where are positive constants. We also assume that (.4) = = or r = ; = constant. (.5) The model (.){(.4) was studied in [4]. Assuming that = > > B + ;

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1 ANALYSIS OF A MATHEMATICAL MODEL FOR THE GROWTH OF TUMORS AVNER FRIEDMAN AND FERNANDO REITICH School of Mathematics, University of Minnesota, 6 Church St. SE, Minneapolis, MN Abstract. In this paper we study a model of tumor which grows or shrinks due to proliferation of cells which depends on nutrient concentration modelled by a diusion equation. The tumor is assumed to be spherically symmetric, its boundary is an unknown function r =. It is shown that there is a unique stationary solution with radius r = R which depends on the various parameters of the problem. Denoting by c the quotient of the diusion time-scale to the tumor doubling time scale, so that c is small, we prove that (i) lim inf t! > (ii) If c is suciently small then! R exponentially fast as t!. (iii) If c is not \suciently small" but is smaller than some constant determined explicitly by the parameters of the problem, then lim sup < ; if however c is \somewhat" larger than then generally does not t! remain bounded, in fact,!exponentially fast as t!. Key words. tumors, parabolic equations, free boundary problems.. The model. In this paper we consider the growth of a tumor, assuming that it has a spherical shape fr <g (r=jxj; x=(x ;x ;x )) at each time t; the boundary of the tumor is given by r =, an unknown function of t. We shall study the model initiated by Byrne Chaplain [4] (see also [] [5]); other models are described in [] [] [7]. Denote by (r;t) the nutrient concentration in the tumor. The time t is measured in minutes so that the diusion time scale R =D is of unit order; here R is the length-scale of the tumor D is the diusion coecient of the nutrient concentration. Denote by B the constant nutrient concentration in the vasculature. Then satises the diusion equation = + ( B ) if r<; t> where is the rate of blood tissue transfer per unit length, is the nutrient consumption rate, c ==T where T denotes the tumor doubling time. Typical values for T are on the order of a day, in which case c is small. The rate of growth of the tumor depends on the number of cells contained in it. Denoting by S() the cell proliferation rate within the tumor, the tumor radius then evolves according to (.) d 4 dt (t) s = We shall consider the case where S() is linear, i.e., S()r sin drd d : (.) S() =( )

2 where are positive constants. We also assume that (.4) = = or r = ; = constant. (.5) The model (.){(.4) was studied in [4]. Assuming that = > > B + ; a steady solution was computed its stability was discussed. In this paper we study in more detail, with rigorous mathematical proofs, the behavior of the time-dependent solution as t!. In the sequel we shall work with Setting ~ = B + instead of : B + ; = = choosing =, the system (.){(.5) reduces to (.6) (.7) (.8) = Finally, we have an initial condition (.) B + ; = + if r<; t> s (t) d dt = ( ~)r dr; = or r = ; > ~ > : (r; ) = (r) if (; )= where s() is given. Our main results are the following: (i) lim inf t! >, can be chosen arbitrarily close to the stationary radius R if c is suciently small. (ii) If c is small enough so that c +ce =c < then is uniformly bounded. (iii) If then, for some initial data, c( ~) > c( ~) >c~! exponentially fast as t!:

3 (iv) If c is suciently small then! R exponentially fast as t!;thus the stationary solution is globally asymptotically stable. In x we establish the uniqueness of the stationary solution in x we establish existence, uniqueness some properties of the solution to (.6){(.). The assertions (i), (ii), (iii) (iv) are proved in sections 4, 5, 6 7, respectively.. The stationary solution. A stationary solution satises is given by (.) where (by (.7)) (.) Substituting s (r) into (.) we nd that (.) where (.4) by (.9) = if r<r p R s (r) = sinh p sinh r R r R ~R = s (r)r dr: tanh = + ; =p R = (.5) < < : Theorem.. There exists a unique stationary solution, i.e., there exists a unique solution ~ ; of (.). Proof. We need to prove that the function (.6) g()=+ coth ( < < ) has a unique zero, >. We compute: Hence it suces to show that the function (.7) We have h() = g() g()! < if! ; g()! if!: is strictly monotone increasing: h () = cosh sinh (sinh ) + (sinh ) :

