腫瘍免疫機構に関する確率モデル : 生体防御に関する理論的示唆 (第8回生物数学の理論とその応用)

A Random Model for Tumor

lmmunobiomechanism:

Theoretical

Implication for Host-Defense Mechanism

Isamu DOKU

Department

_of

Mathematics, Faculty

_of

Education,

Saitama University, Saitama338-8570JAPAN

[email protected] 腫瘍免疫機構に関する確率モデル:

生体防御に関する理論的示唆

道工 勇

埼玉大学教育学部数学教室

The purpose consists in construction ofamathematical model that describesthehost-defense

mech-anismagainstcancer. Roughlyspeaking, therearetwodistinct methodsin mathematicalmodelling

forcancercells, suchasthe methodbyadeterministicmodel and the method byastochastic model.

Inthisarticle,based upon_{the latter}method,weproposean immuneresponsemodel againstcancer

as a mathematical model ofbranching particle system, and grasp the effects ofimmunity as the

reflectiveextinction ofa superprocessarising by takingthe limit of thesystem. Ordinarily, normal

cellsaretransformed intoirregularones bysomereasons, and thetumorigenicprocessproceeds. In concord with that,agroup of immune cellsinvokeimmuneresponse,and insodoing they accomplish their important _{errandof host-defense mechanism. We forcus our mind especially on the immune}

response bothin thetransformationperiod_{ofcell and in theproliferation period ofcancerated cell,}

and propose a stochasticmodel that isabletodescribe thecytotoxicactionsof effectorsagainst

can-cercells. Those effectorsaresupposedtobe NK (natural killer) cells,cytotoxic$T$cells,and activated

macrophages, etc. Analyzing the model mathematically, westudy thequalitative properties of the biologicalphenomenarelatedto immuneresponse,andwe areaimingatexplainingan extraordinary phenomenon such as the suturation of immune effectiveness, from the viewpoint of model theory.

Inourprevious_{research [6] weintroduced the} immigrationrate $q>0$ (apositive constant) asthe

cytotoxicintensity ofeffectorsagainstcancer. In thepresentpaperweimprovethispointandpropose

a moreelaborate model that can describe the effects by thoseeffectors, depending on the location

in accordance with the environmental changes. We finally consider the extinctionproperty of the proposed_{model, which}correspondstoakey physiological phenomenonof immune response relative to the effectors. 本研究の目的はがん細胞に対する免疫反応を記述する数理モデルの構築にある．がん細胞に対する数理モ デリングの手法は，大雑把に言って確定モデルか確率モデルによる記述の2つに大別される．ここではが ん細胞の免疫応答を後者の立場に基づき分枝粒子系の数理モデルとして提案し，免疫能の働きをその極限 の超過程の消滅性の反映として捉える．通常細胞が何らかの要因で形質転換して細胞のがん化過程が進行 する．それに対して免疫細胞群は免疫応答作用することで，生体防御の重要な役割を担っている．細胞の 形質転換期及びがん化細胞の無秩序増殖期における免疫応答に焦点を当て，NK 細胞，キラー

T

細胞，マク ロファージなどのエフェクターによるがん細胞に対する細胞障害性の働きを記述する確率モデルを提案し， そのモデルを数理的に解析することにより，免疫作用に関わる現象の定性的性質や特異現象に対するモデ ル論的な説明を補助的に提供することを目指す．前研究 [6] _{においては，エフェクターのがんへの細胞障害} 性の強さを移入率$q>0$ (正定数) _{として導入した．本研究ではこの点を改良して，場所ごとに異なり個} 体ごとの環境変化に応じたエフェクター効果を記述出来るモデルに関して，キーとなる生理現象に対応す る消滅性について考察する．

1

Introduction

We

are

aimingatmathematicallymodelling theimmuneresonseagainst cancercells. Ordinarily,

some

of normalcellsaretransformed intoirregularonesby severalreasons,suchaschemicals, carcinogens,

car-cinogenic virus and bacteria, DNAreplicationerror, DNA repairdisorder, chromosomal end centromere

disorder, radiation andso on, and the tumorigenic process proceeds. In concord with that, a group of

immune cells invoke the immune response against canceration, accomplishing an important errand of host-defense mechanism in the living body. The effectors are supposed to be NK (naturak killer) cells,

cytotoxic $T$cells, and activated macrophages, etc. We focus_especially on_{the immune response both}

in

the transformationperiodof cell and in theproliferation periodofcancerated cell, and proposearandom

the model mathematically,

we

provide with

a

complementary explanation ofqualitative properties and

peculiar phenomenarelative toimmuneactions from theviewpointof modelling theory.

