Kidney modelling: status and perspectives

PROCEEDINGS OF IEEE, VOL. 94(4):740-752, 2006

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Kidney modelling: status and perspectives S. Randall Thomas, Anita T. Layton, Harold E. Layton, and Leon C. Moore

Abstract— Mathematical models have played an essential role in elucidating various functions of the kidney, including the mechanism by which the avian and mammalian kidney can produce a urine that is more concentrated than blood plasma, quasiisosmotic reabsorption along the proximal tubule, and the control and regulation of glomerular filtration by the myogenic and tubuloglomerular feedback mechanisms. This review includes a brief description of relevant renal physiology, a summary of the contributions of mathematical models at various levels and describes our recent work towards the Renal Physiome. Index Terms— multi-scale systems biology, mathematical modeling, physiome, computational biology, kidney physiology, database, markup language

I. I NTRODUCTION

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N this review, we first give a brief survey of mathematical modeling of mammalian kidney, both past models at several levels of organization and current trends and problems of renal physiology that call for development of new models. We then describe recent work towards the Renal Physiome, which presently consists of the development of web resources to (1) streamline model development and catalog quantitative measurements of renal function (QKDB, a Quantitative Kidney Database), and (2) a website for renal models, which will contain both an interactive repository of published models and several tools for developing epithelial or channel/transporter models. We begin with a brief introduction to relevant renal anatomy. As illustrated in figure 1, most mammalian kidneys have three major sections: the cortex, the outer medulla (OM), and the inner medulla (IM). The outer and inner medullas are collectively referred to as the medulla. The deepest part of the conical IM is referred to as the papilla. Blood enters and leaves the kidney via the renal artery and vein, respectively, which run along the border between cortex and medulla. Within the cortex, the outermost layer of the kidney, fluid in all structures (except the initial part of the distal tubule) is essentially isosmotic to arterial blood. Within the medulla, a cortico-papillary gradient of osmolality serves to concentrate the urine. Within the cortex, the blood is partially filtered in roughly spherical capillary beds, called the glomeruli, and thus enters the renal tubules (radii on the order of 10 µm), called the nephrons. The nephrons are surrounded in all kidney regions This work was supported by INSERM (French National Institute for Medical and Health Research), the CNRS (the French National Center for Scientific Research), the NIH, and the NSF. SR Thomas works at LaMI (Laboratory of Computer Methods), UMR 8042 CNRS/ Univ. Evry-Val d’Essonne, Evry, France. Anita Layton and Harold Layton work at the Department of Mathematics, Duke University, Durham, NC. Leon Moore works at the Dept. Physiology & Biophysics, SUNY Health Sciences Center, Stony Brook, NY.

by a network of blood vessels that recovers the fluid and solutes reabsorbed from the various segments of the nephron. About 75% of the initial nephron flow is reabsorbed into the general circulation by the first part of the nephron, called the proximal convoluted tubule (PCT), situated entirely in the cortex; almost all the rest is reabsorbed from the loop of Henle and the distal parts of the nephron; the remainder (on the order of only 1% of the initial filtrate, during antidiuresis, when urine volume is minimal and its osmolality maximal) is excreted as urine from the tip of the IM (called the papillary tip) into the pelvic space, which empties into the ureter and from there into the bladder. The renal medulla is characteristic of the higher vertebrates, and the IM (which contains the portions of long loops of Henle having thin epithelium) exists only in mammals. The medulla is formed by the hairpin-turn feature of the nephrons (descending and ascending limbs) and blood vessels (called the vasa recta within the medulla), which, by counter-current exchange and counter-current multiplication, allow the generation of a cortico-papillary gradient of increasing osmolality [1] [2] [3] [4] that serves to concentrate the urine, thus permitting mammals to maintain salt and nitrogen balance without excreting a copious urine that would tend to dehydrate the animal. Owing to this adaptation, which is developed to differing extents in different species and corresponds to differences of habitat and diet, some mammals can live in very arid environments with no water beyond what is available in their food, which may consist only of dry seeds. Each mammalian nephron has a hair-pin loop, the loop of Henle. Depending on the species, the loops of Henle extend to various depths of the OM or IM. In rat, for example, about two-thirds of the ∼35 000 loops of Henle (the short loops) extend only to the bottom of the OM, and the remaining third (the long loops) extend to various depths into the IM, with only a small fraction reaching to near the tip of the papilla. Descending and ascending limbs of the loops of Henle have distinct permeabilities and transport properties (see section III.A). Upon returning close to the glomerulus, at the macula densa (see section II.C), the thick ascending limb leads into the distal convoluted tubule (DCT), which then becomes a connecting tubule (in rodents) and finally a cortical collecting duct (CCD). The collecting ducts descend into the medulla, passing through the OM without merging, but in the IM they coalesce successively before emptying their fluid, the urine, from the papillary tip into the pelvic space. II. B RIEF OVERVIEW OF KIDNEY MODELS In the following overview of modeling studies, we present highlights of the main currents of kidney modeling efforts, but we make no attempt to be exhaustive.

