Transport of protein in the abdominal wall
during intraperitoneal therapy. I. Theoretical approach
Michael F.
Flessner
University of Rochester Medical Center, Rochester, New York 14620
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ABSTRACT |
Intraperitoneal therapies such
as peritoneal dialysis or regional chemotherapy use large volumes of
solution within the peritoneal cavity. These volumes increase
intraperitoneal hydrostatic pressure (Pip), which causes
flow of the solution into tissues that surround the cavity. The goal of
this paper is to integrate new experimental findings in a rigorous
mathematical model to predict protein transport from the cavity into
tissue. The model describes non-steady-state diffusion and convection
of protein through a deformable porous medium with simultaneous
exchange with the microcirculation and local tissue binding. Model
parameters are dependent on local tissue pressure, which varies with
Pip. Solute interactions with the tissue in terms of local
distribution volume (solute void space), local binding, and retardation
relative to solvent flow are demonstrated to be major determinants of
tissue concentration profiles and protein penetration from the
peritoneal cavity. The model predicts the rate of fluid loss from the
cavity to the abdominal wall in dialysis patients to be 94 ml/h, within
the observed range of 60-100 ml/h. The model is fitted to
published transport data of IgG, and the retardation coefficient
f is estimated to be 0.3, which markedly reduces the rate of
protein penetration and is far lower than previously published
estimates. With the value of f = 0.3, model
calculations predict that Pip of 4.4 mmHg and dialysis
duration of 24 h result in several millimeters of protein penetration into the tissue.
mathematical model; peritoneum; interstitium; diffusion; convection; dialysis; intraperitoneal immunotherapy
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INTRODUCTION |
THE PERITONEAL
CAVITY is a natural site for introduction of drugs or for
dialytic removal of substances from the circulation (38).
It is potential space contained within the thin tissue called the
peritoneum, which has a large surface area (11) and which
adheres to a variety of well-perfused tissues. In peritoneal dialysis,
2-3 liters of sterile solution are instilled into the peritoneal
cavity via a catheter through the abdominal wall. The 2- to 3-liter
volumes used in typical therapy in the adult peritoneal cavity increase
the intraperitoneal hydrostatic pressure from 0 (when cavity is empty)
to 2-15 mmHg (30, 52). Figure
1 is a horizontal cut through the
peritoneal cavity below the transverse colon, and it illustrates how
the hydrostatic pressure gradient across the abdominal wall
[intraperitoneal pressure
skin pressure (typically atmospheric)]
can change through a decrease or increase of the intraperitoneal
pressure. Fluid loss from the peritoneal cavity into the body of a
human patient has been found to be directly proportional to the
intraperitoneal hydrostatic pressure (24, 63) and amounts
to 60-100 ml/h in healthy patients undergoing peritoneal dialysis
with 2 liters of solution (7, 36, 46). This loss equals or
exceeds the net volume recovery (obtained through the use of hypertonic
solutions) in most patients and can be a major cause of dialysis
failure.

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Fig. 1.
Cross section of the peritoneal cavity below the
transverse colon. Fluid in the cavity causes pressure (P) to rise to
some level intraperitoneal hydrostatic pressure (Pip).
Because the pressure outside the abdominal wall is typically zero,
the intraperitoneal pressure sets up a pressure gradient across the
abdominal wall equal to Pip.
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In contrast to dialysis, intraperitoneal chemotherapy uses an
intraperitoneal dialysis solution as a vehicle for therapeutic agents
to treat infections or metastatic tumors on the peritoneal surface
(53). This form of chemotherapy has recently included intraperitoneal immunotherapy with monoclonal antibodies and other macromolecules, which transport from the cavity into the surrounding tissue primarily by hydrostatic pressure-driven convection of therapeutic proteins into their target within the surrounding tissue
space (16). Fluid flow induced by intraperitoneal
hydrostatic pressure therefore makes up an important part of the
transport of water and large solutes across the peritoneum during this
clinical procedure as well as during dialysis.
A quantitative understanding of the driving forces and parameters that
govern fluid transport may lead to strategies to minimize the fluid
loss to the patient's body and to improve fluid recovery at the end of
dialysis. A knowledge of the same process can also assist in our
understanding and improvement of intraperitoneal therapy designed to
introduce macromolecular medicines into metastatic intraperitoneal
tumors. In prior work, we (17) developed a
non-steady-state, unidirectional model in which the interstitium was
assumed to be a rigid, porous medium. The model simulated diffusion and
a distance-averaged rate of convection in the interstitium, with simultaneous blood capillary uptake in a tissue bed with uniformly dispersed blood capillaries. We fitted the model to in vivo
concentration profiles of a small solute transporting from the
peritoneal cavity into surrounding tissue primarily by diffusion
(18, 20). However, we were unable to fit this simple model
to tissue concentration profiles that resulted from the transport of
IgG from the cavity (16). In subsequent work (61,
62), we found that the properties of the interstitium are highly
dependent on the local tissue pressure [PT, which varies
with intraperitoneal pressure (Pip)] and that the tissue
cannot be modeled as a rigid structure.
Seames et al. (48) developed a more complicated model of
peritoneal transport during dialysis, but because of a lack of tissue-level data, these authors made several assumptions that were
shown subsequently to be in error. Their theoretical model included
diffusive and convective transport across the peritoneum and within the
subperitoneal tissue. They considered the mesothelium of the peritoneum
to be a semipermeable barrier with properties analogous to the
capillary endothelium, and they applied pore theory (43)
to model water and solute flow during dialysis with a hypertonic
solution in the cavity. However, our experimental data
(21) contradict this assumption and demonstrate that the mesothelium is an insignificant barrier to solutes and water flow. As a
result of their assumption of a mesothelial barrier, Seames et al.
predicted that during the first 2 h of dialysis, when the osmotic
pressure in the cavity is high, the tissue pressure is below zero and
the interstitial volume shrinks during hypertonic dialysis. Our data
(62) have demonstrated that the interstitial volume is
dependent on the hydrostatic pressure but independent of the osmotic
pressure in the cavity. In their model, the coefficient of convection
(hydraulic conductivity) was held constant. In contrast, our recent
data in rats (61) have demonstrated that the hydraulic conductivity of the tissue varies fivefold over a clinically relevant range of intraperitoneal pressure. Seames et al. additionally assumed
that molecules would flow at the same velocity as the solvent through
the tissue, despite evidence that solutes are known to be retarded in
their movement through mixtures of interstitial matrix components
(32). Despite the common clinical observation of
significant loss of protein from tissue to the cavity during dialysis
(10) and our experimental data (16, 19) that
demonstrated protein movement from the cavity to the surrounding tissue
in proportion to the flow, Seames et al. assumed that movement of protein in the tissue was insignificant. Because they did not model
protein movement in the tissue, neither the effect of binding nor lymph
flow within the tissue was included in their model. Their model was
able to simulate rates of small-solute transport during dialysis and to
fit our previously published small-solute tissue concentration data
(20), which result primarily from diffusion. However,
their approach is inadequate to calculate the tissue concentrations or
rates of transport of macromolecules, which are transported primarily
by convection during large-volume intraperitoneal therapeutic procedures.
