## Abstract

The process of water reabsorption from the peritoneal cavity into the surrounding tissue substantially decreases the net ultrafiltration in patients on peritoneal dialysis. The goal of this study was to propose a mathematical model based on data from clinical studies and animal experiments to describe the changes in absorption rate, interstitial hydrostatic pressure, and tissue hydration caused by increased intraperitoneal pressure after the initiation of peritoneal dialysis. The model describes water transport through a deformable, porous tissue after infusion of isotonic solution into the peritoneal cavity. Blood capillary and lymphatic vessels are assumed to be uniformly distributed within the tissue. Starling's law is applied for a description of fluid transport through the capillary wall, and the transport within the interstitium is modeled by Darcy's law. Transport parameters such as interstitial fluid volume ratio, tissue hydraulic conductance, and lymphatic absorption in the tissue are dependent on local interstitial pressure. Numerical simulations show the strong dependence of fluid absorption and tissue hydration on the values of intraperitoneal pressure. Our results predict that in the steady state only ∼20–40% of the fluid that flows into the tissue from the peritoneal cavity is absorbed by the lymphatics situated in the tissue, whereas the larger (60–80%) part of the fluid is absorbed by the blood capillaries.

- water absorption
- tissue transport
- hydrostatic pressure
- peritoneal dialysis

one of the objectives of peritoneal dialysis (PD) is to remove excess fluid from the patient's body. This is achieved by infusion of hypertonic dialysis fluid into the peritoneal cavity, which causes an osmotic filtration into the cavity. The infused volume increases intraperitoneal pressure, resulting in inflow of water and solutes into the tissue surrounding the cavity. This reabsorption of water from the peritoneal cavity into the tissue substantially decreases the net ultrafiltration in PD patients by 1–2 ml of water per minute, corresponding to ∼1.44–2.88 liters per 24 h (25–27). The details of this process are not well understood and cannot be easily studied with experimental or clinical protocols (11, 30). Absorption is a dominant fluid transport component after a few hours of peritoneal dwell of (initially) hypertonic dialysis fluid when osmotic pressure disappears because of mostly diffusive absorption of low osmotic agent. Therefore, a theoretical analysis of peritoneal absorption and the main factors that influence its rate may be useful. Actually, absorption occurs in some tissues simultaneously with net filtration into the cavity, and the concomitant protein absorption continues despite use of a hypertonic dialysis solution (23). This approach can also be applied to the description of transport of drugs such as cisplatin and monoclonal antibodies used in intraperitoneal chemotherapy (2, 17).

Fluid loss from the cavity to the body is made up of two components: *1*) direct absorption by diaphragmatic lymphatics (*J*_{DL}) and *2*) absorption to the tissue (*J*). The fluid that enters the tissue layer is later absorbed into the lymphatics (*Q*) or into blood capillaries (*Q*), which are distributed within the tissue at different distances from the peritoneal surface. In some situations, part of this fluid (*J*) can leave the tissue on its other side [as in the intestine or as in experiments in which the impermeable surface is removed (14)]. Figure 1 illustrates these pathways. Only 5–25% of the total fluid loss from the cavity is via lymphatics open to the cavity, and the rest is absorbed into the tissue (13).

The aim of this study was to investigate fluid absorption into the tissue (*J*), taking into account a sophisticated network of factors such as intraperitoneal hydrostatic pressure, interstitial hydrostatic pressure (P), interstitial fluid flow across the tissue described by Darcy's law, lymphatic absorption from the tissue with the rate dependent on P, the dependence of tissue hydraulic conductance and interstitial fluid volume ratio on P as described by experimental data, exchange of fluid through the capillary wall according to Starling forces, and dilution of protein due to the inflow of water into the tissue.

To simulate the process of water reabsorption, we assume that the infused fluid is isotonic, and therefore an osmotic effect is not observed. This approach allows us to focus the investigation on the mechanisms of peritoneal absorption, independent of osmotic fluid transport, which is a more complicated process (14, 41). The mathematical model of fluid transport in the tissue has been developed with two different boundary conditions, which characterize transport in different types of the tissue. The external surface of the tissue may be assumed to be *1*) impermeable, typified by the skin of the abdominal wall, with no flow through it; or *2*) permeable, typified by the mucosal side of the intestine. In each case the peritoneal surface is permeable. The assumptions are an idealization that does not take into account specific characteristics of the subcutaneous layer and mucosa of the intestine. Both of these layers have mechanical and transport properties different from those of the muscle and may absorb water faster than described by the present model (42). The layers should be included in future, more sophisticated versions of the model.

The model uses typical, average parameters for human tissue (53) and is applied to calculate changes in the rate of lymph and blood capillary absorption in the tissue, the influence of Starling forces on water reabsorption, the depth to which tissue is penetrated by water, the time evolution of hydrostatic pressure of the interstitial fluid and the interstitial fluid volume ratio, and the volumetric flux across the peritoneal surface during treatment of PD patients.

