Liz Doman
GSAC talk
March 23, 2004
Multiple Metastatic Tumors
OUTLINE
biological invasions
cancer metastasis
? derive a model
? show some results
BIOLOGICAL INVASIONS
In nature, all organisms expand their ranges to some extent
Many well documented examples
? Muskrat through Europe, 1909-1927
? California sea otter along the West Coast, 1914-present
? cheatgrass in western North America, 1890-1930
Three aspects of range expansion
? random movement
? reproduction
? long distance dispersal
Consider random movement and reproduction only
? random movement ! diusion
? reproduction ! growth term
Skellam Equation: exponential growth
@ n = D @ 2 n + rn
@t
@x2
Fisher Equation: logistic growth
@ n = D @ 2 n + r 1 ? n n
@t
k
@x2
n(x; t) = population density at time t and location x
D = diusion coecient
r = intrinsic growth rate
k = carrying capacity
with the initial condition: n(x; 0) = n0(x)
and...
n0 = 1
r=1
D=1
the results... for t = 2; 4; 6; 8; 10; 12; 14
Solutions of the Skellam Equation
2
Solutions of the Fisher Equation
2
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
1
n(x,t)
n(x,t)
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0
1
0.2
5
10
15
20
25
30
0
0
5
10
x
15
x
20
25
30
2
x
n0 exp rt ?
Skellam: exact solution ! n(x; t) = 4Dt
4Dt
Fisher: numerical solution ! forward Euler nite dierence scheme
Consider long distance dispersal in addition
Scattered Colony Model
? consider an initial colony of size x
... could be radius, or some other measure of size
? assume the species expands its colony size at rate g(x)
... calculated from an equation like Skellam or Fisher
? the colony produces long distance migrants at some rate (x)
? the migrants start new colonies, and so on...
? keep track of the density of colonies of size x at time t
β(x)
x
g(x)
CANCER
Primary Tumor: original location
Angiogenesis: inltration of blood vessels
Metastasis: spread of cancer to another part of the body
? colon ?! liver
? lung ?! brain
? thyroid ?! bone
Problem: no way of telling how many other tumors exist
? a mathematical model is needed!
A Dynamic Model for the Growth and Size Distribution
of Multiple Metastatic Tumors
paper by
K. Iwata, K. Kawasaki, and N. Shigesada
Journal of Theoretical Biology, 2000
! apply the scattered colony model to the spread of cancer
Growth of Tumors:
? cancer cells live in colonies called tumors
? x(t) = the number of cells in a tumor at time t
? describe the growth of a tumor by, dx
dt = g(x)
? assume all tumors start out with one cell, x(0) = 1
? denote the primary tumor xp(t)
... we really care about keeping track of metastatic tumors
Metastatic Tumors:
? (x; t) = density of metastatic tumors with x cells at time t
... density with respect to size rather than space
... dimensions of (x; t) are \# / unit size"
... (x; t)x = number of tumors in size range x to x + x
...
R1
x
(x; t)dx = total number of tumors with x or more cells
? we want to keep track of (x; t)
Conservation Equation:
? it says the tumors are each growing at rate g(x)
@ (x; t) + @ (g (x)(x; t)) = 0
@t
@x
... the way I derived it
? suppose there are n tumors in the class size x to x + x
n = (x; t)x
? they all moving into the next class together...
∆t
x to (x+∆ x)
(x+∆ x) to (x+∆ x)+ ∆(x+∆ x)
? which means after a time t,
n = (x + x; t + t)((x + x))
? which means that,
(x; t)x = (x + x; t + t)((x + x))
x = g(x)
? we also know, dx
=
g
(
x
)
?!
dt
t
(x; t)x = (x + g(x)t; t + t)(g(x + x)t)
? expand with respect to x and t
? divide both sides by x and t
? then, in the limit that x ! 0 and t ! 0,
@ (x; t) + @ (g (x)(x; t)) = 0
@t
@x
Initial Condition:
? no metastatic tumors exist at t = 0
(x; 0) = 0
Boundary Condition:
? metastatic single cells are being created
R1
g(1)(1; t) = 1 (x)(x; t)dx + (xp(t))
LHS: rate at which tumors with 1 cell are entering the system
? g(1)(1; t) has dimensions \size/time #/size" = \#/time"
RHS: rate at which existing tumors are producing metastases
? sum of the rates at which all of the metastatic tumors are producing
new metastatic tumors plus the rate at which the primary tumor is
producing new metastatic tumors
Rate of Metastasis:
? production of metastases is proportional to degree of angiogenesis
(x) = mx
m = colonization coecient (size time)?1
= fractal dimension of blood vessels entering the tumor
... assume blood vessels are distributed on the surface of a tumor
) =2/3 because surface area of a tumor is proportional to x2=3
Full Model:
[1]
@ (x; t) + @ (g(x)(x; t)) = 0
@t
@x
[2]
(x; 0) = 0
[3]
R1
g(1)(1; t) = 1 (x)(x; t)dx + (xp(t))
where,
dxp = g(x ) with x (0) = 1
p
p
dt
(x) = mx
Tumor Growth Rate:
I. Exponential: g(x) = ax
II. Gompertzian: g(x) = axlog( xb )
a = growth rate constant (time)?1
b = tumor size at saturated level (size)
Solve using Laplace Transforms:
I. Exponential: g(x) = ax
? primary tumor ! xp(t) = eat
? Laplace Transform the PDE to get an ODE in x,
s+a ^(x; s) + d ^(x; s) = 0
ax
dx
solve...
