In this paper a stochastic model for the simultaneous growth and division
of a cell-population cohort structured by size is formulated. This probabilistic approach
gives straightforward proof of the existence of the steady-size distribution and a simple
derivation of the functional-differential equation for it. The latter one is the celebrated
pantograph equation (of advanced type). This firmly establishes the existence of the
steady-size distribution and gives a form for it in terms of a sequence of probability distribution functions. Also it shows that the pantograph equation is a key equation for other situations where there is a distinct stochastic framework

Derfel, G

Van Brunt, B

Wake, Graeme C.

English

Massey Research Online

FUNCTIONALDIFFERENTIALEQUATIONSVOLUME 192012, NO 1{2PP. 71{81A CELL GROWTH MODEL REVISITEDG. DERFEL , B. VAN BRUNT y AND G. WAKE zDedicated to the memory of Anatolii Dmitrievich MyshkisAbstract. In this paper a stochastic model for the simultaneous growth and divisionof a cell-population cohort structured by size is formulated. This probabilistic approachgives straightforward proof of the existence of the steady-size distribution and a simplederivation of the functional-dierential equation for it. The latter one is the celebratedpantograph equation (of advanced type). This rmly establishes the existence of thesteady-size distribution and gives a form for it in terms of a sequence of probability distri-bution functions. Also it shows that the pantograph equation is a key equation for othersituations where there is a distinct stochastic framework.Key Words. Steady-size distribution; Asymptotic behavior; Poisson process; Panto-graph equationAMS(MOS) subject classication. 35B40, 35B41, 35 L45, 92C171. Introduction. In this paper we revisit a cell growth modeldeveloped by [7]. This model was originally developed to model plant cells[8], however, it has found applications in tumour growth in humans [2]. Afeature of this model is that a well-known functional dierential equation, thepantograph equation (see [5], [13] for background), arises from a separation ofvariables solution to a Fokker-Planck equation. Specically, let n(x; t) denotethe number density functions of cells of size x at time t i.e.,for 0  a < bthe quantityR ban(x; t) dx is the number of cells of size between a and b attime t, x is \a variable size" of the cells in the cohort, often taken as \DNAcontent." The cell growth process is modelled by a modied Fokker-Planck Department of Mathematics, Ben-Gurion University, Beer-Sheva, Israely Institute of Fundamental Sciences, Massey University, Palmerston North, NewZealandz Institute of Information and Mathematical Sciences, Massey University, Auckland,New Zealand7172 G. DERFEL, B. VAN BRUNT, AND G. WAKEequation, setting the dispersion term to zero for simplicity, of the form@@tn(x; t) =   @@x(gn(x; t)) + 2Bn(x; t)  (B + )n(x; t);(1)where g is the rate of growth,  is the rate of death, and B is the rate atwhich cells divide into  equally sized daughter cells. Here,  > 1 is the\multiplicity of division", that is cells of size x divide to give  cells of sizex=. The rst term on the right hand side of (1) is the growth term; thesecond is the addition to the cohort at size x from division of bigger sizex with frequency B; and the last is the loss term from this cohort due todivision to cells of size x= (also with frequency B), and the death of cellswith a per capita death rate of . For the original model that we study hereg,  and B are positive constants. It is conceded that the assumption thatB, in particular, is constant is not in fact biologically