## Glossary

*x*- Distance, cm
*t*- Time, min
- P(
*t*,*x*) - Hydrostatic pressure, mmHg
*j*_{V}(*t*,*x*)- Volumetric flux through the tissue, cm/min
*j*(*t*)- Volumetric flux through the peritoneal surface,
*x*=*x*_{0}, cm/min *j*(*t*)- Volumetric flux through the external surface,
*x*=*x*_{max}, cm/min *J*(*t*)- Volumetric flow through the peritoneal surface,
*x*=*x*_{0}, ml/min *J*(*t*)- Volumetric flow through the external surface,
*x*=*x*_{max}, ml/min *q*_{V}(*t*,*x*)- Rate of local net fluid absorption from the tissue, min
^{−1} *q*_{B}(*t*,*x*)- Rate of local fluid absorption from the tissue through the capillary wall, min
^{−1} *q*_{L}(*t*,*x*)- Rate of local fluid absorption from the tissue by lymphatics, min
^{−1} *Q*_{AT}(*t*)- Rate of fluid absorption from the whole tissue, ml/min
*Q*(*t*)- Rate of fluid absorption from the whole tissue through the capillary wall, ml/min
*Q*(*t*)- Rate of fluid absorption from the whole tissue by lymphatics, ml/min
*B*(*t*)- Fractional input of absorption from the whole tissue through the capillary wall to total outflow from the whole tissue
*L*(*t*)- Fractional input of lymphatic absorption from the whole tissue to total outflow from the whole tissue
*J*(*t*)- Fractional input of outflow from the tissue through the external surface to total outflow from the whole tissue
*f*_{P}(*t*,*x*)- Rate of local absorption from the tissue through the capillary wall caused by the gradient of hydrostatic pressure, min
^{−1} *f*_{Π}(*t*,*x*)- Rate of local absorption from the tissue through the capillary wall caused by the gradient of oncotic pressure, min
^{−1} - Π(
*t*,*x*) - Oncotic pressure, mmHg
- θ(
*t*,*x*) - Interstitial fluid volume ratio
- θ
^{av}(*t*,*x*) - Average interstitial fluid volume ratio

## METHODS

#### Model formulation.

A detailed description of the fluid transport in the tissue must be taken into account in order to evaluate the main forces that influence water reabsorption. The tissue properties, including the spatial distribution of blood and lymph capillaries, are idealized by the assumption that blood and lymph capillaries are uniformly distributed within the tissue and that interstitium is a deformable, porous medium (16, 45). The interstitial fluid volume ratio (void fluid volume), i.e., the fraction of the total tissue space that is available for interstitial fluid (nondimensional, being the ratio of volume over volume), changes because of the inflow of fluid from the peritoneal cavity. The time evolution of the interstitial fluid volume ratio depends on the volumetric flux across the interstitium (*j*_{v}) and the rate of the net fluid absorption from the tissue by blood capillaries and lymphatics (*q*_{v}). The equation for the interstitial fluid volume ratio (θ) change is: (1) where *t* is time and *x* is the distance measured from the peritoneal surface (*x*_{0} = 0) to the external surface (*x*_{max} = 1 cm, which may be considered as a characteristic thickness of the abdominal wall of adult humans). Note that volumetric flux *j*_{v} is defined as volumetric flow (in ml/min) per unit surface (in cm^{2}) perpendicular to its direction, i.e., the unit of flux is centimeters per minute. The unit of local volumetric flow density *q*_{v} is minute^{−1}, i.e., as for volumetric flow (in ml/min) per unit volume (in ml). According to Darcy's law, the volumetric flux across the interstitium depends on the local tissue pressure gradient and local tissue hydraulic conductivity (*K*) and is equal to *j*_{V} = −*K*(∂P/∂*x*). The tissue hydraulic conductivity *K* is the function of local hydrostatic pressure P such that *K* = *a*_{0} for *P* < *b*_{0} and *K* = *a*_{0} + *a*_{1}(P − *b*_{0}) for P ≥ *b*_{0}, where *a*_{0} is basic tissue hydraulic conductivity level, *a*_{1} is sensitivity of *K* to increase in *P*, and *b*_{0} is basic *P* level for *K* (see Table 1) (54). Fluid exchange between blood capillaries, the tissue, and lymphatics is described according to Starling's law as *q*_{v} = −*L*_{P}*a*[P_{B} − P − σ_{Π}(Π_{B} − Π)] + *q*_{L}, where *L*_{P}*a* is the capillary wall hydraulic conductance times capillary surface area density per unit volume of wet tissue; P_{B} and Π_{B} are blood hydrostatic and oncotic pressure, respectively; and are assumed constant; σ_{Π} is capillary wall reflection coefficient; Π is interstitial fluid oncotic pressure; and *q*_{L} is the rate of lymphatic absorption from the tissue. We assume that oncotic pressure in the tissue, Π, depends on the interstitial fluid volume ratio θ and can be calculated as Π = Π_{0}θ_{0}/θ, where subscript 0 indicates the value of the function at time *t* = 0, i.e., in the state of physiological equilibrium. This functional relationship describes the effect of the dilution of interstitial fluid due to the inflow of protein-free dialysis fluid and the expansion of the interstitial fluid volume ratio. We also assume that the lymph flow is a function of interstitial pressure given by *q*_{L} = *q*_{L0} + *q*_{L1}(P − P_{0}) (4, 5, 24, 43, 52). Because at physiological state *q*_{v} = 0 at *t* = 0, lymphatic absorption is equal to the water flow through the capillary wall due to Starling forces. Thus the rate of lymphatic absorption *q*_{L0} in the steady state of fluid transport (P = P_{0}, at *t* = 0) is (2)