4 Denoting the numerator by k(), it suces to show that k() >. calculation shows that k (4) () =6cosh sinh > But a straightforward whereas k (j) () = for j, so that indeed k() isapositive function. Remark.. Let () denote the solution of (.) set s() = p (). Then g(s(); ) = where g(u; ) = u +u u cosh(u= p ) p sinh(u= p ): One can check that so that g u > ; = g g u < : Also, s()! as!, s() > if.it follows that there exists a unique constant 6 crit such that < crit < s( crit )=; i.e.; ( crit )= p crit : By (.), x == p crit solves tanh x = x=, so that 48 < crit =:77 ::: < 4 48 () > p crit if < < crit ; () < p crit if crit < < : This remark corrects the assertion made in [4] that () > p for < <.. The evolution problem: General properties. Throughout this paper it is assumed that (.) (r) < for r<s(); (r) isacontinuous function. Theorem.. The system (.6){(.) has a unique solution (r;t);, (.) (.) (.4) <(r;t)< if <r<; t>; s()e ~t s()e ( ~)t if t>; ~ _ ( ~) if t>: 4

5 Proof. We rst assume that a solution exists derive the estimates (.){(.4). By the maximum principle, cannot take non-positive minimum in the set fr < g, so that (r;t) > if < r <. Similarly cannot take positive maximum, larger than or equal to, intheset fr <g. Next, from (.7) (.) we get the inequalities in (.4), from which we also deduce the estimates in (.). Local existence uniqueness of solutions to (.6){(.) can be proved by stard arguments as for the Stefan problem [6; chap. 8]; instead of the Stefan condition _s = x we now have the free boundary condition (.7), which actually makes the analysis simpler. Finally, the a priori estimates (.), (.4) enable one to extend the solution step-by-step to all t>. We conclude this section by deriving an integral equation for. We multiply (.6) by r integrate in (r;t) to get Using the relation c c t s() c r (r;t)dr s (t) _dt r (r)dr r< which follows from (.7), setting = t (;t)dt t dt r (r;t)dr = s (t)_+ ~s (t) r (r;t)dr: (.5) (.6) B = c( ~) f(t) = s (t); ; we nd that (.7) where (.8) t cf (t) = =(c Note that, by the maximum principle, (;t)dt + Bf(t) ~ )f() c s() t r (r)dr: f(t)dt (.9) r (;t)> for all t>: 4. lim inf >. t! Theorem 4.. There exist positive constants ; T such that (4.) if t T : Proof. We divide the proof into two steps. Step. We assume that (4.) if t T 5

6 for some T >, show that if is suciently small then we get a contradiction. Introduce the function (4.) v(r;t)= r sinh Mr sinh M for t T where M = ++N N is any positive number. If is small enough then v = M+M r 6 +O(r 4 ) M+ M s 6 +O(s 4 ) =( + O(s )); d dt sinh M= d dt M + M s (t) + O(s 4 ) ) 6 = d M s + O(s M dt 4 = Ms_s 6 ( + O(s )) M ~s ( + O(s )) by (.4); It follows that sinh Mr M v t r ~s ( + O(s )) ~M s ( + O(s )): Choosing small enough so that cv t v + v= cv t (M )v = cv t v Nv c~m s ( + O(s )) ( + O(s )): jo(s )j < 4 if s c ~M < ; we conclude that (4.4) cv t v + v < if r<; t>t : Consider the function w = v + z where It satises z =e (t T ) : w t w + w if r<; t>t ; it is positive onfr=; t>t g on ft = T ; r<s(t )g.by the maximum principle, w> if t>t,i.e., (4.5) (r;t)v(r;t) z(t): 6