realistic, see sectionsI.4 and III.4.2 in [12]. However, we made this assumption so as to explorethe deeper connections with the classical pantograph equation. The partialdierential equation (1) is supplemented by the boundary conditionslimx!1n(x; t) = 0;(2)limx!1@@xn(x; t) = 0;(3)n(0; t) = 0:(4)In fact we need only the boundary condition (4): as (2) and (3) follow asconsequences when n(x; t = 0) = n0(x) satisfy these conditions. The steadysize distributions (SSDs) for the number density function correspond to so-lutions of the form n(x; t) = N(t)y(x) (i.e., separable solutions). Solutionsof this form yieldN 0(t)N(t)=  gy0(x)y(x)+ 2By(x)y(x)  (B + )= ;where  is a constant of separation and 0 denotes dierentiation with respectto the indicated argument. The above relation yieldsN(t) = N0et;(5)where N0 is a constant, and the equation gy0(x) + 2By(x)  (B + ) y(x) = y(x):(6)A CELL GROWTH MODEL REVISITED 73Under suitable scaling (e.g. by choosing N0 =R10n0(x)dx) a solution y 2L1[0;1) to equation (6) corresponds to a probability density function in themodel. The boundary conditions (2)- (4) imply thatlimx!1y(x) = 0;(7)limx!1y0(x) = 0;(8)y(0) = 0;(9)and requirement that y be a probability density function leads to the condi-tions y(x)  0 for all x 2 [0;1) andZ 10y(x) dx = 1:(10)Integrating equation (6) from 0 to 1 gives = (  1)B   ;and equation (6) reduces togy0(x) + By(x)  2By(x) = 0:(11)Equation (11) is a special case of the pantograph equation, which has beenstudied extensively. A detailed analysis can be found in [11]. The pantographequation has found applications ranging from a partition problem in numbertheory to the collection of current in an electric train. The reader is directedto [10] for an overview of the literature and further analysis of the equation.There are two other problems where the pantograph equation plays acentral ro^le that have a distinct statistical avour, viz. the absorption oflight in the Milky Way, [1] and a ruin problem in risk theory, [6]. Althoughthese problems seem distant from the cell growth model, there is nonetheless aconcrete link: all these models are based on the same type of pseudo Poissonprocess; consequently, they have the same limit distribution. The link ismore transparent using an approach of [4] that is based on a probabilistictechnique (see [3]). In the next section we detail this approach and recoversome known results about the cell growth model in a fundamentally dierentframework. Specically, we give a straightforward proof of the existenceof an SSD, the derivation of the pantograph equation for this distribution(invariant measure) and a solution in the form of a sequence of probabilitydistribution functions.74 G. DERFEL, B. VAN BRUNT, AND G. WAKE2. Limit Distribution for Cell Growth. Consider a spatiallyhomogeneous population of cells and suppose that the size x of a cell growslinearly as a function of time. Suppose further that at random momentsdened by a Poisson process a cell of size x splits into  new cells of size x=.Again we concede that in reality, for the most part, cells only divide in twoand the resulting daughter cells are not exactly the same size. Asymmetricaldivision of cells is currently under investigation. Here,   1. Specically,we suppose that the jumps x! x= occur in random moments0 = t0 < t1 <    < tn <    ;where the sequence fng dened byn = tn+1   tn;for n = 1; 2; : : : consists of independently and exponentially distributed ran-dom variables, i.e., for t > 0,Pfn > tg = e t:For simplicity, we assume that the between jumps the cell size x has a unitrate of growth so that after t time a cell of size x grows to size x + t.This assumption