Interstitial fluid volume ratio is a function of the hydrostatic pressure: (3) with parameters θ_{min}, θ_{max}, θ_{0}, β, and P_{0} obtained by fitting *Eq. 3* to the data from Reference 55. *Equation 3* is purely empirical and does not have any theoretical explanation. Because interstitial fluid volume ratio θ depends on interstitial pressure P, and *Eq. 1* can be converted to the following form: (4)*Equation 4* is a nonlinear partial differential equation with one variable P(*t*,*x*), whereas other variables and physical factors, θ, Π, *K*, and *q*_{L}, are functions of P.

#### Initial and boundary conditions.

Hydrostatic pressure in the interstitial fluid is assumed to be equal to 0 at *t* = 0 [P(0,*x*) = P_{0} = 0], i.e., the tissue is in equilibrium before the infusion of fluid into the peritoneal cavity. After infusion, the hydrostatic pressure at the peritoneal surface is equal to the hydrostatic pressure in the peritoneal cavity P_{D} [P(*t*,*x*_{0}) = P_{D}]. At the external surface of the tissue P(*t*,*x*_{max}) = 0 in the case of a permeable external surface, or in the case of an impermeable external surface, i.e., no water flux at *x* = *x*_{max}, *j*_{V}(*t*,*x*_{max}) = 0. Therefore, the boundary conditions (BC) are:

#### Fluid absorption from peritoneal cavity to tissue.

Volumetric flux across the peritoneal surface into the tissue, *j*(*t*) = *j*_{V}(*t*,*x*_{0}), in centimeters per minute, changes with dwell time. The total fluid absorption from the peritoneal cavity to the tissue (*J*, in ml/min) can be calculated as (5) where *A*, in square centimeters, is the surface area of the contact between dialysis fluid and the peritoneum.

#### Fluid absorption from interstitium.

There are two pathways of fluid exchange in the tissue. Fluid may be transported either through the blood capillaries according to Starling forces, *q*_{B} = −*L*_{P}*a*[P_{B} − P − σ_{Π}(Π_{B} − Π)], or it may be directly absorbed by the lymphatics, *q*_{L}. Thus *q*_{V} = *q*_{B} + *q*_{L}, where *q*_{L} and *q*_{B} describe the net flow from the tissue (these quantities are positive if the net flow is from the tissue and negative for net inflow to the tissue). Moreover, the total blood and lymphatic absorption from the tissue, *Q*_{AT}, can be evaluated as *Q*_{AT} = *Q*+ *Q*, where respectively, because the properties of the peritoneal tissue are assumed uniform. Note that local flows *q*_{L}, *q*_{B}, and *q*_{V} are expressed in minute^{−1}, i.e., flow (in ml/min) per unit volume (in ml), (ml/min)/ml, whereas flows *Q*, *Q*, and *Q*_{AT} from the whole tissue layer of the volume *A*(*x*_{max} − *x*_{0}) (in ml) are expressed in milliliters per minute.

In the model for a permeable external surface, there can be a fluid outflow through the external surface, *j*(*t*) = *j*_{V}(*t*,*x*_{max}) ≠ 0, in centimeters per minute, and therefore an additional fluid pathway should be included. In this situation, the total outflow from the tissue is equal to *Q*+ *Q*+ *J*= *Q*_{AT} + *J*, where the total fluid outflow from the tissue through the external surface can be evaluated as *J*= A·*j*, in milliliters per minute. Finally, we may calculate the fraction of each of the pathways in the total fluid outflow from the interstitium as (6) where *B* + *L* + *J* = 1. Note that *J* = 0 in the model with impermeable external surface, *J*= 0.

#### Starling forces.

To evaluate the influence of each of the Starling forces separately, it is useful to split *q*_{B} into two components, *q*_{B} = *f*_{P} + *f*_{Π}. The first component evaluates the density of absorption from the tissue caused by the hydrostatic pressure difference, *f*_{P} = −*L*_{P}*a*(P_{B} − P), whereas the second component measures the density of absorption caused by the oncotic pressure difference, *f*_{Π} = *L*_{P}*a*·σ_{Π}(Π_{B} − Π).

#### Computer simulations.

Computer simulations of the fluid exchange in tissue according to the present model were performed by using a solver for parabolic partial differential equations in one space variable and time from Matlab version 6.5 (3).

