Stochastic Game Formulation for Cancer Radiographical Detection

Copyright © IFAC Control Systems Design, Bratislava, Slovak Republic, 2000

STOCHASTIC GAME FORMULATION FOR CANCER RADIOGRAPHICAL DETECTION H.S.Kang, and K. Kang 470 Carnegie Dr,Milpitas,CA 95035, U.S.A.

ABSTRACT: In this write-up mathematical model of cell osmotic process is developed. Probability density distribution of interval to administer biostimulant for optimal survival is obtained.The traditional cancer therapy problem is formulated as non zero sum cyclic stochastic game employing behavior strategies for optimal survival. Mathematical model is obtained for ion transmitter for hypothermia therapy. Neural supervised training algorithm employing amorphous model for hypothermia therapy is given . Radiation transport model for tumor growth and procedure for forecasting tumor metastasis process is given. Probability density distribution of tumor process termination time is obtained. Copyright @20001FAC

Keywords: Biocontrol , Neural net, Biocybernetics, Monitored control systems, Model management, Medical systems, Differential games.

1.0. INTRODUCTION Mathematical (Kang. 1995) model of cell osmotic process is developed in the following. Clinically measured indices may (Tosteson etl 1960,Sundareshan eH 1986) be employed to access the homeostatic disorder and to compute rate of chronic disease annihilation. Mathematical model for virus growth with bell distribution is obtained in extended probability space. The developed model is applied to analyze cumulative effect of membrane potential and external electrical field (Kang 1995),on chronic disease spread.To obtain time markers for optimal administering ofbiostimulant HJE (Hamilton Jacobi Equation)is employed (Kang 1970)to markov chain system model with jump parameters.The lymphocytes (Sundareshan et! 1986) and tumor (Laired 1964) cell with memory may bind and coalesce on encounter in neural complex. The growth of tumor cells with memory and higher association coefficient is irreversible.The cancer therapy problem is formulated as non zero sum cyclic stochastic game .The proposed method employs propagation ,reflection, scatter of ( Kang et! 1997) ions spherical wave fronts (Kang 1998) for radiographical detection of tumor nodules, lung metastases and sheets. Center average fuzzifier /defuzzifier mapping may be employed to identify grade membership and to implement error back propagation (Kang etl 1997) wavelets training algorithms for hypothermia therapy. Neural supervised training algorithm employing amorphous model for hypothermia therapy is given. Rotational frame field (Kang et! 1997)

585

produced by external field may be employed to control polar orientation of biological cells. Technique proposed in this write-up facilitates heating of deep seated tumors, and may be effective in treating metastatic tumors,which can not be treated by conventional therapies.Mortality dynamics for tumor in mapped cemetery space is obtained 2.0. BIOLOGICAL CELL TO CELL MAPPED NEURON SYNAPTIC WAVELETS Let m be the molecules in inter cellular region (figure 1) with n electron charge, Na+,K+,Cl- be the extra cellular ion field, p be the pump ion exchange rate, q be the ion charge,C Irn be the capacitance ofthe cell membrane, Vc be the volume of the cell. Let ~': ~a'~K,n vel E~ be the membrane potentials , &a,gK'&:1 be the membrane conductance, k, T be the parameters for ideal gas law.From electro neutrality of resting state, neurons ionic current dynamics and membrane potential can be expressed as Ina = &a (~a - (kT/q) log ([Na+]a1[Na+]j ))+ pq IK = gK (~K - (kT/q)log ([K+]J[K+]j)) - pq lel = gel (~eI + (kT/q)log([CI-]J[Cl-]j )) CIrn~' =q(Ve[Na+]j+Ve[K+]j -Ve[Cl-k mn) (1)