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to more global models aimed at calculation of GFR from systemic measurements of blood pressure, oncotic pressure, and clearance measurements. Among the many aspects of glomerular filtration for which modeling studies have been important are: (i) general analyses of factors involved in SNGFR (many references, e.g., [9] [10] [11]); (ii) the relative importance of TGF versus vascular myogenic mechanisms for autoregulation of GFR [12] [13] [14]; (iii) the conclusion that the usual steady-state assumption introduces negligible errors in calculating glomerular membrane parameters from experimental data [15]; and (iv) analysis of defects implicated in glomerular nephropathies [16] [17] [18] [19] [20].

Fig. 1. Basic anatomy of uni-lobed mammalian kidneys. The tip of the conical IM is called the papilla. On the right, a short-looped and a long-looped nephron, together with the collecting duct system (not to scale). Within the cortex, a medullary ray is indicated with a dashed line. G, glomerulus; PCT, proximal convoluted tubule; LDL, long descending thin limb of Henle; LAL, long ascending thin limb of Henle (inner medulla only); MTAL, medullary thick limb of Henle (outer medulla only); DCT, distal convoluted tubule; CNT, connecting tubule; CCD, cortical collecting duct; OMCD, outer medullary collecting duct; IMCD, inner medullary collecting duct. The microvascular system (not shown), including the vasa recta of the medulla, surrounds this nephron system. (adapted from [5] and [6] with permission)

A. Glomerular filtration Blood is filtered into the nephrons at the glomerulus, across the glomerular capillaries, which are open to all small solutes but do not allow blood cells or most plasma proteins to cross into Bowman’s space and enter the tubular fluid. As an indication of the size of the glomerular filter under normal ˚ in diameter) and smaller plasma conditions, hemoglobin (65 A ˚ by 150 A) ˚ pass the glomerular proteins such as albumin (36 A capillaries, but only in small amounts. The glomerular filtration rate (GFR) depends on the number of filtering glomeruli and on the single nephron GFR (or SNGFR). SNGFR is driven by the balance of hydrostatic (blood pressure and intratubular pressure) and osmotic (especially protein oncotic) pressure differences across the glomerular capillaries and basement membrane. The rate of filtration of fluid volume and solutes depends on the net driving force, the rate of blood flow, and on the permeability characteristics (summarized as the ultrafiltration coefficient, Kf ) of the capillaries and basement membrane, usually reduced to a question of equivalent pore size. Net pressure differences and permeabilities both vary along the path from entry (afferent arterioles) to exit (efferent arterioles). Autoregulation of SNGFR is mostly achieved through changes in the resistance of the afferent arteriole; these changes arise in large measure from the arteriole’s intrinsic myogenic response and by means of tubuloglomerular feedback (TGF, section II.C), which regulates the the contractile state of the portion of the arteriole near the glomerulus. Sorting out the relative contributions of all these factors has given rise to a rich literature of mathematical models at various levels of detail, from intimate consideration of 3D architecture of the glomerular capillaries [7], to detailed hydrodynamic models of transport through the glomerular capillary wall [8],

B. Tubular transport After filtration at the glomerulus, the tubular fluid is initially isotonic to plasma and is virtually identical to plasma in the concentrations of small solutes and electrolytes. Very quickly, however, its composition is modified as specialized transport systems, first in the PCT and then in each succeeding nephron segment, carry out the work necessary to regulate plasma pH, recover nutrients, secrete organic cations and other molecules, and maintain the body’s water, salt, and nitrogen balance within narrow limits. Mathematical models have been developed to address many questions related to these processes at the level of epithelial cells and at the level of tubular segments, which sometimes include several different cell types. Model equations frequently involve a set of differential-algebraic equations describing mass balance, macroscopic electroneutrality, transepithelial water and solute fluxes driven by electrochemical gradients, and sometimes buffering reactions. For a comprehensive review up to ten years ago, see [21], and for a more recent account of stillopen issues, see [22]. 1) Models of the proximal tubule: For many years, the proximal tubule was the focus of most tubule modeling work, the main questions being the mechanism of (quasi-)isosmotic reabsorption, the relative importance of cellular and paracellular routes for water and the various solutes, the mechanism of load-dependent bicarbonate reabsorption, and the importance of peritubular oncotic pressure as a driving force for volume absorption. For extensive references to the relevant literature, see Weinstein’s reviews [21] [23]. More recently, attention has turned to more detailed models addressing the role of the apical microvilli [24] and the nature of the tight junctions between proximal tubule cells [25] . 2) Henle’s loop models: Although all models of the renal medulla concerned with the urine concentrating mechanism have necessarily included simple descriptions of the epithelial wall along the thin descending and ascending limbs of Henle’s loop, as well as very simple (usually Michaelis-Menten type) treatment of active reabsorption of NaCl by the thick ascending limbs, there have been very few modeling studies of transport along the loop of Henle giving details of channels and transporters at the cell membrane level. In large part, at least for the thin limbs, this reflects the state of the available experimental data, which come essentially from inflow/outflow in vitro microperfusion studies of microdissected LDL or