Fluid transport in the tissue (*Eq. 4*) was simulated for different values of intraperitoneal pressure and different boundary conditions, using the values of parameters presented in Table 1. It was assumed that before the infusion of fluid into the peritoneal cavity the tissue was in equilibrium and therefore P(0,*x*) = 0. Three different time schedules were simulated: *1*) intact tissue exposed to an instantaneous contact with external isotonic fluid of hydrostatic pressure of 3, 7 or 12 mmHg (simulation was continued until a steady state was obtained); *2*) intact tissue exposed to continuous infusion of isotonic fluid for 15 min, with intraperitoneal pressure linearly increasing from P_{D} = 0 to P_{D} = 7 mmHg (afterwards, P_{D} was maintained at the level of P_{D} = 7 mmHg for 6 h and finally decreased linearly from P_{D} = 7 to P_{D} = 0 mmHg during 15 min of draining procedure); and *3*) four repetitions of the cycle of 15-min infusion, 6-h intraperitoneal dwell, and 15-min drainage, as in *schedule 2*, after which the peritoneal cavity was empty (P_{D} = 0). Model variables such as interstitial hydrostatic pressure P(*t*,*x*), interstitial fluid volume ratio θ(*t*,*x*), fluid fluxes, etc. were functions of time and position in the tissue, measured as the distance from the peritoneal surface to the external surface of the tissue.

## RESULTS

#### Hydrostatic and oncotic pressures.

Figure 2 presents changes in the interstitial pressure and interstitial fluid volume ratio profiles during dialysis with P_{D} = 7 mmHg as a function of distance at different time steps, starting from 10 min after infusion until the steady state is obtained. After infusion of fluid into the peritoneal cavity, interstitial pressure increases, initially only near the peritoneal surface (Fig. 2, *top*). The fluid flows from the cavity to the tissue and expands the interstitial fluid volume ratio in tissue layers close to the cavity, progressing deeper into the tissue. The thickness of the tissue layer, which is expanded by the fluid influx, increases with the rise of intraperitoneal pressure (for P_{D} = 7 mmHg, the layer of ∼0.2 cm thickness is fully filled up in the steady state; Fig. 2, *bottom*). The hydrostatic pressure profiles, as well as the interstitial fluid volume ratio distribution, close to the external surface depend on the boundary condition at this surface (i.e., whether external surface is permeable or not) (see Figs. 2 and 3). The steady-state hydrostatic pressure profiles obtained for different boundary conditions and for intraperitoneal pressures P_{D} = 3, 7, and 12 mmHg are presented in Fig. 3. These profiles depend on P_{D} value as well as on the boundary condition (Fig. 3). In all cases, a substantial drop in hydrostatic pressure within the tissue can be observed. The time needed to obtain the steady state (*t*_{steady,P}) depends on P_{D} and is slightly shorter for a permeable external surface than for an impermeable surface, ∼424 and ∼445 min for P_{D} = 7 mmHg, respectively (Fig. 2, *top*). Time *t*_{steady,P} is defined here as the time after which the changes in the interstitial pressure P for each *x* are <10% of the final value of P. The difference in the pressure and interstitial fluid volume ratio profiles at steady state in Fig. 3 results from the difference in the boundary conditions at the external surface of the tissue and is observed only close to the external surface, which can be either permeable or impermeable. Figure 4 demonstrates how the oncotic interstitial pressure decreases because of the inflow of protein-free fluid from the peritoneal cavity.

#### Fluid absorption from peritoneal cavity.

Fluid loss from the peritoneal cavity into the tissue depends on intraperitoneal pressure (23, 28). According to our simulations, the total volumetric flux across the peritoneal surface, *j*(*t*), decreases in time, and after ∼2 h (*t*) stabilizes at the level close to *j* in the steady state (*j*), i.e., after 2 h the difference between *j*(*t*)and *j* is <10% of *j*(Fig. 5, Table 2). The total fluid absorption from the peritoneal cavity to the tissue at the steady state, *J*, calculated for different BC and P_{D} values, is presented in Table 2. *J* is similar for both permeable and impermeable external surfaces and depends only on P_{D} (Table 2). It stabilizes faster than the pressure profile, at the level dependent on P_{D} value (Fig. 5 and the values of *t* in Table 2).

