## Abstract

Under physiological conditions, interstitial fluid volume is tightly regulated by balancing microvascular filtration and lymphatic return to the central venous circulation. Even though microvascular filtration and lymphatic return are governed by conservation of mass, their interaction can result in exceedingly complex behavior. Without making simplifying assumptions, investigators must solve the fluid balance equations numerically, which limits the generality of the results. We thus made critical simplifying assumptions to develop a simple solution to the standard fluid balance equations that is expressed as an algebraic formula. Using a classical approach to describe systems with negative feedback, we formulated our solution as a “gain” relating the change in interstitial fluid volume to a change in effective microvascular driving pressure. The resulting “edemagenic gain” is a function of microvascular filtration coefficient (*K*_{f}), effective lymphatic resistance (*R*_{L}), and interstitial compliance (*C*). This formulation suggests two types of gain: “multivariate” dependent on *C*, *R*_{L}, and *K*_{f}, and “compliance-dominated” approximately equal to *C*. The latter forms a basis of a novel method to estimate *C* without measuring interstitial fluid pressure. Data from ovine experiments illustrate how edemagenic gain is altered with pulmonary edema induced by venous hypertension, histamine, and endotoxin. Reformulation of the classical equations governing fluid balance in terms of edemagenic gain thus yields new insight into the factors affecting an organ's susceptibility to edema.

- Starling-Landis equation
- mathematical model
- edematogenic

edema, the accumulation of excess interstitial fluid volume (*V*), can be both a cause and an effect (9, 12, 61) of major morbidity such as cardiac, renal, and pulmonary failure. The techniques used to determine the “degree” of edema, however, have limited prognostic potential. For instance, different inflammatory agents, such as histamine and endotoxin, can result in similar degrees of edema, although edema secondary to endotoxin is more likely to worsen with increased microvascular pressure (1, 44). Despite a focus on anti-edema mechanisms, investigators have neglected to address this sensitivity of edema formation to edemagenic challenges. Though the concept of a “gain” has been used to characterize whole-body fluid balance (30), it has yet to be applied to interstitial fluid balance. The potential exists to bridge the gap between basic physiology and clinical practice, as the complexity of interstitial fluid balance results from three relatively simple processes: transmicrovascular filtration, lymphatic return, and interstitial fluid storage.

Interstitial fluid balance is a manifestation of the principle of conservation of mass. Influx of fluid into the interstitium is regulated through transmicrovascular filtration—except for notable exceptions, such as the brain. Transmicrovascular filtration results from an imbalance of the hydrostatic pressure gradient acting to force fluid out of the microvasculature, and the colloid osmotic pressure gradient acting to retain fluid within the microvasculature. The net transmicrovascular fluid flow resulting from these pressure gradients is moderated by microvascular permeabilities to water and plasma proteins (46, 58). Outflow from the interstitium, on the other hand, is determined by lymphatic vessel function and the pressure gradient from the interstitium to the central venous circulation (43). Typically requiring an active process, the net flow depends on effective lymphatic driving pressure and effective lymphatic resistance (15). Interstitial compliance, the change in fluid volume for a given change in interstitial fluid pressure, plays a key role in linking microvascular fluid filtration and lymphatic return. In particular, when inflow rises above outflow, interstitial compliance determines how much interstitial fluid pressure rises. Increases in interstitial hydrostatic pressure not only act to decrease flow into the interstitium, but also increase flow out via the lymphatics (48). Even though these are relatively well-understood phenomena and their interaction is governed by conservation of mass, the resulting behavior can be exceedingly complex.

The difficulty in ascribing changes in *V* to a particular volume-regulating phenomenon is put in bold relief by the example of histamine. Infusion of histamine induces edema within minutes (40, 52, 60). Some reports ascribe this effect to increased transmicrovascular fluid flux due to increased permeability or vascular surface area (6, 40, 49, 56, 60). Lymph flow is reported to increase with histamine infusion (2, 6, 10, 13, 21, 35, 49, 56, 68) either by increased contraction rate (22) or by increased strength of contraction (36, 68, 70). However, inhibition of lymph flow with intravenous infusion of histamine at high concentrations (>5 μM) (22, 62) has also been reported, and thus the lymphatic system may play a role. Yet another mechanism for edema formation has been postulated that does not require changes in permeability or lymphatic function. Histamine has been reported to increase interstitial compliance (2, 65). That is, given the same interstitial fluid pressure, the interstitium can accommodate a much larger volume of fluid. The resulting change in compliance alone is capable of producing edema in skin and skeletal muscle (65, 73). This example illustrates a common difficulty in edema research: multiple mechanisms can lead to edema, and analytical tools are not able to determine their relative contributions.