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LAL segments, sufficient for calculation of global permeability values and reflection coefficients across the tubule epithelium but with no intracellular microelectrode studies, due to the very thin and fragile epithelium of these segments. However, for the medullary and cortical thick ascending limbs of Henle (MTAL and CTAL), there is a very rich experimental literature on the physiology and pathophysiology of the channels and transporters of the apical and basolateral cell membranes and a long tradition of discursive analysis (i.e., not modeling studies) of the pathways for both mono- and divalent ions across the transcellular or paracellular routes, especially since the discovery of the Na-K-2Cl co-transporter (NKCC) [26] [27]. This intense focus on one of the most important segments of the nephron has, curiously, not been accompanied by the development of detailed mathematical models of this epithelium, except for equivalent-circuit-type analyses (e.g., [28] [29] ). To our knowledge, the only modeling study of this segment to date giving detailed descriptions of the channels and transporters is that of Yang et al. [30]. 3) Distal tubule models: The distal convoluted tubule, consisting of an early part (DCT) and a later connecting tubule (CNT), is the portion of the nephron that lies between the mascula densa and the region where the convergence of tubules occurs. The DCT receives a hypotonic fluid from the thick ascending limb and is important mainly for renal Na+ , and HCO− 3 reabsorption, and the main role of the CNT is K+ secretion. Depending on the levels of ADH (antidiuretic hormone), the later portions may become water permeable, allowing osmotic re-equilibration with plasma osmolality. Also, the late distal tubule is an important site for K+ secretion and for the reabsorption of divalent cations. Mutations of certain transporters expressed mainly in this segment of the nephron (especially the thiazide-sensitive Na-Cl cotransporter, TSC, and the apical membrane Na+ channel, ENaC) are key factors in hypertension. Chang and Fujita developed the first mathematical models of the DCT [31] [32] [33], including the first kinetic description of the TSC cotransporter. They simulated water, solute, and acid-base transport along the rat DCT and CNT. Recently, Weinstein [34] [35] contributed two new modeling studies, incorporating new experimental findings (e.g., the presence of a peritubular KCl cotransporter) and treating variable cell volume in response to solute concentrations. These papers present a collection of new models that were used to investigate, among other things, the response of cell volume to changes in solute delivery or peritubular composition, and the homeostatic effect of those volume changes. Many unanswered questions remain about the DCT and CNT, which consist of several major cell types (DCT cells, principal cells, and alpha- and beta-intercalated cells) whose relative prevalence varies along the tube. Each cell type has a specialized role, reflected in the different transporters expressed in the apical and basolateral cell membranes. 4) Collecting duct models: Weinstein has recently published a number of detailed modeling studies of transport along the collecting duct [36][37][38][39][40][41][42]. Each of these addresses particular aspects of collecting duct physiology. Together, these papers comprise a comprehensive resource

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of knowledge concerning this final common pathway of the nephron, the site of the fine-tuning of urine content that is crucial for successful water, salt, and nitrogen balance. Many serious kidney-related pathologies are directly related to disruption or deregulation of some aspect of CD transport, including mutations of the genes coding for transporters specific to this part of the nephron. C. Tubuloglomerular feedback (TGF) The TGF mechanism is a negative feedback mechanism that regulates glomerular filtration rate by balancing the filtration rate through the glomerulus with the capacity of the tubular epithelial cells to transport water and electrolytes. For a review, see [43]. At the site where a thick ascending limb passes near its associated glomerulus, a cluster of specialized cells, called the macula densa, senses changes in luminal NaCl concentration. An elevated concentration promotes a constriction of the smooth muscle of the afferent arteriole, which supplies blood into the glomerulus, thus reducing filtration pressure at the glomerulus and hence reducing the rate of fluid entry into the nephron. The resulting slower fluid flow in the thick ascending limb allows more time for NaCl absorption and lowers its luminal concentration. A depressed NaCl concentration has an opposite effect. In the 1980’s, experiments in rats by Leyssac and colleagues [44] demonstrated that nephron flow and related variables may exhibit regular oscillations [45] with a period of about 30 seconds. Mathematical models, which involve a system of coupled delay-differential and algebraic equations, have indicated that these regular oscillations are TGF-mediated and that they arise from a bifurcation: if feedback-loop gain is sufficiently large, and if the delay in TGF signal transmission at the juxtaglomerular apparatus is sufficiently long, then the stable state of the system is a regular oscillation and not a time-dependent state [46] [47]. Since then, mathematical models of varying degrees of complexity have been built, with some involving detailed representation of the vasculature (e.g., [48]) and others representing internephron coupling (e.g., [49]). These models have been used to address questions such as: Is nephron coupling important? What is the physiological significance of TGF oscillations? What is the origin of the irregular fluctuations in nephron flow in spontaneously hypertensive rats [50]? D. Renal microcirculation Most of the models of the renal cortical microvasculature were developed to address questions related to the important, synergistic interactions between the TGF and myogenic mechanisms in the context of the autoregulation of renal blood flow and GFR. These range from variable, lumpedresistance models [14], to reactive viscoelastic tube models [51], to multi-segment models with explicit representation of the contractile response of the vascular smooth muscle cells [43] [52]. Loutzenhiser and colleagues used a simple kinetic model of afferent arteriole responses to step changes and pulsations in intravascular pressure to explain data that suggests an important effect of systolic blood pressure on renal

