- Email: [email protected]
A mathematical model for the density of malignant cells in the spread of cancer in the uterus
A Mathematical Model for the Density of Malignant Cells in the Spread of Cancer in the Uterus GEORGE W. SWAN Department of Pure and Applied Mathematics, Washington State Universi@, Pullman, Washington Communicated by J. Jacquez
ABSTRACT The solution to a recent mathematical model concerned with the spread of cancer, proposed by Blumenson, is shown to be in error. It is shown how to produce the correct solution to his model. Discussion is presented on extensions of the model to deal with a uterus of finite thickness.
INTRODUCTION attempts to construct mathematical models to describe the processes of tumor formation. One such model has been reported by Blumenson [3]. His model is concerned with the spread of cancer which begins on the inner surface of the wall of the uterus. The first part of Blumenson’s paper deals with the process of cell growth and the justification of a diffusion approach for describing the density of cells at some position in space at some instant of time. By an intuitive argument Blumenson obtains a mathematical expression which forms a key part in the representation to the cancer cell density, which is then used extensively in the subsequent part of his paper, and numerical calculations are based on it. Unfortunately the cell density expression is not correct. The first part of this paper presents highlights from Blumenson’s diffusion model. In the second part we carefully examine the solution to a certain partial differential equation and show that we have an incorrectly posed problem, which is the cause of Blumenson’s error. In the section following we reformulate the basic partial differential equation problem with all physical conditions built in and demonstrate that there is no difficulty in obtaining and interpreting the solution. Further, we give the In recent
years
MATHEMATICAL
there have been various
BIOSCIENCES
25, 319-329 (1975)
QJAmerican Elsevier Publishing Company, Inc., 1975
319
320
GEORGE
W. SWAN
solution of the problem with a general source on the inner wall of the uterus. The remaining sections of the paper deal with the difficulties in obtaining solutions in the cases where the uterus is taken to be of finite thickness, and when the molecular diffusion coefficient is variable. THE MODEL Let c(x,y,z,t) denote the density of cells at the point (x,y,z) The diffusion process is taken to be represented by
ac
at +v.gradc+cdivv=divDgradc+
Q,
at time t.
(1)
where Q symbolizes some appropriate source or sink term. Blumenson examines the use of (1) with a constant coefficient of diffusion D. Changes in the density of cells due to convection are allowed by means of a term uac/ az, where z, (a constant) is the component of velocity in the z-direction. For the problem under investigation, convection of cells in the x- and y-directions is assumed to be of no consequence in comparison to convection in the z-direction; the x- and y-components of the velocity vector are thus taken to be zero in (1). A decision now has to be made on the form of the source term. In the absence of any spatial dependence and convection, Blumenson appears to assume that there is a density c= @‘IT, which means that at each time step the density of cells is double the number at the previous time step. This gives dc/df’=(ln2)c, and it is appropriate now to select the source term Q in (1) to be Q = (In 2)~‘. If we now combine all these details, the differential equation (1) now has the form
-$+$+$
+(ln2)$.
This equation is taken to hold in - cc < x < cc, - cc 0. The quantity c is a representation of the infinitesimal density function for tumor cells in the wall of the uterus. The cancer is assumed to commence with a single cell in the plane z = 0 at t = 0. Since the number of cells in an elementary volume dV is c dV, we see that the initial condition is
where 6 denotes the Dirac delta function. [Recall that 6(x) has dimension (length)-‘.] The problem is to determine the time at which the first few cells cross a threshold which is 20 mm from the inner wall where the cancer starts. Once cells have penetrated past this threshold into adjacent tissues, the surgical removal of the uterus does not leave the patient free of cancer
321
MODEL FOR SPREAD OF CANCER
cells. The tumor is assumed to remain within the uterine wall and not project into the central cavity. Blumenson considers z to extend to infinity, reasoning that by so doing one does not get into additional complications from finite lengths which would detract from the main investigation. The reader should be aware at this point that we are concerned with a very simple approach to an extremely complicated and involved biological problem. Since Eq. (2) is linear, we can apply the technique of separation of variables. Introduce f and g defined by
c(x,r,z,t)=f(x,y,t)g(z,t). Substitution
in (2) gives
af
D$-IJ~-%
%-DVff-(ln2)$= Choose the coefficient zero. This yields
of
g-‘f.