2.1. Modeling Cell Osmotic Process: Let xS)=I,.,d;s=I,.,q; be the state characterizing population of i-th class of molecules in intercellular region, decomposable as martingale semigroup drift

process and Brownian motionprocess .Let s be the organ compartment, Qyjs(yS) be the coefficients in extra cellular ion field, here yS is the molecular migration stimulant ,1:j be the termination time constants ,Wj be the effect of uncertainties on cell population. Let pi() and pij( ) be the probabilities of stimulation and differentiation of cell .Let [u\];i=I,..d; be the control parameter vectors. The compartment cell migration and osmotic effects can be expressed as dx\ = f(xt,t) xSjdt+ [BSj] [USj] dt+[SxJ+ Cwjdwj f(xt,t)= r (fI) fl I'" .f'l1 } {(2pl I2pI2)··(2pqI2P~)} {(2pl IdP 1d)··(2pqldPqd)} 1 I {(2pI2Ipll)··(2p~IPql)} {fl 2fl 2.. • .t
I L{(2pl dlpl 1)··(2pQdIPQ\)}· {(2pl d2 pI2)··(2pQd2P~)}··

.

..... {fI d fl d"

I

J

·f'ld}

martingale {Mxc.loc,.3 t ,0:51:O:;oo}; adapted to progressively measurable filtration .3 Xt= cr(xt; O:O:;~t) ;k=I.2.. ; x(Mt) ERd with borel field in normed variant measurable probability space (Q.3.P) .The bio stimulant Sy: sey~xt is administered at time markers tm=tkAt : inf{Q 0; kAt ~ }[O:O:; ~t <00]. The augmented (XSt,ySt) mapped wavelet is a standard Brownian motion ,Bs=Mtrn ;j=I,2, . , with filtration M'-J s= B trn; O:O:;s < 00. Let Cw: R~R be lipschitz continuous, (Cw)2 be bounded .It may be verified that augmented process (XSt,ySt) is nonexploding and has unique solution. Eigen value computation for proposed wavelet jump process model requires additional effort .

2.2. Optimal Administering ofBiostimulant :

Model developed is applied for determination of time fjj= (-1/·ti j +pij -2I:q.i vilivii) ; i=l...,q ;j=l d; markers to administer biostimulant for optimal survival. xt= [x l \,x 21, •• ,Xql,xI2,x22' .. ,x~, , xl d,x 2d, .. ,xqd]t Let Xj ;i=I,2,.. ,5 be respectively the immune component [u\] = Oyjsl(ysl)[xSj] ; i=I,.. ,d;s=I,.. ,q (2) cells, concentration of plasma cells , immune complexes, monovalent virus, cells free of antibodies. x6 is the virus infected organ, x7 is concentration ofbiostimulant . Dynamics of bio stimulation process can be expressed as dx/dt=A(Sx)xt+ B "t+Sx; Sx=diag[Sxl 00 Sx4 0 0 Sd(4) B =(0 0 0 0 0 0 Wdqsgn[rdqlo)t ;X1 =[xl X2 X3 X4 X5 "6 x7]t A(Sx) = r.II"tI+Cml(PI-2PIPI2) 0 0 0 0 0 01 I 2Cm zP21PI -111:2 0 0 0 0 0 I

Fig 1. Electrical and Osmotic Effects·

-Fig 2. Neural Net Cortical Potential

Here Cwj is the diffusion matrix , and is characterized by the cellular and molecular kinetics, Sxi is the cell population source. Internal controls as cytotoxic T lymphocytes may be expressed by field of matrices of regular pencils of Sxi. which may be reduced to canonical quasidiagonal form Sx=diag {(S\I ",S\I), .. ,(S\d'" Sqxd)} determined by finite elementary divisors of pencil.The indices for Sx are defined by the union of minimal indices of diagonal blocks. In the canonical form parametric control field can be expressed as [BSj][u\] =diag «w ll sat[rlllo .. w1q sat[x\qlo)' .. ,( Wdl sat[rdI1o) .. Wdqsat[rdqlo», here W il ;i=I,. ,q ; 1=1,. ,d; are the synaptic weighting factors.