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vascular resistance [53]. Perhaps the most detailed model of fluid dynamics in the afferent arteriole was developed by Smith et al. [54] to analyze advective and diffusive transport of nitric oxide. Medullary models that use the central core assumption (section III.A) merge the vasa recta, along with the interstitial cells and interstitial spaces, into a single compartment through which the renal tubules interact. However, as more becomes known about the renal microcirculation, increasingly many medullary models have included explicit representation of the vasculature (see the next section). In many of these models, blood flow from descending to ascending vasa recta through shunts or a capillary plexus is taken into account. In recent years, several models have been developed that explicitly represent only the medullary renal vasculature [55] [56] [57] [58] [59] [60] [61] [62]. Because the principal focus of these models is usually the analysis of vascular transport within the medulla, the interactions between blood vessels and renal tubules are ignored for simplicity. Thus, interstitial concentrations of key solutes, or their absorption rates from loops of Henle and collecting ducts, are assumed a priori in these models. There are ongoing efforts to develop vascular models at different scales in several laboratories. Perhaps the most important question to be addressed is related to the dysregulation of GFR and renal blood flow in diabetes, hypertension, and other renal diseases: How do the known changes in the characteristics, anatomical and functional, of endothelial, vascular smooth muscle, and tubular cells contribute to the pathogenic process that leads to progressive renal failure? E. Modeling the renal medulla and the urine concentrating mechanism Despite sustained attention for over 50 years [63] (a recent translation of the often-cited classic paper [64]) [65] [66][67], the mechanism by which the osmotic gradient is created within the IM still has no satisfactory explanation that accounts for measured permeabilities and the lack of active transport within the IM. There have been many modeling studies of this problem, from simple models of the microcirculation (see review by Pallone et al. [68]) and “central core” models that consider the nephrons bathed in a common, ideal vascular space (see review by Stephenson [66]), through multi-nephron models [69][70][71][72], and culminating in 3D models accounting for the lateral separation of structures within, especially, the OM [73][74][75][76][77]. Most of these models share the same basic description of tubular flows and transepithelial transport, differing essentially in the connectivity of the network of tubules and vessels and in the numerical solution techniques, which vary according to the topological complexity (see [78]). An interactive web version of one of our medullary models [79] is available at http://www.lami.univ-evry.fr/∼srthomas/kidneysim. Figure 2 shows a more elaborate model that accounts explicitly for much of the 3D arrangement of tubes and vessels within the medulla. In recent studies, Layton and Layton [76][77] described another highly-detailed mathematical model for the urine con-

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Fig. 2. Schematic diagram of WKM-type models (adapted from [80] with permission). Left: placement of each structure in cross section through each medullary region.Right: tubes in a flattened 2-dimensional projection, approximately in their correct positions. Values in left (e.g., 0.33, 0.25) indicate connection strengths from Henles loops and from descending vasa recta to ascending vasa recta, reflecting their relative proximities in each region. Distances are not to scale. Arrows indicate convective connections.

centrating mechanism of the rat OM. To represent the radial distribution of tubules and vessels, with respect to the vascular bundle, the model uses a region-based configuration (figure 3): the model represents four concentric regions, centered around a vascular bundle, and radial structure is incorporated by assigning appropriate tubules and vessels to each concentric region. That model incorporated, and was used to evaluate, experimental findings not previously included in models, including findings of recent immunohistochemical localization experiments. Outer stripe

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Fig. 3. Schematic diagram of formulation of region-based model [76]. The two panels are cross sections through outer and inner stripes, showing concentric regions and relative positions of tubules and vessels. Decimal numbers indicate relative interaction weightings with regions. R1–R4, concentric regions. Adapted from [77] with permission.

System equations: Without going into the details of connection topology that distinguish these medullary models, the basic descriptions of local flows and exchanges among nephrons and vasculature are common to most medullary models to date. Typically in these models (but not in some recent models that explicitly treat a distributed population of individual nephrons [81][82]), each type of tube is represented

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by a single, lumped tubular structure whose circumference at each depth reflects the total number of such tubes at that depth, and differential equations describe the changes of flows and concentrations with depth in each (lumped) tube, j. System variables are the axial tubular flows of water, Fvj , and solutes, Fij . Concentrations of solutes i in tubes j are calculated from the ratio of solute flow to volume flow, ci = Fij /Fvj . In lumped-tube models, the decreasing number of IM nephrons and vessels, Nj (x), is accomodated by “shunt flows” from descending tube j to the corresponding ascending tube at depth x. For the case of Nj (x) decreasing exponentially at rate ksh , these shunt flows are given by j Fshunt (x) =

Fij (x) d Nj (x) = ksh Fij (x). Nj (x) dx

(1)

Adopting the convention that descending tubule flows are positive and ascending flows are negative, we then have the following system of differential equations: d Fvj (x) dx d Fij (x) dx