$arJmt_{\theta}l$ $7_{C}\mathfrak{g}||$ $C_{:_{:}}^{\sim}O_{\vee::}.)$ ‘ $t|\cdot\cdot|k$ (億岡’$t\aleph$; $macm\alpha\backslash a\Re$ 図 1: Agroupof effectors

Recently the system biological researcheson

cancer

have madea remarkable progress, cf. Wang (2010)

[16]. Accompanying this rapidprogress, the modelling theoreticalresearches, simulation and numerical

analysis have become in full activity now, that

are

driving forilluminating the biological dynamism of

cancer

cells and mechanism of cancer-specific phenomena, based upon the standpoint of mathematical

physiology, cf. Wodarz andKomarova(2005) [18]. In thispaper

we

may adoptstochastic modelling

ap-proach [14] to graspthe immuneresponse against

cancer as

amathematical model of branching particle

system,and to considerthe effectiveness ofimmunity asa reflectionof extinction propertyon

superpro-cesses. In [1] we considereda formulation ofcatalytic processes applicableto filaments and catalysts in

physiologyandbiochemistry,andstudiedasymptoticbehavioursof solutions torelatedequations. While,

in [2] weinvestigatedaspecialclass of stochastic processes related to chemical reaction of the medicinal,

and proved the existence and uniqueness theorem formeasure-valued processeswhich isable to describe

theincreaseordecrease ofabranchingparticle system in number according to whether the environment

isgood or not.

2

Prerequisite from immunobiology

2.1

Network of

immune

system

The immunesystem intheliving bodyis regulatedbytheeffector-induction protocol. It is known that various kindof effectors (such as $T$cells, $B$cells, NKcells, NKT cells, dendritic cells,and macrophages,

etc.) forma very complicated network, and that thereis apossibilitythat it provokesapositive and$/or$

negative immune response for/against cancer cells. Our main concern is antitumor immune response,

and NK cells, NKT cells and $T$ cells have to do with the immune surveillance for cancerated cells. On the otherhand, the samebunch ofimmunecells reveal antitumor immune effects against swollen cancer

cells. The dendritic cell works as an antigen-presenting cell (APC) in the living body, that is to say, it

processesa tumor antigen, activates

an

antitumor $T$cell, and plays animportantrole in urging a CTL

to propagate. The macrophage is animmunecell which possesses a strongcytotoxicity, however in the

living bodywith akindofcancerit worksas animmunesuppressor or

as

anantitumor effectoraccording

tothe physiologicalsituation there. More precisely, in the network of immune system, first of all

a cancer

cell with tumor antigen is taken in by aprofessional APC with phagocytosis, then theAPC presents a

cancer

antigentoaCD$8+T$cellviaaco-stimulatorymolecule with thehelpofCD$4+$helper$T$cell,while

the CD$8+T$ cellwill be able torecognize the antigen viaconjugation with co-receptor, being urged to

CD$8+T$ cell next encounters

a

cancer cellwith the

same

antigen, then the CTL may recognize it

as

a

target and executesakilling of the cancercell by cytotoxicity [13].

