## Abstract

A four-region (capillary plasma, endothelium, interstitial fluid, cell) multipath model was configured to describe the kinetics of blood-tissue exchange for small solutes in the lung, accounting for regional flow heterogeneity, permeation of cell membranes and through interendothelial clefts, and intracellular reactions. Serotonin uptake data from the Multiple indicator dilution “bolus sweep” experiments of Rickaby and coworkers (Rickaby DA, Linehan JH, Bronikowski TA, Dawson CA. *J Appl Physiol* 51: 405–414, 1981; Rickaby DA, Dawson CA, and Linehan JH. *J Appl Physiol* 56: 1170–1177, 1984) and Malcorps et al. (Malcorps CM, Dawson CA, Linehan JH, Bronikowski TA, Rickaby DA, Herman AG, Will JA. *J Appl Physiol* 57: 720–730, 1984) were analyzed to distinguish facilitated transport into the endothelial cells (EC) and the inhibition of tracer transport by nontracer serotonin in the bolus of injectate from the free uninhibited permeation through the clefts into the interstitial fluid space. The permeability-surface area products (PS) for serotonin via the inter-EC clefts were ∼0.3 ml·g^{−1}·min^{−1}, low compared with the transporter-mediated maximum PS of 13 ml·g^{−1}·min^{−1} (with *K*_{m} = ∼0.3 μM and *V*_{max} = ∼4 nmol·g^{−1}·min^{−1}). The estimates of serotonin PS values for EC transporters from their multiple data sets were similar and were influenced only modestly by accounting for the cleft permeability in parallel. The cleft PS estimates in these Ringer-perfused lungs are less than half of those for anesthetized dogs (Yipintsoi T. *Circ Res* 39: 523–531, 1976) with normal hematocrits, but are compatible with passive noncarrier-mediated transport observed later in the same laboratory (Dawson CA, Linehan JH, Rickaby DA, Bronikowski TA. *Ann Biomed Eng* 15: 217–227, 1987; Peeters FAM, Bronikowski TA, Dawson CA, Linehan JH, Bult H, Herman AG. *J Appl Physiol* 66: 2328–2337, 1989) The identification and quantitation of the cleft pathway conductance from these studies affirms the importance of the cleft permeation.

- serotonin
- endothelium
- transport modeling
- pulmonary capillary permeability
- convection-diffusion
- capillary-tissue exchange
- hydroxytryptamine receptors

there is continued widespread interest in vasoactive substrates and how they are removed or modified by vascular endothelial cells (EC). An important step in understanding this process is describing the kinetics of the uptake and diffusion of these substrates out of the capillary blood to their sites of action. A vasoactive substrate of exemplary interest is serotonin (5-hydroxytryptamine). It is involved in the constriction-dilation of pulmonary vasculature as well as in promoting platelet aggregation and blood clotting (51). Serotonin uptake into platelets and endothelial cells (with platelets the main reaction pathway) is used to control plasma serotonin levels and changes in *V*_{max} for platelet uptake is linked to hypertension (8). Pulmonary EC uptake of serotonin may be reflective of the health of endothelial barriers in general and could give early evidence of pulmonary disorders (18). This makes an accurate physiological model to assist investigators in understanding the role of serotonin in the lungs essential.

An important aspect of this study is that 100% of the transport of hydrophilic solutes lacking facilitating transporters (sucrose, amino acids, etc.) is via the clefts to reach the parenchymal cells of the organ. The clefts are vital for glucose passage in the heart and probably in the lung, since endothelial uptake is exceedingly small and virtually unmeasurable by the indicator dilution method, e.g., by comparing d- vs. l-glucose (untransported) (29), and therefore must be accounted for.

Serotonin is an intermediate product of tryptophan metabolism in the brain, blood platelets, and kidneys (8, 51). Plasma serotonin must either traverse ECs or diffuse through interendothelial clefts to get to the parenchymal cells. The transport of serotonin into the pulmonary endothelium occurs through a saturable transporter that can be described by first order nonlinear Michaelis-Menten kinetics (10, 41). Sucrose, a hydrophilic solute that cannot cross cell membranes and has molecular weight similar to serotonin, permeates the clefts in the lung, heart, and liver (35, 14, 22). In the myocardium transcapillary serotonin exchange is mainly via the clefts (34). Structural studies of the pulmonary endothelium show that most of the periendothelial cleft length are closed by tight interendothelial gap junctions with only a small fraction being open (11, 20). For a normal lung then transport through the endothelium is the primary path by which serotonin reaches pulmonary parenchymal cells such as the epithelium and pulmonary smooth muscle cells.

But what about an abnormal lung? With the availability of in vivo multiple indicator dilution analysis and in vitro recombinant transporter protein experiments (47) that characterize the receptor and transporter kinetics, it is possible to look in more detail at pulmonary serotonin kinetics. With the proper study it could be possible to characterize the serotonin receptor binding characteristics on the pulmonary ECs and beyond to the epithelial and smooth muscle cells. This detail of understanding may be important when looking at lung pathologies such as hyperoxia and edema (44, 38), which can increase EC clefts, inflammation where thrombin can cause increases in inter-EC clefts through increasing cytosolic Ca^{2+} concentrations (13), microembolism, and subsequent inflammation, which can block EC transporters (21), pulmonary hypertension (32) and the long term effects of antide pressants on the lungs (47). None of these pathologies can be studied by existing two-region models of pulmonary vascular uptake because they all involve changes in the endothelium that may enlarge interendothelial gaps or inhibit serotonin uptake into the endothelium and do not differentiate between the kinetics of the endothelium and of the surrounding parenchymal cells.

The pioneering analyses of Dawson and colleagues (10, 30, 31, 39, 41) on pulmonary endothelial serotonin uptake kinetics used a two-region blood-tissue exchange model accounting for intracapillary gradients using partial differential equations but did not account for the diffusion of serotonin through clefts into the interstitial space [interstitial fluid (ISF)]. Peeters et al. (39) modeled uptake by two separate carrier-mediated transporters on the luminal side of the EC. The analysis showed that they could not distinguish the possibility that both were saturable processes compared with the possibility that one was saturable and the other was passive. However, they showed that imipramine in high concentrations reduced the total extraction to ∼4% (range 2 to 8%) in their Fig. 3 and chose to consider this extraction so small that cleft permeation was effectively zero. They reported that the secondary transporter, representing passive diffusion, represented, at most, ∼10% of the total uninhibited uptake of serotonin (Table 2 in Ref. 39). Our analysis is based on the complementary assumption, namely there is only one transporter on the luminal endothelial membrane and there is in parallel to it a purely diffusional permeation paracellular route, the cleft.