I I I I

0

-e53

0

.1I1:3

0

0

Co

0

Loo

0

0

0

0

Cjee 0

0

C35Sx4 0 0 C 43 C aa .111:4 -C54CaaSx4 0 Cpe C53 0 .J/~S-C55Sx4 0

-J.-l6

I 0 I 0 I 0I 0

0 .1/t4·CbJ

where SXI is bonemarrow stem cells concentration of immune component source,Sx7 is stimulant administered , Sx4 is the molecular source rate for virus, Caa ,Cb are the coefficient of effect of antibodies on antigens and concentration coefficient for antibodies specific to biostimulant ,J.-l6 is the organ damage time constant ,Co is the infection coefficient . Cpc'C jee, are the rates of antibody production by plasma, and immuno component cells ,C mi ; i = 1,2 are the immuno competent multiplication factors, Pj and Pij ij=I,2; i* j are the induced/differentiation probabilities of antigens. Solution to proposed singular drift matrix formulation is Self Organizing Wavelet Jump Process: Let Sy be determined by minimal indices and elementary divisors administered non_multiplying antigen, biostimulant of the pencil blocks of A(Sx) and [B] .Administered jump process for class of molecular concentration yS. Cjj ; dose toxicity equivalence mapping performance index is i,j= I,..d, be the association! disassociation coefficient . given by PI =E pro (xt'Tt,I(TQyt Q TQv) xt +u'(Tt'IR) Exogenous adapted sernimartingale process ys is given u)dt ; [qjj]= 0; i*j; ij= I, .. ,d" here transformation .Tt is by determined by cell characteristics,TQy: exp(-qQy(x1j)lkT) dyVdt =h(yS,t)ysi(t)+[~]xsi+[S\]; ~r2y(xSd) ; i=l,..,qj=l,..,d, is the cell to cell mapping of (3) hjj(ys,t)= -1/1:i+Cjj-~j..FjjYj j=l,...d neuron grid potential energy; here Qy(xSd) is the [Seyjl=diag[sey1,D..D.. ,SeydO..] ; [Yj]=[yIJ D..D. y1dD..] membrane potential. Dynamics of equivalence mapping transformation biostimulation process is given by here [d j] is the rate coefficient drift matrix. Let molecule population process x\ be continuous local

586

Ldj~1 0 (ii)
Hamilton Jacobi Equation - Administered Biostimulant process: Let Sy be the continuous time markov chain process, Pjj; ij=l,..,n, be the transition probability on (0,3, P). Given S(t)=(Tt-ITov)Sx,it is assumed that system jump parameters are not directly accessible to the controller . Proposition 2.2 : Let M(t,i) be symmetric positive semi definite matrix for given markov chain sequence. Class of admissible biostimulant sequence control action ut is given by solution of following dynamic equations ut=-R-I~iPi(t)(Tov)B(i)t[M(t,i)~~] ; B(i)'={ToJ3(i)} (6) dM(t, i)/dt+{A(Sx)'}tM(t,i)+M(t,i)A(Sx)'+~lijM(tj) +TOvt Q(i) - M(t,i)B(i)'R-l~1 PI(t) Tt B(I) M(t,I)= 0 dS/dt=-[A(Sx)'- B(i)'R-ITtB(I)'t M(t,I)]lSt+ M(t,i)(Tt)-ITQVSX A(Sx)'=(Tt)-IA(Sx); TOvt= (Tt)-ITOvtTOv (7) •

2.3. Bell Distribution Neuronal Virus Growth: Given spherical Set Ss: ri ERd; here ri is the radius. Let Ymik ERd C Ss be the virally infected cells population at neural net link node IIlj,i=I,.,d. Let Ws be the assigned synaptic weights identified by inode perturbation for infected cells moving to left! right daughter net links (figure 2) . Population growth in multiIayer neural network can be expressed as

are holder continuous and bounded (iii) max IV(t,i;,)I$c& (I + 1Ii;,le&) ;c& >1. Using backward Kolmogorov equation termination time ('tf) for annihilation process can be computed from the expression Yt(s)= Ei;[V(sn-)exp{ -Ft k(~,s)ds}]; [O,T)xRd.

