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Additional File 1
Notation:
vi = weight (kg) of organ i
Ci = total concentration (mole/cm3) in organ i.
ci = free concentration in water (moles/cm3)
F = organ perfusion rate (kg/kg/sec)
FA = “apparent” organ perfusion rate for diffusion limited kinetics
P = intrinsic permeability (cm/sec)
a = capillary radius (cm)
N = capillary density (#/cm3)
S = capillary surface area surface area per cm3 tissue (1/cm)
Clr = fraction of solute that equilibrates across capillary in one pass
Kbld-wat = Blood/water partition coefficient
Koil-wat = Oil/water partition coefficient.
Dw and DL = Diffusion coefficient in water and lipid (cm2/sec).
L = thickness (cm) of equivalent aqueous layer corresponding to intrinsic permeability.
I. Quantization of diffusion limitation.
It is assumed that the blood-tissue exchange can be described by an “equivalent”
capillary permeability that characterizes the exchange between the blood and a wellstirred tissue concentration.
Tissue
CT = cT Ktiss-wat
cT
F
CA cA
cB(x)
CB = cB Kbld-wat
cv
x
The upper case indicates the absolute blood (CB, CA, CV) or tissue (CT) concentration
while the lower case (cB, cT) indicates the free aqueous blood (cB) or tissue (cT)
concentration. It is this aqueous concentration which is involved in the diffusional
exchange between the blood an tissue. The solid circles are schematic representations of
the blood lipoproteins and albumin that bind the highly lipid soluble solutes.
Cv
2
It is assumed that the tissue consists of N capillaries/cm3, all with exactly the
same geometry, blood flow, permeability, etc. It will also be assumed that the relation
between the capillary and tissue concentration is in a pseudo steady state. The steady
state differential equation for the concentration in the capillary as a function of position is
(1)
F
dCB
 2 aPN [cB ( x)  cT ]
dx
where F is the tissue perfusion rate (kg/kg/sec) , P is the intrinsic capillary permeability
(defined in terms of the free water concentration) (cm/sec), N is the number of capillaries
per cm3 tissue , and a is the capillary radius (cm). The total blood concentration CB can
be related to the free water concentration (cB) using the blood/water partition coefficient
(CB = Kbld-wat cB). Integrating eq. 1 over the length of the capillary and solving for the
venous concentration leaving the capillary:
cv  cT  [cA  cT ]exp(( PS / Kbld wat ) / F )
(2)
where PS is the permeability-surface area product per tissue weight (1/sec). Equation 2
can be related to fractional clearance or equilibration that occurs in one pass through the
capillary:
Clr  (cA  cV ) /(cA  cT )  1  exp( )
(3)
  ( PS / Kbld  wat ) / F
For >>1, Clr approaches 1 and the solute is flow limited. The clearance depends on
both the intrinsic permeability (P) and the blood-water partition coefficient. Solutes that
have a high intrinsic permeability may be permeability (i.e. diffusion) limited if a large
fraction of the blood solute is bound (i.e. large Kbld-wat).
The following argument supports the above assumption that aqueous diffusion at
the blood/adipose boundary (i.e. “capillary” permeability) is the rate limiting step and
diffusion in the fat cell is not rate limiting. The relative rates of diffusion in the aqueous
and fat tissue are described by:
(4)
Aqueous Diffusion  Dwcw
Lipid Diffusion

DL cL  K oil  wat DL cw
The ratio of the lipid to aqueous diffusion rate is:
(5)
Lipid / Aqueous Diffusion  K oil  wat DL / Dw
Since the lipid (i.e. olive oil) viscosity is about 85 times greater than water, the StokesEinstein relation predicts that the lipid diffusion coefficient is about 85 times smaller than
in water. However, this relation is valid only if the solute is much larger than the solvent,
which is not true for diffusion in olive oil. For small solutes, such as O2 or N2, the
diffusion coefficient in olive oil is only about 3 times smaller than in water [1].
Rogacheva et. al. [2] measured diffusion coefficients of 2-nonanone and benzaldehyde in
oil that were about 10 times less than the water value. As a rough estimate, it is assumed
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that the diffusion coefficient in olive oil for the solutes studied here is about 20 times less
than in water (DL/DW = 0.05). However, since Koil-wat for the solutes considered here is
105 or greater, the aqueous diffusion rate is clearly the limiting step.
II. Flow limited and diffusion limited tissue kinetics.
The general, diffusion limited, differential equation describing the change in the
well mixed tissue concentration (CT) produced by a time dependent arterial input (CA) is:
(6)
dCT
 F[CA (t )  CV (t )]  F Kbld wat [cA (t )  cV (t )]
dt
where CV(t) is the time dependent venous concentration leaving the tissue and cA and cV
are the unbound free blood water concentration. By definition, if the tissue exchange is
flow limited, then the venous concentration equilibrates with the tissue concentration
(cv(t) = cT(t)):
(7)
dCTflow lim
 F Kbld  wat [cA (t )  cT (t )]
dt
Using the definition of clearance (eq. (3)), the general diffusion limited equation (eq. (6))
can be written in the form:
(8)
dCT
 Clr F Kbld  wat [cA (t )  cT (t )]  FA Kbld wat [cA (t )  cT (t )]
dt
FA  Clr F
Equation (8) is identical to the flow limited expresson (eq. (7)), the only difference is that
the “apparent” organ perfusion rate (FA) is reduced proportional to the clearance.