Here we propose to address these issues quantitatively, using a four-region blood tissue model (Fig. 1, with nomenclature defined in Table 1), with vascular heterogeneity, to describe the kinetic interactions between the plasma, pulmonary EC, ISF, and parenchymal cells (PC). It is adapted from the linear exchange model developed by Bassingthwaighte et al. (5) by incorporating saturable Michaelis-Menten transporters and enzyme-facilitated reactions for endothelial and parenchymal cells. There are three bidirectional saturable transmembrane transporters, in the luminal endothelial plasmalemma with permeability surface area product PS_{ecl}, the abluminal endothelial plasmalemma, PS_{eca}, and across the plasmalemma of the parenchymal cell, PS_{pc}, which is in this setting the pulmonary alveolar cell or binding to smooth muscle receptor sites. The model has saturable consumption in endothelial and parenchymal cells, G_{ec} and G_{pc}, and a passive diffusive exchange of serotonin, PS_{g}, via the interendothelial clefts connecting the capillary plasma to the interstitial space. The model is further adapted to account for tracer and nontracer serotonin: these species compete for transporters and enzymes, through concentration-dependent nonlinear competitive binding. This model describes tracer and nontracer serotonin uptake and diffusion over varying serotonin concentrations and handles the competitive transporter fluxes and competitive enzymatic reactions at both steady and during transients in both tracer and nontracer serotonin.

## MATERIALS AND METHODS

### Blood Tissue Exchange Model and Equations

Figure 1 diagrams the axially distributed four-region model; terms are in Table 1. The four regions are described by four partial differential equations over the spatial domain *x* extending from *x* = 0 at the capillary entrance to its exit at *x* = L, the capillary length:

Capillary:

Endothelium:

Interstitial:

Parenchymal:

The subscripts p, ec, isf, pc denote the capillary, EC, ISF, and PC regions, respectively. C_{n}, are the concentrations (mol/ml) of the substrate in the *n* region, dependent on axial position *x* and time *t*. *PS indicates saturable transporters (*Eq. 6*). F_{p} is plasma flow (in ml·g^{−1}·min^{−1}), V_{n} are volumes of distribution (ml/g) within region *n* (in this paper V and “volume” represent volumes of distribution for the substrate in question). A volume of distribution is defined as the sum of the volume in which the free solute resides, e.g., the water space, plus any equivalent volume due to the concentration of any bound substrate within the volume equilibrated with the concentration of free substrate. For this model there are no binding sites so the volume of distribution for serotonin is just the water space of the region. The anatomical volume of plasma water is the reference intravascular volume of distribution, V_{p}, and the others, V_{ec}, V_{isf}, and V_{pc} volumes of distribution, are estimated by fitting experimental data, constrained by the prior estimates obtained by direct techniques [sheep lungs: (46), dog lungs: (33), rabbit hearts: (23)]. D_{n} is the effective axial diffusion coefficient (cm^{2}/s) within region *n*. G_{ec} and G_{pc} are consumption terms (in ml·g^{−1}·min^{−1}) for the EC and PC (defined in *Eq. 7*). PS_{g} is the diffusive permeability-surface area term used to describe the diffusion between the plasma and the ISF and is not saturable. The permeability-surface area products (PS) PS_{ecl}, PS_{eca}, and PS_{pc} are the conductances of saturable transporters (in ml·g^{−1}·min^{−1}) for substrate transport between plasma-EC, EC-ISF, and ISF-PC, respectively. The boundary condition for the capillary entrance is

At *x* = 0: (−F·*L*/V)·(C−C_{in})+D·∂C/d*x* = 0, for all solutes, thus balancing the convective inflow and the diffusive backflow at the entrance. At the capillary exit at *x* = *L*, and at the other regions' upstream and downstream boundaries at *x* = 0 and *x* = *L*, for all solutes: D·∂C/d*x* = 0. The initial condition at *t* = 0 in the capillary is: if *x* = 0, C = C_{in} + C_{init} else C = C_{init}, where C_{init} is the initial concentration within the region (capillary). The intravascular transport of tracer-labeled albumin uses *Eq. 1* with PS set to zero.

The instantaneous extraction, E(*t*), of a solute (serotonin) is given by
*t*) and h(*t*)_{ref} are the transport functions of labeled serotonin and intravascular reference, respectively, C(*t*)_{p} is the capillary concentration of the labeled serotonin, and C(*t*)_{r} is the capillary concentration of the intravascular reference, Dose_{r} is the intravascular reference dose, and Dose_{t} is the dosage of the labeled serotonin injected into the lung.

### Saturable Serotonin Transporters

Endothelial uptake of serotonin is modeled, following Rickaby et al. (41), using Michaelis-Menten parameters *V*_{max} (nmol·g^{−1}·min^{−1}) and Michaelis constant *K*_{m} (μM) defining the conductance PS_{ecl} (ml·g^{−1}·min^{−1}), across the luminal surface of the EC
*C*_{p} is the total concentration of serotonin in the capillary plasma. PS_{ecl} and PS_{eca} are assumed identical in *V*_{max} and *K*_{m}, but the governing concentration for PS_{eca} is C_{isf}. We modeled the effect of imipramine (C_{i}) on serotonin transport across the endothelial membrane by use of a simple, fully competitive inhibitor model of the binding of imipramine to the serotonin transporter. The dissociation constant for the inhibitor reaction, *K*_{i}, is the ratio of the Off over the On rates of imipramine binding to the serotonin transporter (19). The transporter into the PC from the ISF [PS_{pc}(*t*,*x*)] is assumed to be identical, kinetically, to the EC transporter, except for the absence of the imipramine inhibitor term, since the concentration of imipramine in the ISF is assumed to be zero. The use of a simplified transporter is based on two assumptions: the serotonin binding and unbinding reactions with the membrane transporter are fast compared with the movement of the serotonin-transporter complex across the membrane, and the transporter concentration is small compared with that of the substrate. The latter assumption means that this is a noncapacitive transporter. Neither assumption is exactly true, and in tracer-only experiments one should probably consider that the transporter binds some serotonin (40).