Even though the equations governing interstitial inflow and outflow can be expressed algebraically, two complications arise when they are combined. First, fluid volume regulation results from classical negative feedback: any increase in interstitial fluid volume acts to limit further increases in volume [unless tissue compliance becomes infinite (29)]. Second, the principle of conservation of mass introduces a derivative (*Eq. 4*) into the equations that must be solved. To deal with these complications, investigators typically develop detailed mathematical models requiring numerical solutions (7, 8, 11, 33, 34, 64, 79). Because results cannot be expressed analytically (i.e., by an algebraic formula), they must be expressed as separate plots of fluid volume as a function of any one of a long list of parameters and variables: microvascular pressure, plasma colloid osmotic pressure, water permeability, interstitial compliance, effective lymphatic resistance, and venous pressure. Each resulting plot is only valid for a particular set of assumed parameters, limiting the generality of the results; each organ system, disease state, and pharmacological intervention must be simulated independently. Furthermore, the inability to view the effect of more than one parameter on fluid volume at any one time makes it impossible to grasp the complex interaction between microvascular filtration and lymphatic return. The purpose of the present work is therefore to present a simple, analytical formulation of edemagenic gain (the change in fluid volume resulting from changes in effective microvascular driving pressure) in terms of microvascular permeability, effective lymphatic resistance, and interstitial compliance.

## METHODS

### Theory

The Starling-Landis Equation (*Eq. 1*) characterizes microvascular fluid filtration, relating the transmicrovascular water flow rate to an effective microvascular driving pressure (46, 58). The difference between microvascular (*P*_{c}) and interstitial (*P*_{int}) hydrostatic pressures tends to force fluid into the interstitium. The difference between microvascular (Π_{c}) and interstitial (Π_{int}) colloid osmotic pressures tends to draw fluid in the opposite direction, from the interstitium into the microvessels. Unlike hydrostatic pressure, which is a force arising only from mechanical properties, colloid osmotic pressure is a force that arises from physicochemical properties of solutions and is proportional to the concentration of plasma proteins. The reflection coefficient (σ) characterizes the relative permeability of the microvasculature to plasma proteins (having a value between 0 and 1) and thus modulates the effective colloid osmotic pressure. The microvascular filtration coefficient (*K*_{f}) determines the amount of transmicrovascular fluid filtration rate (*J*_{V}) that results from the net microvascular driving pressure gradient. (1) The value of *K*_{f} depends on microvascular surface area and permeability to water.

The Drake-Laine Model (*Eq. 2*) characterizes lymphatic function by relating lymph flow rate (*J*_{L}) to an effective lymphatic driving pressure (15). The difference in *P*_{int} and systemic venous pressure (*P*_{sv}) tends to retard lymph flow, since *P*_{sv} is typically greater than *P*_{int}. The effective lymphatic resistance (*R*_{L}) is the slope of the relationship between net effective lymphatic driving pressure and the resulting lymph flow. The value of (*P*_{int}+*P*_{p}) represents the effective lymphatic driving pressure composed of *P*_{int} and lymphatic pumping pressure (*P*_{p}). (2) In this formulation, *P*_{p} and *R*_{L} are empirically derived parameters used to describe the lymphatic pressure-flow relationship and are not necessarily equivalent to pressure developed by lymphatic vessel contraction or resistance to lymph flow (15).

Although the relationship of *P*_{int} and *V* is nonlinear (29, 66), there is no widely accepted empirical model describing this relationship. A simple piecewise linear relationship between *P*_{int} and *V* is therefore assumed (*Eq. 3*). The slope of this relationship is the reciprocal of interstitial compliance (*C*), (3) where *P*_{o} is an empirical constant. Typically, under normal conditions, interstitial fluid pressure is sensitive to changes in interstitial fluid volume (i.e., *C* is small). With overhydration, the compliance can become much larger, and the sensitivity of interstitial fluid pressure to interstitial fluid volume is lost. To capture this behavior, interstitial fluid pressure-volume relationship can be approximated to be “piecewise linear,” a commonly used technique to deal with the effect of hydration on interstitial compliance (7, 11, 33).