= −Jvj (x) ∓ ksh Fvj (x) j (x), (2) = −Jij (x) ± ksh Fij (x) − Ji,pump

where Jvj and Jij represent transmural fluxes of volume and solutes, resp., at medullary depth x; the subscript i refers to NaCl, urea, glucose, or lactate; and j refers to individual tubes or vessels. The symbol ∓ is taken to mean “-” for descending j tubes and “+” for ascending tubes. Ji,pump is transmural flux due to active (i.e., metabolism-dependent) solute flux across certain segments, especially the thick ascending limbs of Henle and the collecting ducts. In these equations, transmural fluxes of volume and solute i out of tube j are given by: Jvj (x)

=

Jij (x)

=

j Ji,pump (x)

=

4   X R Aj Lpj RT σij cAV (x) − cji (x) i  i=1  R Aj Pij cji (x) − cAV (x) + (3) i   cj (x) + cAV R (x)  i i Jvj (x) 1 − σij 2 j Vm,i cji (x) . (4) Aj j Km,i + cji (x)

These equations incorporate the assumption that interstitial concentrations are identical to concentrations in AVR (the ascending vasa recta). This is related to the central core assumption, which assumes in addition that AVR and DVR are also perfectly equilibrated (an assumption not made here). As shown in eq. 4, a simple Michaelis-Menten type relation is assumed for the active solute pumps, saturable as a function j of solute concentration, with a maximum rate of Vm,i and j half-maximal concentration of Km,i . Conservation of mass for the medulla as a whole in the steady state says simply that, at any depth x, the algebraic sum of flows of type i in all tubes j (taking flows to be positive toward the papilla and negative away from the papilla) must equal the exit rate of i from the terminal collecting duct (the only exit at the bottom of the system) minus (for metabolized

components such as glucose and lactate) the total amount of i synthesized from x to the papillary tip, x = L: Z L X j CD Fi (x) = Fi (L) − Sij (x) dx, (5) j

x

where flows at level x are taken to be zero in tubes that do not extend all the way to x. The Sij (x) term (amount of solute i synthesized in tube j at depth x, per unit depth) is of course zero everywhere for conserved solutes such as NaCl and urea and applies, for example, only in the interstitial space for metabolized solutes (e.g., it is negative for glucose and positive for lactate in models treating the conversion of glucose to lactate by anaerobic glycolysis [62] [79]). In general, it is considered that there are no sources or sinks (except chemical reactions), that hydrostatic pressure plays a negligible role compared to osmotic pressure forces, and that axial diffusion is negligible relative to convective flow of solutes [6][83]. Boundary conditions and system inputs: Boundary conditions at the bends of loops are based on tube connectivity. The inputs to the system are the volume flows and solute concentrations at the entry into the long and short descending limbs and into the vasa recta. Rather than including distal tubules explicitly, the entry to the collecting ducts is commonly calculated from flows and concentrations at the top of the short and long ascending limbs, based on constraints deduced from the literature. Numerical solution methods: Owing to the high water permeabilities of some of the renal tubules, the differential equations of the medullary models are frequently stiff, and standard numerical methods become insufficient. Thus, a variety of numerical techniques have been adopted. For central core models and multi-nephron, shunted models [72][79] with a common interstitial compartment for all tubes (so-called ‘flat’ models), Stephenson and colleagues [84] introduced a very robust, nested Newton-Raphson method that uses the hairpinturn arrangement of tubes and vessels to good advantage. In a six-nephron flat model, Lory [69] successfully used multipleshooting. For the more complicated 3-D ‘WKM’ models [73][74], quasilinearization was used for the initial development but proved troublesome during model modifications and was replaced [85][86][87][75][80] by the excellent collocation package COLNEW (available in the netlib archives on the internet at http://netlib.bell-labs.com/netlib/ode/), which gives much better convergence for this difficult multiple-boundarypoint problem. More recently, the SLSI-Newton method has been developed to allow one to rapidly compute steadystate solution to urinary concentrating mechanism (UCM) models while avoiding the numerical instability with which one frequently must contend with when solving the steadystate model equations. In the SLSI-Newton method, one first applies the semi-Lagrangian semi-implicit (SLSI) method to the dynamic formulation of model equations and generates an approximate steady-state solution using a large time step. This intermediate solution is then used as an initial guess for the Newton solver, which generates a more accurate steady-state solution by solving the steady-state formulation of the model