### Serotonin Consumption

Consumption of serotonin within the EC is assumed to follow Michaelis-Menten enzyme kinetics (5) and so the consumption term is similar to *Eq. 6*
_{ec}(*t*,*x*) is the rate of consumption in the EC. Consumption in the PC, G_{pc}, is assumed to behave, kinetically, the same as in the EC. This assumes no influence of imipramine on G_{ec}.

### Endothelial Cleft Diffusion

Diffusion through interendothelial clefts is by passive diffusion (25) with little steric hindrance for a molecule the size of serotonin (6, 16). Serotonin permeates myocardial endothelial clefts in rabbit hearts (34) with a mean PS_{g} of 2.16 ± 0.93 ml·min^{−1}·g^{−1}. Pulmonary smooth muscle cells are activated by intra-arterial serotonin so it is clear that it crosses the capillary wall. Imipramine, an endothelial serotonin transport blocker, diminished the transpulmonary instantaneous extraction of serotonin by ∼90% (18, 43). This result implies that up to 10% of the serotonin traverses the clefts, even though their experiments lacked an extracellular reference to measure cleft permeability explicitly. The studies of Yipintsoi (52) and of Tancredi and Yipintsoi (50) demonstrated that cleft permeability to small hydrophilic solutes was moderately high, with PS_{g} values of 1 to 5 ml·min^{−1}·g^{−1} for sodium, and also with clear evidence of recruitment of PS_{g} with increasing flow and pressure, in accord with the recruitment of extravascular water at increasing pressures and flows (49).

For the pulmonary endothelium, sucrose, a hydrophilic molecule, provides evidence of diffusion through the clefts (9, 35). Sucrose has a molecular weight of 342 compared with 176 for serotonin and so we expect, from the square root of the ratio of molecular weights, serotonin PS_{g} to be ∼1.4 times sucrose permeability (PS_{gsuc}) (34). In an anesthetized dog lung, Neufeld et al. (35) found that PS_{gsuc} was 2.1 ± 0.6 ml/s at a plasma flow of 10.5 ± 1.6 ml/s, a PS-to-flow ratio of ∼0.2. In human lungs Neufeld et al. (35) observed a PS-to-flow ratio of 4.1 ± 1.9 ml/s to 29.9 ± 1.9 ml/s or ∼0.14, which is in the same range. In the data used here (Table 2), the flows ranged from 2.3 to 3.9 ml·g^{−1}·min^{−1}. If the ratios are the same in the data we analyze here, PS_{g} will be ∼0.5–0.8 ml·g^{−1}·min^{−1}. For our analysis we will fix the PS_{g} value at 0.5 ml·g^{−1}·min^{−1} and fit to the data followed by allowing PS_{g} to be free. Fixing PS_{g} allows for better estimates of the EC transporter parameters because there is one less free parameter to optimize against and will generate narrower estimated confidence intervals and allow us to realistically maximize the effect of cleft diffusion.

### Four-Region Model Variables and Parameters

For fitting our model to serotonin outflow data we use three sets of concentration curves: Mother CM(*t*,*x*), Tracer CT(*t*,*x*), and Intravascular reference CR(*t*,*x*). “Mother” is the unlabeled serotonin; Tracer is labeled serotonin at trace levels (∼10^{−4} times mother). The intravascular reference concentration curve is described by a version of *Eq. 2* without permeation:

The intravascular reference is assumed to not permeate either the ISF or the EC region. Since the intravascular reference is injected simultaneously with the mother and tracer serotonin, its outflow curve is used to represent the dispersion and delay of the serotonin in the vascular space and thereby serve as a reference to determine the intraorgan serotonin uptake. The input functions were not recorded, so the outflow curve was also used to reconstruct the input concentration-time curve by a forward approach instead of a deconvolution. The experimenter's original analyses had used Bronikowski's deconvolution method (10) as we have generally done (45), but as this is essentially a derivative technique and is not inherently stable, we chose a less complex, more stable method. A long-tailed input function composed of a lagged normal density function terminated in a multiexponential tail was used for the Rickaby 1981 and 1984 analyses (41, 43) and a pulse input function was used for the Malcorps analysis (31) to represent the putative input function. Automated optimization to adjust the parameters of the input function was used to minimize the differences between the intravascular outflow curve to the observed intravascular reference (albumin) curve. This was then used to drive the model for each injection.

### Flow and Dispersion

Flow is modeled as constant velocity in the axial direction in the capillary. The EC, ISF, and PC spaces have no flow but have axial diffusion. The serotonin plasma diffusion term, D_{p}, set for D_{p} = 1 × 10^{−4} cm^{2}/s (17), is a combination of molecular diffusion, axial velocity distribution, flow disturbances by branching, etc. (40). The values for D_{ec}, D_{isf}, and D_{pc} were set to 1 × 10^{−6} cm^{2}/s for this model. Radial diffusion is assumed to be instantaneous within each region since the radial distances are 10 μm or less (4); i.e., each extravascular region is considered to be radially “well stirred” but has axial gradients.

### Flow Heterogeneity

The modeling of pulmonary capillary flow heterogeneity was based on a probability density function (PDF) of relative regional flows, *w*(f), represented as seven discrete flows, with a mean of 1.0 and area 1.0. Seven paths were chosen to shorten computational time but still give a relatively smooth transport function. Using more than seven paths for the analysis adds little information since each of these has a dispersed output function and their sum covers the range provided by a larger number of paths (27, 28). Each path consisted of an arteriole, a capillary, and a venule, as described by King et al. (27) and by Audi et al. (2):
*w*_{i} is the frequency of occurrence, the fraction of the mass of the organ with flow in the range Δ*f*_{i} about the relative flow *f*_{i}. Based on transpulmonary relative dispersion (RD) observed by Knopp et al. (28), a random walk PDF was used with a RD of ∼0.5 (varied slightly with different data sets) and skewness of 1.6. The relative flows ranged between 0.5 and 2.0 times the mean flow. (The RD of a PDF is its coefficient of variation.) We used equal widths, Δ*f*_{i}, to cover the range.

The heterogeneity in arterioles and venules was set identically to that in the capillaries, by using the same PDF, but the estimates of vascular volumes depended on which of the three data sets was being analyzed. Arteriolar volumes were 0.04 to 0.05 ml/g and venular volumes were ∼50% larger (Table 2). The arterioles and venules are impermeable “pipes” in that there is no uptake or metabolism of serotonin, only dispersion and delay between input and output.