Transmicrovascular inflow into the interstitium (*Eq. 1*) and lymph flow out of the interstitium (*Eq. 2*) are both functions of interstitial fluid pressure, and thus, interstitial fluid volume (*Eq. 3*). Assuming conservation of mass, the rate of change of interstitial fluid volume (i.e., *dV*/*dt*) is equal to the difference in rates of interstitial inflow and outflow. (4)

With known values of parameters (*P*_{p}, *C*, *R*_{L}, *K*_{f}) and inlet and outlet pressures (*P*_{c}-σ·Π_{c} and *P*_{sv}), *Eqs. 1*–*4* can be solved simultaneously for four unknown variables (*J*_{V}, *J*_{L}, *P*_{int}, and *V*) (see appendix). In the present approach, variations in interstitial colloid osmotic pressure (Π_{int}) are neglected. Previously published reports were reviewed to characterize the possible range of parameter values and summarized in Table 1.

### Experiment

The experimental preparation has been described previously (25). Experiments were conducted in accordance with protocols approved by the University of Texas Medical School Animal Welfare Committee. Briefly, sheep (*n* = 20) were anesthetized with halothane. Using a sterile technique, we placed polyethylene catheters into the pulmonary artery and left atrium through a left thoracotomy. A 30-cc Foley balloon catheter was introduced into the left atrium, and fluid filled catheters were introduced into the right atrium via either the azygous vein or the left femoral vein. The chest was closed, and the sheep were allowed to recover for ∼1 wk before the acute experiments. Postoperative antibiotics and analgesics were administered by veterinarians as clinically indicated.

This preparation allowed the measurement of pulmonary arterial (PAP) and left atrial pressures (LAP). A pressure control system was used to regulate the size of the left atrial balloon and control LAP in a subgroup of sheep (*n* = 5) (25). Pulmonary *P*_{c} was estimated as the average of PAP and LAP (25). Solid-state pressure transducers, amplifiers, and a chart recorder were used to record all pressures. We chose to use the olecranon as the zero pressure reference level because it is near the level of the left atrium and was easily identified (25). Π_{c} was measured with a membrane osmometer. Sheep were euthanized after a 3-h period, and the extravascular fluid weight/blood free dry weight ratio (EVF) was determined using a modification of the method of Pierce, as described by Gabel et al. (25).

#### Control experiments.

We have previously determined the control relationship between EVF and *P*_{c} − Π_{c} in the lungs of nine anesthetized sheep (25). To confirm the consistency of our experiments, we determined EVF in two additional control sheep. In the first experiment, EVF was 3.9 after a 3-h period with no elevation in microvascular pressure (*P*_{c} − Π_{c} = −12.5), and in the second experiment, EVF was 4.2 with *P*_{c} − Π_{c} = 5 mmHg. These EVFs were consistent with the EVF of 4.0 ± 0.2 obtained in sheep without *P*_{c} elevation, and the EVF of 4.3 ± 0.1 with *P*_{c} − Π_{c} = 5 mmHg. Therefore, we combined the new data with the previously determined data to establish the relationship between control EVF and *P*_{c} − Π_{c}.

#### Histamine experiments.

We infused 4 μg/kg/min of histamine phosphate into a total of nine sheep for a 3-h period. In seven sheep, histamine was infused into the systemic venous circulation, while in two other sheep, it was infused into the pulmonary venous circulation. In five of these experiments, we used the pressure control system to control LAP, so that *P*_{c} was either 0 or 5 mmHg higher than Π_{c} for the 3-h period. In four experiments, we did not elevate LAP. Two protocols for histamine infusion were used to evaluate whether different histamine infusion locations affect EVF with or without LAP elevation.

### Analysis of Previously Reported Data

#### Compliance estimation.

To compare results of histamine infusion to increased systemic venous pressure (SVP), we analyzed data previously reported by Laine et al. (42). Briefly, the effect of systemic venous pressure elevation on lung edema formation was determined by elevating superior vena caval pressure (SVCP) of anesthetized sheep (*n* = 8) by inflating a balloon occluder placed above the level of the azygos vein. Left atrial pressure was then controlled by partially inflating a balloon occluder inserted into the left atrium with and without elevating SVP to 10 mmHg for 3 h. The amount of fluid present in the lung was determined from wet-to-dry weight ratios, similar to the current study.