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equations. The SLSI-Newton method has been successfully applied to UCM models of various complexities [81], [88], [82], [89], [76]. For a more complete review of modeling assumptions and numerical methods for medullary models, see [6]. III. C URRENT I SSUES A. Urine concentrating mechanism (UCM) Background: When, in 1951, Hargitay & Kuhn [64][63] introduced the countercurrent multiplication hypothesis, they carried out their formal analysis using a hydrostatic pressure difference but carefully explained that in the kidney the actual driving force was more likely to be “electro-osmotic”. Later in the 1950’s, Kuhn & Ramel [90] settled on active salt transport from ascending to descending limbs as the most feasible single effect1 , and then Niesel & R¨oskenbleck [91] briefly considered the idea that interstitial “external” osmoles might also supply a single effect. The idea that IM glycolysis might participate was investigated once by in vivo micropuncture [92], but the idea was abandoned in favor of active transport from the ascending limbs. During the 1960’s, it gradually became clear that although vigorous active salt transport occurs from the OM thick ascending limbs, this is not the case in the IM. Thus emerged the enigma that the steepest and major portion of the medullary osmotic gradient is established in the IM with no apparent means of support. The “passive” hypothesis, introduced in 1972 by Stephenson [93] and by Kokko and Rector [65], astutely proposed that the metabolic effort spent in the outer medullary MTAL could serve indirectly for the establishment of the IM osmotic gradient if not one but two solutes were involved, namely, NaCl and urea. Permeabilities of individual nephron segments were unknown at the time, but the passive hypothesis made specific predictions that must obtain if urea in fact serves the proposed external osmole role. In particular, IM descending limbs of Henle (LDL) must have very low urea and salt permeabilities and high water permeability and ascending thin limbs (LAL) must be more permeable to NaCl than to urea. Under these conditions, they predicted that the urea that enters the deep medullary interstitium from the collecting ducts will draw water from LDL, thereby concentrating their luminal solutes, especially NaCl, which will then diffuse passively out of the water-impermeable ascending limbs, thus providing an osmotic single effect with no local expenditure of metabolic energy. Subsequent measurement of tubular permeabilities by in vitro microperfusion were in conflict with these predictions — e.g., urea permeability of LDL was found to be low in rabbit, which does not develop a highly concentrated urine, but quite high in species that concentrate well, such as the chinchilla [94] and the rat [95]. Thus, there is still no adequate explanation for the steep cortico-papillary gradient of osmolality in the IM. We briefly summarize three possibilities that are currently under investigation. All three are controversial, in that they 1 The term “single effect” was introduced [63] to describe a small effort, applied locally at each depth between neighboring counterflowing tubes, that is then multiplied by means of countercurrent recycling to achieve a very much larger global gradient.

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are based on well-established experimental findings but also contradict at some level certain generally accepted principles or pose other problems of a thermodynamic or biophysical nature that remain to be resolved. A full explanation may involve all of these possibilities, plus perhaps presently unidentified factors. 1) Transport heterogeneity along descending limbs of long loops of Henle: Through a combination of the techniques of immunohistochemical localization and computerized threedimensional reconstruction, Dantzler and collaborators conducted experimental studies that provide surprising new insights into the transport properties of long loops of Henle in the rat IM [96][97]. The textbook description of the loops of Henle is that throughout the IM, the LDL are highly water permeable and moderately urea permeable, whereas the LAL are highly NaCl and urea permeable. However, the careful studies by Dantzler et al. suggest that in the IM, LDL have terminal segments of significant length (approximately 60% of the inner medullary portion of a given descending limb) that do not label for aquaporin-1, a water channel, and thus likely have limited water permeability. In addition, these LDL do not label for CLCK1 (a chloride channel), except for a short prebend segment, and are thus likely NaCl impermeable. Finally, the long loops of Henle do not label for several common urea transporters, thus raising the question of the explanation for high reported urea permeability in LDL and LAL. The implications of these findings have been investigated in a modeling study by Layton et al. [81]. They identified two modes that could produce a significant axial osmolality gradient. One mode, suggested by the immunohistochemical findings by Dantzler et al., assumes that the water-impermeable portions of loops of Henle have very low urea permeability. Another mode, suggested by perfused tubule experiments from the literature, assumes that these same portions of loops of Henle have very high urea permeabilities. Both hypothetical models appear to be capable of producing a significant IM osmolality gradient while maintaining reasonable urine flow. Both modes are similar to the passive hypothesis previously described by Kokko and Rector [65] and by Stephenson [93] (see section III.A for more detail), in that passive NaCl absorption from loops of Henle is driven by interstitial urea that diffuses into the interstitium from the collecting duct system, and the mixing of this NaCl and urea raises the interstitial osmolality. However, both modes depend on locally high rates of passive NaCl absorption around the bends of the long loops of Henle, and both modes depend on active NaCl absorption along the collecting ducts, which ensures that urearich fluid is absorbed from the portion of the collecting duct system that is deep in the IM. 2) Papillary peristalsis: The kidney—the renal papilla, in particular—exhibits complex dynamics. In unipapillary kidneys such as those in hamster and rats, a muscular sheath in the wall of the renal pelvis milks the papilla with peristaltic contractions that are coupled with the peristalsis of the ureter. These peristaltic contractions interrupt tubular fluid flow intermittently by forcibly collapsing the loops of Henle, collecting ducts, and blood vessels. In the loops of Henle and vessels, fluid is trapped at the turn of the loops in the capillaries closest