This paper presents a model that is based on experimental observations
and attempts to correct the inconsistencies of previous theoretical
models of fluid and solute transport across the peritoneum. The model
focuses on the complex problem of macromolecular delivery from the
cavity to the surrounding tissue and includes the processes of
diffusion, hydrostatic pressure-driven convection, tissue binding, and
macromolecular removal via lymphatics. The interstitial space and its
transport parameters are variables, dependent on the local tissue
interstitial pressure; they are derived from experimental work of our
own laboratory and from the work of others. The model equations are
solved numerically, and the sensitivity of the model to major
parameters is tested by varying the parameters from their baseline and
calculating the output. The model output is further compared with
existing interstitial concentration profiles for immunoglobulin in
normal tissue to demonstrate the need for more experimental data on
specific parameters. Extensions of the model to predict events in
longer-term intraperitoneal immunotherapy and dialysis applications are
also discussed.
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MODEL FORMULATION |
Our research has emphasized the study of these forces in the
abdominal wall, because it receives 30-50% of the total fluid flowing from the cavity and because the pressure forces exerted on the
tissue can be easily controlled (15). Figure
2 is a conceptual model of the abdominal
wall (see Fig. 1 for anatomic position of the tissue), across which the
pressure can change through a decrease or increase of the
intraperitoneal pressure (Pip). The abdominal wall of the
experimental animal is very accessible for the determination of tissue
pressures and tissue concentrations of various marker substances
(13, 16).

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Fig. 2.
Tissue model illustrating the theory of flow through
porous media. The tissue cells and blood vessels are surrounded by the
interstitium or "tissue interstitial volume," which is also termed
if, the fraction of total tissue to which water
distributes. Flow occurs when a pressure difference
(Pip 0) exists across the tissue bed and is
proportional to the hydraulic conductivity (K), the surface
area in contact with the fluid (A), and the pressure
gradient (dPT/dx, where x is
distance). See Eq. 1 in text. Likewise, diffusion occurs
because the exogenous solute in the solution within the cavity sets up
concentration profiles within the tissue; the slope of the
concentration profile (dCs/dx, where
Cs is the solute concentration in the solute void volume)
is the driving force for diffusion. Note that the diffusivity is a
function of the specific solute void volume ( s) within
the tissue that is available to the solute. K is a function
of local hydrostatic tissue pressure (PT). Cip,
intraperitoneal concentration.
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According to Darcy's law, fluid flow through tissue (Q) has been shown
to be directly proportional to the hydraulic conductivity (K), the cross-sectional area of tissue (A), and
the tissue pressure gradient (dPT/dx, where
x is distance) (33)
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(1)
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Rather than using Pif, the true interstitial
pressure, we designate the tissue pressure by PT, which is
determined in rat experiments by a micropipette mounted on a precision
manipulator that is tunneled through the muscle from the skin side of
the abdominal wall (13). The micropipette is connected to
a servo-null system and is used to create successive small chambers of
free interstitial fluid (3- to 6-µm diameter × 50-µm length)
within the tissue. Within each small chamber, the pipette-servo-null device is allowed to come to equilibrium with the local fluid pressure,
and this is assumed to be equivalent to Pif. Further discussion of this approximation is contained in papers by Gilanyi and
Kovach (28) and Wiig et al. (59).
Our theoretical formulation takes the practical approach that a model
should include parameters that are either published or attainable from
current experimental techniques. The mathematical model is based on our
previous work (17) with modifications to take into account
theories of convection (1, 48) and experimental findings
(19, 22, 61, 62) in the transport process. The model
focuses on hydrostatic pressure as the chief driving force for
convection. Thermodynamic analyses (28) have suggested the importance of local colloid osmotic pressure to interstitial flow, and
some researchers have included the effects of local colloid osmotic
pressure in their mathematical models (34, 51). However, there are few in vivo data available for testing the significance of
these concepts, and the determination of local changes of colloid osmotic pressure at multiple sites within the rat abdominal wall has
not been performed.
The interstitium has been demonstrated to be compliant in several
tissues (2). Interstitial compliance (
) is defined as the "ratio of the change in the interstitial fluid volume divided by
the corresponding change in interstitial hydrostatic pressure" (2)
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(2)
|
where
if is interstitial fluid volume. To
estimate
(PT) for a given tissue,
if is
measured versus the interstitial pressure and the slope of the curve is
determined. Typically, the shape of these curves is linear in
dehydration (57, 58) but highly nonlinear above zero
pressure (relative to atmospheric pressure) (42, 62). The
compliance of abdominal wall muscle during hydration has been
determined to be similar to that of skeletal muscle with magnitudes of
1.4-4.3 ml · mmHg
1 · 100 g
tissue
1, depending on the average tissue pressure
(62).
For intraperitoneal pressures above 3 mmHg in the rat, we have
determined that the pressure profile is nearly linear across the entire
abdominal wall (~2 mm thick) (13). Figure
3 illustrates these profiles and their
initial slopes (dPT/dx) at the peritoneal edge,
which are equivalent to the convective driving force from the cavity
(13). Note that the slopes in the tissue are approximately the same with magnitudes of
20 to
25 mmHg/cm of tissue (13, 61). Across the abdominal wall, therefore, pressures will be high at the peritoneum and low at the skin side (see also Figs. 1 and
2). This variation in pressure may cause the interstitial volume to be
higher in the vicinity of the peritoneum and to decrease across the
abdominal wall in accordance with the pressure profile in the tissue
and the tissue compliance. We have also found (61) that
the hydraulic conductivity of the abdominal wall tissue increases linearly with the mean tissue pressure. If we impose an Pip
of 4.4 mmHg (6 cmH2O) across the abdominal wall, Fig.
4, using data from our previous studies
(13, 61, 62), illustrates the pressure gradient and the
corresponding extracellular space for the initial 2 mm of tissue
adjacent to the peritoneum. This variation of PT and
if in the tissue results in variation of the tissue hydraulic conductivity and in the effective tissue diffusivity (Deff)
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(3)
|
where Dv is the solute diffusivity in the tissue
void and includes a correction for the tortuosity of the path and
s is the solute void fraction (proportion of the total
tissue that is available to the solute) and is 
if.

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Fig. 3.
Tissue pressure profiles for Pip (in
mmHg) = 0 ( ), 3.3 ( ), 6 ( ), and 8 ( ). Note the equivalent
slopes at the peritoneal surface for the pressures >3 mmHg. Data are
derived from Ref. 13.
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Fig. 4.
Hypothetical pressure profile and the corresponding
interstitial volume profile based on data from Refs. 13
and 62.