#### Endotoxin infusion.

Similarly, to compare results of histamine infusion to endotoxin infusion, we analyzed data previously reported by Gabel et al. (25). Briefly, the effect of endotoxin infusion (1 μg/kg) on lung edema formation was determined in anesthetized sheep (*n* = 6). LAP was varied by inflating a balloon occluder placed in left atrium. The amount of fluid present in the lung was determined from wet-to-dry weight ratios, as in the current study. The resulting edemagenic gain for the endotoxin infusion group was compared with histamine and SVCP elevation groups.

## RESULTS

#### Edemagenic gain.

Simultaneously solving *Eqs. 1*–*4* and rearranging (see details in appendix) results in a ratio of Δ*V* to Δ(*P*_{c}-σ·Π_{c}). We term this new equation “Edemagenic Gain” (EG). (5) Edemagenic gain can be expressed in the form of a classical feedback system relating the tendency to store excess interstitial fluid volume to the parameters *C*, *R*_{L}, and *K*_{f}. Edemagenic gain is represented in the form of a “transfer function” relating an input variable, Δ(*P*_{c}-σ·Π_{c}), to an output variable, Δ*V* (Fig. 1*A*). The possible ranges of parameters derived from the literature are listed in Table 1.

#### Compliance-dominated and multivariate gains.

*Equation 5* degenerates into an even simpler form when the combination of *R*_{L}·*K*_{f} is either much greater or less than 1. (6) When *R*_{L}·*K*_{f} is much larger than 1, EG is “compliance-dominated,” and when *R*_{L}·*K*_{f} is much less than 1, EG is “multivariate”. Although *K*_{f} is smaller than 1, *R*_{L} can have fairly large values (see Table 1). Therefore, the value of *R*_{L}·*K*_{f} is larger than 1 when *R*_{L} is elevated, even if *K*_{f} varies markedly. These results are summarized in Fig. 1*B*.

#### Effect of histamine.

Figure 2 illustrates the change in EVF plotted as a function of *P*_{c} − Π_{c} for each experiment. The histamine EVFs were significantly higher than the control EVFs at baseline *P*_{c} − Π_{c} and for *P*_{c} − Π_{c} = 5 mmHg. The location of histamine infusion had no effect on EVF. In the two experiments with histamine infusion into the pulmonary venous circulation, EVFs were 4.7 without LAP elevation and 4.9 with LAP elevation. These EVFs were consistent with EVFs from the experiments with histamine infusion into the systemic venous circulation (with EVFs 4.5 ± 0.2 without LAP elevation and 5.1 ± 0.3 with LAP elevation). The estimate of EG for the control group is 0.018 ml·g^{−1}·mmHg^{−1}, and 0.034 ml/g^{−1}·mmHg^{−1} subsequent to histamine infusion.

##### Compliance estimation.

Figure 3 illustrates the change in EVF resulting from a change in effective microvascular driving pressure in both control and the SVCP elevation groups. When LAP was elevated above control, a greater amount of pulmonary fluid accumulated in animals with elevated SVCP levels than the control group with normal SVCP. The estimate of EG of the SVCP elevation group is 0.069 ml·g^{−1}·mmHg^{−1} and that of the control group is 0.018 ml·g^{−1}·mmHg^{−1}. Assuming a high *R*_{L} (Fig. 1*B*), the *C* of the SVCP elevation group is therefore 0.069 ml·g^{−1}·mmHg^{−1}.

#### Comparison between effects of histamine and endotoxin infusion.

Figure 3 illustrates the change in EVF in response to a change in effective microvascular driving pressure in the control case, as well as with histamine infusion, endotoxin infusion, and SVCP elevation. EG was the lowest in the control case (0.018 ml·g^{−1}·mmHg^{−1}). EG of the endotoxin group (0.112 ml·g^{−1}·mmHg^{−1}) was greater than the histamine (0.04 ml·g^{−1}·mmHg^{−1}) and SVCP elevation groups (0.069 ml·g^{−1}·mmHg^{−1}).