! ^(x; s) = c(s)x?s=a?1
? Laplace Transform the BC to get
g(1)^(1; s) = R11 (x)^(x; s)dx + R01 (xp(t))e?stdt
? plug in the expressions for ^(x; s) and xp(t), solve for c(s)
! ^(x; s) = a(a??ms+m) x?s=a?1
? take the inverse Laplace Transform to get...
(x; t) = ma e(a+m)tx??m=a?1
1e–09
Density of Metastatic Tumors with x cells after t days
8e–10
6e–10
p(x,t)
4e–10
2e–10
0
2e+10
4e+10
6e+10
8e+10
x (cells)
t=950
t=1000
t=1050
a = 0:00286 (day)?1
m = 5:3 10?8 (cells day)?1
= 2=3
1e+11
? the total number of tumors can be found by integrating,
R
! N (x; t) = x1 (x; t)dx = # of tumors size x or larger at time t
N (x; t) = am+m e(a+m)tx??m=a
50
Total Number of Tumors of Size x or Larger after t days
40
30
N(x,t)
20
10
0
2e+10
4e+10
6e+10
x (cells)
t=950
t=1000
t=1050
8e+10
1e+11
II. Gompertzian: g(x) = axlog( xb )
Gompertzian growth rate, g(x)=axlog(b/x)
8e+07
7e+07
6e+07
5e+07
g(x) 4e+07
3e+07
2e+07
1e+07
0
1e+10 2e+10 3e+10 4e+10 5e+10 6e+10 7e+10 8e+10
x (cells)
? primary tumor ! xp(t) = b1?e?
at
Growth of the Primary Tumor
8e+10
7e+10
6e+10
5e+10
Xp (cells) 4e+10
3e+10
2e+10
1e+10
a = 0:00286 (day)?1
b = 7:3 1010 cells
0
500
1000
1500
time (days)
2000
2500
3000
...using LT in this case is not quite as easy...
taking the inverse LT we get
?1 P
log
(
x
)
1
n Res[f (z ); zk ]
(x; t) = alog(b)x 1 ? log(b)
k=1
? where f (z) =
m
z
1? mz F
z log
a
e
(x)
1? log
log(b)
+zt
? and zk are the roots of 1 ? ms F = 0
? YIKES!!!
... Solve it Numerically
Upwind type of scheme
? forward dierence in t to approximate @t@
? backward dierence in x to approximate @x@
PDE ! Ujn+1 = (1 ? xt g(j ))Ujn + xt g(j ? 1)Ujn?1
BC !
hP
i
n
+1
1
n
J
U1 = g(1) j =1 (j )Uj x + (xp(n))
CFL condition: xt >> g(x)
? characteristics of PDE must lie within characteristics of numerical scheme
Results...
−9
16
Density of Metastatic Tumors with x cells, after t days
x 10
t=250
t=500
t=750
t=1000
14
12
10
p(x,t)
8
6
4
2
0
−2
0
2
4
6
10
x (10
8
cells)
a = 0:00286 (day)?1
b = 7:3 1010 cells
m = 5:3 10?12 (cells day)?1
= 2=3
10
12
total number of tumors of size x or bigger is given by (Nb)jn =
PJ
k=j
Uknx
Total Number of Tumors of Size x and Bigger, after t days
200
t=250
t=500
t=750
t=1000
150
100
b
N (x,t)
50
0
−50
0
2
4
6
x (1010 cells)
8
a = 0:00286 (day)?1
b = 7:3 1010 cells
m = 5:3 10?12 (cells day)?1
= 2=3
10
12
CONCLUSIONS
Cancer metastasis can be thought of as a biological invasion
Derived a model
? growth of individual tumors
? long distance dispersal
Analyzed the model
? analytically
? numerically
If parameter values a, b, m, and could be quantied
! the model could be a useful tool
REFERENCES
K. Kawasaki, N. Shigesada
Biological Invasions: Theory and Practice, Oxford Series in Ecology
and Evolution, Oxford University Press (1997)
K. Iwata, K. Kawasaki, N. Shigesada
A Dynamical Model for the Growth and Size Distribution of Multiple
Metastatic Tumors, J. Theor. Biol., 203, pp.177-186 (2000)
J.G. Skellam
Random Dispersal in Theoretical Populations, Biometrika, 38, pp.196213 (1951)
J.A. Lubina, S.A. Levin
The Spread of a Reinvading Species: Range Expansion in the California Sea Otter, American Naturalist, 131, pp. 526-543 (1988)
R. Mack
Invasion of Bromus Tectorum L. into Western North America: An
Ecological Chronicle, Agro-Ecosystems, 7, pp.145-165 (1981)