(
1
f on the right hand side of this equation
ag yD$+,
t>O,
z
>o,
to be
(3)
g(z,O)= B(z),
(4)
and $=D($+$)+(lnZ)$ with
f (x,y, 0) = 8 (x)6 (_v). If we set f(x,u,t)=2”Th(x,y,t),
+(S+$),
t>O,
-co
y>oo,
h(w,O)= Qx)KY). Introduce
(5)
(6)
the double Fourier
transform
@(a;p,t)=Jm Jmh(x,y,t)ei’“Z+By)dxdy. --m
Application
of this transform
-co
to (5) gives d@/dt
+ D(a*+ /3*)@=0,
with
322
GEORGE
W. SWAN
solution (P(cu,p,t)=~(,,p,O)e-~‘~*+~*)* =
e-D(a2+fi’)r
IIm --m
O”8(x)8(y)ei(ax+fi)dxdy -02
=e -D@+j3*)r
Inversion
gives the result: h(x,y, t) = (4rDt)-‘exp[
By “an intuitive with (4) is g(z,t)=
(47rDt)-“*
argument”
(exp[ -(z
- (x2+y2)/4Df].
(7)
Blumenson
claims that the solution
- ct)*/4D1]
+ exp[ -(z
+ ot)*/4Dt]).
to (3)
(8)
Direct differentiation of this expression reveals that g, # Dgzz - ug,,and we must therefore reject (8) because it cannot be a solution of (3) with initial condition (4).
THE INCORRECTLY
POSED PROBLEM
We now proceed to examine the problem posed by (3) and (4). Introduce the following Laplace transformation of g(z, t): @~Q(s,z)=
Jmg(z,t)e-“df.
(9)
0 Application
of this transform d*@ ----dz*
to (3) gives u da D dz
;a= - is(z).
For z < 0 we have @= 0, and for z > 0 the solution Q)=Ae’l’+ where A and B are arbitrary
h=t(u/D)+(p/D)“*,
constants
(10)
of (10) is
(11)
BeAs ? and
~42=t(4’)-(p/D)“*,
p =
s +
0*/4D. (12)
MODEL FOR SPREAD OF CANCER
There are two further conditions:
323 across .z= 0
[@]=O,
[
g 1
c-k,
(13)
where [] indicates the change in the enclosed quantity; see Friedman [7], p. 158, Theorem 3.3. Use of (13) with the two solutions for Cpallow us to determine A and B. However, A, > 0, and this indicates that Cp becomes unbounded as z-+00. This solution must be rejected. On physical grounds we anticipate that the cell density approaches zero for z large, and this supplies an additional boundary condition namely g(z,t)+O,
z+CO.
(14)
If we apply this condition, then cP+O as z-+00, from (1 l), forces the selection of A to be zero. The dilemma now is that there remain two conditions (13) for the determination of B. This is not viable. DERIVATION
OF THE SOLUTION
In this section when the problem above come from from a single cell cancer start from (3) with the initial
we show how to include all the boundary conditions for g(z, t) is reformulated. The difficulties encountered the boundary condition which asks that the cancer start on z =O. We alter this condition slightly and ask that the a single cell at z = z0 (> 0). The problem now is to solve condition g(z,O) = S(z - za).
(15)
It turns out that no difficulties in developing the solution are experienced. Since progress can be made for a general source we provide the solution to that problem and then specialize to a delta function. Our objective is to solve (3) with the initial condition g(z, 0) = 4(z)* Application
of the Laplace transform d=@ _-__dz2
(16)
(9) to (3) gives
u d@ D dz
(17)
The middle term on the left is removed by the transformation @(s,z) = q(s,z)eoz/2D,
(18)
324
GEORGE W. SWAN
where (19) A general expression for cp can be found using a Green’s function construction. However, it turns out that the selection of the boundary conditions on z = 0 requires care, and we now illustrate this. Let G denote the desired Green’s function. Multiply (19) by G and integrate the result by parts to give
1J m
+
0
mcpLGdz.