#### Fluid absorption from interstitium.

The fluid that enters the tissue can leave it by three different pathways. Thus the total net fluid outflow from the tissue is obtained as the net result of total absorption to blood (*Q*), lymphatic absorption (*Q*) from the tissue, and outflow through the external surface (*J*, only for permeable external surface). A comparison among the total net outflow from the interstitium, the fluid inflow to the tissue (*J*), and the total lymphatic absorption from the tissue (*Q*) is presented in Fig. 6. Because at the steady state the total net fluid outflow from the tissue and the inflow into the tissue are in equilibrium, *Q*_{AT} + *J* converges with time to *J*(Fig. 6, *left*). Therefore, *Q*+ *Q*+ *J*= *J*. It should be noted that for an impermeable external surface *J*= 0 and, in equilibrium, *Q*_{AT,steady} = *J*. Moreover, there are changes in the direction of the fluid exchange through the capillary wall, *Q*, from the net inflow into the interstitium before the beginning of the intraperitoneal dwell to the net absorption afterward (Fig. 6, *right*). At the beginning of the peritoneal dwell, the fluid flow through the capillary wall is from blood to tissue, i.e., negative according to our system of flow signs, and therefore the net absorption from the tissue, *Q*_{AT}, is lower than lymphatic absorption (Fig. 6, *right*). Later on, the flow through the capillary wall changes its direction and passes from tissue to blood, i.e., has a positive magnitude, and the net absorption from the tissue, *Q*_{AT}, is higher than lymphatic absorption (Fig. 6, *right*). The switch time (*t*_{switch}), at which the total (for the whole tissue layer) fluid flow through the blood capillary wall changes its sign, is short (<10 min) and depends on P_{D} (Table 2).

The changes in the net fluid outflow from the tissue (Fig. 6, *left*) are caused by the local changes in each of the three component flow rates in the tissue (*Q*, *Q*, *J*). The fractional contribution of each of the pathways to the total fluid outflow from the interstitium, evaluated as parameters *B*, *L*, and *J* (see *Eq. 6*) for different BC and P_{D} = 7 mmHg, is shown in Fig. 7. On the other hand, the comparison of each of the pathways of fluid removal and the inflow to the interstitium from the peritoneal cavity can be performed by using the ratios *Q*/*J*, *Q**J*, and *J*/*J*(Fig. 8). It should be noted that in the steady state the ratio of the total outflow from the interstitium over the fluid absorption from the cavity is equal to 1. Evaluation of these ratios at the steady state for different BC and P_{D} values is presented in Table 2.

The contribution of each pathway to the fluid removal from the interstitium changes in time. At the beginning of the process, the overall (for the whole tissue layer) ultrafiltration from blood capillaries prevails over capillary absorption. Therefore, until the switch time, fluid that enters the tissue is absorbed mainly by lymphatics. The length of this period depends on P_{D}, but not on the external boundary condition, and ranges from 0.5 to 8.5 min (Table 2).

The share of total absorption to blood, *Q*/*J*, increases with time faster than the share of lymphatic absorption, *Q*/*J*, and fluid outflow, *J*/*J*(Fig. 8). Therefore, the fractional input of absorption to blood into the fluid removal is the highest at the steady state (Figs. 7 and 8; Table 2), and its value (within the range 60–80%) depends on the P_{D} value and BC (Table 2). It should be also noted that the total fluid outflow from the tissue, *J*, stabilizes much slower than the fluid inflow to the tissue, *J*(Fig. 5).

#### Starling forces.

The model was also used to evaluate the influence of Starling forces on water reabsorption through the capillary wall. Figure 9 presents the fluid flux caused by the hydrostatic pressure difference (*f*_{P}) and oncotic pressure difference (*f*_{Π}) and lymphatic absorption (*q*_{L}) at each point of the tissue at the steady state. For *x* < 0.6 cm there is a substantial decrease of oncotic pressure in the tissue, Π, caused by the dilution of proteins (Fig. 4). Therefore, oncotic pressure difference between blood and tissue is higher in this layer but drops toward physiological values in deeper layers.

The depth at which the absorption of the interstitial fluid is high depends on the P_{D} value and is equal to ∼0.3, ∼0.7, and ∼1 cm at the steady state for P_{D} = 3, 7, and 12 mmHg, respectively (see Fig. 9 for the case of P_{D} = 7 mmHg). Close to the cavity, the capillary flow component triggered by the oncotic pressure difference *f*_{Π} is higher at the steady state than hydrostatic pressure component *f*_{P} (although initially *f*_{P} > *f*_{Π}). Within the tissue, at the steady state, *f*_{Π} may also be higher than the absolute value of *f*_{P} (Fig. 9), but this relationship depends on the permeability of the external surface and the P_{D} value. Nevertheless, the contribution of *f*_{Π} to the absorption to blood capillaries decreases with the distance from the cavity, while the negative contribution of hydrostatic pressure difference increases (Fig. 9).

#### Consecutive peritoneal exchanges.

The computer simulation was also used to evaluate the changes in the interstitial pressure during consecutive peritoneal exchanges. We assume that during infusion, which takes 15 min, intraperitoneal pressure linearly increases from P_{D} = 0 to P_{D} = 7 mmHg. Afterwards, P_{D} is maintained at the level of P_{D} = 7 mmHg for 6 h and finally decreases linearly (from P_{D} = 7 to P_{D} = 0 mmHg) during 15 min of draining procedure. The drainage process influences the interstitial pressure distribution close to the cavity, while the pressure in the deeper layer remains unchanged (Fig. 10).

Time evolution of the average interstitial fluid volume ratio and volumetric flux across the peritoneal surface (*j*) during four repetitions of the cycle of 15-min infusion, 6-h intraperitoneal dwell, 15-min drainage, after which the peritoneal cavity is left empty (P_{D} = 0), are presented in Fig. 11 for an impermeable external surface and different P_{D} values.