corresponds to choosing B and g such that B=g = 1 in theFokker-Planck equation. The results detailed below follow mutatis mutandisfor a more general choice of constants.Lemma 1. There exists a limit distribution (invariant measure) for the sizeof a cell. This distribution is independent of the initial cell size x0.Proof. Consider a cell of initial size x0 that splits (jumps) at randommomentst1; t2; : : : ; tn; : : :. Let tn  denote the moment immediately before the nthsplitting. The size of a cell at t1  isx(t1 ) = x0 + 1:The cell then splits into  equal parts at t1 so that at t2  the size isx(t2 ) =1(x0 + 1) + 2:Similarly, at t3 x(t3 ) =11(x0 + 1) + 2+ 3=x02+ 3 +2+12;A CELL GROWTH MODEL REVISITED 75and in generalx(tn ) =x0n 1+ n +n 1+   + 1n 1:(12)Equation (12) shows that there exists a limit distribution for cell size x asn!1 and that this distribution is independent of the initial cell size x0. In-deed, the limit distribution function coincides with a probability distributionfunction of the random variableZ = 0 +1+22+   + nn+    ;(13)where the k are independently, exponentially distributed random variables.LetF (x) = Fn(x) = 1  Pfn > xg =1  e x; x > 00; x  0;(14)p(x) = pn(x) = F0(x) =e x; x > 00; x < 0:(15)Denote the probability distribution function (pdf) for (13) by z(x) and lety(x) = z0(x). The function y thus corresponds to the probability densityfunction for (13). The next theorem shows that the probability density func-tion dened by (13) satises the pantograph equation (11) with B=g = 1and y(0) = 0.Theorem 1. The probability distribution function for (13) satisesz0(x) + z(x) = z(x)(16)z(0) = 0;the probability density function for (13) satisesy0(x) + y(x) = y(x)(17)y(0) = 0:Proof. The proof follows a method developed by [4], which is based on theself-similarity of Z. In particular, equation (13) can be recastZ = 0 +11 +2+32+   = 0 +1Z1;(18)where Z1 has the same distribution as Z.76 G. DERFEL, B. VAN BRUNT, AND G. WAKEGiven random independent variables w1 and w2 with pdfs G1(x) andG2(x) respectively, the pdf of their sum w1 + w2 is given by the Stieltjesconvolution (see [9])(G1#G2)(x) =Z 1 1G1(x  t) dG2(t):In addition, for any  > 0 the pdf of G1 is G1(x=). The pdf for Z1= istherefore z(x) and the pdf for 0 is F (x). Equation (18) thus impliesz(x) = z(x)#F (x):(19)The Stieltjes convolution (19) can be expressed as a Laplace convolution.Since z(x) = 0 for x  0 and dF (x) = p(x) dx,z(x)#F (x) =Z x0z((x  t))e t dt;consequently,z(x) = z(x)  p(x);(20)where  denotes the Laplace convolution. Now,z0(x) = z(0)e x + Z x0z0((x  t))e t dt;and noting that z(0) = 0 integration by parts yieldsz0(x) =  e tz((x  t))x0  1Z x0z((x  t))e t dt= z(x)  z(x):We thus see that z satises (16). Equation (17) follows immediately from(16) by dierentiation noting that z0(0) = 0.3. A Solution Method. [11] showed that problems such as (16) and(17) do not have unique solutions. Indeed, there are an innite number ofsolutions to these problems. The requirement that solutions to (16) are alsoprobability distribution functions, however, resolves this uniqueness problem.In essence, there is only one solution to (16) such thatlimx!1z(x) = 1:(21)A CELL GROWTH MODEL REVISITED 77A similar comment applies to problem (17) if condition (10) is imposed.These uniqueness results can be found in [6] and [7].Problems such as (16) and (17) can be solved using Dirichlet series or,what leads to the same thing, Laplace transforms (cf. [6], [10] and [11]).Solutions to these problems can thus be expressed in the formz(x) =1Xn=1ane nx;y(x) =1Xn=1 nane nx:The probabilistic interpretation detailed in Section 2, however, brings tothe