7 This inequality can be used to estimate _s from below: s (t)_= ((r;t) (v ~)r dr ~)r dr ( ~)s + O(s 5 ) z(t)r dr e (t T ) s (t) where jo(s 5 )j 6 ( ~)s if is small enough. It follows that for some large enough T (T >T ) _> if t T ; i.e., is monotone increasing. This implies (by a stard result for parabolic equations [6; Chap. 6]) that (r;t) converges to a stationary solution. By Theorem. we must then have lim t! =R, which is a contradiction if we initially choose <R. Step : Choose < < < (say = ; = ; < < ), as in Step. Without loss of generality we may assume that is not for all t>. Then, by Step, there exists a t = t such that s(t )=. Weshall prove that (4.6) for all t>t ; this establishes the theorem. Suppose (4.6) is not true. Then there exist such that s(t )= ; s(t )= t <t <t (4.7) (4.8) << if t <t<t ; _s(t ) : In order to derive a contradiction we need to construct a subsolution to, this requires us to rst obtain a lower bound on (r;t ). Since _= ~, wehave, similarly, The domain contains the domain D = fr <; t t t ~ log t t ~ log : D = f(r;t); r< ~ ( + e ~ ); t <t<t g <t<t g. Weintroduce the solution W to cw t =W W in D ; W ( ~ ;t) =; t <t<t ; W(r;t ) =; r< ~ : 7

8 We can represent w = e t=c W by Green s function for the heat equation cw t =w, Using the ; t) w(r;t)= t t r=~ : C t =e jx j t (C; positive constants) (which can be obtained by comparison with Green s function for a rectangular domain constructed by a series of reections [6; p. 85]), we nd that (4.9) W (r;t )" > where " depends only on ; ~ ;c, i.e., only on ; ;~; c;. By the maximum principle W in D thus, in particular, (4.) (r;t )" ; r<s(t ): Next we introduce the domain D = fr <; t <t<t g a comparison function in D : (4.) v(r;t)=(t) r sinh Mr sinh M where (4.) so that (t )=. As in Step wecompute (4.) (t) =e N(t t ) " ; t <t<t ; N= t t log " ; (t) cv t v + v = c (t) v (t) c _ (t) + v (t) (M )v + O(s ) if M = ++N. Thus v is a subsolution. (We need to note here that if we take = ; = with xed, < <, then ; are uniformly bounded from above, " is uniformly bounded from below. Therefore N is uniformly bounded from above the same holds for M. Hence the O(s ) term in (4.) is negligible if is small enough.) In view of (4.) the denition of (t); v on t = t on r =. Hence, by the maximum principle, >vin D, in particular, (r;t )>v(r;t )= s(t ) r sinh Mr sinh Ms(t ) : Using this in (.7) we deduce that _s(t ) >, a contradiction to (4.8). 8

9 Theorem 4. does not give a sharp bound on. Such a bound can be obtained by going more carefully over Step. This is done in the following lemma. Lemma 4.. let >be dened by (4.4) = R + c~: Then the inequality (4.5) < cannot hold for all t suciently large. Proof. We assume that (4.5) holds for all t>t derive a contradiction; for simplicity we take T =. Let v be dened as in (4.). We want to prove that (4.6) cv t v + v < (4.7) s s vr dr > ~ s provided (4.4) holds; this will lead to a contradiction as in Step. Since vr = (4.7) is satised if only if h r s cosh Ms rsinh Mrdr =s sinh Ms Mssinh Ms (Ms) i (4.8) cosh Ms Mssinh Ms (Ms) > ~ =: But since the left-h side is strictly monotone increasing in s tanh = + ; or cosh sinh =; (4.8) holds if only if (4.9) Ms < = p R : Note that the function f(x) = xcosh x sinh x satises: f()=; f (x)>. Since also _s ~s, Ms <, cv t = c _svh sinh Ms Mscosh Ms ssinh Ms i <c~vh cosh sinh i; the right-h side is equal to c~v. It follows that cv t v + v < ( + c~ M )v< provided (4.) + c~ <M : 9

10 Thus it remains to choose M which satises both (4.9) (4.). Since s<, this is possible if + c~ = which is precisely the relation (4.4). Theorem 4.. For any ">there holds: (4.) lim inf t! R ( ") provided c is suciently small. " Proof. We proceed as in the proof of Theorem 4. but choose = R ( ). Then Step " follows from Lemma 4.. To proceed with Step wechoose = R ( "); = R ( ). We now need to take a larger constant M in order to control the derivative of (t) so that v remains a subsolution. Indeed we take M = + C for some large enough constant C, then require that c is so small that (4.) + C c<m where C is another constant. The constants C C actually depend on c. Indeed the proof of (4.9) shows that " = " c where " is a positive constant independent ofc,, recalling the denition of N in (4.) we nd that C = ~ C log c where ~ C is a constant independent of c. The same holds for C : C = ~ C log c where C ~ is a constant independent of c. In order to be able to choose M which satises co (4.9), (4.) (4.) it remains to show that + c maxn~ ; C ~ log R < R( ") : But this inequality is clearly satised if c is suciently small. (5.) 5. Boundedness of for small c. Theorem 5.. If then there exists a constant C such that c +ce =c < (5.) C for all t>: We rst need a lemma which estimates r (;t): In view of Theorem 4. we may assume that > ) for all t>. Take t > set K = s( t). Since _s=s ~, we have, for any, <K+K( t t) t <t< t