0
The most plausible boundary condition on z = 0 appears to be that one which states that there be no flux of malignant cells across this boundary into the interior of the uterus. Mathematically this requires that
!L,
(21)
onz-0,
aZ
which, in turn, implies that 3g/ az =O, dQ/dz = 0 on z = 0, and thus from (18), dq/dz +(u/2D)q1=0 on z =O. Since the condition (14) implies that cp+0 as z+cc, we can now make the contribution in the square brackets of (20) vanish when dG ;i;++&G=O
onz=O,
G+O
(22)
asz+w.
Now
q(w)=
f$-pG= The solutions
(23)
-a(~-{),
(24)
of (22), (24) are
G,(z,S)= G,(d)
$10mG(z,S)e-“S/2DIC1(S)d3,
=
P+ v/2D
e-P{
p-v/2D
2p
P+v/~D
e-p2
p-vu/2D
2p
z < i-2
.?>s.
(25)
(26)
MODEL FOR SPREAD OF CANCER
325
From (18) and (23) and these last results, Q is determined evz/2D @C&z)=
as
z
G2(z,I’)e-““/2D$(Z)dS
71 0
(27)
and inversion
of (9) gives
(28) The solution to (3) with the initial condition (19) and the boundary conditions (14) and ag/ 3z = 0 on z =0 is given by the formulae (27) and (20 We now present the explicit solution when the initial condition is a delta function, namely (18). Choose q(z)= 6(z - zo). In this event (27) now becomes @(s,z) =
D -‘eD(L-Z~)~2DG1(z,zO),
=
e
20
P
where p=(s + u2/4D)‘12/D
e v(z - r,)/2D dz,r)=
e-P(z,+z)
,-drg-r)
dz - z,)/ZD
z < zo
-
+
P
‘j2 . The inversion
I
formula
epp ~,V2t’/2
2D,,2
2e-p(ro+r) p-v/2D ’ (28) gives
$+&,&.t
)I ,
where, for convenience, p = v2t/4D + (z,, - z)2/4Dt, q = v2t/4D + (z. + z)‘/ 4Dt and the function F is defined by (35). On using (37) the expression for g now becomes e 00 - z,,,/ZD
- 04/4D [e-_(r,-r)‘/4Dt+
g(z,
t)
=
20
e-(~o+d1/4Dq
1/2&2t’/2
+ -!?-e-“o”/Derfc 20
z,+z 2D i/2t’/2
(29)
GEORGE W. SWAN
326 For
z >
z,,,
(27) now becomes e--Po+zo)
,-dz-“0)
,dz-r,)/ZD
w&z)=
2e-p(z+zo) p-v/2D ’
+ 2D
P
-
P
I
and inversion now produces the result (29). The representation (29) is the solution to (3) with the initial condition (18) and the boundary conditions ag/ at = 0 on z = 0 and (14). Indeed, it is this latter condition which guides in choosing (37) over (36). Also, the representation (29) is valid for 0 < z < co, z0 > 0. Note, however, that there does not appear to be any apparent difficulty in setting z,=O in (29). Finally, the density of malignant cells is
(30)
where g(z,f) is given by (29), and this represents the solution of (2) under the initial condition c(x,y,z,O)= G(x)S(y)S(z - zs) and the boundary conditions: c+O as z+cc and the flux &/az =0 on z = 0. Inspection of the nature of the solution (30) reveals that this is precisely the same representation as would have come from solving the axially symmetric form of the basic equations: the equation plus initial and boundary conditions for g are the same as before; the equation for h is D -r
Application gives
ah
a
ah
at-r-=%3 ar
of a Hankel
h(r,t)=
transform
which is readily identified THICKNESS
T.
to this equation
~-m~e~~2D’Jo(~r)d~=
0
S(r)
h(r,O)=
and inverting
the result
%?!?, 4rDt
with (7).
OF UTERUS
FINITE
In the previous section the thickness of the uterus was taken as being infinite. We now examine the central problem of the previous section with the thickness taken to be a finite quantity, z*, say. Accordingly, g, @, cp and the Green’s function G are required to vanish on z =z* instead of at
327
MODEL FOR SPREAD OF CANCER
infinity.