fore a dierent solution method. This method entails a sequence gener-ated by convolutions. We focus exclusively on the probability distributionfunction.Let fzng be the sequence dened byz0(x) = 1  e xzn+1(x) = zn(x)#F (x);(22)where n  0 and x  0. We show that this sequence converges to a probabil-ity distribution function z that is a solution to problem (16). One advantageof this method is that the approximations to the solution preserve the statis-tical structure of the problem. Each term of the sequence is a pdf, and it isclear from the denition of the sequence that zn corresponds to the pdf forthe random variable at the nth splitting.Lemma 2. The sequence fzng converges uniformly on intervals of the formI = [0; a], where a > 0.Proof. We note rst thatzn+1(x) = zn(x)#F (x)=Z x0zn((x  ))e  d:(23)Since z0 is continuous on [0;1) it is clear that zn is also continuous on thisinterval for all n  1. For any continuous function f : [0;1)! R and b > 0letkfkb = sup2[0;b]jf()j:78 G. DERFEL, B. VAN BRUNT, AND G. WAKEIt is sucient to show that the series1Xn=0(zn+1(x)  zn(x))(24)is uniformly convergent on I. For x 2 I,jzn+1(x)  zn(x)j Z x0jzn((x  ))  zn 1((x  ))j e  d kzn   zn 1kx;where = 1  e a:The above calculation can be repeated to show thatjzn+1(x)  zn(x)j  kz1   z0knan:(25)We havez1(x) = z0(x)  e x   e x  1 ;(26)so that for all x 2 [0;1)jz1(x)  z0(x)j  1  1 :(27)Inequalities (25) and (27) thus givekzn+1   znkna  1  1n:(28)Since 0 <  < 1, the Weierstrass M test can be used to show that the seriesconverges uniformly on I.Lemma 3. Each term of the sequence fzng is a pdf that is dierentiable on[0;1). The limit of the sequence is also a pdf.Proof. The sequence fzng is dened by a convolution with a pdf F (x). Sincez0(x) is a pdf, z0(x) is also a pdf. Now, z1(x) = z0(x)#F (x). Since z1is dened by the convolution of two pdfs, z1 must also be a pdf (cf. [14,p. 37]). The argument can be repeated to show that zn must be a pdf for allA CELL GROWTH MODEL REVISITED 79n  1. Each zn is continuous on [0;1). Equation (23) and the FundamentalTheorem of Calculus therefore imply that the zn are dierentiable andz0n+1(x) = Z x0z0n((x  ))e  d:(29)Lemma 2 shows that there is a z such that zn(x) ! z(x) as n ! 1for x 2 [0;1). To show that z must be a pdf we study the characteristicfunctions associated with the zn. The characteristic function of zn is givenbyn(t) =Z 10eitz0n() d:Equation (22) implies thatn+1(t) = Qn(t) (t);where Qn is the characteristic function for zn(x) and =11  itis the characteristic function for F . Now,Qn(t) =Z 10eitdzn()=Z 10eit=z0n() d= n(t=);therefore,n+1 =11  itn(t=);and consequentlyn+1 =n+1Yk=01  itk 1:The product1Yk=01  itk 180 G. DERFEL, B. VAN BRUNT, AND G. WAKEconverges uniformly on all compact intervals of R;hence,n(t)! (t) =1Yk=01  itk 1;where  is continuous on R. We can now appeal to a standard result inprobability theory (cf. [14, p. 42]) that guarantees the existence of a uniquepdf z corresponding to ; moreover, zn ! z as n!1.Theorem 2. Let z denote the limit of the sequence dened by (22). Then zis the unique solution to equation (16).Proof. Integrating the right hand side of equation (29) by parts givesz0n+1(x) = zn(x)  zn+1(x):(30)Now,jz0n+1(x)  z0n(x)j = jzn(x)  zn+1(x)  (zn 1(x)  zn(x))j jzn(x)  zn+1(x)j+ jzn 1(x)  zn(x)j;and since fzng is uniformly convergent on I, the above inequality shows thatfz0ng is uniformly convergent on I. We thus have that z is dierentiable onI and that z0n ! z0 as n ! 