INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL FORMULATION MODEL PREDICTIONS AND... MODEL SIMULATION OF PREVIOUS... DISCUSSION OF THE MODEL... APPENDIX REFERENCES

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.

View larger version (50K): [in this window] [in a new window]   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.

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 [P_T, which varies with intraperitoneal pressure (P_ip)] 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.

	MODEL FORMULATION

TOP ABSTRACT INTRODUCTION MODEL FORMULATION MODEL PREDICTIONS AND... MODEL SIMULATION OF PREVIOUS... DISCUSSION OF THE MODEL... APPENDIX REFERENCES

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 (P_ip). 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).

View larger version (32K): [in this window] [in a new window]   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.

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 (dP_T/dx, where x is distance) (33)

(1)

Rather than using P_if, the true interstitial pressure, we designate the tissue pressure by P_T, 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 P_if. 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)

(2)

where _if is interstitial fluid volume. To estimate (P_T) 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 · mmHg¹ · 100 g tissue¹, 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 (dP_T/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 P_ip 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 P_T and _if in the tissue results in variation of the tissue hydraulic conductivity and in the effective tissue diffusivity (D_eff)

(3)

where D_v 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.

View larger version (20K): [in this window] [in a new window]   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.

View larger version (22K): [in this window] [in a new window]   Fig. 4.   Hypothetical pressure profile and the corresponding interstitial volume profile based on data from Refs. 13 and 62.

Solute Balance

(4)

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 = D_eff(dC_s/dx). The second term is derived as follows. Within the interstitial space, from Eq. 1, the velocity of the solvent, v_f, equals Q/(A_if) or (K/_if)(dP_T/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 v_s = fv_f. The mass flow equals C_sv_s = C_s(fv_f) = C_s(fK/_if)(dP_T/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 (R_i) are defined by separate rate equations. P_T 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 D_eff, K, f, _s, and _if are variable and are functions of the local P_T.

Volume Balance

(5)

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 C_s 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, C_tissue, would equal C_free = C_ss. Because there is either specific or nonspecific binding for all proteins, the more general definition for C_tissue is as follows

(6)

where C_free is concentration of free solute in tissue = C_ss and C_bound is concentration of bound solute in tissue, where C_bound = g(C_free, C_bound), a function defined by experiment.

(7)

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.

(8)

Lymphatic transport from tissue. Lymph will be assumed to flow from the tissue at a constant rate equivalent to F_lymph, which must be derived by experiment. The rate of solute flow from the tissue space is given by

(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 R_metab and R_gen. 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

at x = 0, t = 0: C_s = C_ip(0), the intraperitoneal concentration at time "0," P_T (0) = P_ip = intraperitoneal pressure. These two conditions presume that there is no boundary layer on the peritoneal surface that affects concentration or pressure.

(10)

At x = x₁, t > 0: dC_s/dx = 0 and dP_T/dx  0 at x₁. Based on experimental data (see Fig. 3), dP_T/dx at the abdominal wall edge (x₁  0.2 cm) at high pressures (P_ip > 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 P_ip 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

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 B_R = 0 and B_F = 1.08 × 10⁴ s¹ (22).

From our previously published results (61, 62), expressions for _if and K will be derived. The expression for K is as follows

(11)

where hydraulic conductivity coefficients A₀ and A₁ are 0.15 × 10⁶ cm² · s¹ · mmHg¹ and 0.18 × 10⁶ cm² · s¹ · mmHg² respectively, the units for K are square centimeters per second per millimeter of mercury, and the units for P_T 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

(12)

where B₀ = 0.19 ml/g tissue and B₁ = 0.046 ml · g¹ · mmHg¹. B₁ is equivalent to the compliance [(P_T)] 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 P_T 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 P_ip was obtained. We observed that P_T(x) did not vary significantly with time, and we assume that a constant P_ip will result in a constant P_T(x) for t > 0. We therefore compute the tissue pressure profiles by the following