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Solute Balance
The general balance equation for a solute transporting via
unidirectional, non-steady-state diffusion and convection in the interstitium is given in the following
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(4)
|
where Cs = Cs(x,t) is free solute concentration
within its tissue void; t is time;
s =
s(PT) is solute void volume, which is the
portion of the tissue volume within which the solute distributes, dependent on the local interstitial pressure (PT);
Deff = Dv(
s), where
Dv is the solute diffusivity in the solute void volume and includes effects of tortuosity of the solute path; f is the
solute retardation factor (ratio of solute to solvent velocity);
K = K(PT) is the tissue
hydraulic conductivity; Rcap is local solute exchange with distributed blood capillaries;
Rbind is net rate of binding of free solute to
local tissue; Rmetab is local uptake of agent by
cells; Rlymph is local rate of lymphatic uptake
and removal from tissue; and Rgen is local rate
of solute generation in tissue.
Equation 4 is derived from Eqs. 1 and 3 and from the model concept in Fig. 2. The first term of
the right-hand side of Eq. 4 follows directly from the
diffusion equation: diffusion velocity =
Deff(dCs/dx). The second term is
derived as follows. Within the interstitial space, from Eq. 1, the velocity of the solvent, vf, equals
Q/(A
if) or
(K/
if)(dPT/dx).
Because large solutes may not move with the same velocity as the
solvent, the solvent flow is multiplied by the solute retardation
factor f to obtain a solute velocity
vs = fvf. The mass
flow equals Csvs = Cs(fvf) = Cs(
fK/
if)(dPT/dx).
To relate the mass velocity within the interstitium to movement in the
tissue as a whole, the interstitial velocity is multiplied by
s, the fraction of the total tissue volume that the
solute can occupy. The result is the second expression in the brackets
of the right-hand side of Eq. 4.
Equation 4 encompasses several assumptions. In the abdominal
wall, groups of muscles in tissue planes and separated by fascia (see
Fig. 1) are assumed to form a single tissue unit with isotropic characteristics. It is assumed that the abdominal wall is equivalent to
a thin-walled shell, which can be modeled as an infinite plane with
isotropic properties including uniform tissue compliance and a uniform
density of blood and lymphatic microvessels. Thus the model transport
can be constrained to one direction (x), and the location of
blood vessels can be simplified to a uniform density throughout the
tissue. As mentioned above, the driving force for convection is assumed
to be hydrostatic pressure alone. The various rates
(Ri) are defined by separate rate equations.
PT is assumed to be less than intracapillary hydrostatic
pressure. [In therapeutic situations, pressures above the portal vein
pressure collapse the vein and cause hemostasis in the gastrointestinal
tract (unpublished observations).] The solute retardation factor
f is defined to have a range of 0-1 and must be derived
from model fits to experimental data after all other parameters are
established. Parameters Deff, K, f,
s, and
if are variable and are functions
of the local PT.
Volume Balance
The volume balance on the interstitium is as follows
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(5)
|
where
if is interstitial void volume, a function
of PT, as shown in Eq. 2, Fcap is
local volume flow to the blood capillaries, and Flymph is
local volume flow to lymph capillaries.
Rate Equations
Solute binding.
To model the movement of proteins through tissue, binding of the solute
must be taken into account in the model. Actual tissue data typically
consist of the total amount of labeled solute per unit volume of
tissue, including bound and free species. The model variable
Cs equals the free concentration in the solute void volume, which is essentially a virtual space within the tissue in which the
solute distributes within the total tissue volume. If there was no
binding of the solute, the concentration measured in the tissue,
Ctissue, would equal Cfree = Cs
s. Because there is either specific or
nonspecific binding for all proteins, the more general definition for
Ctissue is as follows
|
(6)
|
where Cfree is concentration of free solute in
tissue = Cs
s and Cbound is
concentration of bound solute in tissue, where Cbound = g(Cfree, Cbound), a function defined by experiment.
The general mass balance for the bound species is (8, 9)
|
(7)
|
where BF is the forward (association)
rate coefficient and BR is the reverse
(dissociation) rate coefficient. Both of these coefficients must be
derived from experimental data.
Transendothelial transport.
The expressions for solute and water transport across the blood
capillary barrier are taken from the multiple-pore theory of Rippe
(47, 49). The details of this theory are given in the
APPENDIX. Because this theory has been used extensively in
simulations of the subperitoneal tissue (18, 20), and most
of the blood capillary coefficients have been defined by Rippe
(45, 49) for the muscle that surrounds the mammalian
peritoneal cavity, the theory will be used in simulations for normal
abdominal wall muscle. However, the author is well aware of new theory
that supports the glycocalyx as a major barrier in the endothelium
(25, 26). When those new models mature, they can be
incorporated into this model, the chief goal of which is to simulate
the interstitial convection of proteins.
If the solute of interest is a protein or macromolecule larger than
albumin (~58,000 Da), pore theory constrains all transport to
convection through the "large pore" in the endothelial barrier. Because the intracapillary pressures are generally higher than the
interstitial pressures, transport is generally one way, from the
capillary lumen to the interstitium. Equation A1 (see
APPENDIX) can be simplified to
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(8)
|
where RLP is solute transport via large
pores, CP is plasma concentration of protein,
FLP is water transport via large pores, and
LP is the reflection coefficient for the large pore for
the protein. For the short-duration simulations used in this paper, the
solution of Eq. 4 for macromolecules does not depend
significantly on the expression for Rcap (see below).
Lymphatic transport from tissue.
Lymph will be assumed to flow from the tissue at a constant rate
equivalent to Flymph, which must be derived by experiment. The rate of solute flow from the tissue space is given by
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(9)
|
Other rate equations.
If the solute of interest can be taken up by cells, metabolized, or
endogenously produced in the tissue, experimental data must be
collected to define Rmetab and
Rgen. In our experimental transport studies,
exogenous solutes that are not produced in the experimental subject and
that do not undergo significant metabolism are chosen to obviate the
need for these additional data; these rate functions are therefore set
to zero in the simulations in this paper.
Initial and Boundary Conditions After an Intraperitoneal
Injection
Figure 2 illustrates the key initial and boundary conditions
at x = 0, t = 0:
Cs = Cip(0), the intraperitoneal
concentration at time "0," PT (0) = Pip = intraperitoneal pressure. These two
conditions presume that there is no boundary layer on the peritoneal
surface that affects concentration or pressure.
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(10)
|
At x = x1,
t > 0: dCs/dx = 0 and
dPT/dx
0 at x1.
Based on experimental data (see Fig. 3),
dPT/dx at the abdominal wall edge
(x1
0.2 cm) at high pressures
(Pip > 6 mmHg) may not = 0
In the general case of a tissue that undergoes expansion, the
value of the x1 boundary is a variable. In the
abdominal wall, the expansion of the whole tissue between
Pip of 0 and 8 mmHg is <20% (62). If we
assume that the expansion is equal in all directions, the
x1 value could possibly change by ~6%.
Practically speaking, this magnitude of change cannot be measured in
quantitative macro-autoradiographic data (16). For the
model simulations in this paper, x1 is assumed
to be constant.