鵜まAn$\phi\grave$に議融婁れ激 -燃鶏細謝と灘編餓$\sim$ 叢礁1れ桑 縢編編総 丁織纏 $’$ $-$ 灘蕩鱒轟駒$-$ $-$ 仮殴$a*C$乙 L が $-$ 灘購綴騰]繕謙 $arrow$ $-$ $-$

図2: _{Induction and effectual}phase ofCTL

2.2

Monoclonal

antibody and

antitumor immune response

Generally, the monoclonal antibody is produced by an immortalized hybridoma $(=$ a sort ofhybrid

cell)which isanantibody-formingcell ($=B$_cell)of_{target antigen-immunizedmouse, amalgamated with}

aspecial myeloma cell. The monoclonalantibody is much more specific than theblood serum antibody

(or polyclonal antibody),

moreover

there is a merit that it is possible to produce largely uniform and

identically specific antibodies. This cell fusion technique applied to this production of hybridomawas

established in 1975 by G. K\"ohler and C. Milstein, and they won the Nobel Prize in Physiology and Medicine in 1984 for this exploit. Recently plenty ofattempts have been made that the human-type monoclonalantibody produced_{from human cells is usedtothe}

_cure

_{forcancer. Forinstance, outstanding}

antitumor effects have been recently confirmed for two exceptional antibodies among monoclonal

anti-bodiesproducedfromhuman-cancer-cell-immunizedanimalsofdifferentspecies, suchastheantibodyof Her2 of breast

cancer

and antibody ofCD20 oflymphoma, see e.g. the report ofJACI (2011) [11].

The elucidation of molecular biologicalmechanism for antitumor immune response has been rapidly

promotedin thesedays. However, it is certain that the actualsituation is really complicated, and also

that the unknown parts are not few. A group ofimmune cells (suchas dendritic cells, NK cells, NKT

cells, and macrophages) carries the innate immune responseon itsback in the early stage, and induces

the acquired (adaptive) immune response of$T$ cells and $B$ cells being a system with high output rate

by antigen-specified _{proliferation, through the secretion of}cytokines and the presentation of antigen.

Although the antitumor effect is observed in the administration of monoclonal antibody via mouse, it is not clear unexpectedly what the antitumorimmunity via antibodies produced by the patient

means

in fact. The $T$ cell plays an extremely important role for tumor rejection in plenty of animal tumor models and human malignant melanoma [15]. In the immune response of$T$ cell, the $T$ cell receptor specifically recognizes the tumor antigen peptide-MHC complex on the

cancer

surface, and the $T$ cell secretes cytokines and injures the cancer cell directly. There are two kinds of $T$ cells in the class of tumor-reactive $T$ cells; one is the CD_$8+T$ cell that recognizes the MHC class I-peptide complex, _and

the other is the CD$4+T$ cellthat recognizes the MHC class II-peptide complex. TheCD$8+T$ cell has

to do directly with the recognition ofcancer cell. On the other hand, the CD$4+T$ cell has todo with

macrophage and the collection or wandering interception of antitumor CD$8+T$ cell within the tumor

area,

see

also e.g. Murphy et al. (2008) [12].

r$\zeta$

細胞 _Ct)

$3$

がん細胞

$|$

図 3: Antitumor cytotoxicity of$T$cell

2.3

Tumor escape mechanism

It is reported that the

cancer

cell possesses the so-called escape mechanism $hom$ various kinds of

immune responses. Sincethe cancercell has malfunction inthemolecule that is concerneddirectly with

antigen recognition by$T$cell, the

cancer

cell is capable to escape$hom$ theimmunesurveillance without

recognition by $T$ cell. For example, the malfunction in the molecule can be found in tumor antigen,

MHC, $\beta 2$ micro-globulin, and various molecules related to the antigen processing.

$?r\wedge\tilde{\kappa*}$ 嫁

$l1\mathscr{K}^{\aleph}\nearrow^{\prime\infty}It^{\gamma}arrow$

immune suppressors. Those suppressors are, for instance, $TGF-\beta$ and IL-10secreted from the cancer,

andIL-6 and PGE2emitted from the macrophage (whichisurged tosecrete bythe cancercell). Except

the above avoidance,

we can

list below someother factors: weakening of Thl responseby Th2

displace-ment, signaltransduction disorder of$T$cell,induction oftumor-antigen-specific immunologicaltolerance,

induction of antitumor immune suppressive $T$ cell, $T$ cell apoptosis induction by_$FasL$ appearanceon a

cancercell, local environmentthat intercepts collection of$T$cellsinthetumor tissue,and so on. See e.g.