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to the papillary tip as the peristaltic wave approaches. In the papillary collecting ducts, the peristalsis of the papilla propels fluid toward the bladder as a bolus. Reinking and Schmidt-Nielson [98] were perhaps the first to hypothesize that peristalsis plays a fundamental role in the UCM. However, experimental studies have so far failed to lend consistent support to that hypothesis: although some studies (e.g., [99], [98], [100]) have found that disabled peristalsis or excision of the pelvic wall [101], [102], [103] reduces urine osmolalities, results from other studies (e.g., [102]) have failed to support that peristalsis is crucial to the generation of a concentrated urine. It is also not clear whether a significant contribution of peristalsis to the concentrating mechanism is thermodynamically plausible. Nonetheless, the peristalsis hypothesis has been revived by various investigators [104], [105], [106], [107]. The effects of peristaltic contractions on tubular flow and transepithelial transport may be studied using a mathematical model, which may be formulated using a modified (and yet to be developed) version of the immersed boundary method [108] that represents transmural water transport. 3) Inner medullary metabolic osmole production: In a study using a “flat” model, Hervy & Thomas [79] (following previous work [62][75]) were the first to reconcile the high urea permeabilities measured in LDL with an appreciable IM NaCl gradient. The central feature is that metabolic osmole production (MOP) plays the role previously attributed to urea. Because the loops of Henle and collecting ducts are essentially impermeable to glucose and lactate, the external osmoles contributed by anaerobic glycolysis (AG) (two lactates produced for each consumed glucose) in the hypoxic IM (well established experimentally) can exert their full osmotic effect across the epithelium of the LDL. The effective accumulation of lactate in the deep IM would be favored by reduced IM blood flow (known to be the case in antidiuresis [109]) and high lactate permeability of the descending vasa recta (plausible but not yet measured). Attractive as it seems, this hypothesis still presents some problems. For example, AG produces not merely lactate but lactic acid (i.e., lactate and protons), and the fate of the H+ ions is crucial to the question of whether there is a net production of osmotically active particles. To the extent that the protons get buffered by HCO− 3 (producing CO2 and water), the osmotic advantage is lost. However, their buffering by NH3 (producing ammonium ions) would maintain the osmotic + advantage. Both HCO− 3 and NH3 /NH4 are present in the IM, their relative importance being dependent on physiological conditions. Experimental evaluation of this question would be complicated. IV. P HYSIOME PROJECTS UNDERWAY FOR THE KIDNEY Initial progress has been made towards the development of web resources for quantitative renal physiology. In particular: (i) We have built a quantitative kidney data/knowledge database, QKDB (http://www.lami.univevry.fr/∼srthomas/qkdb). To date, we have developed a relational data model (entity-relationship model) for QKDB, have

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implemented this under MySQL, and we have built (using PHP) a web GUI with basic functionality [110]. (ii) With Peter Harris and Andrew Lonie in Australia and with Carey Stevens, from the group of Peter Hunter in New Zealand, we have developed two proof-of-concept web interfaces for exploration of published renal models, including limited customized simulations: one (KSim) uses a Java applet for one of our renal medulla models [79] and the other (Kidneysim3D) presents a published model of transport along the distal tubule [31], implemented via translation into the markup language CellML and superimposed on a zoomable, 3D representation of the kidney (a “virtual kidney”). We next describe the present state of these initial developments. A. QKDB Future model development driven by new hypotheses will be considerably streamlined by the QKDB, because a large part of the development time for each new modeling project has always involved an extensive literature search for relevant measurements of transport parameters, flow rates, concentrations, etc., carried out under appropriate experimental conditions and on appropriate animal models. As the contents of the QKDB mature through the contributions of the renal community, it is intended that it become the definitive resource for quantitative information about the kidney. To this end, a major criterion in the development of the data model behind the database was that it be readily extensible with no re-programming of the user interface, because any list we can contrive today of all the types of information that should be included must necessarily be incomplete in the near future as new approaches are developed. The degree to which this objective is satisfied will be assessed during the seeding of the database by ourselves and interested colleagues in the development phase, and corresponding adjustments will be made as necessary. In the interest of the perenity of these resources, they will be mirrored in France, in the USA, and in Australia and/or New Zealand. Furthermore, far from being a resource of interest only to the renal modeling community, both the QKDB and the (future) modeling website will be useful for the whole renal research community and should also become a valuable source for basic concepts of renal physiology and, eventually, an aid to the teaching of renal physiology. In the interest of this, it may later be extended to include timely summaries of basic renal physiology and physiopathology, complementing textbook treatments, which are necessarily published with a greater lag time. By facilitating access to legacy data and published modeling studies (incidentally making it easier to identify gaps and open questions), the transparency of the resource will ease entry of new researchers into mathematical modeling of the kidney and could contribute to a transformation of the perception of modeling studies. Instead of being the domain of a few experts, modeling could become a part of the normal toolbox of renal research labs and perhaps even part of the standard curriculum, because the teaching of renal physiology would be greatly facilitated, and renal physiology de-mystified, by interactive

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access to the principal modeling studies that have contributed significantly to the field. B. Two modeling site prototypes 1) KSim present state: We have developed a prototype Java applet to display simulation results (http://www.lami.univevry.fr/∼srthomas/kidneysim). To ground the development in a realistic context, we implemented this prototype for one of our own models of the renal medulla [79]. On this website, separate web-pages present: the history leading to the development of this particular model; a description of the model itself (including a table of the basic parameter values); and a window giving entry into a Java applet for presentation of simulation results. The visitor may choose among a small set of previous simulations (typical of those presented in the published article) or may launch a new simulation (server-side execution) based on the user’s modifications of a selected set of model parameter values. The applet loads the chosen set of simulation results in the form of an XML file and permits the web visitor to display them as x-y plots or as a color-gradient diagram of the medullary structures. These are illustrated in the screenshots of figure 4 showing the applet GUI, one under Windows XP and the other under Mac OS X.