3.0. VIRUS MULTIPLICAnON IN EXTENDED PROBABILITY PRODUCT SPACE Given random process consisting of virally infected cells, antibodies, plasma in product space (Rd,Ql~' .~nd' B(Rn) , 3 1 ~' .03 d,3 0'P10..0Pd), and disjoint sets of viral sequence vooi=IYi~Vand nooi=l Vj=O. Extension {~+l (xsm+l ,Ym+1) +Lmi=l fi(x\'Yi)}~ f(xs,V) of finite virus sequence in product space may be employed to analyze growth dynamics of virus. Let P be the probability measure on (C[O,oo)d, B(C[O,oo)d» with M vj= ~(xSi' V j )~(xSo)-fo 4(A(Sx) Ve(~»de; .Jt, 05t<00 a continuous martingale. It may be verified that weak solution {x\'Vt ,rr,.Jt-,P"'""} -+x\- in extended probability space exists.

Formulation

0/ External Biostimulant Problem: Let N

={N~n;3GE300

with N~G,P(G)=O} be the collection of

Null sets. Let 3=cr(3tvN); O:O:;t~vmx};j=I,2 .. Let MC,x: {M\=B\j;j= Boundary condition can be decomposed as absorbing, reflecting and refracting subgroups. Let Ni be the total 1,2,.. ;3t; 0 $t ] random process k(8Yt)E35 be the rate of annihilation ~ E( 0; tj$t 5tj+ I' The distribution of for virally infected cells. Density dynamics of Bell distribution genetic traits and epidemics virally infected stimulant markers tj j= 1,2,.. is given by P[trlmxEdt]= S cell population growth Yt(S)ECt.2 ([O,t)xRd ) is given by «xS-(9-xS-mx)/( -J21tt/» exp{ _(XS-(~). X - mx)2 /2~}dt ; s XSt-:O:;x - mx ; tj:o:;~tj+ I'

se

Iac,?

avt(S )/8t=4{A(sx)Vt(i;)+k(t,i;)Vt(i; ) ; 0$t$jM:<;oo;j=1,2,.. LlA(SX>Vli;» =( 1/2)Ldi~lLdj=1ll;lt,i;)02Vli;)/aslasj (10) here k(s)V is the brownian motion. Assuming (i)

3.1. Control by Aggravation o/Chronic Disease:

Let x (yS,Sxj) be the collection of convex functions of semimartingales in extended probability space ,with

~di~l

587

parametric controls Sxj' Let Llt°(ys,Sxj); (t,yS,XS)E [0,00 )x Rdx Rd ,XsEn be the measure of enhanced immune response to administered dose and is characterized by life span of recovered normal cell state XheRd. Given x(~)=xSo(f2vo), here f2vo is the membrane bias potential. Let PO(x\-XSo) be the linear variety of the probability distribution function P(~S). For optimal survival it is desired to maximize immune response LltOf miI;(l/2){ IXs-t-Xhtl +Ixso-xhol +roSy sat(xs-Cxht) dM"t}; j=1,2,.. measure in stable periodic canonical space C[O,oo) .

press

pump emit photon counter

Fig 3. Oximeter

Fig 4. Cureability brownian trajectory

Proposition 3.2 (Curability): Let [lljj(t,cP)]c D be the bounded open subset. Define cP ={ Oaj/Ox\}, let aD be the support plane, with closure mapping D-~R~RdID .Given the state (xSo,vo)~aD (figure 4) and class of parametric controls consisting of sequence of administered dose (immuno globulin receptors and B Lymphocyte), the recovered state D\{XSo} of disease is contained in translated cone xSo+C(xS(cPoM); C(xS(cPoM) ~ RdID~ IIxStll IIxS(cPo) llcoscP; with aperture ~mn$cP$~mx ,and XS(cPo) axis of the cone. •