### Two-Region Model

The two-region model described by Linehan et al. (30) is a simplification of that of Peeters et al. (39) consisting of three components that describe the serotonin kinetics between the capillary and the extravascular space: a one-way facilitated transporter from capillary into an extravascular space, PS_{xv}, a passive backdiffusion term from extravascular space to capillary, PS_{ev}, and an extravascular linear constant consumption term, G_{xv}. Vascular axial dispersion, D_{p}, was set to zero, to match their usage. Capillary heterogeneity is accounted for by relating a distribution of capillary transit times to points on a capillary with the longest transit time, thereby simplifying the number of equations needed to solve the system. The equations used for the two-region model are

Capillary region (p):

Extravascular region (XV):
*Eq. 11* with PS_{ev} = PS_{xv} = 0.

PS_{ev} (ml·g^{−1}·min^{−1}) is the backdiffusion into the capillary, and G_{xv} (ml·g^{−1}·min^{−1}) is the extravascular linear consumption term. PS_{xv}(*x*,*t*) is the apparent PS for the transport of serotonin from the capillary to the extravascular space with Michaelis-Menten kinetic parameters *V*_{max_xv} and *K*m_{xv}.

Intraorgan flow heterogeneity was accounted for by Audi et al.'s (2) transit time variant on King et al.'s (27) model for the PDF of regional flows.

### Experimental Conditions

The experimental data used to analyze the model came from Rickaby et al. (41, 43), and Malcorps et al. (31). These experiments consist of perfusing the dog lung with plasma, injecting a bolus containing a vascular reference tracer along with the tracer-labeled serotonin. Unlabeled serotonin (mother) is either constantly perfused, or injected along with the tracers; this latter approach is known as the bolus sweep method, because the tracer competes with mother solute for the transporters as the mother substance concentrations sweep through a large range to inhibit the uptake. The inhibition is small at the front end of the bolus, reaches a peak, and diminishes. The modeling has to account for tracer and nontracer serotonin separately as the degree of transporter saturation changes the fluxes.

### Computational Methods

The models were developed under JSim, a Java-based simulation environment for building and analyzing numeric models (7). The PDE solver used was the McCormack solver. Extensive on-line manuals and methods used in JSim are at http://www.physiome.org/jsim.

The three sets of equations representing the three serotonin dosage levels from each set of experimental data were fitted simultaneously with the model solutions. Optimization of kinetic parameters (*V*_{max_ecl}, *K*m_{ecl}, PS_{g}, Gec_{max}, *K*m_{Gec}) was done through minimizing the root mean square error (RMS) of the differences between model solution and the data at each time point. The differences were uniformly weighted; from the RMS error the coefficient of variation was calculated for comparison with previously published analyses of the same data (30). Optimizations were done by using the JSim's Simplex and SENSOP optimization routines (12). SENSOP was used first to get a visual goodness of fit to data and Simplex was then employed to fine tune the optimization. Parameter confidence limits and their interdependences (correlation) were evaluated by using the normalized covariance matrix of the parameters. The models used in this paper are available at http://www.physiome.org/jsim/models/webmodel/NSR/JardineSerotonin_FourRegion/.

Sensitivity analysis of model parameters was accomplished by making a finite differences approximation to the rate of change of the model with respect to each parameter. If Cout(*t*) represents a model function dependent on *N* parameters, *p*_{i}, then the sensitivity function S_{pi}(*t*) is defined as

For our model analysis, the function used is the organ outflow curve C_{p}Out(*t*) and Δ*p*_{i} = 1%.

## RESULTS

Two variations of the four-region model and the two-region model were fit to three sets of data [Rickaby 1981 (41), Malcorps 1984 (31), Rickaby 1984 (43)]. These data sets use the same multiple-indicator dilution technique but with different conditions to help characterize serotonin uptake and transport. The Rickaby 1981 data were obtained from bolus injections of serotonin and reference indicators, the Malcorps 1984 data included bolus injections during constant infusion of serotonin, and the Rickaby 1984 data included bolus injections while the serotonin transporter is blocked by constant infusion of imipramine. The variants on the four-region analysis were *1*) PS_{g}, the permeability × surface area product for the interendothelial clefts, is fixed at 0.5 ml^{−1}·g·min^{−1} (referred in the text as model “4PS_{g}”) and *2*) PS_{g} is not fixed (referred in the text as model “4PS_{g} free”) and allowed to vary during model optimization.

Table 2 gives the physiological values used in the four- and two-region model fits to the three data sets of Rickaby 1981, Malcorps 1984, and Rickaby 1984. Vascular volumes based on estimates given by Linehan et al. (30) and Audi et al. (1). The Linehan analysis used data originally obtained by Rickaby et al. (41). The arteriole and venule volumes are assigned values based on the capillary volume representing 48% of the total organ vascular volume and the venule representing 32% of the organ vascular volume (1). The arteriole and venule volumes are treated the same computationally, the only difference being one volume is in front and the other behind the capillary. The volumes of the arterioles and venules were adjusted to obtain an input curve that matched the experimental produced output curve.

The parameter estimation results from the four- and two-region fits, along with the original reported values, are in Tables 3, 4, and 5 for Rickaby et al. 1981, Malcorps et al. 1984, and Rickaby et al. 1984, respectively. Figure 2 displays the four- and two-region model fits to the Rickaby 1981 data. Figure 2*A* shows outflow curves for the model variant 4PS_{g} free as described above to data from Rickaby 1981. The Rickaby 1981 data consisted of three serotonin outflow curves at 10-, 30-, and 100-nmol doses of serotonin in the injectate and an intravascular reference curve. The reported units are fractions of dose per milliliter. Figures for model variant 4PS_{g} fit are not shown, since its fit to the data is very similar to the 4PS_{g} free results.

A plot of extraction ratios can give a better view of changes in the shape of the substrate outflow curve vs. the reference curve that are not always apparent in the outflow curves. Figure 2*C* shows the instantaneous extraction ratio fits for model. Specifically, the extraction curves reveal backdiffusion into the capillary and evidence of flow heterogeneity. The effect of backdiffusion is recognized by the drooping of the tails of the E(*t*) curves. If there were no backdiffusion or transportation back into the capillary then E(*t*) would remain constant or rise gradually with time as tracer outflow comes increasingly from the slow-flow pathways. But backdiffusion of unconsumed serotonin raises the concentration in the capillary blood, lowering the apparent extraction. Flow heterogeneity changes the peak extraction times at the different dosages. Figure 2*C* shows that the peak extraction occurs earliest at the lowest dose extraction and latest at high dose. At low doses, the peak concentrations of mother solute are much less then *K*m_{ecl} so the extraction curve reaches a peak close to the maximum. At high doses the earliest emerging tracer has its uptake blocked by competition from the nontracer mother solute and it is only after the nontracer concentrations fall to low levels, long after the peak, that the extractions rise as competition diminishes.