(13)

where P_ip 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 (F_lymph) to equal 1.33 × 10⁶ cm² · s¹ · mmHg². 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, R_cap 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 · min¹ · 100 g tissue¹ or 5 × 10⁶ ml · s¹ · g tissue¹. 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 P_P (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 P_T, 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 (L_Pa) = 6 × 10⁵ ml · mmHg¹ · s¹ and _LP = 0.07, the resulting F_cap,LP is ~6 × 10⁶ ml · s¹ · g tissue¹ (45, 49). With the estimated value of F_cap,LP and C_P(t), the contribution of the "backflow" of IgG from the plasma to the tissue can be calculated. In simulations with F_cap,LP = 6 × 10⁵ to 6 × 10⁴ ml · s¹ · g¹, no significant change in tissue concentration was observed.

As shown in Eq. 3, the effective diffusivity, D_eff, equals D_vs. For a given tissue the diffusivity within the tissue distribution space of the solute, D_v, is assumed to be a constant. D_v also takes into account the tortuosity of the path. Our previous experiments (22) and those of others (3) produced an estimate for D_v of 2 × 10⁷ 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 P_T = 0 and mean P_T = 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 P_T = 0, _s = 0.043-0.050, whereas at P_T = 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 P_T = 0 or 0.21 ml/g at a mean P_T 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 C_s(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

	MODEL PREDICTIONS AND SENSITIVITY

TOP ABSTRACT INTRODUCTION MODEL FORMULATION MODEL PREDICTIONS AND... MODEL SIMULATION OF PREVIOUS... DISCUSSION OF THE MODEL... APPENDIX REFERENCES

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 P_ip, 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.

View larger version (22K): [in this window] [in a new window]   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 C_s(x,t), C_bound(x,t), and C_tissue(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, P_ip, _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 D_v 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.

View larger version (13K): [in this window] [in a new window]   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.

Figure 7 demonstrates the effect of time on output of the model. As time increases beyond 60 min, a greater proportion of the total C_tissue is due to C_bound. C_free at any distance from the peritoneum can be found by subtraction: C_tissue  C_bound. Binding of protein to tissue is a major determinant of C_tissue (=C_free + C_bound). Figure 7 illustrates the effect of time on the concentration of bound IgG (C_bound) and the total concentration (C_tissue). 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 P_ip 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.

View larger version (21K): [in this window] [in a new window]   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 C_tissue when the binding within the tissue varies from 20% of the baseline to five times the baseline value. As binding increases, C_tissue 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).

View larger version (19K): [in this window] [in a new window]   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.

Because of the dominance of convection in the transport of IgG, factors that directly affect the rate of water flow (K, P_T) 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 P_ip is varied from 2.0 to 8.8 mmHg. The P_ip is transmitted to the tissue as demonstrated in Fig. 1 and sets up the pressure profile of P_T (Fig. 2), which drives fluid flow from the cavity into the tissue. When P_ip 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 P_ip 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.

View larger version (19K): [in this window] [in a new window]   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.

View larger version (19K): [in this window] [in a new window]   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.

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.

View larger version (21K): [in this window] [in a new window]   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 D_v (Table 1) do not result in large changes in C_tissue (see Fig. 12). This would likely change if the _s were higher and the magnitude of D_eff (D_vs) 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.

View larger version (17K): [in this window] [in a new window]   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.

View larger version (22K): [in this window] [in a new window]   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 C_tissue and likely result in a greater depth of penetration and higher tissue concentrations close to the peritoneum.

View larger version (21K): [in this window] [in a new window]   Fig. 14.   Calculated IgG concentration profiles for t = 60 min: effect of interstitial volume (if) on model output. See text for discussion.

	MODEL SIMULATION OF PREVIOUS DATA

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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.

View larger version (16K): [in this window] [in a new window]   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).

As an initial test of the model as a quantitative tool, we have fitted the model to the data in Fig. 15. The P_ip 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 × 10⁴ (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

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Theoretical Approach to Convection in Subperitoneal Tissue

In the model presented here, the solute retardation factor <