Weinberg (2007) [17]; see also [15].

$t^{\eta}\backslash A^{\backslash l}.\#$

.

$\backslash jt^{f}\theta t\backslash r::$,

図4: Mechanism of antitumor effectors

3

Random

model

for

immune

response

3.1

Proliferation

process of

cancer

cells

When the tumorigenic process proceeds, normal cells are transformed into irregular ones by some

reasons and are cancerated, and they repeat disorder proliferation peculiar to the cancer because of

continualemission of falseproliferation signals bymalfunctioned oncogenes andtumor suppressorgenes.

On the other hand, the

cancer

cell is preyed or destroyed by effectors (agroup ofimmune cells such as

NK cells and

so

on) byvirtue of the immune surveillance mechanism in alivingbody. Then,taking them

allinto consideration, we introduce the natural number valuedrandomvariable$N_{n}$ : $\Omegaarrow \mathbb{N}$ for each_$n$,

whichmeansthe total number ofcancercells inthe n-th generation. We assumethat thereisa sequence

$\{\gamma_{n}\}_{n}$ofpositivenumbers such that

$\gamma_{n}arrow\gamma\in \mathbb{R}+$ $(narrow\infty)$

and also that

$E[ \xi_{n}]=1+\frac{\gamma_{n}}{n}$, $Var(\xi_{n})=\sigma_{n}^{2}arrow\sigma^{2}$ $(narrow\infty)$

where$\xi_{n}$ isthenumber of offsprings generated bythe n-th generation. This implies that thebranching

particle system has a clear tendency to increase in number. When we suppose that for each cell, the

proliferation or division

occurs

independently at a random time, we introduce the branching rate $n\lambda$

$(\lambda>0)$,which meansthe accelerated increase rate for the number_ofcancer _{cells. We adopt amodelby}

$*\succeq u$ $\hat{*}\overline{|\prime},\phi\searrow$ ...$u_{l}-\sim$ $|.\overline{\prime}\overline{\prime x\backslash *}$ 図 5: hajectories ofbranchingprocess

3.2

Spatial

movement

of

cancer

Sincewehave onlyto describe the immuneresponse inalocally limitedtissue,theregioninquestion

isrestricted toacomparatively small

area.

Sothat, itsuffices to consider the model in

a

bounded domain

$D\subset \mathbb{R}^{d}$ with$d=3$

.

For $N_{n}$ pieces of

cancer

cells in the n-th generation, each

cancer

cell is sopposed to

start at the initial point$x_{i}^{(n)}\in \mathbb{R}^{d}(i=1,2, \ldots, N_{n})$. While, it isconsidered that the targetcell $(=$ the

cancer

cell)

moves

little in theearly stage, namely in the transformation period ofcell, and also that in

theproliferation periodof cancerated cell it may diffuse and expand

as

if the liquid should seep through

aleatherbagbecauseofasuperfluityof proliferated

cancer

cells. Hence,weregard it

as

adiffusion with diffusion coefficient $k(\epsilon)$ dependingon

a

small parameter$\epsilon(>0)$

.

Thediffusionoperatorisdefinedas$L_{\epsilon}$

$=k(\epsilon)\Delta$, where $\Delta$is the Laplacian.

図6: Various kinds of

cancer

cells

3.3

Cytotoxicity of effectors

In

our

model the effectors

are

supposed to be NK cells, killer $T$ cells, macrophages among

a

group

ofimmune cells, and we will takethe cytotoxicityof these effectors against cancer into account. Inthe

previouspaper [6], the previous report [4] orthe previous announcements [3] (seealso [5]), weintroduce

adeterministic emigration rate$q(>0)$ (apositiveconstant) in the terminology of the theory of stochastic processes,whichexpressesthe intensityof cytotoxicitybyeffectorsagainstcancer. Although

one

may find it interesting

as

the first randommodel, it is not necessarily desirable to treat it like asimpleand poor

model, inorder to imitate the effects of immune responsebyeffectors against

cancer

from theviewpoint

of the modelling theory

as

well

as

$hom$ thestandpoint of future simulation analysis. In this articlewe

depending onthe location in accordancewith the environmentchanges. Thereare three methods inthe improvement. That is, it

means

that instead of thepositiveconstant $q$,we adopt a (random) function $q$

like:

(i)$q(x),$$x\in D$; (ii)$q(\omega)$ or$q(\omega, x),\omega\in\Omega$; (iii)$q(t, \omega),$$\omega\in\Omega$

.

In the model (i) the intensity of cytotoxicity $q$ depends on the location $x\in D$, which means that the

intensity $q(x)$ varies as _{the environment changes, and it strengthens or weakens according to the good}

or bad environment. In the second new model the parameter $\omega$ expresses the environmental change

independentof the sample$\omega’$ whichcomes _$hom$

the original stochasticity of the branching model. The

latter

case

$q(\omega, x)$just correspondsto the

case

$q(\omega)$dependingonthe location. In the model (iii)thetime

evolution of$q(\omega)$ can alsobedescribed. As amatterof fact, we canrealizeit asthe choice of

branching

rate $\alpha(x)$ and branching mechanism$\beta(x)$ dependingon the location $($or$\omega,$$t)$, forexample.

3.4

_{Superprocess under the limiting}

procedure

Under theabove-mentioned settings, weproposearandom model for the target

cancer

cells:

$X_{t}^{(n)}= \frac{1}{n}.\sum_{i=1}^{N_{n}(t)}\delta_{x_{t}^{(n)}(t)}$ (1)

where $x_{i}^{(n)}(t)$ is the location of the

i-th

cancer

cell in the n-th generation at time $t$, and _{$N_{n}(t)$}

denotes

the total numberof

cancer

cells alive at time $t$

.

Eq.(l) is the quantityrelated to an empirical measure,

expressing the stateof the cancer at time $t$. For instance, the qualitative property ofa random

walk is

wellreflectedby its limiting process,say, theBrownian motion. Likewise, the qualitative property ofan

aggregateofcancer cells canbe thought tobe reflected by its limiting process $X_{t}$. On this account, we

will analyze thesuperprocess $X_{t}$ in what follows.

4

Analysis

on

the limiting

process

Let $C=C(\mathbb{R}^{d})$ be the space of continuous functions on $\mathbb{R}^{d}$.

When $C_{b}$ denotes the set of bounded

continuous functions on $\mathbb{R}^{d}$,

then $C_{b}^{+}$ is the set of positive members

$g$ in $C_{b}$

.

Let $\langle\mu,$$f \rangle=\int fd\mu$,

and $M_{F}=M_{F}(\mathbb{R}^{d})$ is the space of finite

measures

on $\mathbb{R}^{d}$

.

We denote an $L_{\epsilon}$-diffusion process by $\Xi=$

$\{\xi, \Pi_{s,a}, s\geq 0, a\in \mathbb{R}^{d}\}$. Then _{$K\equiv K(dr)$} is the _{associated continuous additive functional}

(CAF),

and we

assume

that $K$ liesin the Dynkin locally admissible class of_{CAF, and} we _{write it}

as

_{$K\in$ IK}$\eta$

(some $\eta>0$). Then asuperprocess $X=\{X, P.,\mu, S\geq 0, \mu\in M_{F}\}$ with _{branchingrate functional}$K$ (

or $(L, K, \mu)$_{-superprocess)}

can

_{be characterizedas acontinous}$M_{F}$-valued time-inhomogeneousMarkov process $X=\{X_{t}\}$ with Laplacefunctional

$\mathbb{P}_{s,\mu}e^{-\langle X_{t},\varphi\rangle}=e^{-(\mu,v(s,t)\rangle}$, _{$0\leq s\leq t$}, _{$\mu\in M_{F}$},

$\varphi\in C_{b}^{+}$

.