1. Equation (30) therefore implies that z is asolution to equation (16).The uniqueness of solutions to equation (16) satisfying the given bound-ary conditions has been established in [6] and [7]. For completeness, however,we give a proof.Suppose that there are two distinct solutions z and w to the boundary-value problem. Let  = z   w. Then0(x) = (x)  (x)(31)(0) = 0(32)limx!1(x) = 0:(33)By hypothesis, the solutions are distinct and therefore (x) 6= 0 for some x >0. Without loss of generality we can assume that (x) > 0 for some x > 0.Now,  is dierentiable, a fortiori, continuous for x  0, and the boundaryconditions (32) and (33) imply that there exists a global maximumM at somepoint 0 < xm <1. We thus have M = (xm) > 0 and 0(xm) = 0. Equation(33) implies that (xm) = (xm); therefore, the global maximum must alsobe achieved at xm. The arguments can be repeated to show that the globalmaximum is achieved at nxm for all n  0; consequently, (nxm) = M .Since M 6= 0, we have the contradiction that limx!1 (x) 6= 0. We thusconclude that (x) = 0 for all x > 0.A CELL GROWTH MODEL REVISITED 81Acknowledgements. This research was started when the rst author(GD) visited Massey University and the publication was completed whenthe third author (GCW) was a Visiting Fellow at OCCAM (Oxford Centrefor Collaborative Applied Mathematics) under support provided by AwardNo. KUK-C1-013-04, made by King Abdullah University of Science andTechnology (KAUST). Both authors thank them for their support.REFERENCES[1] V.A. Ambartsumyan,, On the uctuation of the brightness of the Milky Way, Dokl.Akad. Nauk SSSR,44 (1944), 223{226.[2] B. Basse, B. Baguley, E. Marshall, W. Joseph, B. van Brunt, G. Wake, D. Wall,Modelling Cell Death in Human Tumour Cell Lines Exposed to the AnticancerDrug Paclitaxel, J. Math. Bio., 49 (2004), 329{357.[3] L. Bogachev, G. Derfel, S. Molchanov, and J. Ockendon, On bounded solutionsof the balanced generalized pantograph equation,IMA , 145 (2008),Topics instochastic analysis and nonparametric estimation, 29-49.[4] G. Derfel, Probabilistic method for a class of functional dierential equations,Ukrain. Mat. Zh. vol. 41 (1989), 1322-1327. English translation Ukrainian Math.J.,41 (1990), 1137{1141.[5] L. Fox, D. F. Mayers, J. A.Ockendon, ,A.B. Tayler, On a functional dierentialequation, J. Ins. Math. Appl. 8 (1971), 271{307.[6] D. P. Gaver, An absorption probablility problem, J. Math. Anal. Appl., 9 (1964),384{393.[7] A. J. Hall, & , G.C. Wake, A functional dierential equation arising in themodelling of cell-growth, J. Aust. Math. Soc. Ser. B, 30 (1989), 424{435.[8] A. J. Hall, G. C. Wake, & P. W. Gandar, Steady size distributions for cells in onedimensional plant tissues, J. Math. Bio., 30 (1991), 101{123.[9] I. I. Hirshman, & D. V. Widder, The Convolution Transform, Princeton UniversityPress, Princeton, 1955.[10] A. Iserles, On the generalized pantograph functional dierential equation, Euro.Jour. Appl. Math., 4, (1993), 1{38.[11] T. Kato & J.B. McLeod, The functional-dierential equation y0(x) = ay(x) +by(x), Bull. Amer. Math. Soc., 77 (1971), 891{937.[12] J.A. J. Metz and O. Diekmann, editors. The dynamics of physiologically structuredpopulations. Springer-Verlag, 1980.[13] J. R. Ockendon and A.B. Tayler. The dynamics of a current collection system foran electric locomotive, Proc. Royal Soc. London A, 332, (1971),447-468.[14] H.R. Pitt. Integration Measure and Probability. Oliver & Boyd, 1963.MASSEY UNIVERSITYMASSEY RESEARCH ONLINE http://mro.massey.ac.nz/Massey Documents by Type Journal ArticlesA cell growth model revisitedDerfel, G2012http://hdl.handle.net/10179/975720/01/2020 - Downloaded from MASSEY RESEARCH ONLINE

A cell growth model revisited

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A cell growth model revisited

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