Because of the short time of drainage, the total interstitial fluid volume ratio of the tissue does not come back to its initial state after the first exchange (Fig. 11). It increases further during the second exchange, and during the third dwell period it stabilizes at a level that depends on P_{D} value (Fig. 11, *left*). Afterwards, only small changes during infusion and draining procedure in the total interstitial fluid volume ratio are observed. On the other hand, the total fluid flow from the peritoneal cavity into the tissue, *j*, reaches its steady-state level during the first dwell period (Fig. 11, *right*). The changes in *j* caused by the draining and infusion procedure, as well as its steady-state level, depend on P_{D} value and are the highest for P_{D} = 12 mmHg. The total removal of fluid from the peritoneal cavity, leaving it empty (P_{D} = 0), at *t* = 1,560 min causes a fall of interstitial pressure. This decrease of P and the fast drop of the total interstitial fluid volume ratio stabilize at the initial steady-state values after ∼8 h from the moment of fluid drainage (Fig. 11).

## DISCUSSION

In this study of the transperitoneal water transport, a distributed model was applied as proposed by Dedrick, Flessner, and colleagues (8, 19). A similar approach has been applied to describe such phenomena as exchange of gases between blood and artificial gas pockets within the body (38) and exchange of matter and heat between blood and tissue for the intratissue source of solute or heat (36, 37). The first application of the distributed model for solute transport was proposed by Patlak and Fenstermacher (34) to describe diffusion of small solutes from cerebrospinal fluid to the brain. Later, the model was applied to the description of peritoneal transport of small, middle-sized, and macromolecules in rats (18, 20, 21) as well as in continuous ambulatory peritoneal dialysis (CAPD) patients (8, 45, 48). Moreover, the model was also used for other applications such as intraperitoneal transport of drugs in chemotherapy (7, 17) or drug transport in the bladder wall after intravesical infusion (50, 51).

Despite the variety of studies on solute transport, only a few attempts at describing peritoneal water transport by means of distributed modeling have been undertaken. The first model, proposed by Seames et al. (41), was developed to study peritoneal fluid and solute transfer. However, their assumption that the mesothelium is an important barrier for solute and water transport, yielded negative intratissue hydrostatic pressure that did not correspond to experimental data (14, 15). Another description of solute and water transport was proposed by Flessner, who initially assumed that the interstitium is a rigid, porous structure (see Refs. 19, 45). Later, this assumption was modified according to the results of experiments demonstrating that the tissue is a compliant, porous structure (16).

Recently, some studies have provided information about the details of water transport in the tissue. Our model combines the latest experimental data, the assessment of fluid transport parameters for humans, and previous formulations of theoretical models of fluid transport (16, 42). The applied model does not take into account the variability of anatomic and physiological parameters among patients and tissue types, because all simulations were done for the “average” patient and the “average” tissue surrounding the peritoneal cavity. We propose a theoretical model of peritoneal water transport in order to evaluate clinically practical correlations rather than describing details of transport that occurs in a particular patient and tissue. Although the model is generalized, it may be helpful in finding a cause for PD failure in some patients. Moreover, the model does not describe the process of combined fluid and solute transport and therefore should be treated as the first step toward more complex modeling of combined transport induced by hypertonic dialysis fluid. Although isotonic fluid is not used in clinical practice (see, however, Ref. 2), it is sometimes applied in animal experiments or as a vehicle for intraperitoneal or intravesical drug delivery (9, 49, 50, 56).

Peritoneal absorption is a dominant fluid transport process after 3–4 h of dwell time, when dialysis fluid is practically isotonic and ultrafiltration from blood usually becomes negligible. Detailed description of water reabsorption is still mathematically complex even for isotonic solution (i.e., without osmotic fluid transport). Nevertheless, some clinical aspects of the fluid transport may be studied with this model, because the absorption occurs simultaneously with and separately from the osmotic transfer.

To apply a mathematical model, we idealized and simplified the tissue characteristics. The assumption that interstitium and capillaries are uniformly distributed within the tissue does not take into account some details of the real structure of the tissue. The thin layer of peritoneum was also not included in the model, under the assumption that its contribution to the peritoneal transport is negligible (12, 15). Tissue planes and fascial layers of connective tissue have been neglected in the model as well as the fact that lymphatics are typically located at tissue planes (16). Moreover, possible changes in the local structure and the size of the tissue layer are not provided for, as well as the detailed structure of the external surface of the tissue layer, such as subcutaneous tissue in the abdominal wall or mucosa in the viscera. Therefore, this model cannot be used for description of some other important phenomena, such as the local tissue edema that may be caused by high intraperitoneal pressure in a patient with an abdominal wall hernia.