Here the function $v$ isuniquelydetermined bythe log-Laplaceequation

$\Pi_{s,a}\varphi(\xi_{t})=v(s, a)+\Pi_{s,a}\int_{s}^{t}v^{2}(r, \xi_{r})K(dr)$, $0\leq s\leq t$, $a\in \mathbb{R}^{d}$.

We need Dynkin’s Historical Superprocess. $\mathbb{C}=C(\mathbb{R}_{+}, \mathbb{R}^{d})$ denotesthespace of continuouspathson$\mathbb{R}^{d}$

with topology ofuniform convergence on compact subsets of$\mathbb{R}+\cdot$ To each $w\in \mathbb{C}$ and $t>0,$ $w^{t}\in \mathbb{C}$

expresses the stopped path of$w$, and$\mathbb{C}^{t}$ is the totality of

all these paths stopped at time $t$. To every

$w\in \mathbb{C}$, putting $\tilde{w}_{t}=w^{t}$, _{$t\geq 0$}, we associate the corresponding stopped path trajectory

$\tilde{w}$

.

The

$\mathbb{C}_{R}^{\cross}\equiv \mathbb{R}+\cross \mathbb{C}\wedge=\{(s, w) : s\in \mathbb{R}_{+}, w\in \mathbb{C}^{\epsilon}\}$

.

We consider the set $M(\mathbb{C}_{R}^{x})\equiv M(\mathbb{R}+\cross \mathbb{C})\wedge$ of

measures

$\gamma$

on

$\mathbb{R}+\cross \mathbb{C}\wedge$ whichare finite, ifrestricted to afinite time interval. Suppose that $K$ is a positive CAF of$\xi$.

Then Dynkin’s historical superprocess (1991)

$\tilde{X}=\{\tilde{X},\tilde{\mathbb{P}}_{s,\mu}, s\geq 0, \mu\in M_{F}(\mathbb{C}^{8})\}$

is defined

as

a time-inhomogeneous Markov process with state $\tilde{X}_{t}\in M_{F}(\mathbb{C}^{t}),$ $t\geq s$, with Laplace functional

$\tilde{\mathbb{P}}_{s,\mu}e^{-(\overline{X}_{l},\varphi\rangle}=e^{-(\mu,v(\epsilon,t)\rangle}$ _{$0\leq s\leq t$}, $\mu\in M_{F}(\mathbb{C}^{\epsilon})$, $\varphi\in C_{b}^{+}(\mathbb{C})$

where $v$ isuniquely determined bythe log-Laplace typeequation

$\tilde{\Pi}_{s,w_{\delta}}\varphi(\tilde{\xi}_{t})=v(s, w_{s})+\tilde{\Pi}_{\epsilon,w_{s}}l^{t}v^{2}(r,\tilde{\xi}_{r})K(dr)$, $0\leq s\leq t$, $w_{8}\in \mathbb{C}^{8}$

.

Theorem 1. Let$K\in K^{\eta}$ and$\mu\in M_{F}$ with compact support. Then there enists an$(L, K, \mu)$-superprocess $X=\{X, \mathbb{P}_{s,\mu}, s\geq 0, \mu\in M_{F}\}$

with branchingrate

_functional

$K$

.

Theorem 2. There exests a Dynkin’s historical superprocess

$\tilde{X}=\{\tilde{X},\tilde{\mathbb{P}}_{\epsilon,\mu}, s\geq 0, \mu\in M_{F}(\mathbb{C}^{8})\}$.

In thepreviouswork[6] (see also [3-5])wehaverecognizedthat the extinctionpropertyofsuperprocesses

is very importantin the model theory. Especially

as

far

as

local extinction isconcerned, it isofextreme

interest and importance because itjust corresponds to the situation that the

cancer

cells

are

expelled

locally from the cancerated

area

bythe immune effects of effectors.

Since the initial

measure

$\mu\in M_{F}$ has

a

compact support, it follows from the argument of compact

support property (cf. Dawson-Mueller : Ann Prob 23 (1995)) that the range $\Re(X)$ of$X$ is compact.