4.0 FORMULATION OF CYCLIC STOCHASTIC GAME FOR CANCER THERAPY Consider metastasis tumor process in finite countable space. The dimension of malignant cell space is same as the degree of its generator minimal polynomial . The cell space may be decomposed into cyclic subspaces . Let Ilj be the rate of transformation to malignant cells,Aj; j=l, 2.. be the rate of phase replication. Define (xSi)'= {xlii xri z)' ;i= l,..,ne; here Xhj is the normal component of cells, and Xrj is the resting components of cells ,and Zj ;i=l, .. ,H e is the malignant component of cells. The i th stage cell population during conversion process can be e;\:pressed as dXj(t)/dt= A;(u,t)Xjct) ;i=I,2, .., here A;= diag . Ah j- (11. 1 h {Ahi' Ar j, AZ}.' j, IS the quaSI. di agon al matrIx, / h r Il j ) : N i= (Arj+ Il j ) ; are the drift matrices for cell recycle, and resting phase; AZ i= _(AZj+ IlZj ) is the tumor cell replication and mortality phase. Survival index is determined by union of minimal degree of diagonal blocks of A(u,t) . Let Cl' c 2 be the rate constants characterized by the individual effector and tumor cell structures, Ne be the total number of effector cells. llzmx.z is the maximum number of cells available for transforma tion to tumor cells before process termination. Player I employs progressively measurable process of clinical indices as neoplastic radiographical detectors and set of

588

actions uI(c],ciz) EU;as radiation ,hypothermia, immuno, traditional cancer therapies, and radical surgery. Player II employs set of actions u 2(1lz, yz) El as infiltrating lymphocytes,epitheticallayers, lymphoid organs. Target tumor cells (z) a submartingale process for organism in cemetery space may be expressed dzldt=n mIX_z(u 2(l!zIz )+u2("-z!z )VZmx]-Ne(l-U I (Cl ,C 2·I )Z]U I (C I )2 (11 )

4.1 Nonzero Sum Game with Behavior/Pure Strategies The traditional cancer therapy in phase I may not detect initial tumor cell growth . By employing radiation transport model (section 6) neoplastic nodules volume is given by Vz=(7tl6)IIj 3Clxi ,where Clxi are the major/minor axis and may be obtained by radiographical detection. For non zero sum game characterized by tumor target cell dynamics (section 4.0), Players IIII employ pure / behavior strategies respectively at the initiation of game. Conditional probability for malignant cell state zk+ I is given by P(zk+l/ zk,Nek,u1k' u2k)' Let r(zk,u1k,u2k) be the game return, and is dependent on parameters Ne,Zmx, let Pk be the discount factor, a measure of spread of cancer. Let vk be the value of game with stationary strategy . Stochastic game may be reformulated as a game with payoff PI = r (zk,U1k,U2k) +L:k=l"" PkP(zk+l/zk, Nek,ulk, u2k)vk vk= inful ""sup u2"" r(zk ,ul k, U2k ) ,k=l,2,.. ; ztn=zts (12)

Behavior Strategies for Optimal Survival: Let B, U be nonempty borel sets. After initial radiographical detection of malignant cells,cancer therapy stochastic game problem is reformulated in the phase 11 . It is assumed that in phase 11 players I /II employ finite countable behavior strategies. The strategies U1,U2EU are optimal for the cyclic stochastic game, if are also optimal for the game formulation as given in equation (12) with ~o=Ztt1. The return r (zk, u1k' u2k) is minimized by player I and is maximized by player 11 over the survival period. The transition probability distribution matrix for stochastic game can be expressed as p(\,llj)= diag [pi]; here probabilities related to malignant cell transformation to i+ 1th stage is given by pi ll = njj=l (A/t)+llj (t)) ,conversion in resting state is pl[2= Aj(t) nl-'j=l("-i
5.0. SYNAPTIC WAVELET SPIN LINKAGE