In particular
we now have
G,(z,~)=p-l(Ye~Z*+e-~z*
) - ‘( YePZ+ e-PZ).sinhp(z*
G2(z,{)=p-‘(Yepz*+e-pZ’)-‘(
- J),
YeP~+e-P~)*sinhP(z*-z),
z<5,
(31)
z>r,
(32)
where Y = (p - u/2D)/(p + u/2D). The expression for g(z, t) in terms of the general condition (16) can be written down. In the situation where we replace this general condition with the point source condition (15) for 0 < z,, < z*, we find that
z <
(33)
zo,
where &=
(P--uPD)(
epzl - ep+) + (p+ u/2D)(e-pz2-
evpl) ,
W
w=(p-u/2D)ep’+(p+o/2D)e-Pz’,
Z1=z+z*-zo,
P=(s/D+u~/~D~)“~,
(34)
z2=z-z*+zo.
Unfortunately it does not appear that we are able to perform the inversions analytically, and resort needs to be made to numerical techniques, such as described by Bellman, Kalaba and Lockett [2]. CONCLUSIONS At this stage in the understanding of clinical developments in cancer of the uterus [3], it appears that the useful contribution from the present work of this paper is that given by (30). Equation (30) can be used to numerically determine the density of malignant cells at various distances measured from the inner wall. The drawback to implementation of this formula is that only ranges of the parameters D and u are known [3]. Computations, based on (30), produce temporal plots which differ only slightly from those presented by Blumenson [3] for small values of u (0.06 to 0.08). These values are taken from Blumenson et al. [4]. Larger values of u produce more marked differences between the temporal plots, but there is some question as to the biological significance of these u values [5]. Also at this stage there does not appear to be much advantage in working out the details of the problem when the thickness of the uterus is finite.
328
GEORGE W. SWAN
APPENDIX We demonstrate
here that
F@,a,& t>=
s
c+im
&
c-i00
e-0'
=
e-AG+o)“2+sr
(s+d’*-D
e-u/41 ~ ,wtw
I
G!s
(35)
-peS'-merfc
e-h=/4t
e-"'
I
,Wt'P
(36)
(
+Pe
A
fi21-Bherfc
2t’/2 -Pt’j2 (
)I ’
(37)
Cut the complex s-plane from s = - u to s = - cc. It is straightforward convert the contour integral into t) = fe-O’-(, *a ( ~2+p2)-‘e-“4(arcosha-~sinha)d~,
F&a,&
which, on rearrangement,
(38)
can be written as
F(A,o,/3,t)=~e-“(P-/32Q-/lR),
(39)
where 77v2
o"e-a'4cosXada= -e 2t'i2
p=
-Al/4t
,
,
by Erdelyi et al. [4], p. 15, No. 15, and 00
R=
J 0
=Tefi4
(ye-& sin ha!
da
(Y2+p2
I e
-flAerfc
to
(
/3t1/2- A)
- eaherfc( pt’/2+
+$
, 11
MODEL
FOR
SPREAD
329
OF CANCER
by Erdelyi et al. [6], p. 74, No. 26. Substitution of these results into (39) gives (36). We note, however, from the definition of F in (38) that F( - A, (I, - /I, t) = F (A, u, /?, t), from which we readily deduce (37). The author thanks Dr. L. Blumenson for his assistance in clarifying various points and for a copy of [4]. REFERENCES 1
M. Abramowitz
2
tional Bureau of Standards, Applied Mathematics R. E. Bellman, R. E. Kalaba and J. A. Lockett,
and
I. A. Stegun
(Eds.),
Handbook
of Mathematical
Functions,
Na-
3
Transform: Applications to Biology, Engineering and Physics, Elsevier, New York, 1966. L. E. Blumenson, Random walk and the spread of cancer, J. Theor. Biol. 27, 273-290
Series, Washington, D.C., 1964. Numerical Inversion of the Laplace
(1970). 4
L. E. Blumenson, LD. J. Bross and N. H. Slack, Application of a mathematical model to a clinical study of the local spread of endometrial cancer, Cancer 28, 735-744
5
(1971). L. E. Blumenson,
6
A. Erdelyi
personal
communication.