## DISCUSSION

We have developed a new concept, referred to as “edemagenic gain,” relating changes in effective microvascular driving pressure to changes in interstitial fluid volume, and we related edemagenic gain to the three most important phenomena affecting fluid balance-microvascular filtration, lymphatic function, and interstitial fluid storage capacity (*Eq. 5*). By making simplifying assumptions, we were able to derive a simple algebraic formula for edemagenic gain. The resulting first-order approximation was rearranged to separate parameters characterizing structure (i.e., microvascular permeability, effective lymphatic resistance, and interstitial compliance) from input and output variables characterizing function (i.e., change in effective microvascular driving pressure and change in interstitial fluid volume). The resulting formulation presents the solution to classical fluid balance equations in a form recognizable as a classical negative feedback system (Fig. 1*A*). Thus, by accepting the cost of losing a degree of accuracy, we were able to reap the benefit of conceptual clarity that is not currently available from complex numerical simulations.

#### The concept of gain as an integrational approach to interstitial fluid dynamics.

Mathematically modeling the dynamics of interstitial fluid balance has had a long history, (7, 8, 11, 30, 33, 34, 64, 79) and resulted in concepts such as an edema “safety factor” that has played a significant role in our current understanding of edema formation (31). Typically, these approaches focused solely on hemodilution or hemorrhage, and the role of vascular permeability. Given how often the concepts of vascular permeability and interstitial compliance are invoked, it is perhaps overlooked that *K*_{f} is a “black box” parameter relating hydrostatic and colloid osmotic pressure gradients to microvascular filtration and that *C* is a black box parameter relating interstitial fluid volume to pressure. The black box approach to interstitial fluid balance has further been refined *K*_{f}, for instance, has been further reduced into smaller black boxes, such as “hydraulic conductivity” and “surface area” (50, 51). The introduction of molecular biology analytical techniques has provided further opportunities to provide mechanistic interpretations, especially in the understanding of changes in interstitial compliance (63, 67). Although this process of reductionism has led to a fundamental new understanding of the individual physiological processes involved in edema formation and resolution, it has been unclear how these mechanistic elements interact with each other and thus affect interstitial fluid volume as a global variable, the necessary step to bridge basic science to clinically relevant research. The approach used in the present work therefore provides an integrational approach that complements reductionist approaches, providing the means to *1*) predict the implications of changes in any of the currently identified mechanistic elements on interstitial fluid volume, *2*) determine the dominant mechanisms that have the greatest impact on interstitial fluid volume, and *3*) identify which parameters must first be determined before ascribing edema formation to any one process. Our integrational approach is complementary to mechanistic studies, and in the present case, we have at least identified what remains unknown and what requires further study.

#### Necessary components to describe a physiological process as a gain.

To derive an algebraic formula that characterizes a complex physiological process in terms of a constant “gain,” it is necessary to formulate the system as *1*) linear, *2*) first-order, and *3*) time-invariant (23, 41). Even though most physiological processes are highly nonlinear, approximating them as linear is permissible when changes in the variables of interest are small. When variables change sufficiently to make nonlinear effects significant, it is common practice to treat the system as “piecewise linear.” For example, the interstitial fluid pressure-volume relationship can be approximated as a combination of two linear relationships corresponding to two different levels of hydration (7, 11, 33). Similarly, although most physiological processes can be overwhelmingly complex, approximating them as first-order (i.e., eliminating secondary effects) is permissible when the secondary effects are small. For instance, to arrive at the relationship between lymphatic driving pressure and lymph flow in Drake-Laine formulation (15), reabsorption of the lymphatic fluid back into circulation in lymph nodes is neglected. The final criterion, time-invariance, requires that all parameters, except the input and output variables (i.e., interstitial fluid volume and microvascular pressure), have constant values. That is, parameters such as *K*_{f}, *C*, *R*_{L}, and σ are relatively constant and do not vary appreciably from moment to moment. If any particular intervention alters one of these parameters, then the gain is said to have changed. Taken together, the assumptions that the system is linear, first-order, and time-invariant make it possible to avoid using a highly complex convection-diffusion transport model, which must be solved numerically. Although numerical solutions can include many relevant details (7, 8, 11, 33, 34, 64, 79), they cannot be expressed as a simple algebraic formula. In fact, such solutions can only be plotted graphically. Plotting volume as a function of just four parameters (*K*_{f}, *C*, *R*_{L}, and σ) cannot be accomplished in a single graph. Numerical solutions thus suffer from a loss of conceptual clarity, since they lack an explicit relationship of cause and effect and require the knowledge of numerous variables that cannot be retrieved experimentally.