Table 3 lists parameter values and goodness of fit calculations for the four-region model variants, the two-region model, Linehan's 1998 model, and Rickaby's 1981 parameter values. Estimates of *V*_{max_ecl} and *K*m_{ecl} were similar for both four-region variants with the estimates from the model variant 4PS_{g} giving tighter confidence intervals for *V*_{max_ecl} and *K*m_{ecl} due to less free parameters in the model. For model variant 4PS_{g} free, the estimated PS_{g} is 0.3 ± 0.1 ml·g^{−1}·min^{−1}, lower than our fixed estimate based on sucrose permeation of 0.5 ml·g^{−1}·min^{−1} for flow of 2.26 ml·g^{−1}·min^{−1}. The estimates for the extravascular transporter (PS_{xv}) parameters *V*_{max_xv} and *K*m_{xv} from our two-region model were approximately half of those reported by Linehan and the ratio *V*_{max_xv} to *K*m_{xv} was about the same (see two-region model description and results below). Curve fits for the four-region and two-region models are similar, with the two-region model fits by Linehan and us reporting a smaller coefficient of variation, 7% (8.9% for our two-region model) vs. the 11% obtained from our four-region model fit. The four-region model has many more constraints and will give a better fit to experimental data only when the serotonin concentration in the endothelium is large enough to break the assumption in the two-region model that transporter uptake is unidirectional and approximating backtransport as backdiffusion is no longer concentration independent.

The four-region model parameters for the EC transporter are similar to the extravascular transporter parameters *V*_{max_xv} and *K*m_{xv} reported by Rickaby for their original fit. Rickaby's two-region model analysis used a one-way transporter with no consumption or backdiffusion; heterogeneity was taken into account (41). Because both of the four-region models variants have similar EC transporter values and are similar to Rickaby's results it appears that the results from Linehan's two-region model overestimated the amount of backdiffusion and compensated for it by increasing *V*_{max_xv} and *K*m_{xv}.

The correlation matrix (obtained through normalizing the covariance matrix) for the 4PS_{g} free model reveals important information about parameters' relationship with each other. It gives an estimate of their negative or positive interdependence. The relationship between *K*m_{ec} and *V*_{max_ecl} is positive, whereas PS_{g} decreases as *V*_{max_ecl} increases. Pairs of parameters with correlation near ±1 are considered irresolvable and are codependent based on the model used and experimental data obtained. The correlation matrices show that model variants 4PS_{g} free and 4PS_{g} give us two independent parameters: *V*_{max_ecl} and the ratio of Gec_{max} to *K*m_{g_ec} (correlation between Gec_{max} and *K*m_{g_ec} is 1). Fixing PS_{g} (model 4PS_{g}) does bring down, a bit, the correlation between *V*_{max_ecl} and *K*m_{ecl} from 0.96 to 0.87, as expected, but at the expense of increased uncertainty in the consumption parameters Gec_{max} and *K*m_{Gec} as shown in the SDs for model PS_{g} in Table 3. The two-region correlation matrix reveals high correlation (coefficient of 0.90) between G_{xv} and PS_{ev}, and for the final two data sets we will not report the consumption parameters for all models.

### Sensitivity Analysis

To compare parameter sensitivities, we normalize S_{pi}(*t*) by multiplying S_{pi}(*t*) with the ratio of *p*_{i} to C_{p}Out(*t*). Figure 2*E* shows the normalized S_{pi}(*t*) for 4PS_{g} for five parameters: *V*_{max_ecl}, *K*m_{ec}, Gec_{max}, *K*m_{g_ec}, and PS_{g}. Besides model sensitivity to a parameter, an important characteristic of a parameter can be ascertained from S_{pi}(*t*): whether a model parameter is separable from another. If evaluation of S_{pi}(*t*) for each parameter reveals values that are the same or are a product of a scalar of another parameter, then those two parameters are not separable based on the model. In addition, if these two parameters appear as a product in a model equation they will be inseparable, and if these same two parameters form a quotient, S_{pi}(*t*) for each will be mirror images of each other. Combining the sensitivity information with the correlation matrix gives us a good feel for which parameters are separable in our model and therefore which parameters are extraneous when fitting the model to the data. The timing of the sensitivity function's maximum or minimum also gives some insight into importance of that parameter at that particular point in the outflow curve (3).

We see, from Fig. 2*E* for a high dose of 100 nmol, that S_{p}(*t*) values for *V*_{max_ecl} and *K*m_{ecl} are not different by a scalar but appear as a quotient in the modeling of PS_{ecl} (when *K*m_{ecl} ≫ C_{p}) and so are inseparable at low serotonin concentrations, as noted above by the high correlation between the two parameters. The S_{p}(*t*) for *K*m_{ecl} shows a maximum after the reference outflow peak revealing the effect it has on the tail of the outflow curve. The S_{p}(*t*) for PS_{g} shows little change in sensitivity throughout the time course, relative to the serotonin outflow concentration, and tells us that, for the four-region model, PS_{g} has only a minor role in the kinetics of serotonin uptake in the lung. This is not to say there is no effect; the sensitivity of PS_{g} at times past the reference peak outflow for the low dose (10 nmol) gives a relative sensitivity of ∼0.02 vs. ∼0.16 at high dose (100 nmol) (Fig. 2*E*), which is expected due to saturation of the EC transporter.

The next experiments to be analyzed are two experiments of Malcorps et al. (31) using trace bolus injections: one, labeled *method B*, used a trace bolus injection into a dog lung receiving a constant infusion of serotonin, and the other, labeled *method C*, used the bolus sweep method, as described in the Rickaby (1981) experiment, with no background infusion of serotonin. A third experiment, *method A*, used constant infusion of serotonin without tracer injection and was thus not suited for analysis. Malcorps modeled the experiment using just a one-way transporter with no backdiffusion or consumption terms and noted that the fit was only done to data near the peak outflow where backdiffusion was negligible (31). Table 2 shows the physiological values used for the model fits. Behavior of the four-region model during constant infusion of serotonin into the capillary may reveal the importance of accounting for diffusion through interendothelial clefts.