Let Wjj~IO, I) ; WjjcW be the fuzzy set membership function with fuzzy implication ~\j!t = [L'.\j!e L'.\j!v1 representing grade membership L'.\j!eE~ , here \j!e(t) is the external field, ; L'.\j!v= [S,,:!wi{ XSt~Z ;\j!vE~ is the neuron synaptic activated transient membrane potential field signaling nerves and muscles (figure 2) . Let q be the ion charge, and 1be the muscle segment length. The molecular ions energy gradient is given by L'.u(t)= q( L'.\j!v -L'.\j!e(t) l/kT, where k and T are constants. Transition probability of cells to resting and tumor replication! mortality phase is given by pZlI=W II exp (L'.U(t» /{ exp ( L'.u(t» +exp(-L'.U(t»}, pZ22=w 2/m{ exp (L'.u(t» +exp (L'.u( t» }.The probability of tumor bonding affinity and tumor unbonding is given by pZ2'= W21 exp ("zmx-z)/ {exp (L'.u(t» +exp(-L'.u(t»}, pZ'2=w l/m{ 1+ exp (L'.u(t»}, where m=6 for tumor cubic lattice structure of neighboring organ compartments,wij ; ij= 1,2 are the neural synaptic weights. It may be verified that [pZij( \j!e' \j!v)]n =wn[ l/{ 1+exp (-MU(t»)}] is the probability of energy level increase of n cell markov chain and daughter cells carried by lymph to enter malignant cell state . Probability that a multiplying pathogenic malignant cell may recover to normal cell is given by [Pz('1'e,\j!v)]n = w nz exp(-Mu{t» .

5.1. Forecasting Tumor Metastasis Growth: The tumor metastasis process is inaccessible for identification, causing failure of traditional growth forecasting methods. Given spherical set Ssc: rERIO,CO), with r~t)EB{Rd) ;~tE[O,OO) ,k= 1,2,.. ; the smallest afield containing all sphere sets Ssc . Let Cm be the mean value of tumor cell concentration in lymph, Cva be the constant characterized by environment , p be the bonding coefficient of tumor cells,Vm be the molar volume of tumor nodule, Az(I-.,J.l.,u) be the diffusion coefficient of tumor cell lymph . Concentration of collection of tumor cells carried by lymph at sphere surface with radius r p is given by Cp=Cvaexp (2p Vm/ rpRT), here R is a constant, T is temperature. From mass balance, number of tumor cells z(rp,t) at sphere wave front r $ rp$ r+8r ,carried by lymph can be expressed as

an (mVt,mat)/8t= farnxrxoY (m\,h\) hat mat "t(hvt,hat)"t( m\,mat)8mVt8mat + (1/2) fam"o rm"oY {mVt-mVt_1 ,hvt)hat [(hat {h\)2/3+mat_, (m\,)213 -m\l{mVt)213 )/(m\)213] n(h\, hat) n[mVCmVt_l,({hat (hVt)213+m\,(mVt_I)213-mat_l(m\) 2/3)/(mVt )213] [(mVt)213/«mVCm\I)213»)8mVt8m\ (14)

6.0. RADIATION TRANSPORT AMORPHOUS HYPOTHERMIA MODEL In the following Radiographical detection technique is developed for early detection of tumor. Using Monte carlo simulation for transport model , reflectance/ refractance may be employed to identify objects as lung metastatic nodules and neoplastics. Proposed technique may be extended to compute blood oxygen (figure 3) levels. Let dV be the volume of the cell slab, (r,e ) be the coordinates of incidence vector orientation of cell slab ion field, c be the speed oflight, Ac(vce,vce') be the scattering cross section, here vce',vce are the velocity and unit normal velocity component. Let SI,Sa be the attenuation from scattering and absorption, and ScecSsc be the unit sphere. By employing the conservation of energy, transport model for intensity radiance function I(r,e,vce) can be expressed as (l/c)8I{r,e,vce )/01+ vce .VI{r,e,vce) =(l/47t XSI) f sce -\:{vce ,vce' ) I(r,e,vce )dvce ' - (SI+Sa)I(r,e,vce ) (15)