Model Solution and Parameter Estimation
Equations 4 and 5, along with rate
equations for solute and water transport across the capillary and
lymphatic endothelia (Eqs. 8 and 9), rate
equations for binding (Eq. 7), appropriate transport
parameters, and boundary conditions, can be solved numerically for
values of
if and Cs. If the intraperitoneal
solution is isotonic, intraperitoneal concentration
[Cip(t)] is essentially constant for molecules
>50,000 Da (15, 19, 23), whereas
CP(t) rises to 10% of
Cip(0) over several hours (15,
19). We have collected data on
if(PT), which allows us to concentrate on
the solution of Eq. 4 alone. The solution of Eq. 4 depends on the following: solute-specific parameters
s, f, Deff, binding rate
coefficients, and capillary large-pore reflection coefficient; the
tissue-specific parameters K,
if, lymph flow
rate, and capillary large-pore flow rate; and the local hydrostatic
pressure gradient (dPT/dx). Our laboratory has
published data on several of these parameters as functions of
Pip or PT (21, 61, 62).
To make use of existing data, the solute was chosen to be IgG with no
specific binding to the abdominal wall muscle. The equation describing
nonspecific binding of IgG to muscle is represented by Eq. 7
with BR = 0 and
BF = 1.08 × 10
4
s
1 (22).
From our previously published results (61, 62),
expressions for
if and K will be derived. The
expression for K is as follows
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(11)
|
where hydraulic conductivity coefficients A0 and
A1 are 0.15 × 10
6
cm2 · s
1 · mmHg
1
and 0.18 × 10
6
cm2 · s
1 · mmHg
2
respectively, the units for K are square centimeters per
second per millimeter of mercury, and the units for PT are
millimeters of mercury. A1 will be varied to
determine how sensitive the model output is to K.
The expression for
if is as follows
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(12)
|
where B0 = 0.19 ml/g tissue and
B1 = 0.046 ml · g
1 · mmHg
1.
B1 is equivalent to the compliance [
(PT)]
of Eq. 2 and is varied in the sensitivity analysis to
demonstrate the importance of this factor to macromolecular transport.
From Fig. 3, we observe that the average slope of the three
PT curves is
20 to
25 mmHg/cm. Our previous
determinations were made in anterior abdominal wall tissue, 15 min to
4 h after a steady-state Pip was obtained. We observed
that PT(x) did not vary significantly with time,
and we assume that a constant Pip will result in a constant
PT(x) for t > 0. We therefore
compute the tissue pressure profiles by the following
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(13)
|
where Pip is the intraperitoneal pressure (held
constant at 2, 4.4, or 8.8 mmHg for this study) and x is in
centimeters. Previously, we measured the rate of lymph flow
(Flymph) to equal 1.33 × 10
6
cm2 · s
1 · mmHg
2.
This value is varied in the analysis below to show that it does not
significantly affect the convective flow of protein through the
abdominal wall.
According to pore theory, solute transport of IgG occurs only through
large pores in the capillary barrier. Therefore,
Rcap can be calculated from Eq. 8.
During conditions of normal lymph flow, the rate of flow through the
"large pores" has been estimated by Rippe and Haraldsson
(44) to be on the order of 0.03 ml · min
1 · 100 g tissue
1
or 5 × 10
6
ml · s
1 · g tissue
1. If we
assume that the large pore radius is 25 nm, the estimated
LP for IgG (molecular radius = 5.7 nm) is 0.20 (44). Assuming values for PP (14.8 mmHg)
(31), osmotic pressure in plasma (
P; 19.2 mmHg) (6), and osmotic pressure in the interstitium
(
T; 6.3 mmHg) (31) and substituting in the
local PT, the driving force was calculated to be on the
order of
1 to
2 mmHg (flow out of the capillary). With
overall transcapillary hydraulic conductivity (LPa) = 6 × 10
5
ml · mmHg
1 · s
1 and
LP = 0.07, the resulting Fcap,LP is
~6 × 10
6
ml · s
1 · g tissue
1
(45, 49). With the estimated value of Fcap,LP
and CP(t), the contribution of the
"backflow" of IgG from the plasma to the tissue can be calculated.
In simulations with Fcap,LP = 6 × 10
5 to 6 × 10
4
ml · s
1 · g
1, no
significant change in tissue concentration was observed.
As shown in Eq. 3, the effective diffusivity,
Deff, equals Dv
s. For a given
tissue the diffusivity within the tissue distribution space of the
solute, Dv, is assumed to be a constant. Dv
also takes into account the tortuosity of the path. Our previous
experiments (22) and those of others (3)
produced an estimate for Dv of 2 × 10
7
cm2/s for IgG. This value is used as the baseline value in
the sensitivity analysis and is varied to determine its effect on the
transport of IgG.
There is still uncertainty concerning the fraction of tissue that makes
up the distribution space for the solute. Expansion of the interstitial
space from fluid influx, which occurs in the abdominal wall muscle
during dialysis, only complicates the estimation of this parameter. We
have estimated the apparent
s under conditions of
PT = 0 and mean PT = 4 cmH2O (22). We did this by injecting labeled
IgG 24 h before a second injection of IgG with a different radioactive label. Ten minutes after the second injection, the animal
was euthanized and the abdominal wall tissue was sampled and counted
for the concentrations of each tracer. If it is assumed that the first
tracer is in equilibrium with its volume of distribution within the
tissue, its tissue concentration divided by the plasma concentration
provides an estimate of the total distribution space volume (including
the intravascular space). The second tracer is assumed to remain within
the vascular space, and therefore the ratio of its tissue concentration
to the plasma concentration provides an estimate of the intravascular
space. The total distribution space minus the intravascular space
produces an estimate of
s. At PT = 0,
s = 0.043-0.050, whereas at
PT = 4 cmH2O,
s increases to 0.07-0.08 (unpublished data).
These estimates may in fact be much lower than the actual values,
because Witte (see Ref. 60) has shown that it is unlikely that the interstitial protein concentration ever reaches a true equilibrium with the plasma. Wiig et al. (54, 55) have
defined our result as an "apparent IgG distribution space," because
an equilibration between interstitium and plasma is unlikely and some
of the tracer is bound in the tissue. They have carried out continuous
infusions of labeled proteins and attempted to account for tracer
binding and for a lack of equilibration between plasma and interstitial
fluid. They have demonstrated in nonexpanded skeletal muscle that
60-65% of the interstitial space is available to IgG. If we
assume that their ratio of IgG space to interstitial space (0.6) also
describes the abdominal wall,
s would be 0.11 for the
abdominal wall at PT = 0 or 0.21 ml/g at a mean
PT of 3 mmHg. Therefore, our values likely underestimate
the true value;
s is therefore varied to test its
importance to the model output.
The parameter f is equal to the ratio of the solute velocity
to that of the solvent. It is dependent on the specific mathematical formulation for convection and is constrained to lie between 0 and 1 in
our model. There exists one correlation in the literature for this
parameter (50), which is not derived from actual tissue data but is based on previously published sedimentation studies of
particles and proteins in solutions of hyaluronan (32). We have measured the hyaluronan concentration in the abdominal wall as
0.26 mg/g wet tissue (61). If we assume that the
hyaluronan makes up ~70% of total glycosaminoglycan (GAG) in this
tissue (33) and
if = 0.35 ml/g tissue
(62), then the total GAG content in the interstitial space
is 0.26/(0.35 × 0.7) or 1.06 mg/ml of interstitial fluid.