#### Reinterpreting the lymphatic “effective resistance”.

The Drake-Laine lymphatic system model characterizes the lymphatic flow as a function of two lumped parameters: one describing the slope of the pressure-flow relationship and one describing the intercept. Using a convenient analogy to an electric circuit, we refer to these quantities as “effective lymphatic resistance (*R*_{L})” and “lymphatic pump pressure (*P*_{p})”. Values for these parameters have not yet been predicted from first principles but are estimated from linear regression of measured data. They are therefore empirical (descriptive) parameters (15). To date, *R*_{L} and *P*_{p} have been used to describe lymphatic function in a number of organs (16, 18, 20, 43, 45) but have yet to be assigned particular physiological interpretations. Much like *K*_{f}, a black box parameter relating hydrostatic and colloid osmotic pressure gradients to microvascular filtration and *C*, a black box parameter relating interstitial fluid volume to pressure, *R*_{L} and *P*_{p} are similar black box parameters, which relate lymphatic outflow to interstitial fluid pressure. Since transserosal flow, flow across the serous membrane, which encloses several organs, is ultimately collected by lymphatics and returned to systemic circulation, transserosal flow is accommodated by the Drake-Laine model. The historical use of the term “effective resistance” to describe the empirically derived slope of the line of the pressure gradient-flow relationship is unfortunate and has caused some consternation, since this empirical relationship clearly arises from an active pumping of lymph, rather than a purely passive process as implied by the term “resistance” (3).

#### Edemagenic gain suggests a novel structure-based classification of edema.

The equation used in the present work to approximate edemagenic gain (*Eq. 5*) assumes an even simpler form (*Eq. 6*) when specific parameters are either very large or small. In the first case, when effective lymphatic resistance is significantly elevated, *R*_{L}·*K*_{F} can become much larger than 1. The value of 1+*R*_{L}·*K*_{F} thus becomes approximately equal to *R*_{L}·*K*_{F}, and EG can be approximated by *C*. Under such conditions, *C* alone determines edemagenic gain, and thus the susceptibility to edema, irrespective of the particular value of the microvascular filtration coefficient (Fig. 1*B*). In other words, edemagenic gain becomes compliance dominated. In most other cases, however, *R*_{L} is relatively small, including edema caused by enhanced microvascular permeability. Edemagenic gain then depends on the combination of *C*, *R*_{L}, and *K*_{f} (Fig. 1*B*). In this case, edemagenic gain is multivariate. This analysis has, therefore, revealed two types of gain, compliance-dominated gain, approximately equal to *C* and multivariate gain, dependent on *C, R*_{L}, and *K*_{f}. In both cases, *C* plays a significant role in edema formation. This structure-based classification of edema provides a novel approach to estimate *C* on one hand and a novel insight into the causes of edema on the other.

#### Edemagenic gain provides a novel approach to estimate interstitial compliance.

Conventional techniques to estimate *C* require recording changes in both *V* and *P*_{int}. Whereas methods to measure changes in tissue fluid volume are fairly accurate and stable, measuring *P*_{int} is problematic. Wiig et al. (72–75) reported significant deviation when comparing *P*_{int} measurements derived from micropipette techniques, chronic perforated and porous capsule techniques, and wick methods. The concept of edemagenic gain provides an alternative approach that eliminates the need to measure interstitial fluid pressure. By increasing *R*_{L} (or alternatively increasing lymphatic outflow pressure), edemagenic gain becomes equal to *C* (*Eq. 6*). Only changes in interstitial fluid volume and microvascular hydrostatic pressure therefore must be measured to estimate *C*. This method can be particularly useful to determine chronic changes in tissue compliance, especially in the myocardium and lungs, where interstitial fluid pressure measurement is confounded by tissue motion (54). Indeed, because the edema in the SVCP elevation group described in Fig. 3 was caused by an intervention that decreased lymphatic flow (comparable to impaired outflow edema), edemagenic gain is expected to depend only on the *C* (Fig. 1*B*). *C* estimated using our novel method (*C* = 0.069 ml·g^{−1}·mmHg^{−1}) is within the range previously estimated by conventional methods (Table 1).