6.1. Amorphous Model For Hypothermia Therapy:

Let z be the malignant cells state, rj be the position of the i th cell, rij be the distance of adjacent normal and malignant cells, ~ be the mass of the i th cell ,Tz be the therapy environment temperature, k b is the boltzmann constant. The Brownian dynamics of tumor particles can be expressed as IDj82r/8t2= FPi+Fhi +Fbi; i= 1,2,.. , here FPi, Fhi, Fbi are the interparticle , hydrodynamics ,and Brownian forces .Let ~ be the inter particle potential energy. Let (ay/8t) be the shear rate .The Pecblet number is given by Pe=(8y/8t)rp2/ Az,(A, fL), where Az{I-.,J.l.) is the diffusion coefficient. For the case Pecblet number Pe
589

verified that larger magnitude of potential energy in hypothermia therapy may adversely effect normal cells. A delay in malignant cells detection may reduce probability of containment of cancer spread. Further for Izt-Zml:$; Izt-Zmx I, recovery state is nonattainable.

velocity, Vrnj be the velocity of surrounding medium, d.i be the drag function, g be the gravity. From conservation of momentum dynamics of ion penetration velocity, required external electrical field, and external voltage (gradient) for ion transmitter may be expressed as

Conditional Survival Index- Cancer Therapy:

~dvP/dt=dj(vrnrvPj)+Qj4>+~g;lj>=-vue ;V .4>=( 47t1E)~i

Amorphous model is employed to obtain neural training algorithm for hypothermia therapy. Given cylindrical Set Scyj: 1tf2 1 , "ej(Tj)c Scy; where "ej is the i th cell with orientation 8, T j is the environment temperature. Define Lie derivative in tangent space Tlli,Tl(xSj) ~ Tit lli,Ti) XSj . Consider fuzzy rule grade membership set J.1r ("ej,Tj)C [0,1] with differential mapping TIi Tlli,Ti f(xSj) ~Tttlli,Ti) XSj ERn xRl, in product tangent space .The acceptance criterion for fuzzy implication rule is given

(lR)

It may be verified that penetration in objects is deeper if higher voltage gradient is employed .

7.0 TUMOR GROWm MORTALITY DYNAMICS

To include uncertainty for survival and uncertain fertility characteristics of cells, a stochastic model for tumor growth is introduced. Let 1z;0:$;1z[Na+J o )' ([K+Ji computation,NC'98,sept 1998,Vienna,Austria >[K+l o )' ([CI-J?[CI-J o)' introduced external extra cellular 3. Kang. H.S,(1995)"Canonical transfonnation in modelling ion electrical field may result in swelling on cells, and bio systems", IFAC-17th IFIP conf. on systems may result in extended! collapsed conformation of cells . modelling and optimization,July,Prague,Czech rep 4. Kang.H.S., (1970) "On the lyapWlov design of systems 6,2. Hypothermia Radiation Therapy Transmitter: with zeros in the right halfplane",IEEE,Trans auto cntr, no 2,feb 1970 Proposed hypothermia therapy transmitter employs disk 5. Sundareshan, M.K.,Fundakowski.( 1986), "Stability and and pattern controller to create electrical field for control of class of compartmental system with aptransmitting ions. Transmitter employs lines of field to plication to cell proliferation and cancer therapy" carry electrically charged ions from ion source to the ,IEEE Trans on auto contr,AC-31(l1), plO22 object . In the following mathematical model for hypothermia radiation therapy transmitter is obtained in 6.1Aired,A.K..(1964), "Dynamics oftumor growth", British Journal of Cancer 18,490-502. the following . Let &be the permittivity of the medium , 7. Tosteson,D.C,and HojJman(J960);"Regulation of cell volume qj be the particle ion charge, lj> be the electrical field, by activation transport in high and low potassium Ue(t)EOe(t) be the external voltage (gradient) employed sheep red cells", J.ofGeneral Physiology 44:169·194 . Let mj be the particle mass,vPj be the ion particle

Pi

590