11 where =(+ )~; later on we shall need to take >. Lemma 5.. For any < <there exists a constant C such that (5.) r (s( t); t) c( + )~( + e =c )s( t)+c : Proof. We shall construct functions W;V such that cw t W W; cv t =V V in D = fr <K+( t t)k; t <t< tg; W = ; V = or r = K +( t t)k; (W + V )(r; t ) n (r; t ) if r<s( t ) if r>s( t ); V (r; t ) if r<s( t ): By the maximum principle it then follows that W + V in D = fr <; t <t< tg, since = = W + V at (s( t); t), (5.4) r (s( t; t) (Wr + V r )(s( t); t): where or (5.5) We take W simply as W =e N(r+(t t)k K) N = ck ckn = N ; We shall compare V with the solution v(x ;t) to +h ck i= + : cv t v v in f <K+( t t)k; t <t< tg; v= if x = K +( t t)k; v V at t = t v(x ; t ) = if x >s( t ); v x (;t)=. Then v V in D, since v = V at (s( t); t), (5.6) V r (s( t); t) vx (s( t); t): In order to estimate v x (s( t); t) we introduce a function z, z(; t) =v(x ;t)e (t t+)=c ; = p cx :

12 Then p z t z = if <b( t t)+k c; t <t< t; p z = if = b( t t)+k c; p z(; t ) V z(; t ) = if >s( t ) c where b = K p c. By Lemma of [8] (5.7) jz t ( p cs( t); t)j (b +C ) sup jv (; t )j: The proof of that lemma requires the assumption that the distance from the support of z(; t ) to the point (b+k p c; t ) is uniformly bounded from below, that is, (5.8) ~K p c const. = c > ; it is here that we need to choose >. In a similar way one can prove that jz ( p cs( t); t)j (b+c ) sup jv (; t )j; this implies that (5.9) jv x (s( t); t)j p c(k p c + C )e =c provided (5.8) is satised. Note that if (5.8) does not hold then the slope b is uniformly bounded, by stard parabolic estimates, jz jc sup jv (; t )j so that (5.9) is again valid. By the maximum principle V r atr=s( t), so that from (5.6) we obtain the bound (5.9) also for V. Recalling (5.4) the denition of W, the estimate from above for r, as asserted in (5.), follows. Proof of Theorem 5.. From (.7) it follows that for any <t <t, cf (t ) therefore, using (5.), (5.) cf (t ) t cf t (t )= s r (;t)dt + B[f(t ) f(t )] ~ t cf t (t ) c( + )~( + e =c ) f(t)dt + C t + B[f(t ) f(t )] ~ t t f(t)dt: t t t f(t)dt f(t) = dt Suppose that the theorem is not true. Then for arbitrarily large M we can nd t = t such that f(t )=M : Wetake the smallest such t dene t to be such that M <f(t)<m if t <t<t ; f(t )=M;

13 such a t exists if M >f(). We then have t f = (t)dt t t M = t f(t)dt: Substituting this into (5.) choosing small M large ( recalling that >c), we nd that cf (t ) cf t (5.) (t ) < f(t)dt + B[f(t ) f(t )] t for some >. By assumption B < (Bis dened in (.5)) so that the right-h side is On the other h the left-h side is < jbj(m M): cf (t )= cs (t )_s(t ) c( ~) s (t ) = c( ~)f(t )= c( ~)M; which is a contradiction if M is large enough. 6. Unboundedness of for c not small. In x5 weproved that if c is small enough, i.e., if c < B< (Bas in (.5)) then remains bounded as t!. In this section we show that if B > then may not be bounded. More precisely, we shall prove that is unbounded if (6.) (6.) (6.) B > ( + ) B > : c~ for some > s() s() c [( + ) (r) ]r dr > c~ Theorem 6.. Under the assumption (6.){(6.), : (6.4) s (t) f()e B c(+) t for all t>: (6.5) Proof. Since r (;t), (.7) yields We claim that the inequality t cf (t) >Bf(t) ~ f(t)dt : (6.6) t Bf(t) > ( + )(~ f(t)dt + )