Swabb's correlation yields an estimate of 1.07 for f,
outside the defined range of the parameter. Because there are no real
experimental data on this parameter in tissue, the parameter will
therefore have to be fitted to data once all other parameters are
measured. In this paper, f is varied in a sensitivity
analysis to demonstrate its importance to the values of
Cs(x,t). Because f
directly modifies the rate of convection, it should theoretically have
a significant effect on the rate of protein movement, which is
dominated by this form of transport.
Numerical Solution
Because of the nonlinear nature of the partial differential
equations, a numerical solution is necessary. A finite-difference technique that employs the "tri-diagonal matrix" was used
(4). Variable coefficients are quasi-linearized by holding
the entity constant through one iteration and then recalculating. The
solution scheme was written in Fortran 77 and solved with an Intel
Pentium II-based personal computer.
 |
MODEL PREDICTIONS AND SENSITIVITY |
The model was exercised for a realistic range of a parameter or a
realistic variation in each parameter. Table
1 lists the parameters and the baseline
values with the perturbed values. Figure
5 compares the diffusive and convective
fluxes at the peritoneal surface that result from the perturbation of
the values from baseline. As can be seen in Fig. 5, increasing
Pip, f, K, and
s
results in a significant increase in the convective flux. On
the other hand, increasing the local binding or the void space
diffusivity or decreasing f results in significant increases
in the diffusive flux. Because the flux of IgG is dominated by
convection, this dependence of the flux on the parameters directly
affecting the rate of solute flow through the tissue would be expected.

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Fig. 5.
Convective and diffusive fluxes at the peritoneal surface
for the base values of parameters and for the perturbed values listed
in Table 1. The dominance of convection over diffusion in the case of a
large protein such as IgG is apparent. See text for full discussion.
Downward arrows represent low values for the parameter. Upward arrows
pertain to high values for the parameter. BF,
forward (association) rate coefficient; f, solute
retardation factor; s, solute void fraction;
Dv, solute diffusivity in tissue void.
|
|
To test the sensitivity of the model predictions for concentrations in
the tissue space, the changes observed in the magnitudes of the
concentration profiles will be calculated with the following algorithm.
The concentrations of Cs(x,t),
Cbound(x,t), and
Ctissue(x,t) will be calculated with
the model baseline parameters. Each of the parameters in Table 1 will
then be assigned either the low or high value listed, and the
concentration profiles will be calculated for 30, 60, and 180 min over
a tissue thickness of 0.2 cm. Each concentration profile will then be
sampled at x = 0.01, 0.03, 0.06, and 0.1 cm from the
peritoneum. The sensitivity factor,
, will be calculated as follows
|
(14)
|
The value
therefore reflects the normalized change in model
output divided by the normalized change in a particular parameter. By
sampling at three time points and at four points in the tissue, we
attempt to define an index that reflects overall changes in the output.
Figure 6 illustrates the model
sensitivity to perturbations in the parameters listed in Table 1. The
one parameter that did not affect the concentrations in the tissue
significantly was the lymph flow rate. The results are not plotted, but
was <0.1. For f, Pip,
s, and
K, the resulting
values are intuitive, given the
structure of Eq. 4 and the fact that the macromolecule
transports chiefly via convection. The sensitivity to Dv
demonstrates that the diffusion still plays a role in the tissue
penetration of IgG and cannot be eliminated from Eq. 4. From
Fig. 5, the diffusive flux across the peritoneal surface varies in
magnitude from 3 to 30% of the convective flux. As shown below,
binding has major effects on the solute diffusion and the shape of the
tissue concentration curves. With decreases in binding, there are
higher free IgG concentrations in the tissue and the diffusive flux
actually reverses direction to diffusion from the tissue toward the
peritoneal cavity. On the other hand, an increase in binding provides a
"sink" in the tissue that decreases the free concentration within
the tissue and speeds up the rate of diffusion into the tissue. The
values for
if are negative because a decrease in the
interstitial volume will increase the solute velocity (see Eq. 4), and, provided the
s remains constant (somewhat
artificial in this analysis because it will likely parallel the changes
in
if), the rate of solute movement into the tissue will
increase. Analogously, an increase in
if with a constant
s results in a decrease in solute velocity and a
decrease in the solute flux into the tissue.

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Fig. 6.
Model sensitivity ( ) to parameter variation:
normalized change in tissue concentration profiles sampled from
different times and distances from the peritoneum. Downward arrows
represent low values for the parameter. Upward arrows pertain to high
values for the parameter.
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|
Figure 7 demonstrates the effect of time
on output of the model. As time increases beyond 60 min, a greater
proportion of the total Ctissue is due to
Cbound. Cfree at any distance from the
peritoneum can be found by subtraction: Ctissue
Cbound. Binding of protein to tissue is a major determinant
of Ctissue (=Cfree + Cbound).
Figure 7 illustrates the effect of time on the concentration of bound
IgG (Cbound) and the total concentration (Ctissue). Most of the observed IgG in the tissue after
3 h is bound through interaction with nonspecific sites in the
tissue, as demonstrated in our previous 4-h binding studies of IgG to muscle slices (22). With longer time of the peritoneal
solution at a constant Pip of 4.4 mmHg, the concentration
rises at each point in the tissue and the protein gains access to
deeper portions of the tissue. This points to the importance of dwell
duration in the clinical delivery of macromolecules to the
subperitoneal interstitium.

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Fig. 7.
Calculated IgG concentration profiles for
Pip = 4.4 mmHg: effect of time on solute concentration
measured in tissue (Ctissue; solid line) and bound solute
concentration in tissue (Cbound; dotted line). Free solute
concentration in tissue (Cfree) can be found from
Ctissue Cbound. The effects of time on
the penetration of IgG are clear: the longer the duration of transport
at a given Pip, the greater the delivery to deeper portions
of the tissue.
|
|
Figure 8 shows what happens to the 60-min
curves of Ctissue when the binding within the tissue varies
from 20% of the baseline to five times the baseline value. As binding
increases, Ctissue near the source (the peritoneal surface)
increases dramatically and less solute is transported further into the
tissue. At the extreme end of the binding spectrum, the concentration
of IgG at the surface would be very high, with almost no penetration into the interior of the tissue. This has been termed the "binding site barrier" and may be present in some antibody-tumor interactions (27).

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Fig. 8.
Calculated IgG concentration profiles for
Pip = 4.4 mmHg: effect of binding on
Ctissue. The forward binding coefficient
(BF, Eq. 7) was varied to produce
change in output. Increased local binding of IgG to the tissue results
in higher tissue concentrations close to the peritoneum but results in
less transport deeper into the tissue.