Three sets of data were obtained for *method B* and *method C*. *Method B* had constant infusion levels of unlabeled serotonin at 0.12, 0.81, and 4.50 nmol/ml followed by a bolus injection containing a trace amount of labeled serotonin (bolus 6, 7, 8, respectively). *Method C* had no constant infusion and three bolus injections containing equal amounts of tracer serotonin and 0, 27, and 110 nmol of unlabeled serotonin (bolus 5, 3, 4, respectively). The reported Malcorps 1984 data are in the form of a ratio of intravascular reference curve to tracer (labeled serotonin). Figure 3 shows the Malcorps *methods B* and *C* data with the four- and two-region fits. Table 4 shows parameter fits and RMS error for the four- and two-region model fits to *methods B* and *C* data as well as Malcorps' results. The parameters for consumption in the endothelium [*Gmax*_{ec}, *K*m_{Gec} (four-region model) and extravascular region G_{xv} (two-region model)] are not reported for Malcorps 1984 and Rickaby 1984 because the estimates had very large SDs (greater than the value itself) due to the high correlation between parameters mentioned earlier.

Looking at the two model fits to the constant infusion (*method B*) we see, from Fig. 3, *A* and *B*, and Table 4 that the curve fit to the 0.12 nmol/ml infusion was not as good in the two-region as it was in the four-region model fit. This is confirmed by the higher RMS error of 1.65 vs. 0.35 for four-region fit (Table 4). The parameter fits for the EC consumption terms are not meaningful since at such low concentrations for *method B* very little serotonin enters the cell. Also the *V*_{max_ecl} value is much smaller than the estimated *V*_{max_xv} for the two-region model (Table 4). This is directly related to the influence of PS_{g} at these low constant concentrations. The four-region model fits to the constant infusion (*method B*) show, from Table 4, that the parameter estimates are nearly the same, with 4PS_{g} free estimating PS_{g} to be 0.4 ml·g^{−1}·min^{−1}. This result is evidence of the influence of PS_{g} under constant infusion conditions. Since the infusion of serotonin at 4.50 nmol/ml (bolus 8) would saturate the EC transporter (based on actual *V*_{max_ecl} of around ≤10 ml·g^{−1}·min^{−1}), we cannot obtain a good estimate of EC transporter parameters but it does provide another method for estimating the maximum value of PS_{g}. Assuming the EC transporter is saturated [the outflow ratio of vascular reference (CR) to labeled serotonin (CT) is flat] then we can use the Crone expression for calculating the instantaneous extraction at the outflow peak: *E* = 1 − *e*^{−1PS/Flow} (15). Since the EC transporter is saturated, PS = PS_{g}. From the data, at 6 s, the ratio of CR to CT is 1.32, giving a CT extraction of 0.24. Flow was 2.60 ml·g^{−1}·min^{−1}. This gives a PS_{g} value of 0.70 ml·g^{−1}·min^{−1}. Our 4PS_{g} free fit to the *method B* data gave us a PS_{g} value of 0.4 ± 0.1 ml·g^{−1}·min^{−1}, similar to the 4PS_{g} fixed value of 0.5 ml·g^{−1}·min^{−1}. PS_{g} value for 4PS_{g} free fit to *method C* data was 0.1 ± 0.6 ml·g^{−1}·min^{−1}, lower than any of our estimates but given the error in the fit, within the expected range (Table 4). Another clue to the significance of PS_{g} under constant infusion is the correlation coefficient between *V*_{max_ecl} and PS_{g}. The 4PS_{g} free *method B* correlation coefficient between *V*_{max_ecl} and PS_{g} is −0.75, indicating that under constant infusion conditions, diffusion through the clefts should be taken into account. The correlation coefficient between *V*_{max_ecl} and PS_{g} is −0.91 for the 4PS_{g} free fit to *method C*, confirming the interendothelial clefts lack of influence on serotonin uptake when the EC transporters are not saturated. The two-region fit to *method C* data had a RMS error of 0.64, better than the four-region fit RMS error 0.80, as expected given the results from the Rickaby 1981 data fits. Note that the total flux out, *V*_{max_ecl}/*K*m_{ecl}, of the vascular space and into the EC for both four-region model fits to *methods B* and *C* are approximately the same and similar to the values reported by Malcorps (Table 4). This gives more indirect evidence of small PS_{g} influence on extravascular serotonin uptake as Malcorps calculated *V*_{max} and *K*_{m} around the outflow peak so as to remove the influence of backdiffusion (31).

Another set of data to examine is that obtained when the serotonin transporter is blocked. Ideally, if the serotonin transporter is completely blocked, we can ascertain whether interendothelial clefts do allow serotonin diffusion into the ISF. One dog lung experiment involving imipramine as the serotonin transporter inhibitor was done by Rickaby et al. (43). We fit the four-region models to two data sets: one control set, using bolus injections of vascular reference and serotonin, and one in which imipramine was constantly infused along with bolus injections of reference and serotonin. Rickaby et al. (43) used a model that assumed serotonin backdiffusion was negligible and only used data obtained near the peak outflow for their parameter estimation. The concentration of imipramine infused, 3.5 μM, should have been enough to block almost all the serotonin transport, so the reported instantaneous extraction would reflect the cleft permeation. Rickaby et al. noted that the experimental results, estimating an increase in *K*_{m} and a decrease in *V*_{max}, would point to imipramine as a mixed inhibitor (19) rather than a fully competitive inhibitor where only the *K*_{m} would change (43).

The four-region models were fit to the instantaneous extraction of bolus injections of 6.6, 30, and 100 nmol of serotonin after the isolated lung lobe had been infused with imipramine, C_{i}, at 3.5 μM. Figure 4 shows the Rickaby 1984 data with the four- and two-region model fits to it. Table 5 is a summary of the model parameter fits and RMS error for the curve fits for comparison. For *Eq. 6*, we initially used an estimated imipramine *K*_{i} of 50 nM based on *K*_{d} for the serotonin transporters on porcine platelet membranes: these depended on plasma sodium concentration and ranged, as reported by Talvenheimo et al. (48), from 137 ± 38 nM at 50 mM Na^{+} down to 9.1 ± 1.3 nM at 200 mM Na^{+}; 50 nM was chosen as an approximate value at typical plasma concentrations of Na^{+} of 140 mM.

