Appendix
Equations:
Tumor dynamics
MI
T
T  rT( 1- )-aT
(
K
eT  M I
aT ,β  Fβ  eT,β
Fβ  eT,β
)CT
hT
hT  T
(1)
Equation (1) describes the tumor cell dynamics, with T representing tumor cell
numbers at any moment. The first term on the right hand side (RHS) of Eq. (1) stands
for tumor growth with no immune intervention. This classical logistic expression uses
the concept of ‘‘carrying capacity’’, i.e., maximal tumor cell burden, K. The term r
stands for maximal tumor growth rate. The second term on the RHS of Eq. (1)
represents tumor elimination by CTL, C, based on the assumption that it is
proportional to both T and C, with saturation for large T. The saturation is
represented by the factor
hT
. The maximal efficiency of a CTL is denoted aT .
hT  T
Two other multiplicands in the elimination term represent the effect on CTL efficiency
of MHC class I receptors ( M I ) and of TGFβ ( F ), which is assumed to be a major
immunosuppressive factor for CTL activity. Both effects are assumed to follow
Michaelis–Menten saturation dynamics. The dependence on M I is increasing from 0
to 1 with a Michaelis constant eT . The dependence on F is decreasing from 1 to aT , 
with Michaelis constant eT ,  .
CTL dynamics
C  -μ C  C  S
(2)
CTL (C) dynamics are described by Eq. (2). Although the cancerous process
stimulates also regulatory T-cells to end the immune response, for simplicity we
assumed a constant death rate,  C for the CTLs. The term S describes the rate of
infusion of primed CTLs. In the absence of immunotherapy S was set to 0.
Cytokine dynamics
Fβ  a β,T  T -μ β Fβ
Fγ  aγ,C  C - μγ  Fγ
(3)
(4)
Cytokine dynamics are described by Eqs. (3, 4). Equation (3) describes the dynamics
of TGFβ, F . The only source of F in our system is the tumor [5]. We assume F
secretion to be proportional to the tumor size, a  ,T being the release rate per tumor
cell. The last term accounts for the degradation of F , with constant rate,   .
Equation (4) describes the dynamics of IFNγ, F . the first term on the RHS is the
production of IFNγ, where a ,C is the release rate per single CTL. We assume that
the only source of F is CTL. The second term is the degradation of Fγ with constant
rate,   .
MHC class I dynamics
M I  g MI 
a MI,γ  Fγ
Fγ  eMI,γ
-μ MI  M I
(5)
Equation (5) represents the dynamics of the number of MHC class I receptor
molecules, M I , on a single tumor cell. The first term on the RHS of Eq. (5) is the
basal rate of M I receptor production per tumor cell, g MI . The second term represents
the stimulation by IFNγ of M I expression on the surface of a tumor cell.[1] We use a
Michaelis–Menten type saturated function, where the maximal effect of IFN-c is a MI ,
and the Michaelis parameter is denoted eMI , . The last term in Eq. (5) is the
degradation of MI with constant rate,  MI .
Parameter Estimation
In this section we present a list of all evaluated model parameters, the detailed
methods and the literature sources for their evaluation (Table A1). The methods for
evaluating most of the parameters that we use here have been published in Kronik et
al. (2008).[2] We elaborate only for parameters for which new values were estimated
in order to apply our approaches to melanoma cellular therapy and IL-2 injections.
Table A1. Parameter estimation used in the current model.
Parameter
r
Value
0.0001 - 0.001
K
aT
Units
h-1
Reference
[3] and [4]
[5]
[6] and [9]
eT
1012
1.54  10 7
50
aT , 
0.69
cells
cell-1 ·h-1
rec·cell-1
none
eT , 
104
pg
[2]
hT
C
5.2  10 5
6.65  10 for all
cell
h-1
[6]
[7]
[11]
a  ,T
Or 0.0077 and 0.0014
for Yee et al.
5.75  10 6
pg/cell-1·h-1
[2]

6.93
h-1
[2]
a  ,C
1.02  10 4
0.102
pg·cell-1·ml-1 ·h-1
[2]
h-1
[2]
4
[2]
[2]
simulations but Yee’s.

gMI
1.44
-1
-1
[2]
-1
-1
[2]
rec·cell ·h
a M I ,
2.89
e M I ,
3.38 10 5
pg
[2]
M
0.0144
h-1
[2]
I
rec·cell ·h
Maximal growth rate of the tumor, r. Carlson reports the ranges of tumor doubling
times for primary and metastatic melanoma.[4] He reports median of 94 days (2256
h) for the former and 33 days (792 h) for the latter. We deduce r in the following way
ln 2
 0.0003 h 1 ,
2256
ln 2
rmetastatic 
 0.0008 h 1 .
792
rprimary 
Plesnicar et al. found even faster doubling times up to 24 days which is translated to
576 h, giving us an r value of 0.001 h-1. Therefore we chose to take a wide range of r
from 0.0001 h-1 to 0.001 h-1[3].
Tumor carrying capacity (maximal tumor burden), K. Lagerwaard et al. report
several sizes of tumors, the largest is 14.8 cm in diameter. Assuming near spherical
form, this is equivalent to a volume of about 1600 cm3 [5]. Since we are interested
mainly in order of magnitude of maximal tumor burden, and assume that the tumor
volume may include other non cancerous cells, we take this volume to be 1000 cm3.
Arciero et al. takes the carrying capacity of tumor cells to be 109 cells / cm3 [8].
Multiplying maximal tumor cell number per cm3 by the rough volume of the tumor
yields:
10 9
tumor cells
 10 3 cm 3  1012
3
cm
tumor cells
Maximal efficiency of CTL, aT. In Kogan et al. (2010) [6] we used a mean value of
CTL kill rate reported by Wick et al. [9], and deduced aT to be 1.54  10 7 cell-1 h-1.
Here, we have observed from our simulations that the appropriate rate is double that
value, 3.08  10 7 cell-1 h-1. This value is well justified because Regoes et al. (2007) [10]
measured T-cell killing rate in infection settings and estimated the killing rate to be
much higher probably depending on the phase of the infectious disease.
Death rate of CTLs,  C . For all simulations we used the same value based on
Dudley et al. (2005) [7] Figure 5B in the paper reports absolute number of CD8 + T
cells at 800/µl multiplying that by blood volume of 4500 ml we get a typical number of
CD8+ T cells at 3.6  10 9. Dudley et al also report about 3% survival rate for MART-1
cells was measured after 146 days. Assuming that this persistence rate was also true
for day 300 in our simulation we used these data to deduce typical CD8 + death rate:
C (0)  3.6  10 9 cells
C (300day )  3%  3.6  10 9  1.08  10 8 cells
1.08  10 8
 e  C 300day24h / day
9
3.6  10
 C  6.65  10  4 h 1
For Yee et al. (2002) simulations we used the following data:
They report an average of 1.47% of peripheral blood CD8+ cells at day 1 and a drop
to 0.48% by day 7. This was without the presence of IL-2. With the presence of IL-2
the corresponding rates were 1.52% at day1 and 0.97% at day 14.
0.048  0.0147e 0t
t  6days  24h / day
 0  0.0077h 1
0.0097  0.0152e  2t
t  13day  24h / day
 2  0.0014h 1
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