14 holds for all t. Indeed for t =this follows from (6.). If (6.6) does not hold for all t> then there is a smallest t = t > such that (6.6) holds for all t<t,but (6.7) t Bf(t ) = ( + )(~ f(t)dt + ): It follows that Bf (t ) ( + )~f(t ): However by (6.5), (6.7) (6.), cbf t (t )> B f(t ) B(~ f(t)dt + ) = B f(t ) which is a contradiction. Having proved (6.6) we now deduce from (6.5) that (6.4) follows. B + f(t )= + B f(t ) c~ ( + )f(t ) cf (t) B + f(t); 7. Stability of the stationary solution for small c. In this section we prove that the stationary solution is globally asymptotically stable if c is suciently small. This result was suggested by a formal two-scale asymptotic analysis in [4]. Theorem 7.. Let ((r;t);) denote the solution of (.6){(.). Then, there exists a number c > constants B such that if c c then j R jbe t : In particular, the stationary solution is (nonlinearly) stable. Remark 7.. Theorem 7. includes Theorem 4., except for the size of smallness of c: In Theorem 4. the number can be chosen to be anywhere between, the corresponding range for c then depends on ; if is near, then c is arbitrary (by Theorem 4.). The proof of Theorem 7. is based on the following lemma. Lemma 7.. Let ((r;t);) denote the solution of (.6){(.) ( s (r);r ) the stationary solution. Assume that (7.) j R j; j _j j(r;t) s (r)j for some > all t. Then, there exists a number c > constants A, independent of c, such that if c c j R ja c + e t ; j_j A c + e t (7.) j(r;t) s (r)ja c + e t : Proof. Let v = v(r;t)bedened by v(r;t)= sinh( p sinh( p r) ) r 4

15 so r Then, using sinh( r ) + v = c _ sinh ( p ) + v Ac cosh( p ) p where here, in the remainder of the proof, A denotes a generic constant independent of c. This, in turn, implies + c + Ac) + (v + r Ac) + (v Ac) r Recalling c<, it follows that for any constants K > ; < + Ke t + Ac + Ke t ) r (7.4) Ac Ke t Ac Ke t ) + (v + Ac + Ke t ) + (v Ac Ke t ) : : Since (7.5) we have j s (r) v(r;t)jaj R j j(r; ) v(r; )j j(r;) s (r)j + j s (r) v(r; )j +Ajs() R j+a; taking K = A in (7.), (7.4) we get, by comparison, (7.6) j(r;t) v(r;t)ja(c + e t ): Next, note that (v(r;t) where we have set (7.7) ~)r dr = sinh( p ) = sinh( p ) = r sinh( p r) dr cosh( p ) p = (t) (t) coth((t)) (t) (t) = p : ~! sinh( p ) ~ 5

16 Then, letting (7.8) using (.7), we obtain = E(t) = ((r;t) v(r;t)) r dr (7.9) (t) _(t) = = ((r;t) ~)r dr = (t) (t) coth((t)) (t) + E(t) where, from (7.6){(7.8), t (t) (7.) je(t)j A c + e : Thus, the dierential equation (7.9) for can be written as (7.) where (7.) (7.) Now consider the functions (7.4) _(t) =G((t)) + E(t) G((t)) coth((t)) (+(t) ) (t) (t)a c + e t E(t) E(t) (t) (t)a c + e t : G c () =G()Ac: It is easy to show that G c, i.e. G c is convex, for all c ( all ). Moreover, G c () = G c() = Ac since, for > Ac, lim G c =! we conclude that, for c suciently small, there exists a unique number c > such that (7.5) G c ( c )=: We further have that (7.6) (7.7) Then, from (.) we obtain (7.8), for some constant c, G c( c ) < G c () > if only if << c : c = p R c (7.9) c c Ac for <cc : 6