Analysis of the imipramine data verified that serotonin PS_{g} through the interendothelial clefts was low. The 4PS_{g} free control fit gives a PS_{g} of 0.1 ± 0.9 ml·g^{−1}·min^{−1} (Table 5). Fitting the 4PS_{g} free model to imipramine data by using our initial value for *K*_{i} of 0.05 μM resulted in a *V*_{max_ecl} of 18.7 ± 16.7 nmol·g^{−1}·min^{−1}, *K*m_{ecl} = 0.4 ± 0.4 μM, and a PS_{g} of 0.01 ± 0.76 ml·g^{−1}·min^{−1} with a RMS error of 0.063 (Table 5). Although the PS_{g} drops to essentially zero, the values for *V*_{max_ecl} and *K*m_{ecl} are within 1 SD of the control fit, suggesting that, for this data set, interendothelial cleft permeation is very small. This is confirmed when we look at the 4PS_{g} model fit with PS_{g} fixed at 0.5 ml·g^{−1}·min^{−1}: The control fit is very similar to the 4PS_{g} free fit but the imipramine data fit shows a 30-fold decrease in *K*m_{ecl} (from 0.6 to 0.02 μM), indicating that the cleft added too much serotonin flux into the ISF. This forces the EC serotonin transporter to compensate by drastically increasing uptake to increase overall serotonin extraction where it can be consumed within the endothelium and parenchymal cells (PC transporter uses the same kinetic parameters as the EC).

What if our initial estimate of the imipramine dissociation constant *K*_{i} is incorrect? If we constrain *V*_{max_ecl} to values within one standard deviation of 28 nmol·g^{−1}·min^{−1} and fix *K*m_{ecl} at 0.9 μM and PS_{g} at 0.1 ml·g^{−1}·min^{−1}, values found in the control fit to fit the imipramine data, we get a *V*_{max_ecl} value of 16.9 ± 5.2 nmol·g^{−1}·min^{−1}, a *K*_{i} value of 2.1 ± 1.6 μM, and a RMS error 0.063 for model variant 4PS_{g} free (Table 5). For the model variant 4PS_{g} (PS_{g} of 0.5 ml·g^{−1}·min^{−1}) we get a *V*_{max_ecl} of 18.4 ± 7.7 nmol·g^{−1}·min^{−1} and a *K*_{i} of 0.81 ± 0.61 μM with a RMS of 0.073. These values for *K*_{i} suggest that imipramine is a partial inhibitor of the serotonin transporter (*K*_{i} is > *K*m_{ecl}) and that the interendothelial clefts do contribute measurably to serotonin uptake.

The two-region model fit to the Rickaby 1984 imipramine data (Fig. 4*D* and Table 5) gives a better overall fit (RMS error of 0.031 for the control and 0.045 for the imipramine data vs. 0.056 and 0.063 for four-region model), but the imipramine fit requires huge changes in the parameter values that do not make sense physiologically. *K*m_{xv} goes from 1.0 to 0.026 μM and PS_{ev} goes from 0.8 to 150 ml·g^{−1}·min^{−1} (Table 5). If we fix our parameters as we did for the four-region models and solve for *K*_{i} then we get a *K*_{i} value of 1.6 ± 0.5 μM and a RMS error of 0.067, similar to the four-region fits. From the analysis the of Rickaby 1984 imipramine data we can make two observations: if the *K*_{i} value of 50 nM is reasonable then uptake through interendothelial clefts is small and the two-region model cannot adequately describe the data, and, second, if *K*_{i} is actually much larger (on the order of *K*m_{ecl} or larger) then interendothelial cleft diffusion is small, but significant (0.1 ml·g^{−1}·min^{−1} or greater).

## DISCUSSION

The Linehan analysis of Rickaby 1981 data shows that the two-region model can fit extravascular pulmonary serotonin uptake as well as the four-region for the specific conditions of the experiment. The Malcorps 1984 data looked at the case of constant infusion of serotonin and the Rickaby 1984 data looked at constant infusion of a serotonin transport inhibitor. Both cases revealed issues with the two-region model: incorrectly accounting for the saturation of the transporter resulting in high extractions at low concentration and the inability for the model to be physiologically consistent when describing inhibited uptake.

A final condition that can affect serotonin uptake is flow through the capillary as measured through the PS/flow ratio. Previous experiments have shown that *V*_{max} for the endothelial transporter is flow independent, *K*_{m} is independent at plasma flows near or above 6 ml·g^{−1}·min^{−1} for isolated dog lung and for low-flow cases the calculated *K*_{m} will increase as backdiffusion was not taken into account in the model fit (43). This is in contradiction to the understanding that *K*_{m} is a property of the specific chemical reaction and should always be flow independent. Although we do not use any experimental data to compare changes in flow we can compare the two-region and four-region model to see whether the added interendothelial cleft can influence the extraction ratio, in particular, affect *K*_{m} of the two-region model endothelial transporter with respect to plasma flow changes.

Figure 5 shows the influence plasma flow has on the extraction ratio of the two- and four-region models. The two-region model parameters were obtained by fitting the outflow curves to the four-region model (4PS_{g}) for three dosages (10, 30, 100 nmol) at a plasma flow rate of 2.3 ml·g^{−1}·min^{−1}, which was the flow used in the Rickaby 1981 (41) data fit. All physiological parameters were those used in Table 2. No flow heterogeneity or axial diffusion was used, just a single capillary path. For Fig. 5, *top*, dosage was 100 nmol and plasma flows used were 2.0 and 10 ml·g^{−1}·min^{−1}, where 10 ml·g^{−1}·min^{−1} is approximate pulmonary plasma flow in a dog at rest (42). Figure 5, *top* shows the effect of the cleft at both flow rates and high dosage. At the low flow the instantaneous extraction ratio in the tail is slightly larger and longer for the two-region model. At high flow it is reversed as the PS_{g} term of the four-region model has less of an impact on extravascular uptake and back diffusion. At the intravascular reference peak, when serotonin levels are highest in the capillary, we see a higher extraction ratio with the four-region model at the low flow as the EC transporter is saturated and the effect of PS_{g} is more noticeable. This is evidence that serotonin uptake is less saturated in the four-region model owing to the effects of the interendothelial cleft since it is a passive process dependent on flow and volume and not concentration. The result of this is that the two-region model would tend to overestimate *V*_{max} to compensate for the absence of the cleft and this should be experimentally noticeable at low flows. Figure 5, *bottom* shows the effect of flow on low concentration (2-nmol dose). At low flow the two models show practically identical extractions over the full time course. This is expected because neither transporter is saturated, reducing the small effect of PS_{g} even further. Notice for high flow that the four-region model gives a higher extraction ratio about the front of the reference peak compared with the two-region model. This is due to the transporter briefly reaching saturation exposing the effect of PS_{g} on uptake. *K*m_{xv} must be lowered to match the four-region *K*m_{ecl} and changes with flow. On the basis of this and the earlier discussion that *K*_{m} is increased when fitting low dosage curves at low flows, we see that backdiffusion (either through two-way transporter or through interendothelial clefts) is important in obtaining a *V*_{max} and *K*_{m} that reflects actual physiological situations.