Under the historical superprocess setting$\tilde{X}_{t}(dw)$,

we

define

$\mathbb{C}_{M}=\{w\in \mathbb{C}:|w_{8}|<M, \forall s\geq 0\}$

for $M\geq 1$. By the compact support property, wehave

$\lim_{Karrow\infty}\inf_{t\geq 0}\tilde{\mathbb{P}}_{0,\mu}(supp(\tilde{X}_{t})\subseteq \mathbb{C}_{M})=1$, $\mathbb{P}-$a.a.$\omega$

.

Proposition 3. For$K\in K^{\eta}$

$\lim_{tarrow\infty}\overline{\mathbb{P}}_{0,\mu}(\tilde{X}_{t}\neq 0$, and $supp(\tilde{X}_{t})\subseteq \mathbb{C}_{M})=0$

.

Finally, through the projection technique (cf. Dawson-Perkins (1991); D\^oku (2003)) we obtain Theorem 4. (Extinction property) Let$d=1$ and$\mu\in M_{F}$ with compact support. Then

$\mathbb{P}_{0,\mu}$($X_{t}=0$ for some $t>0$) $=1$

.

Acknowledgements This work is supported in part by JapanMEXT Grant-in Aids SR(C)

20540106

References

[1] D\^oku, I. : Exponential _{moments of solutions for nonlinear equations with catalytic noise and large}

deviation, Acta Appl. Math. 63 (2000), 101-117.

[2] D\^oku, I. : _{A limit theorem of homogeneous superprocesses withspatially dependent parameters, Far}

East J. Math. Sci. 38 (2010), 1-38.

[3] D\^oku, I. : A mathematical model for immune response to the cancer cells, the twentieth conference

of JSMB, Abstract Collection, DiseaseI, O-S (2010), p.66.

[4] D\^oku, I. : An application of random model to mathematical medicine, ISM Cop. Res. Rept. 262

(2011), 108-118.

[5] D\^oku, I. : On mathematical modelling for immune responseto the cancer cells, J. SUFE Math. Nat. Sci. 60 (2011), no.1, 137-148.

[6] D\^oku, I. : On

a

random model for immune response: toward

a

modelling of antitumor immune

responses, RIMS K\^oky\^uroku,

1751

(2011), 18-24.

[7] D\^oku, I. : A remarkontumor-induced angiogenesis fromthe viewpointofmathematical cell biology: mathematical medical approach via stochastic modelling, J. SUFE Math. Nat. Sci. 60 (2011), no.2,

205-217.

[8] D\^oku, I. : Onextinction property ofsuperprocesses, ISM Cop. Res. Rept. 275 (2012), 34-42.

[9] D\^oku, I. : _{A random model for tumor immunobiomechanism arising in mathematical physiology,}

preprint.

[10] D\^oku, I. : A stochastic model for tumor immune response from the standpoint of mathematical medicine: immunomechanismagainst cancercells andextinctionproperty ofsuperprocesses, preprint. [11] JACI (Ed.) : The 15th JACI Cancer Immunology Annual Meeting 2011, Abstract Collection, Senri

Life Science Center, Osaka, June 30–July 1, (2011).

[12] Murphy, K., Travers, P. and Walport, M. : Immunobiology, the seventheedition, Garland Science, Taylor

&

Flrancis Group, NewYork (2008).

[13] Peters, G. andVousden,K.H. : OncogenesandTumour Suppressors, Oxford UniversityPress,Oxford

(1997).

[14] Preziosi, L. : Cancer Modelling and Simulation, Chapman

&

Hall/CRC Press, BocaRaton (2003).

[15] Tannock, I.F.et al. : The Basic Science of Oncology,thefourthedition,McGraw-Hill Co. Inc., New

York (2005).

[16] Wang, E. : Cancer System Biology, CRC press, Boca Raton (2010).

[17]Weinberg, R.A. : The Biology of Cancer, Garland Science, Taylor

&

Rancis Group, New York(2007).

[lS] Wodarz, D.andKomarova, N.L. : Computational Biology ofCancer, WorldScientfficPubl., London