Another assumption behind the model is the conservation of protein amount locally in the tissue. This assumption results in a relatively simple model of fluid transport but, on the other hand, may limit its applications. The local amount of protein in the interstitial fluid during steady-state normal physiological conditions is a result of a balance between inflow from capillaries and uptake by lymphatics. The process of absorption of protein-free fluid through the tissue surface disturbs this local protein mass balance. Protein concentration decreases by the effect of dilution, and, at the same time, lymphatic absorption increases, because increased interstitial fluid volume, described by θ, increases and causes the increase in interstitial fluid hydrostatic pressure; this in turn increases lymphatic absorption from the tissue. Because absorption, as a volume flow, increases but protein concentration decreases, protein lymphatic absorption may decrease, increase, or stay relatively constant, depending on the parameters for fluid and protein transport. Another process that may change the local amount of protein is the flow across the tissue. This flow convectively carries protein from the layers close to the surface into the layers deep in the tissue. This flow results in the depletion of protein from subsurface layers and its accumulation in deeper layers. The apparent movement of hyaluronan from the peritoneum toward the skin has been experimentally measured and described (55). The intensity of this effect depends on the particular values of the transport parameters and the magnitude of the flow through the tissue, which is high close to the surface, whereas the concentration of protein is low there. The third process that affects protein amount is transport across the capillary wall. In the presence of fluid absorption from the surface the direction of fluid (and protein) transport may, and does, change, especially close to the tissue surface. Nevertheless, protein (especially large proteins of the size of albumin or larger) in this reversed flow is sieved by small pores in the capillary wall, and its absorption is lower than the absorption of fluid. All three processes together may be expected to decrease the protein amount close to the surface. In deeper layers, protein accumulation or its conservation may be expected. The net result is, however, difficult to predict without numerical simulations of these complex interactions. Our modeling may be considered as a possible approximation of the combined fluid and protein transport in the tissue during fluid absorption through the surface and should be critically evaluated with a complete model. Nevertheless, one may expect quantitative changes in the profiles in space and time of hydrostatic pressure, interstitial fluid volume ratio, fluxes, and flows, which are presented in this article, rather than qualitatively different profiles.

Intraperitoneal pressure P_{D}, which is a major driving force for water transport, varies with body position and with the site in the peritoneal cavity. In general, for the standard 2-liter fluid infusion, it ranges from 2–7 mmHg in the supine position to 5–27 mmHg in the standing and 7–28 mmHg in the sitting position (29, 44). Our simulations were done for P_{D} between 3 and 12 mmHg, the values reported for patients in the recumbent position by Imholz et al. (29). For higher values of intraperitoneal pressure such as 20 or 30 mmHg, complex processes such as edema, collapse of the lymphatics and blood capillaries, and compression of the tissue, which are not taken into account in this model, may occur.

The values of physiological parameters such as blood hydrostatic and oncotic pressures are taken for a typical human and do not take into account variability among patients, organs, and physiological states. According to the measurements, interstitial pressure in the physiological state and in the intact peritoneal cavity is slightly negative, whereas we assumed that before the fluid infusion in the tissue the interstitial pressure is equal to 0 mmHg. This assumption reflects the situation of clinical and experimental peritoneal dialysis with an inserted catheter or, as in many animal experiments, opening of the peritoneal cavity before infusion of fluid. In such a case, pressure in the cavity, and consequently in the subperitoneal tissue, equilibrates with atmospheric pressure. The functional relationships between the interstitial fluid volume ratio θ, the tissue conductance *K*, and the interstitial pressure P were taken according to animal experiments in the abdominal wall (54, 55). The particular parameters may differ among humans and animals, but the type of relationship should remain the same. However, the differences in numerical values of the parameters are typically not high; for example, the value of *L*_{P}*a* used for modeling of water transport in humans, *L*_{P}*a* = 0.53 × 10^{−4} min^{−1}·mmHg^{−1} (53), is close to, although half as much as, the value 1.0 × 10^{−4} min^{−1}·mmHg^{−1} (i.e., ∼0.01 ml/min per mmHg per 100 g of muscle) measured for animal skeletal muscles; in fact, *L*_{P}*a* values estimated or measured for human tissue are consistently lower than values for animal skeletal muscles (32).

Transport of water through the capillary wall was modeled in the present study by using the classic concept of Starling forces. This concept was recently criticized because of the inability to describe correctly some experimental data, and alternative proposals were formulated (31). In particular, the sieving effect for proteins in small pores of the capillary wall may result in local changes in protein concentration close to the capillary wall surface compared with the average protein concentration in the interstitium and plasma. As a consequence, the fluid transport rates calculated for the average oncotic pressures may not be correct (31). Modeling of these phenomena needs to be done with microscopic description of the capillary bed and the structure of pores in the capillary wall (1, 33), and their contribution to the distributed models of fluid transport in the tissue is not clear (e.g., whether these new concepts would yield only new values of “average” parameters, they would need a new mathematical structure of the distributed model, or the distributed approach would be discarded entirely).

