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Tzafriri et al., Modeling and Optimization of Intratumoral Drug Delivery
SUPPLEMENT A: PARAMETER ESTIMATES
Morphological parameters. Modeling the cells as tightly packed spheres with an
equivalent radius Rc = 7.5μm (1,2) and assuming that the tumor radius is RT = 1cm
we find
c 

6
 0.52 ,
(A1)
4RT3 3
Nc 
 1.24  10 9 cells ,
3
8Rc
nc 
Nc
1
cells
cells

 3.0 10 8
 3.0 1011
.
3
VT 8Rc
ml
l
(A2)
(A3)
Note that the last result is independent of the tumor volume. Moreover, this picture
implies that the surface area of a single cell is 4Rc2 , so that the surface fraction of
cells in a tumor of tightly packed cells is
S c  4Rc2 n c 

2 Rc
 2094cm -1 .
(A4)
According to Lankelma et al. (2) Sc=4700cm-1 and, extrapolating the results of Hilmas
and Gilette (3) to large tumors we find Smv=100cm-1. Although the microvascular
volume fraction is highly variable in mammary tumors,  mv  0.004  0.35 (4),
estimates of its average values are rather consistent. Using morphometry Vogel (5)
estimated  mv  0.15  0.18 and Hilmas and Gilette (3) estimated  mv  0.17 .
Recently Bogin et al. (4) used MRI to estimate  mv  0.142  0.002 . We will therefore
assume
 c  0.52 ,  mv  0.13 ,  i  1   c   mv  0.35 .
(A5)
This estimate of  i agrees well with published estimates:  i  0.35  0.55 by Jain (1)
and  i  0.5 by Krol et al. (6).
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Tzafriri et al., Modeling and Optimization of Intratumoral Drug Delivery
Transport parameters. Several groups estimated the effective (apical to basolateral)
permeability of Caco-2 cells to paclitaxel and found rather consistent results. Walle
and Walle (7) estimated Pc  4.4  0.410 6 cm/sec in buffer. Walgern and Walle
(8) estimated Pc  3.3  01410 6 cm/sec in buffer and Pc  4.2  1.410 6 cm/sec
in plasma. Stephens et al. (9) estimated Pc  2.3  0.210 6 cm/sec in buffer (and
Pc  1.0 10 5 cm/sec for the passive component). These estimates are consistent
with the estimate of Lankelma et al. (2) for the effective permeability of MDA-468
cells to doxorubicin, Pc  4.0 10 6 cm/sec . We therefore estimate

PC S C
i

3 10 6  2100
sec 1  0.018 sec 1  64.8h 1 .
0.35
(A6)
Estimates of the vascular permeability are harder to come by. Moreover, uptake by
capillaries also involves convective effects. Lovich et al. (10) measured the
partitioning and effective diffusion coefficient of paclitaxel in calf carotid arteries
immersed in calf serum. These authors found Dm v,eff  2 10 9 cm 2 sec 1 . This
effective diffusion was estimated in the absence of convection, but incorporates
binding and partitioning in addition to simple hindered diffusion in the arterial tissue.
Since the average microvessel thickness in solid tumors is approximately 1.0 m (11)
this estimate implies Pmv  Dmv 1m  2.0  10 5 cm/sec for paclitaxel. Combining
this with our estimates of  m v and S m v yields   Pmv S mv i   55.5 h 1 . Saikawa et
al. (12) injected phenol red (MW=378) into tissue isolated tumors and estimated
γ=(74.4.4±13.2) h-1 for well perfused tumors regions and γ=(32.4±10.8) h-1 for poorly
perfused regions. Moreover, they noted that larger tumors corresponded to smaller γ,
which is consistent with observation of increased necrosis in large tumors. Similar
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Tzafriri et al., Modeling and Optimization of Intratumoral Drug Delivery
experiments using mitomycin (MW=334) yielded γ=(38.04±4.44) h-1 (13). As a
baseline estimate we used
  Pmv S mv i   36 h 1 .
(A7)
Several theoretical models exist for estimating diffusion coefficients in porous
media like the interstitum. El-Kareh et al. (14) reviewed past efforts and used
homogenization theory to obtain
2i
2
,
i  Di D0 
3
3  i
(A8)
where D0 is the diffusion coefficient in water. Substituting D0  6.5 10 6 cm 2 / sec
(15) and i  0.35 in (A8) we obtain Di  1.7 10 6 cm 2sec 1 . For comparison, the
correlation obtained by Swabb et al. (16) implies Di  1.1 10 6 cm 2sec 1 for
paclitaxel (MW=854), which is compatible with (A8) for the value i  0.24 . We
took the baseline estimate
Di  1.0 10 6 cm 2sec 1 .
(A9)
Butler et al. (17) estimated that the fluid velocity at the periphery of small tissue
isolated solid mammary tumors was 0.1  0.2 m sec . Baxter and Jain (18) used
mathematical modeling to extrapolate these estimates for a subcutaneous tumor of the
same size that is surrounded by normal tissue and estimated that
ui  0.016 m sec
(A10)
at the periphery. Moreover, fluid velocity decreases sharply from its maximal value at
the periphery towards the center of the tumor where it is identically zero.
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Tzafriri et al., Modeling and Optimization of Intratumoral Drug Delivery
Binding Parameters. We could find no direct measurements of paclitaxel binding onto
extracellular elements. Kuh et al. (17) reanalyzed the data of Song et al. (18) for
paclitaxel binding onto culture medium that contained fetal bovine serum and found
KD=(0.781±0.012) μM and bmax=(3.94±0.16) μM. Wild et al. (19) measured the
uptake of paclitaxel by platelets at 37º C and employed Scatchard analysis to estimate
a KD=(0.8±0.1) μM. We therefore used the baseline values
K D ,i  k i ,off k i ,on  0.8M , bi,max  5.0M .
(A11)
The results of Song et al. (18) can be used to estimate the baseline values
ki ,off  14.4 h 1 .
(A12)
and (consequently)
ki ,on  14.4h 1 0.8 M  1.8 107 h 1M -1 .
(A13)
We used the estimates of Kuh et al. (17) for the equilibrium binding parameters
K D ,c  k i ,off k i ,on  4.9 nM , bc,max  60.0 M ,
(A14)
and the estimate of Caplow et al. (20) for the on rate of paclitaxel binding onto
microtubules
k c,on  7.2 1012 h 1M -1 .
(A15)
Consequently
kc,off  K D,c k c,on  35500 h 1
(A16)
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Tzafriri et al., Modeling and Optimization of Intratumoral Drug Delivery
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