4 Supplementary Information
In this section we investigate the e ects of a heterogeneous sensitive cell population. Specif-
ically assume that the sensitive cell population at time t is
Z0(t) =
NX
j=1
Z0;j(t);
where for each j, Z0;j is a sub-critical binary branching process with birth rate r0, death
rate d(j) and net decay rate r(j) = jr0  d(j)j. Lastly the initial conditions are given by
Z0;j(0) = pjn where pj 2 (0; 1). De ne r = minfr(j) : 1  j  Ng, then de ne tn = 1r log n
and for  > 0
 n = n
 bv=r+  1 log n:
We can then write the Laplace transform of the rescaled and sped up resistant cell process
as
E exp (nZ1(vtn)) = E exp
0
@ 
r0 
n 
Z b
0
g(x)
Z vtn
0
 
1 ~xvtn s( n)
 NX
j=1
Z0;j(s)dsdx
1
A
 exp
0
@ 
r0 
n 
Z b
0
g(x)
Z vtn
0
 
1 ~xvtn s( n)
 NX
j=1
e r(j)spjndsdx
1
A ;
where we replace Z0 by its mean. De ne
~I1(n; v) = r0 n
1 
Z b
0
g(x)
Z vtn
0
 
1 ~xvtn s(n)
 NX
j=1
e r(j)spjdsdx
Then note that the derivation for the approximation of I1(n; v) in (2) is independent of
the value of r in the exponent in the integrand, and therefore
~I1(n; v)  
 ro rg(b)
v
NX
j=1
pj
r(j) + b
;
which gives us the approximation
Z1(vtn)  n
1+bv=r ro rg(b)
v
NX
j=1
pj
r(j) + b
:
Comparing with the approximation in the setting of homogenous sensitive cell population
we see only a change in the values of the constant.
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