All calculations for fluid absorption from the peritoneal cavity into the tissue (*j*) were done by using an effective peritoneal surface area equal to 0.5 m^{2} (6). In general, the surface area of the peritoneal membrane was reported to be ∼0.8–1.3 m^{2} for the average adult (10, 35, 40), whereas according to computer tomography measurements only ∼50% of anatomic peritoneal surface area is effectively in contact with infused fluid (6). Furthermore, the rate of water reabsorption may vary among tissue types (9). The thickness of the tissue layer assumed in our computer simulations was 1 cm, which is a characteristic (rather as a lower limit) thickness of the abdominal wall of adult humans. Some parts of the gastrointestinal tract, such as the stomach, may have a similar thickness, but most of the tract involved in PD is the small intestine, the thickness of which may be estimated as 2–3 mm. Some results of computer simulations for tissue of low thickness may be found in References 16 and 42. The relative role of different organs in fluid and solute transport in vivo is, however, not known because of the problem of the contact area between different organs and dialysis fluid (22).

In CAPD patients only ∼10–20% of total water absorption is directly absorbed from the cavity via lymphatics, whereas the larger part of the fluid is absorbed into the tissue (26, 39). Assuming that 90% of total absorption from the cavity goes to the tissue, one can calculate *J* from clinical data for the total absorption. Thus *J* varies from ∼1 ml/min for 1.36% glucose to ∼1.5 ml/min for 3.86% glucose in the recumbent position (26, 47). Somewhat lower values of *J*(0.82 and 0.88 ml/min for 2.27% glucose in the recumbent and upright positions, respectively) were also reported (29). Much higher values of ∼4.5 ml/min were found in patients with permanent ultrafiltration capacity loss caused by high lymphatic absorption rate (27). The values of peritoneal absorption quoted above were obtained with a volume marker method, which is at the same time used for the estimation of peritoneal absorption (25–27). Practically the same values of the constant rate of dialysis fluid volume decrease are observed during the final part of peritoneal dwell (after 2–4 h, depending on the initial osmolality of dialysis fluid), when the osmotic effect of dialysis fluid disappears because of diffusive absorption of osmotic agent from the peritoneal cavity. The values of peritoneal absorption measured in clinical studies with (initially) hypertonic dialysis fluids are constant during the final part of the study (i.e., one may assume that the absorption is approximately at the steady state during the final part of the dwell), and therefore they may be compared with the values calculated from our model for pure absorption during the steady state. This does not mean that the fluid transport process during the initial part of the dwell with hypertonic fluids is described by our model. The water absorption from the peritoneal cavity into the tissue obtained in our simulations is slow for low values of intraperitoneal pressure P_{D} and increases with the increase in P_{D}. This prediction is in agreement with animal experiments and clinical data, in which the rate of fluid absorption from the cavity was found to be directly proportional to the intraperitoneal hydrostatic pressure (23, 28, 57). However, the values from our numerical simulations tend to be higher then those observed in the clinic, especially for hydrostatic pressure >7 mmHg (26, 29, 47). A possible explanation for this can be that parameters describing tissue conductance *K* were originally measured in the abdominal wall of the rat, whereas the real *K* values for the “average” tissue may be lower. In fact, the rate of water absorption in the intestine and other abdominal organs (∼90% of total peritoneal surface) is lower than in the abdominal wall (<10% of the total peritoneal surface) (9, 10, 35, 40). To investigate the possible reason for this high water absorption, we analyzed the changes in *J* due to lower transport parameters *K* and *L*_{P}*a* than those used in the simulations (Table 3) (other parameters do not contribute substantially to the decrease of *J*). In all cases the scale of change in the absorption of water from the peritoneal cavity into the tissue (*J*) depends on the intraperitoneal pressure (Table 3). For example, if the external surface is impermeable and P_{D} = 3 mmHg, a twofold decrease in all *K* parameters results in 22% reduction of *J*, whereas for analogous change in *L*_{P}*a* the decrease in *J* is 28%. On the other hand, the obtained high values of *J* may describe the fluid transport in patients with permanent ultrafiltration failure due to high peritoneal absorption (46). A possible explanation for this treatment failure could be a high hydraulic conductivity of the interstitium, *K*.

In summary, computer simulations based on the distributed model show that increased intraperitoneal pressure changes substantially the fluid exchange process in the tissue. Interstitial hydrostatic pressure decreases with distance from the peritoneal cavity, in agreement with the experimental profiles obtained for isotonic solution in the rat abdominal wall (14). Fluid absorption from the peritoneal cavity into the tissue achieves its steady state faster than interstitial pressure, whereas the time period necessary for the stabilization of tissue hydration is rather long, because of the slow expansion of the interstitial fluid volume ratio under increased interstitial hydrostatic pressure. The percent contribution of blood and lymphatic routes of absorption from interstitium in the removal of the inflowing water quickly stabilizes at a rate that depends on the intraperitoneal pressure. Computer simulations show that ∼20–40% of the fluid that flows into the tissue from the peritoneal cavity is absorbed by the lymphatics located in the tissue, whereas the larger (60–80%) part of the fluid is absorbed by the blood capillaries.

## Footnotes

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