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|
Because of the dominance of convection in the transport of IgG, factors
that directly affect the rate of water flow (K,
PT) and the force of convection on the solute
(f) should demonstrate significant effects on the tissue
concentrations of IgG in the abdominal wall. Figure
9 demonstrates the effects of doubling or
halving the hydraulic conductivity coefficient A1 of
Eq. 11, which changes the hydraulic conductivity
(K) proportionately; the higher the K, the
further protein penetrates into the tissue. The effects of pressure are
shown in Fig. 10, in which
Pip is varied from 2.0 to 8.8 mmHg. The Pip is
transmitted to the tissue as demonstrated in Fig. 1 and sets up the
pressure profile of PT (Fig. 2), which drives fluid flow
from the cavity into the tissue. When Pip is
2 mmHg,
there is no significant change in the interstitial volume or in
K (61, 62). With the increase in pressure to
4.4 mmHg, the tissue undergoes a linear expansion with doubling of the
interstitial volume and proportionate increase in K. With the increase in Pip from 4.4 to 8 mmHg, the interstitial
volume does not increase, but K continues to increase; the
lack of tissue expansion explains the similar y-intercept at
x = 0, whereas the higher conductivity increases the
rate of flow of water and protein through the tissue, resulting in
deeper IgG penetration. Without experimental data, this nonlinear
response of the tissue could not have been anticipated.

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Fig. 9.
Calculated IgG concentration profiles for
Pip = 4.4 mmHg: effect of hydraulic conductivity
(K). The coefficient A1, which determines
K(PT) in Eq. 11 is varied to change
K. Ctissue and the depth of penetration are
directly dependent on the value of K.
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Fig. 10.
Calculated IgG concentration profiles at 60 min: effect
of Pip on Ctissue. As illustrated,
Pip has nonlinear but direct effects on Ctissue
and the depth of IgG penetration.
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|
Less subtle are the effects of f on protein transport.
Figure 11 illustrates the effect of
increasing rates of solute velocity through the tissue. Pure diffusion
is shown at f = 0 in a curve that can be contrasted
with almost any degree of convection. This parameter has a major effect
on protein transport through the tissue and the depth of macromolecular
penetration in the subperitoneum.

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Fig. 11.
Calculated IgG concentration profiles for
Pip = 4.4 mmHg demonstrate the direct effects of
solute retardation factor f on Ctissue and the
depth of penetration.
|
|
Significant variations in Dv (Table 1) do not result in
large changes in Ctissue (see Fig.
12). This would likely change if the
s were higher and the magnitude of Deff
(Dv
s) were correspondingly higher. The large
exclusion space in the tissue is a major factor in the rate of
transport, as illustrated in Fig. 13.
The nonexcluded volume is varied over three alternate ranges (low to
high in ascending order). The larger the space available to protein,
the more will transport, provided the driving forces and other
coefficients do not change. Note that with the increase of
s, the concentration at the peritoneum and throughout
the tissue increases. A pure change in this variable, however, does not
increase the depth of penetration for a given amount of transport time.
Because
s is assumed to be a function of
if, the isolated variation of
s is likely
an unrealistic representation of the changing environment of the
tissue.

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Fig. 12.
Calculated IgG concentration profiles for
t = 60 min: effect of solute void diffusivity
(Dv). This demonstrates the lack of effect that diffusion
has on transport of large-molecular-weight solutes (such as IgG) where
convection dominates.
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Fig. 13.
Calculated IgG concentration profiles for
t = 60 min: effect of solute void fraction
( s) on model output. This demonstrates that, as the
tissue expands and the space available to the solute increases,
transport of IgG increases. Because other parameters such as
f, K, and Pip are held constant, the
depth of penetration does not vary.
|
|
Figure 14 demonstrates the effects of
variation of
if from low to higher values. As the
interstitial volume increases, transport increases. However, these
simulated curves are likely artifactual and represent unrealistic model
output because
s and K have been held
constant. With the expansion of the interstitium, the volume available
to the solute and the hydraulic conductivity would surely increase. As
demonstrated in Figs. 9 and 13, the cumulative changes in K
and
s would bring about further changes in
Ctissue and likely result in a greater depth of penetration
and higher tissue concentrations close to the peritoneum.

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Fig. 14.
Calculated IgG concentration profiles for
t = 60 min: effect of interstitial volume
( if) on model output. See text for discussion.
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|
 |
MODEL SIMULATION OF PREVIOUS DATA |
In previous work (14), we examined the penetration of
IgG from an isotonic solution in the peritoneal cavity into the tissues surrounding the cavity of RNU-nude rats. These experiments were carried
out with MAb 96.5, a monoclonal antibody specific for receptors on the
FEMX2 human melanoma cell line. Because the rats had no tumor implants,
the binding of the antibody was assumed to be nonspecific.
Intraperitoneal volumes were scaled to produce pressures of
2.2-3.0 mmHg in the cavity throughout the 180-min experiment.
Concentrations in the peritoneal cavity were found to be nearly
constant. After euthanasia at the end of the experiment, the abdominal
cavity was rapidly drained and the carcass was frozen to preserve the
concentration profile in the surrounding tissue. Tissue samples were
sliced with a cryomicrotome, dried, and placed against autoradiographic
film. The autoradiograms were subsequently analyzed with a computerized
densitometer to determine the profile (n = 44)
displayed in Fig. 15.

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Fig. 15.
Model fits (solid line) to published concentration
profiles (solid symbols with ±SD) obtained after a 3-h dialysis in an
athymic rat with MAb 96.5 (specifically binds to human melanoma FEMX-2
cells) seen at Pip = 3.0 mmHg (Ref.
14).
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|
As an initial test of the model as a quantitative tool, we have fitted
the model to the data in Fig. 15. The Pip was set equal to
3 mmHg, and the parameters used in the simulations were tissue void
diffusivity, hydraulic conductivity, interstitial volume, lymph flow,
transendothelial transfer, and compliance of the tissue as displayed in
Table 1. Three parameters required adjustments to improve the fit. The
minimum tissue void volume (
s) was set to 0.07, with the
variation from the high-pressure side to the low-pressure side of 0.10 to 0.07; this is close to our unpublished observations. The solute
retardation (f) was set to 0.3 to produce a better fit to
the curve. We did not have binding data from these experiments and
found that setting the binding coefficient to 8.66 × 10
4 (8 times the number in Table 1) improved the fit
considerably. The fitted curve is displayed in Fig. 15. The necessary
adjustments in the three parameters demonstrate the importance of good
experimental data on binding characteristics and the interstitial space
available to the solute. The parameter f must typically be
fitted to concentration profile data, as in Fig. 15, once all other
parameters have been determined. In reality, there is likely a family
of parameter values that could produce reasonable simulations of the
data. However, this exercise demonstrates the utility of the
mathematical approach in discovering data gaps and in planning
experiments to fill these gaps.