(Ed.), Tables of Integral
7 B. Friedman, 1956.
Principles
Transforms,
and Techniques
Vol. 1, McGraw
of Applied
Mathematics,
Hill, New York, Wiley,
New
1954. York,
ABSTRACT The solution to a recent mathematical model concerned with the spread of cancer, proposed by Blumenson, is shown to be in error. It is shown how to produce the correct solution to his model. Discussion is presented on extensions of the model to deal with a uterus of finite thickness.
INTRODUCTION attempts to construct mathematical models to describe the processes of tumor formation. One such model has been reported by Blumenson [3]. His model is concerned with the spread of cancer which begins on the inner surface of the wall of the uterus. The first part of Blumenson’s paper deals with the process of cell growth and the justification of a diffusion approach for describing the density of cells at some position in space at some instant of time. By an intuitive argument Blumenson obtains a mathematical expression which forms a key part in the representation to the cancer cell density, which is then used extensively in the subsequent part of his paper, and numerical calculations are based on it. Unfortunately the cell density expression is not correct. The first part of this paper presents highlights from Blumenson’s diffusion model. In the second part we carefully examine the solution to a certain partial differential equation and show that we have an incorrectly posed problem, which is the cause of Blumenson’s error. In the section following we reformulate the basic partial differential equation problem with all physical conditions built in and demonstrate that there is no difficulty in obtaining and interpreting the solution. Further, we give the In recent
years
MATHEMATICAL
there have been various
BIOSCIENCES
25, 319-329 (1975)
QJAmerican Elsevier Publishing Company, Inc., 1975
319
320
GEORGE
W. SWAN
solution of the problem with a general source on the inner wall of the uterus. The remaining sections of the paper deal with the difficulties in obtaining solutions in the cases where the uterus is taken to be of finite thickness, and when the molecular diffusion coefficient is variable. THE MODEL Let c(x,y,z,t) denote the density of cells at the point (x,y,z) The diffusion process is taken to be represented by
ac
at +v.gradc+cdivv=divDgradc+
Q,
at time t.
(1)
where Q symbolizes some appropriate source or sink term. Blumenson examines the use of (1) with a constant coefficient of diffusion D. Changes in the density of cells due to convection are allowed by means of a term uac/ az, where z, (a constant) is the component of velocity in the z-direction. For the problem under investigation, convection of cells in the x- and y-directions is assumed to be of no consequence in comparison to convection in the z-direction; the x- and y-components of the velocity vector are thus taken to be zero in (1). A decision now has to be made on the form of the source term. In the absence of any spatial dependence and convection, Blumenson appears to assume that there is a density c= @‘IT, which means that at each time step the density of cells is double the number at the previous time step. This gives dc/df’=(ln2)c, and it is appropriate now to select the source term Q in (1) to be Q = (In 2)~‘. If we now combine all these details, the differential equation (1) now has the form
-$+$+$
+(ln2)$.
This equation is taken to hold in - cc < x < cc, - cc
where 6 denotes the Dirac delta function. [Recall that 6(x) has dimension (length)-‘.] The problem is to determine the time at which the first few cells cross a threshold which is 20 mm from the inner wall where the cancer starts. Once cells have penetrated past this threshold into adjacent tissues, the surgical removal of the uterus does not leave the patient free of cancer
321
MODEL FOR SPREAD OF CANCER
cells. The tumor is assumed to remain within the uterine wall and not project into the central cavity. Blumenson considers z to extend to infinity, reasoning that by so doing one does not get into additional complications from finite lengths which would detract from the main investigation. The reader should be aware at this point that we are concerned with a very simple approach to an extremely complicated and involved biological problem. Since Eq. (2) is linear, we can apply the technique of separation of variables. Introduce f and g defined by
c(x,r,z,t)=f(x,y,t)g(z,t). Substitution
in (2) gives
af
D$-IJ~-%
%-DVff-(ln2)$= Choose the coefficient zero. This yields
of
g-‘f.
