentheaffimityof$T$cellreceptorstoatumorpeptideisweakeror$T$cells
are more
repressed byTregsin comparison with the$\infty$ndition of$T$cells where
an
enoughhighreliabilityis achieved andthelocal ignition
can
becaused,the number of the$T$oelk with thesame
receptor, whichhas theweakaffimity,must beincreasedtoaverylargeextent to have thesamereliability. References
1. Takase, M. (2010) Induction and application of
an
equation to analyzea
local ignition of the immune system fora
$\infty$mplete deletion of a cancer mass. Theory of Biomathematics and itsapplications VI.RISM1704,53-60Kyoto University.
2.Janeway,C.$A$,Jr.etal.Immunobiology:the imnmunesystemin health and disease. Garland. 3. Zhang,$Y$ etal. (2006)Thl celladjuvant therapycombined with tumor
vaccination.
IntemationalImmunology 19,151-161
4. $N\infty rth,$ $R$ $J$., and Bursuker, I. (1984) Generation and decay of the immmune response to a
progressive fibrosarcoma $J.$$Exp$
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Med. 159,$12\Re$)$-1311$5.Takase,M.(2009)Cancer and immunesysteminteractionmodel like a neural network
model, analysis of
cancer
mass effect and meaningofvaccine. Theoryof Biomathematics and its applications V.RISM 1663,35-40Kyoto University.$T$cellactivation $T$cellactivation
Probability$p$ Probability$p$
receptorandantigen receptorandantigen
Fig.1 Thegraphsof$T$cellactivationprobabihtyvs.affinitybetween$T$cellreceptorandantigen
Asmeaningsof $\alpha$ the$f_{0}u_{ow\dot{m}g}$twocasesare$\infty$nsidered.
(1) When theaffimityis1and
near
toapeptideofhealthycells, $\alpha$means
theinhibitionby Ikegcells.(2) $\alpha$
means
alittledisplacementof$T$cellreceptorfrom themaximumfittingwhenwe assumetheylocation $y$location
A schematic tumor cell peptide A schematic healthy cell peptide pattemwith a mutationon MHCI or patteminhibitedstrongly bylbeg MHCII
ylocation
A pattem made by mutation thought to be extractedthrough Treg
Fig. 2 Mechanism to make the difference between a tumor peptide and that ofhealthy cells by deleting thecommonandsimilarpart
Aboutwhyeigenvalue
can
beappliedFig.3Anexampleofmechamism to
cause
thelocalignition Thact activated$CD$4$T$cellqbact activated$CD$8$T$cell
Thm
.
memory$CD$4$T$celllbm memory$CD$8$T$cell
It
is
considered that
each$T$cell worksthroughtheprobabihty expressed by $\alpha\cdot\beta\cdot\gamma$?keg Tcell Whenan antigen becomes more similar to ahealthy cell peptide,the mhibitionby‘Ilregbecomesstronger.
$\square$
Fig. 4 Relationshipof$T$cellactivationprobabihties by affimity$Z1$,
z2
$T$ cell activation probabilities by affimity $Z1$, z2 are
$p_{1}$ and p2 respectively, but the contact