17 Next, using the convexity of G c we have, from (7.) (7.) _(t) =G((t)) + E(t) =G c ((t)) + Ae t (t)+ E(t) Ac Ae t that is (7.) G c ((t)) + Ae t G c( c )((t) c )+Ae t ; ((t) c ) _ G c( c )((t) c )+Ae t ( c )+A c e t : Integrating (7.) using (7.6) we get that (7.) (t) c Ae t ; <= minf G c( c );g: Recalling the denitions of (t), c making use of (7.8), t (7.9), it follows that R A c + e, using (7.) to bound the right h side of (7.), _ A c + e t : Similarly, using the the lower bound for E(t) in (7.), one can prove that A c + e t R A c + e t _ thereby establishing the validity of the rst two inequalities in (7.). Finally, the bound on j(r;t) s (r)j immediately follows by combining (7.5), (7.6) with the rst inequality in (7.). Proof of Theorem 7.. We may now establish the stability of the stationary solution by repeated application of the preceding lemma. Indeed, combining (.4) (5.), we know that for c small the hypotheses of the lemma hold true that is, for c<c, there exists an > such that (7.) holds. Then, by Lemma 7., we have j R ja c + e t Ac for t T where, for any given c such that Ac < wedene T by e T = c: Similar estimates hold for j _j j(r;t) s (r;t)j for t T. Iterating this result we obtain (t (n )T (Ac) n for t nt ; j R ja(ac) n c + e ) with similar estimates for j _j j(r;t) s (r;t)j. Finally, dene > by (Ac) =e T (< ), given t>, let n be the largest integer that satises nt t<(n+)t. Then j R j(ac) n = e nt = e t e (nt t) e T e t = Be t as desired. 7

18 8. Conclusions. In this paper we have considered radial growth of nonnecrotic tumors in the absence of inhibitors. The parameters of the problems are such that a unique stationary solution, with radius R, exists. We proved rigorously that according to this model the tumor will never totally disappear. Furthermore, in the case where the tumor doubling time is large compared to the time scale of the diusion of nutrient (within the tumor), the radius of the tumor converges to the stationary radius R, the convergence is exponentially fast. On the other h if the tumor doubling time is small compared to the diusion time scale, then the stationary solution is generally unstable the tumor size increases exponentially fast to innity, for a large set of initial data. Acknowledgement. The rst author is partially supported by the National Science Foundation Grant DMS 94{5. The second author is partially supported by AFOSR through contract number F496-95{{ by NSF through grant number DMS{ Acknowledgement Disclaimer. Eort sponsored by the Air Force Oce of Scientic Research, Air Force Materials Comm, USAF, under grant no. F The US Government is authorized to reproduce distribute reprints for governmental purposes notwithsting any copyright notation thereon. The views conclusions contained herein are those of the authors should not be interpreted as necessarily representing the ocial policies or endorsements, either expressed or implied, of the Air Force Oce of Scientic Research or the US Government. REFERENCES [] J.A. Adam, A simplied mathematical model of tumor growth, Math. Biosciences, 8, 9{44, 986. [] J.A. Adam, A mathematical model of tumor growth. II. Eects of geometry spatial nonuniformity on stability, Math. Biosciences, 86, 8{, 987. [] N.M. Byrne, The eect of time delays on the dynamics of a vascular tumor growth, Math. Biosciences, 44, 8{7, 997. [4] N.M. Byrne M.A.J. Chaplain, Growth of nonnecrotic tumors in th presence absence of inhibitors, Math. Biosciences,, 5{8, 995. [5] N.M. Byrne M.A.J. Chaplain, Growth of necrotic tumors in the presence absence of inhibitors, Math. Biosciences, 5, 87{6, 996. [6] A. Friedman, Partial Dierential Equations of Parabolic Type, Prentice-Hall, Englewood Clis, N.J., 964. [7] H.P. Greenspan, Models for the growth of a solid tumor by diusion, Studies in Appl. Math., 5, 7{4, 97. [8] P. Van Moerbeke, An optimal stopping problem with linear reward, Acta Math.,, {5,

A Proof of the EOQ Formula Using Quasi-Variational. Inequalities. March 19, Abstract

A Proof of the EOQ Formula Using Quasi-Variational Inequalities Dir Beyer y and Suresh P. Sethi z March, 8 Abstract In this paper, we use quasi-variational inequalities to provide a rigorous proof of the