 |
DISCUSSION OF THE MODEL APPROACH |
Theoretical Approach to Convection in Subperitoneal Tissue
This paper presents an integration of extensive in vivo data with
theory to produce a new mathematical approach toward protein movement
through the subperitoneal interstitium during dialysis or during a
procedure designed to regionally treat the peritoneum. Previous
predictive models have been based on assumptions that have subsequently
been shown to be incorrect in experimental studies. Our model
incorporates the nonlinear compliance of the abdominal wall muscle
interstitium to predict the volume expansion over a clinically relevant
range of intraperitoneal pressure (56, 62). The major
driving force for convection has been shown to be PT, which
is dependent on the pressure in the cavity (24). Because
the mesothelium does not act as a membranelike barrier (21), the model is based on flow through porous media, and
all parameters are varied on the basis of the hydrostatic pressure profile in the tissue. With a rise in PT, the tissue is
predicted to expand, which causes
if and
s to increase in magnitude (22, 62). The
model incorporates this extracellular expansion and appropriately
increases the magnitude of the major transport coefficients of
diffusion (Deff) and convection (K) as well
(22, 61). By varying model parameters and calculating the
changes in concentration profiles and solute fluxes, the model
sensitivity to specific parameters has been tested. We have
demonstrated that IgG transport is most sensitive to factors that
influence convection: Pip, f, K,
if, and
s. We have also included effects
of capillary transport, lymphatic transport, and local binding of
protein to the tissue. We have demonstrated that for IgG, much of the
total tissue protein concentration is bound, even if that binding is
nonspecific. Although diffusion is not insignificant (especially in the
early period of transport when the protein is just entering the tissue
and the concentration gradient is large), we have demonstrated that it
has only small effects on the concentration profile after 1 h of transport.
In the model presented here, the solute retardation factor f
has been demonstrated to be a major factor influencing the convective movement of proteins through tissue (see Figs. 5, 6, and 11). In our
earlier model (17), f was assumed to equal 1 because of the lack of data. The factor is not even mentioned in the
work of Seames et al. (48). In this paper, with other
major parameters defined, the model has been fitted to previously
published IgG concentration profiles, and f was estimated to
be 0.3. This estimate means that 70% of the protein is retarded in its
passage through the tissue, relative to the solvent flow. Although
certain assumptions were made in these calculations, it is unlikely
that the velocity of large solutes within the abdominal wall
interstitium equals that of water flowing through the tissue. This
result does not match the calculated value for f given by
Swabb's formula (50), which produces an estimate of
f that is >1 for the interstitial hyaluronan concentration
in the abdominal wall. Swabb's formula was derived from the data of
others (32) who studied the sedimentation of proteins in
pure solutions of hyaluronan. Although hyaluronan is an important
component of the interstitial matrix, other substances such as
proteoglycan and collagen make up the interstitial matrix. Collagen is
thought to form the scaffolding between cells via
1-integrins (41) and has recently been
shown to be a major determinant of IgG transport resistance in solid
tumors (37). The characteristics of the in vivo
interstitium are therefore unlikely to be properly characterized by
correlations involving only one of the components. There is a clear
need for more detailed experimental data concerning the interstitial
structure of various tissues and their transport characteristics.
Parameters such as the solute void space have been determined properly
in only a few tissues (55) and are very dependent on the
degree of hydration of the tissue.
Model Limitations
To simplify the conceptual approach to the abdominal wall, the
tissue has been assumed to have uniform properties of the interstitium and microvessel distribution. Tissue planes and fascial layers of
connective tissue are neglected in the analysis of the data and in the
mathematical model. Local tissue data (i.e., measurements vs. position
in the data) are used to implement the model, but there has been no
attempt to correlate irregularities in different concentration or
pressure profiles with different layers of muscle groups. Variations in
local tissue data in Fig. 15 have been smoothed out by averaging the
profiles together. Lymphatic vessels are typically located at tissue
planes (39), but these are modeled as uniformly
distributed. All of these assumptions are in keeping with the practical
approach of keeping the mathematics within the boundaries of the data.
Although we are working toward better definition of the microvascular
perfused surface area and distribution, we do not have the means at
this time to specify the necessary parameters to justify a more
detailed and complex approach.
A major assumption in the mathematical model is that transport
parameters are solely dependent on interstitial hydrostatic pressure.
This neglects possible effects of osmotic or oncotic pressure within
the interstitium. Some authors (51) have proposed inclusion of the local osmotic pressure within the expression for the
convective driving force. However, there is a lack of local
(
T vs. position) data to implement the proposed model
scheme, and it is uncertain how to calculate the effects of osmotic
gradients within the interstitium. Based on our experimental data at
Pip > 3 mmHg (62), the interstitial
space doubles and the concentration of interstitial albumin and large
interstitial matrix molecules must change. Indeed, experiments in which
we dialyzed rats for 2 h at Pip = 6 mmHg
demonstrated an apparent movement of hyaluronan from the inner layer of
abdominal wall muscle to the subcutaneous space (62). We
are actively pursuing the question of how the loss or addition of
hyaluronan will affect the transport of fluid and large solutes within
the interstitium. When more data are available, the model will be
updated to include these effects.
Implications for Dialysis
Despite the fact that this model does not include the tissue-level
effects of a hypertonic solution in the cavity, it does provide
insights into the effects of hydrostatic pressure in the peritoneal
cavity. The pressure in the cavity will cause fluid and solutes to move
from the cavity into the tissue surrounding the cavity, provided that
there exists some pressure gradient. Because the major goal for
peritoneal dialysis is to remove excess fluid and solute from the body,
it is instructive to calculate an estimate of the rate of fluid
movement into the abdominal wall in a human being with Eq. 1. Assuming a mean Pip of 3.0 mmHg, Eq. 11
produces an estimate of K of 0.47 × 10
6
cm2/(s · mmHg) and
dP/dx = 20 mmHg/cm (61) and the fluid flux across the abdominal wall
peritoneum = 9.5 × 10
6
ml/(s · cm2). To provide an estimate of the flow
into the abdominal wall, it will be assumed that 50% of the area of
contact is abdominal wall. The active area of fluid contact for the
human peritoneal cavity during dialysis with a 2-l volume has been
estimated recently to be 5,500 cm2 (5).
Multiplying the fluid flux times 50% of this area yields an estimate
of fluid loss to the abdominal wall of 94 ml/h [9.5 × 10
6 ml/(s · cm2) × 2,750 cm2 × 3,600 s/h]. This estimate is of the same order
of magnitude as the measurements in patients of 60-100 ml/h
(7, 30, 35, 46); unfortunately, none of these authors
measured Pip and therefore the 3 mmHg value was assumed on
the basis of our own measurements in normal patients (12).
This calculation implies that the abdominal wall, because the full
force of the Pip is exerted across it, is the major
recipient of fluid loss from the cavity. Although we do not know the
pressure profiles in other peritoneal tissues that are inaccessible to
direct measurement, we can speculate that the hollow viscera, which are
within the peritoneal fluid, may not experience the hydrostatic
pressure gradients that exist in the abdominal wall and may be the
major sources of osmotic ultrafiltration when hypertonic dialysis
solutions are infused into the peritoneal cavity.