#### Edemagenic gain provides a novel insight into the causes of edema.

Determining how a physiological intervention causes edema is particularly difficult, because it is not possible to directly measure microvascular permeability, effective lymphatic resistance, and interstitial compliance in a single experiment. The present work uses the example of histamine-induced edema to illustrate how the concept of edemagenic gain can reconcile disparate reports. First, several reports suggest that increases in the microvascular filtration coefficient are responsible for histamine-induced edema (6, 40, 49, 56, 60). Second, inhibition of lymph flow by histamine (22, 62) suggests changes in lymphatic function could be responsible for histamine-induced edema. Third, interstitial fluid pressure measurement studies have demonstrated that interstital compliance increases significantly after histamine infusion, suggesting changes in compliance alone may be responsible for histamine-induced edema, as it is for edema induced by other inflammatory agents (2, 4, 5, 27, 37–39, 57, 65, 69, 77). From an experimental standpoint, our results (Fig. 2, slope) clearly indicate that histamine increases edemagenic gain by approximately a factor of 2. However, our theoretical formulation (*Eq. 5*) suggests that all three parameters *C*, *R*_{L}, and *K*_{f} may determine the extent of edema formation. In this case, there is not enough information contained in the data to exclude any of the three postulated edemagenic mechanisms.

#### Active regulation of C, R_{L}, and K_{f}.

Until recently, *C* and *R*_{L} were considered as passive properties (29, 31). Recent studies, however, have suggested that these properties are actively regulated. *R*_{L}, for instance, has been shown to be a regulated property, since it is affected by several neurohormonal factors that modulate lymphatic function (21, 22, 35, 36, 62, 70). Furthermore, recent studies have suggested that interstitial compliance is not only passively affected by the level of hydration but also by modulation of β1-integrin adhesions (39, 63, 67, 76, 78). The present work presents a framework to study the effect of acute to chronic regulation of the structural properties *C, R*_{L}, and *K*_{f}. Not only can the edemagenic gain be used to identify changes in critical properties that produce edema, such as permeability, it can conversely identify structural changes resulting from edema, such as changes in interstitial compliance following fibrosis.

#### Generalizing edemagenic gain: incorporating changes in σ and Π_{int}.

In the formulation of EG given by *Eq. 5*, only three parameters, *C*, *R*_{L}, and *K*_{f}, determine the relationship between the change in effective microvascular driving pressure and the change in fluid volume. In this particular formulation of EG, there are two constants: σ, which characterizes the relative microvascular permeability to proteins, and Π_{int}. Changes in the σ, as well as Π_{int}, however, can affect interstitial fluid volume. The values of σ and Π_{int} were not determined in the present study, although there is evidence that histamine and endotoxin decrease σ and may increase Π_{int} (10, 25, 49, 60). To take advantage of measurements of σ and Π_{int} in future studies, EG could be reformulated (see appendix), such that σ and Π_{int} are variables (*Eq. A6*). In this case, the EG is expressed as ΔV/Δ[*P*_{c}-σ·(Π_{c} − Π_{int})]. Not only can this more general formulation incorporate changes in σ or Π_{int}, it can capture such phenomena as “protein washdown.” This more general formulation degenerates into ΔV/Δ(*P*_{c} − σ·Π_{c}), as in *Eq. 5*, when variables σ and Π_{int} are constant. If all the parameters change (e.g., *P*_{c}, Π_{c}, Π_{int}, and σ) with a particular intervention, then they would all need to be independently measured. The measurement of σ and Π_{int} would allow more accurate estimation of the edemagenic gain. The process of simplification used in the present work could therefore be used as a guide for investigators developing new approximations of edemagenic gain. Caution is advised, however; a previous attempt to relate interstitial volume to effective microvascular driving pressures (25) oversimplified the equations, and the critical contribution of interstitial compliance was neglected (see appendix).