(
1
f on the right hand side of this equation
ag yD$+,
t>O,
z
>o,
to be
(3)
g(z,O)= B(z),
(4)
and $=D($+$)+(lnZ)$ with
f (x,y, 0) = 8 (x)6 (_v). If we set f(x,u,t)=2”Th(x,y,t),
+(S+$),
t>O,
-co
y>oo,
h(w,O)= Qx)KY). Introduce
(5)
(6)
the double Fourier
transform
@(a;p,t)=Jm Jmh(x,y,t)ei’“Z+By)dxdy. --m
Application
of this transform
-co
to (5) gives d@/dt
+ D(a*+ /3*)@=0,
with
322
GEORGE
W. SWAN
solution (P(cu,p,t)=~(,,p,O)e-~‘~*+~*)* =
e-D(a2+fi’)r
IIm --m
O”8(x)8(y)ei(ax+fi)dxdy -02
=e -D@+j3*)r
Inversion
gives the result: h(x,y, t) = (4rDt)-‘exp[
By “an intuitive with (4) is g(z,t)=
(47rDt)-“*
argument”
(exp[ -(z
- (x2+y2)/4Df].
(7)
Blumenson
claims that the solution
- ct)*/4D1]
+ exp[ -(z
+ ot)*/4Dt]).
to (3)
(8)
Direct differentiation of this expression reveals that g, # Dgzz - ug,,and we must therefore reject (8) because it cannot be a solution of (3) with initial condition (4).
THE INCORRECTLY
POSED PROBLEM
We now proceed to examine the problem posed by (3) and (4). Introduce the following Laplace transformation of g(z, t): @~Q(s,z)=
Jmg(z,t)e-“df.
(9)
0 Application
of this transform d*@ ----dz*
to (3) gives u da D dz
;a= - is(z).
For z < 0 we have @= 0, and for z > 0 the solution Q)=Ae’l’+ where A and B are arbitrary
h=t(u/D)+(p/D)“*,
constants
(10)
of (10) is
(11)
BeAs ? and
~42=t(4’)-(p/D)“*,
p =
s +
0*/4D. (12)
MODEL FOR SPREAD OF CANCER
There are two further conditions:
323 across .z= 0
[@]=O,
[
g 1
c-k,
(13)
where [] indicates the change in the enclosed quantity; see Friedman [7], p. 158, Theorem 3.3. Use of (13) with the two solutions for Cpallow us to determine A and B. However, A, > 0, and this indicates that Cp becomes unbounded as z-+00. This solution must be rejected. On physical grounds we anticipate that the cell density approaches zero for z large, and this supplies an additional boundary condition namely g(z,t)+O,
z+CO.
(14)
If we apply this condition, then cP+O as z-+00, from (1 l), forces the selection of A to be zero. The dilemma now is that there remain two conditions (13) for the determination of B. This is not viable. DERIVATION
OF THE SOLUTION
In this section when the problem above come from from a single cell cancer start from (3) with the initial
we show how to include all the boundary conditions for g(z, t) is reformulated. The difficulties encountered the boundary condition which asks that the cancer start on z =O. We alter this condition slightly and ask that the a single cell at z = z0 (> 0). The problem now is to solve condition g(z,O) = S(z - za).
(15)
It turns out that no difficulties in developing the solution are experienced. Since progress can be made for a general source we provide the solution to that problem and then specialize to a delta function. Our objective is to solve (3) with the initial condition g(z, 0) = 4(z)* Application
of the Laplace transform d=@ _-__dz2
(16)
(9) to (3) gives
u d@ D dz
(17)
The middle term on the left is removed by the transformation @(s,z) = q(s,z)eoz/2D,
(18)
324
GEORGE W. SWAN
where (19) A general expression for cp can be found using a Green’s function construction. However, it turns out that the selection of the boundary conditions on z = 0 requires care, and we now illustrate this. Let G denote the desired Green’s function. Multiply (19) by G and integrate the result by parts to give
1J m
+
0
mcpLGdz.