From the comparison between the four- and two-region models we see that the presence of the interendothelial cleft has only a small effect on the EC kinetics but that backdiffusion, of some type, into the capillary is important and needs to be accounted for. The two-region model, not accounting for two-way fluxes at low concentrations, at or below *K*m_{ecl}, and at low plasma flows (less than 3 ml·g^{−1}·min^{−1} in dog), gives underestimates of *K*m_{xv} and overestimates *V*_{max_xv}. However, to verify the four-region model experimentally, as there are too many free parameters to get accurate parameter fits, it will be necessary to run experiments using an extravascular reference, such as l-glucose, whose molecular weight and hydrophilic nature are similar to serotonin, but which remains extracellular, to gather information about diffusion rates and volumes of distributions in the ISF region. This will help in separating the effects of endothelial cleft diffusion from endothelial uptake, which was a big issue in trying to fit the four-region model to the data in the paper. Between an extravascular reference and the use of an endothelial transporter inhibitor the existence of serotonin diffusion through endothelial clefts should be quantifiable. To get confirmation and a more precise value for kinetic parameters PS_{g}, *V*_{max_ecl}, and *K*m_{ecl}, the experiments should be run at varied flows and low concentrations. It will be necessary to take data for longer periods (30 s for plasma flows of 2.5 ml·g^{−1}·min^{−1}) to capture the tail of the curve because this has important information about serotonin backdiffusion and consumption.

### Conclusion

From our analysis and comparison of serotonin pulmonary uptake data we conclude that interendothelial clefts contribute measurably but not greatly to the transport of serotonin into the interstitium, as one would expect from structural studies (20, 36, 37). The overall extraction, at high dosage, of serotonin due to the clefts is ∼7–20% of the total extraction, based on our three data set fits with PS_{g} = 0.5 ml·g^{−1}·min^{−1} and a unidirectional flux simulated by a very large extravascular space, For the Malcorps et al. *method B* (31) constant-infusion study at high concentration, the overall serotonin extraction was 32% with the fraction due to the clefts representing 60% of it. The overall extraction of serotonin in the imipramine study of Rickaby et al. (43) went from the control of 72 to 40% when imipramine inhibited, with the interendothelial clefts representing 11 and 27%, respectively, of the total extraction. The increase in the influence of the clefts on the overall extraction in the presence of an inhibitor confirms the possible influence of the EC gaps in pathological lungs. This contrasts sharply with data on rabbit hearts, where the clefts are the main mode of serotonin transport to the ISF (34). The use of the four-region model for serotonin kinetics allowed us to measure cleft permeation and to compare kinetic differences between organs, these results implying that the clefts are more blocked by gap-junctional “zippers” in the lung than in the heart.

These conclusions on the quantitative contribution of the interendothelial clefts are reinforced by other observations from the same laboratory in later years even though they did not specifically identify the clefts as the pathway for passive transport. Peeters et al. (39) found that their *model 3* fitted “best.” It had two paths, one saturable and one with such low affinity that it was in the linear regions, i.e., far below the *K*_{m} and therefore indistinguishable from a purely passive cleft and with a low PS. The PS estimate for this path was ∼10% of the saturable path (39). Dawson et al. (18) used a very high level of imipramine in the perfusate, ostensibly blocking the transporter completely. The serotonin extraction on the upslope of the indicator dilution curve in their Fig. 1 was a few percent, indicating a passive permeation. An analysis of these data with the four-region model gives a PS_{g} of 0.15 ml·g^{−1}·min^{−1}: the interendothelial clefts then represented ∼1% of the control and 85% of the total serotonin PS during the imipramine-inhibited uptake.

The four-region model presented here along with the bolus sweep method with the multiple indicator dilution technique is a useful way to understand the structure and kinetics of the vasculature and reveals differences in endothelial structure from organ to organ, as a whole, without changing the behavior and structure of the cells and organ itself. Nevertheless, it should be remarked that data on the cleft permeability would have been provided directly had the authors included an extracellular reference indicator in their experiments. Probably the best one is tracer-labeled sucrose, as used by Moffett et al. (34), since it has an appropriate molecular weight (342 vs. 176 for serotonin, so the ratio of expected permeabilities can be pretty accurately calculated) and is strictly extracellular so its extraction represents cleft permeation. Cleft widths are governed by the structuring of gap-junctional connections and interstitial glycocalyx and are thus probably similar to functional cleft sizes in the heart, ∼7 nm (26). The clefts are an important avenue of exchange for solutes of molecular weight in the hundreds to the few thousands, the range for most substrates and signaling molecules, and therefore important to lung function.

## GRANTS

This research has been supported by National Institutes of Health Grants from NHLBI T15 HL88516 and from NIBIB EB08407.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## AUTHOR CONTRIBUTIONS

B.E.J. and J.B.B. conception and design of research; B.E.J. performed experiments; B.E.J. analyzed data; B.E.J. and J.B.B. interpreted results of experiments; B.E.J. prepared figures; B.E.J. drafted manuscript; B.E.J. and J.B.B. edited and revised manuscript; B.E.J. and J.B.B. approved final version of manuscript.

## ACKNOWLEDGMENTS

The authors appreciate the efforts of Dr. Said H. Audi in the Department of Bioengineering at Marquette University in supplying us with the original data sets, the work of Erik Butterworth in developing JSim capabilities for these studies, and the work of Gary Raymond in refining the methods for the multicapillary, heterogeneous flow modeling.

- Copyright © 2013 the American Physiological Society