From the theoretical concept in Figs. 1 and 2 and from the dependence
of convection into the abdominal wall on the tissue pressure profile,
it was hypothesized that pressure applied to the skin side of the
abdominal wall would decrease the overall driving gradient and decrease
fluid movement. This maneuver successfully reduced fluid loss by 50%
in rats (12), in which the abdominal wall is approximately
the thickness of the pressure profile (~2 mm). However, it was
unsuccessful in humans (12), presumably because the human
abdominal wall is over a centimeter thick. Abdominal counterpressure
would only compress the first few millimeters of external skin and
subcutaneous space of the abdominal wall and would not eliminate the
pressure profile on the inside of the wall.
Implications for Regional Immunotherapy
Figures 8 and 11 illustrate how interaction of the protein with
the tissue may decrease penetration. In Fig. 8, increased protein binding results in higher concentrations close to the surface but
decreases penetration into the deeper portions of the tissue. Fujimori
and colleagues (27) have called this the "binding site barrier," which is even more pronounced in tumors that overexpress the ligand for the antibody. Antibodies with less binding will transport further into tumor but demonstrate lower total concentrations close to the peritoneal surface of the tumor. Although binding affects
total tissue antibody concentration closest to the cavity, f
has significant effects on transport throughout the tissue. As seen in
Fig. 11, marked retardation (f = 0.1) results in a
concentration profile very close to that of pure diffusion. With no
solute retardation (f = 1), the concentrations
throughout the tissue are markedly higher and the solute penetration
distance doubles in 3 h from pure diffusion. This parameter has a
major influence on IgG delivery to the interior of targeted tissue, but
there is a dearth of data to properly define its value in normal or
neoplastic tissues. We have demonstrated that the single correlation in
the literature (50) is based on in vitro data and does not
fit our in vivo data. Defining the values of f for tumor
tissue will be a major effort of our laboratory in the future.
The influence of convection in the delivery of IgG to the interior of
tissue is demonstrated by Figs. 9-11. Although f is
determined by the protein interaction with the interstitial matrix and
cellular components in the tissue, Pip is determined by the
volume infused and the size of the patient. Pip directly
affects PT, which controls the magnitude of
if,
s, and K. The expansion of
the abdominal wall has been shown to be a nonlinear function of
PT (62), but K is a linear function
of PT (61). To promote the penetration and
delivery of macromolecules to subperitoneal tissue, the most practical
method is to use maximal Pip for the longest duration possible.
The model simulations illustrated in Fig. 7 demonstrate how increasing
the time of a therapeutic dwell from 1 to 3 h at constant pressure
in the peritoneal cavity improves the protein delivery to deeper
portions of the tissue. The total tissue concentration rises
dramatically and the penetration distance of the protein improves as
time increases. Our previous work demonstrated little penetration after
the same period of time for purely diffusive delivery of IgG
(22). If an isotonic salt solution is infused into the
peritoneal cavity, it would be absorbed into the tissues with time and
Pip would decrease, which would slow delivery of the
protein. However, starch solutions are now available that are able to
maintain their volume for up to 48 h (29). This type
of solution could potentially maintain Pip at a relatively constant value and facilitate the delivery of macromolecular agents. Figure 16 demonstrates the potential
for antibody delivery of such solutions, assuming that Pip
is maintained nearly constant for 24 h. In this model simulation,
f = 0.3 and Pip = 4.4 mmHg;
PT and
if are as in Fig. 4, and the other
coefficients are as given in Table 1. Cip will decrease as
the protein transports into the tissue and therefore must be varied
over the longer dwell period. If we assume a clearance (k)
of ~90 ml/h (see estimate of fluid loss above) and a 3-l volume, the
half-life (T1/2) of the antibody in the cavity
is ~23 h (k = 90 ml/h
3,000 ml = 0.03/h;
T1/2 = 0.693/0.03 h = 23 h). When Cip is varied accordingly, the model
predicts the concentrations of Fig. 16. The tissue pressure gradient
causes convection of the antibody to ~2 mm, where transport becomes
primarily diffusive. This causes a concentration "wave" to appear
because of the buildup of protein in this region. The model predicts
that, with sufficient time and pressure, the penetration is potentially
several millimeters of tissue.

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Fig. 16.
IgG tissue concentration profiles in the abdominal wall
over 24 h. Pip and intraperitoneal concentration
(Cip) are held constant with the PT and
if profiles of Fig. 4 and with f = 0.3. The "wave" of concentration occurs at x = 2,000 because convection decreases to zero at this point but the slow process
of diffusion is unable to dissipate the protein as rapidly as it is
delivered. This demonstrates the therapeutic advantage of time in the
delivery of large proteins.
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|
 |
APPENDIX |
Transendothelial Transport
The expressions for solute and water transport across the blood
capillary barrier are taken from the multiple-pore theory of Rippe
(47, 49)
|
(A1)
|
and
|
(A2)
|
where a is capillary surface area-density (area/unit
volume of tissue),
j is Peclet number for
pore j
o is the ratio of pore-to-bulk solute
diffusivity,
o is the sieving coefficient of the pore,
D
is the solute diffusivity in free solution,
Ap/(lAc) is the ratio of
total pore surface area per unit capillary area to thickness of the
capillary wall,
j is the fraction of
LP that is due to pore j, CP is plasma concentration, Lp is the overall filtration
coefficient for capillary, PP is capillary pressure,
i,j is the solute
reflection coefficient of the ith solute in the jth pore,
T,i is osmotic pressure
in interstitium of the ith solute, and
P,i is
osmotic pressure in the plasma. Details of these coefficients can be
found in our previous publications (18, 20) and in those
of others (40). Most of the blood capillary coefficients
have been defined by Rippe (45, 49) for the muscle that
surrounds the mammalian peritoneal cavity and will be used in
simulations for normal abdominal wall muscle.
The following is an additional constraint
|
(A3)
|
where
j is the fraction of
LP that is due to pore j and the subscripts
refer to the pore type: transcellular pore (CP), small pore (SP), or
large pore (LP).
If the solute of interest is a protein or macromolecule larger than
albumin (~58,000 Da), the theory constrains all transport to
convection through the large pore. Equation A1 can be
simplified to
|
(A4)
|
where RLP is solute transport via large pores,
CP is plasma concentration of protein, FLP is
water transport via large pores, and
LP is reflection
coefficient for the large pore for the protein.
 |
ACKNOWLEDGEMENTS |
This work was supported by a grant from the Whitaker Foundation and
Public Health Service Grants R29-DK-48479 and R01-CA-85984.
 |
FOOTNOTES |
Address for reprint requests and other correspondence: M. F. Flessner, Div. of Nephrology, Dept. of Medicine, Univ. of
Mississippi Medical Center, 2500 N. State St., Jackson, MS
39216-4505.
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 27 September 2000; accepted in final form 26 February 2001.
 |
REFERENCES |
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An, K-N,
and
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