0
The most plausible boundary condition on z = 0 appears to be that one which states that there be no flux of malignant cells across this boundary into the interior of the uterus. Mathematically this requires that
!L,
(21)
onz-0,
aZ
which, in turn, implies that 3g/ az =O, dQ/dz = 0 on z = 0, and thus from (18), dq/dz +(u/2D)q1=0 on z =O. Since the condition (14) implies that cp+0 as z+cc, we can now make the contribution in the square brackets of (20) vanish when dG ;i;++&G=O
onz=O,
G+O
(22)
asz+w.
Now
q(w)=
f$-pG= The solutions
(23)
-a(~-{),
(24)
of (22), (24) are
G,(z,S)= G,(d)
$10mG(z,S)e-“S/2DIC1(S)d3,
=
P+ v/2D
e-P{
p-v/2D
2p
P+v/~D
e-p2
p-vu/2D
2p
z < i-2
.?>s.
(25)
(26)
MODEL FOR SPREAD OF CANCER
325
From (18) and (23) and these last results, Q is determined evz/2D @C&z)=
as
z
G2(z,I’)e-““/2D$(Z)dS
71 0
(27)
and inversion
of (9) gives
(28) The solution to (3) with the initial condition (19) and the boundary conditions (14) and ag/ 3z = 0 on z =0 is given by the formulae (27) and (20 We now present the explicit solution when the initial condition is a delta function, namely (18). Choose q(z)= 6(z - zo). In this event (27) now becomes @(s,z) =
D -‘eD(L-Z~)~2DG1(z,zO),
=
e
20
P
where p=(s + u2/4D)‘12/D
e v(z - r,)/2D dz,r)=
e-P(z,+z)
,-drg-r)
dz - z,)/ZD
z < zo
-
+
P
‘j2 . The inversion
I
formula
epp ~,V2t’/2
2D,,2
2e-p(ro+r) p-v/2D ’ (28) gives
$+&,&.t
)I ,
where, for convenience, p = v2t/4D + (z,, - z)2/4Dt, q = v2t/4D + (z. + z)‘/ 4Dt and the function F is defined by (35). On using (37) the expression for g now becomes e 00 - z,,,/ZD
- 04/4D [e-_(r,-r)‘/4Dt+
g(z,
t)
=
20
e-(~o+d1/4Dq
1/2&2t’/2
+ -!?-e-“o”/Derfc 20
z,+z 2D i/2t’/2
(29)
GEORGE W. SWAN
326 For
z >
z,,,
(27) now becomes e--Po+zo)
,-dz-“0)
,dz-r,)/ZD
w&z)=
2e-p(z+zo) p-v/2D ’
+ 2D
P
-
P
I
and inversion now produces the result (29). The representation (29) is the solution to (3) with the initial condition (18) and the boundary conditions ag/ at = 0 on z = 0 and (14). Indeed, it is this latter condition which guides in choosing (37) over (36). Also, the representation (29) is valid for 0 < z < co, z0 > 0. Note, however, that there does not appear to be any apparent difficulty in setting z,=O in (29). Finally, the density of malignant cells is
(30)
where g(z,f) is given by (29), and this represents the solution of (2) under the initial condition c(x,y,z,O)= G(x)S(y)S(z - zs) and the boundary conditions: c+O as z+cc and the flux &/az =0 on z = 0. Inspection of the nature of the solution (30) reveals that this is precisely the same representation as would have come from solving the axially symmetric form of the basic equations: the equation plus initial and boundary conditions for g are the same as before; the equation for h is D -r
Application gives
ah
a
ah
at-r-=%3 ar
of a Hankel
h(r,t)=
transform
which is readily identified THICKNESS
T.
to this equation
~-m~e~~2D’Jo(~r)d~=
0
S(r)
h(r,O)=
and inverting
the result
%?!?, 4rDt
with (7).
OF UTERUS
FINITE
In the previous section the thickness of the uterus was taken as being infinite. We now examine the central problem of the previous section with the thickness taken to be a finite quantity, z*, say. Accordingly, g, @, cp and the Green’s function G are required to vanish on z =z* instead of at
327
MODEL FOR SPREAD OF CANCER
infinity.