#### Extrapolation of edemagenic gain estimations to different organs.

The EG formulation is general—no assumptions of a specific organ or parameter values were necessary to define EG in *Eq. 5*. To use the EG formulation for a specific organ, the model should be evaluated and validated with particular attention to the specific assumptions made for the organ under study. In the present study, for instance, values of *C*, *R*_{L}, and *K*_{f} listed in Table 1 are used to characterize interstitial fluid balance in lung. The lung model has been well studied over the last four decades, and the measurements for all three parameters (*C*, *R*_{L}, and *K*_{f}) have been determined (Table 1). A dilemma may arise, however, when extrapolating the outcomes to other organ systems, especially those for which the measurements of the parameters are not available. In this case, either the unknown parameters must be determined before predicting the edemagenic gain or the ranges of the permissible values of the parameters should be determined to estimate the range of edemagenic gain that can result. The sensitivity analysis presented in Table 2 provides some guidance to determine how critical a particular parameter value is. In some specific cases (such as when the edemagenic gain is compliance dominated), the particular values of parameters (such as *R*_{L} and *K*_{f}) have a relatively small influence. Extrapolation of the present model implementation to other organ systems must be evaluated on a case-by-case basis.

#### Inflammatory agents have disparate effects on edemagenic gain.

The concept of edemagenic gain can be used to characterize the extent to which specific inflammatory agents can sensitize tissues to edemagenic conditions. Both histamine and endotoxin are proinflammatory agents that can result in similar degrees of edema. However, pulmonary edema induced by endotoxin does not regress, even after significant reduction in effective lymphatic resistance (44). Analysis presented in Fig. 3 provides a potential interpretation. The endotoxin group has a larger edemagenic gain (0.112 ml·g^{−1}·mmHg^{−1}) than either the histamine group (0.034 ml·g^{−1}·mmHg^{−1}) or the superior vena caval pressure elevation group (0.069 ml·g^{−1}·mmHg^{−1}). The endotoxin group is therefore approximately twice as sensitive to changes in effective microvascular driving pressure. Although these results are likely dose dependent, this particular example suggests that endotoxin-induced edema is potentially more serious, less stable, and more likely to present as acutely fulminating edema. The concept of edemagenic gain does not provide a measure of edema but instead provides an index of an organ's susceptibility to edema formation.

## APPENDIX

#### Derivation of edemagenic gain.

In this formulation, *P*_{int}, *J*_{V}, and *J*_{L} are considered to be functions of *V*. The microvascular σ is considered to be constant, and the effect of changes in microvascular filtration on interstitial colloid osmotic pressure is neglected. A piecewise linear relationship between *P*_{int} and *V* is assumed (*Eq. 3*). *C*, reciprocal of the slope of this relationship, determines the effect of change in *V* on *P*_{int}. Substituting this relationship (*Eq. 3*) into the Starling-Landis Equation (*Eq. 1*) indicates how transmicrovascular flow is affected by changes in interstitial fluid volume. (A1) By substituting *Eq. A1* and the Drake-Laine Model (*Eq. 3*), into *Eq. 4*, the effect of changes in interstitial fluid volume on lymphatic flow can be reformulated into an integral, (A2) where *V*_{o} is the initial volume.

Integrating *Eq. A2* results in interstitial fluid volume as an analytical equation that includes a function of time (*t*). (A3) Assuming all parameters are constant, the difference between two volumes is obtained as a result of changes in effective microvascular driving pressure (*P*_{c}-σ·Π_{c}) and expressed as EG (*Eq. 5*)

In previous attempts to characterize the change in *V* in response to a change in the effective microvascular driving pressure (25), *P*_{int} was considered constant. The negative feedback provided by changes in *P*_{int}, therefore, was neglected. The change in interstitial fluid volume given by the time integral of the inflow and outflow rate (*Eq. 4*) would thus be (A4) The relation between change in fluid volume and change in effective microvascular driving pressure is obtained from *Eq. A4*. (A5) The formulation of EG presented in the current work (*Eq. 5*) is preferred over *Eq. A5*, because *Eq. 5* does not require the assumption that *P*_{int} is constant, and thus it requires fewer assumptions.

The formulation of EG represented by *Eq. 5* can be modified to characterize the relationship of Δ*V* to changes in the microvascular driving pressure determined by *P*_{c}, Π_{c}, Π_{int}, and σ. The modified edemagenic gain thus becomes a function of σ as well as Π_{int}. (A6)

## Acknowledgments

Portions of this work were supported by National Institutes of Health Grants K25HL070608 (to C. M. Quick), AHA-0565116Y (C. M. Quick), AHA-0365127Y (R. H. Stewart), CDC-620069 (G. A. Laine), and CDC-623086 (G. A. Laine).

## Footnotes

↵† Deceased October 23, 2006.

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