The general approach used in this paper is to determine the “apparent” adipose
blood flow for a series of solutes that all have similar intrinsic permeabilities (P) but
whose clearance varies because of differing values of Kbld-wat. The ratio of the “apparent”
flow over the actual blood flow should have the following dependence on the value of
Kbld-wat:
(9)
FA / F  Clr  1  exp( )

PS
K bld  wat F
III. Mathematical representation of the time dependent plasma concentration.
The procedure used to determine the “apparent” organ perfusion rate was to fit the
experimental plasma concentration data points with a smooth function and then
determine the perfusion rate that gave the best fit to the experimental adipose tissue
concentration for this arterial input. The smooth plasma function was determined as
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follows. It was assumed that the plasma concentration (C(t)) resulted from a constant
input of duration T into a 1, 2 or 3 compartment system:
t
(10)
C (t )   A *[1  H (  T )] rfunc( ) d
0
Where A is the constant infusion rate, H is the Heaviside unit step function, and rfunc is
the response function for a delta input to a 1, 2 or 3 compartment system. For example,
for 2 compartments, the response function is.
(11)
rfunc(t )  a1 exp(t / a2 )  a3 exp(t / a4 )
In addition, a constant term (a0) was added to the C(t). For example, for 2 compartments,
the final expression for C(t) has 5 adjustable parameters (a0 .. a4):
(12) C(t) = a0 – A[-a1a2+a1a2exp(-t/a2)+a1a2H(t-T)-a1a2H(t-T)exp((-t+T)/a2
-a3a4+a3a4exp(-t/a4)+a3a4H(t-T)exp((-t+T) /a4)]
Non-linear Powell minimization was used to find the parameter set that gave the best fit
to the experimental plasma concentration. The procedure that was used to find the
parameters was to fit plasma concentration sequentially to 1, then 2, then 3
compartments, using the parameters from the previous lower order fit as starting values
for the higher compartment fit.
Some of the experiments used intravenous input for which the value of T = 0.5
minutes was assumeed, while others used oral input for which T = 260 minutes was
assumed.
IV. Estimate of “equivalent” thickness of rat adipose aqueous diffusion limiting
layer.
One can estimate the thickness of aqueous layer that would produce the
experimental PS. The permeability P is equal to:
(13) P  DW / L
where DW is the aqueous diffusion coefficient and L is thickness. Thus, relating L to PS:
(14)
L  DW S / PS
where is S is the rat adipose capillary surface area. For example, for the value of PS =
750/min determined from the data plotted in fig.6, and using estimated values of DW of 5
x 10-6 cm2/sec [3] and S for the rat of 35 cm2/cm3 [4], the value of L = 1.4 x 10-5 cm.
V. Prediction of oil/water partition coefficient using octanol/water coefficient.
For most of the solutes discussed here, there are no available experimental values
of the oil/water (e.g. olive oil/water) partition coefficient. However, it will be shown here
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that for solutes with these relatively simple structures one can predict (within a factor of
about 2) the oil/water partition from the experimental values of the octanol/water
partition. Figure 1 (below) shows a plot of (log octanol/water - log oil/water partition)
versus the log octanol/water partition for a series of non-polar solutes. Figure 2 (below)
shows a similar plot for solutes with 1 or 2 aliphatic or aromatic hydroxyl groups and no
other polar groups. The oil/water data in these figures is based on values from the
literature [3-14]. The octanol/water data are the recommended values from the web site
of James Sangster (http://logkow.cisti.nrc.ca/logkow/intro.html).
It can be seen in fig. 1 that for the non-polar solutes, the oil-water and octanolwater partition are nearly identical (octanol-water partition about 25% less then oil-water)
over a very wide range of structures. As shown in fig. 2, the addition of 1 aliphatic
hydroxyl makes the oil-water partition about 1.03 log units (factor of 10.5) smaller than
the octanol-water partition and one aromatic hydroxyl makes the oil-water about 0.82 log
units (factor 6.6) smaller. The larger partition of the hydroxyl compounds in octanol
versus olive oil results from the hydrogen bond forming potential of octanol. The data in
fig. 2 also show that the effect of adding additional hydroxyls is roughly additive.
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References
1.
2.
3.
4.
Davidson D, Eggleton P, Foggie P: The diffusion of atmospheric gases through
fats and oils. Q J Exp Physiol Cogn Med Sci 1952, 37:91-105.
Rogacheva S, Espinosa-Diaz MA, Voilley A: Transfer of aroma compounds in
water-lipid systems: binding tendency of beta-lactoglobulin. J Agric Food
Chem 1999, 47:259-263.
Werner D, Ghosh U, Luthy RG: Modeling polychlorinated biphenyl mass
transfer after amendment of contaminated sediment with activated carbon.
Environ Sci Technol 2006, 40:4211-4218.
Paaske WP: Absence of restricted diffusion in adipose tissue capillaries. Acta
Physiol Scand 1977, 100:430-436.
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Figure 1. Plot of (log octanol/water – log oil/water) versus log octanol/water for nonpolar solutes. Pure hydrocarbon molecules (*) and halogenated hydrocarbons (x) are
indicated. The dashed line is the least square fit to all the data.
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Figure 2. Plot of (log octanol/water – log oil/water) versus log octanol/water for solutes
with 1 or 2 hydroxyl groups.