In particular
we now have
G,(z,~)=p-l(Ye~Z*+e-~z*
) - ‘( YePZ+ e-PZ).sinhp(z*
G2(z,{)=p-‘(Yepz*+e-pZ’)-‘(
- J),
YeP~+e-P~)*sinhP(z*-z),
z<5,
(31)
z>r,
(32)
where Y = (p - u/2D)/(p + u/2D). The expression for g(z, t) in terms of the general condition (16) can be written down. In the situation where we replace this general condition with the point source condition (15) for 0 < z,, < z*, we find that
z <
(33)
zo,
where &=
(P--uPD)(
epzl - ep+) + (p+ u/2D)(e-pz2-
evpl) ,
W
w=(p-u/2D)ep’+(p+o/2D)e-Pz’,
Z1=z+z*-zo,
P=(s/D+u~/~D~)“~,
(34)
z2=z-z*+zo.
Unfortunately it does not appear that we are able to perform the inversions analytically, and resort needs to be made to numerical techniques, such as described by Bellman, Kalaba and Lockett [2]. CONCLUSIONS At this stage in the understanding of clinical developments in cancer of the uterus [3], it appears that the useful contribution from the present work of this paper is that given by (30). Equation (30) can be used to numerically determine the density of malignant cells at various distances measured from the inner wall. The drawback to implementation of this formula is that only ranges of the parameters D and u are known [3]. Computations, based on (30), produce temporal plots which differ only slightly from those presented by Blumenson [3] for small values of u (0.06 to 0.08). These values are taken from Blumenson et al. [4]. Larger values of u produce more marked differences between the temporal plots, but there is some question as to the biological significance of these u values [5]. Also at this stage there does not appear to be much advantage in working out the details of the problem when the thickness of the uterus is finite.
328
GEORGE W. SWAN
APPENDIX We demonstrate
here that
F@,a,& t>=
s
c+im
&
c-i00
e-0'
=
e-AG+o)“2+sr
(s+d’*-D
e-u/41 ~ ,wtw
I
G!s
(35)
-peS'-merfc
e-h=/4t
e-"'
I
,Wt'P
(36)
(
+Pe
A
fi21-Bherfc
2t’/2 -Pt’j2 (
)I ’
(37)
Cut the complex s-plane from s = - u to s = - cc. It is straightforward convert the contour integral into t) = fe-O’-(, *a ( ~2+p2)-‘e-“4(arcosha-~sinha)d~,
F&a,&
which, on rearrangement,
(38)
can be written as
F(A,o,/3,t)=~e-“(P-/32Q-/lR),
(39)
where 77v2
o"e-a'4cosXada= -e 2t'i2
p=
-Al/4t
,
,
by Erdelyi et al. [4], p. 15, No. 15, and 00
R=
J 0
=Tefi4
(ye-& sin ha!
da
(Y2+p2
I e
-flAerfc
to
(
/3t1/2- A)
- eaherfc( pt’/2+
+$
, 11
MODEL
FOR
SPREAD
329
OF CANCER
by Erdelyi et al. [6], p. 74, No. 26. Substitution of these results into (39) gives (36). We note, however, from the definition of F in (38) that F( - A, (I, - /I, t) = F (A, u, /?, t), from which we readily deduce (37). The author thanks Dr. L. Blumenson for his assistance in clarifying various points and for a copy of [4]. REFERENCES 1
M. Abramowitz
2
tional Bureau of Standards, Applied Mathematics R. E. Bellman, R. E. Kalaba and J. A. Lockett,
and
I. A. Stegun
(Eds.),
Handbook
of Mathematical
Functions,
Na-
3
Transform: Applications to Biology, Engineering and Physics, Elsevier, New York, 1966. L. E. Blumenson, Random walk and the spread of cancer, J. Theor. Biol. 27, 273-290
Series, Washington, D.C., 1964. Numerical Inversion of the Laplace
(1970). 4
L. E. Blumenson, LD. J. Bross and N. H. Slack, Application of a mathematical model to a clinical study of the local spread of endometrial cancer, Cancer 28, 735-744
5
(1971). L. E. Blumenson,
6
A. Erdelyi
personal
communication.
(Ed.), Tables of Integral
7 B. Friedman, 1956.
Principles
Transforms,
and Techniques
Vol. 1, McGraw
of Applied
Mathematics,
Hill, New York, Wiley,
New
1954. York,