Fig. 4. Screenshots from java applet for display of simulation results from the ‘flat’ medullary model of [79]. Upper panel shows color gradient of urea concentrations in medullary structures. Lower panel shows the graphing interface (total osmolality along collecting ducts).

Development of this demonstration gave us valuable experience concerning the possibilities and limitations of existing

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graphics packages for Java applets and revealed key development objectives to be met in order to develop a more flexible and generic approach for use with the variety of models projected for the Renal Physiome site. 2) Kidneysim3D – Virtual Kidney interface: The screenshots in figures 5 and 6 show a first version of our ‘virtual kidney’ interface [111][112][113]. This preliminary version (funded by the Victorian Partnership for Advanced Computing), constructed from casts of a rat kidney, is still far from complete. We propose to use this interface not only for didactic exploration of kidney anatomy but also as an interface to the collection of curated, interactive models at various scales. For exploration of kidney anatomy, in the top screenshot of figure 5, the model in the left panel can be zoomed, rotated, and even disassembled using the computer mouse and its buttons. This is illustrated in the bottom screenshot, where the capsule, OM, and blood vessels have been removed, revealing an anatomically correct superficial nephron whose path through the OM can be examined by rotating and zooming.

Fig. 5. 3D-Virtual Kidney interface. (Upper) For exploration of kidney anatomy, the model in the left panel can be zoomed, rotated, and even disassembled using the mouse and its buttons. The right panel displays graphs of simulation results from a model selected from a roll-down list. (Lower) The capsule, OM, and blood vessels have been removed (checkboxes between the left and right panels), revealing a superficial nephron whose path through the OM can be examined by rotating and zooming.

To illustrate the implementation of a model of transport along the distal tubule [31], Figure 6, left panel, shows a closeup view of the nephron; a graph, right panel, contains colored lines that represent profiles of ion concentrations (Na+ , K+ ,

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and Cl− ) along the length of the tubule. A slider at the bottom of the panel moves through the simulated time-steps. In the nephron shown in the left panel, at this scale, one can clearly see a ‘rod’; this represents the distal tubule in this preliminary version and the colors along its length change to indicate changing concentration of a selected variable. In the next phase of development, this will be mapped onto the distal tubule (instead of onto the representative rod), and many additional simulations of the other nephron segments will be implemented.

Fig. 6. Closeup view of superficial nephron (left panel) and graphs of distal tubule transport model results (right panel); the colored lines represent profiles of ion concentrations (Na+ , K+ , and Cl− ) along the length of the tubule. A slider at the bottom of the right panel moves through the simulated timesteps. In the left panel, at this scale, the spurious ‘rod’ represents the distal tubule in this prototype, and the colors along its length change to indicate concentration of a selected variable.

This GUI is being built using the XUL environment of the Mozilla group (open source development tools) to provide a web front-end for the 3D visualization environment developed by the group of Peter Hunter in Auckland. Their work has long been applied to models of cardiac physiology, and they are now working with us to adapt these tools for the kidney. They aim to provide a general GUI for physiome models across all organ systems (at all spatial scales) via the ontology database. It is intended both as a means of navigating the model databases and as a means of running model simulations and viewing simulation results. V. C ONCLUSION We have only begun to realize the potential of modeling to obtain a comprehensive understanding of renal function. The new Physiome resources will provide general access to both new and legacy models while facilitating communication and collaboration between modelers and the rest of the renal research community. They will also streamline future model development and, hopefully, ease the entry of new people into this field. ACKNOWLEDGMENTS We are grateful to our colleagues Peter Harris and Andrew Lonie in Melbourne and Carey Stevens in Auckland for their

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[98] L. N. Reinking and B. Schmidt-Nielsen, “Peristaltic flow of urine in the renal capillary collecting ducts of hamsters,” Kidney Int, vol. 20, no. 1, pp. 55–60, 1981. [99] J. P. Pennell, F. B. Lacy, and R. L. Jamison, “An in vivo study of the concentrating process in the descending limb of Henle’s loop,” Kidney Int, vol. 5, pp. 337–347, 1974. [100] B. Schmidt-Nielsen, B. Graves, and H. MacDuffie, “Effect of peristaltic contractions of the renal papilla in hamsters, mesocricetus auratus,” Bulletin of the Mount Desert Island Biological Laboratory, vol. 25, pp. 70–72, 1985. [101] E. L. Chuang, H. J. Reineck, R. W. Osgood, and J. R. T. Kunau, “Studies on the mechanism of reduced urinary osmolality after exposure of the renal papilla,” J Clin Invest, vol. 61, pp. 633–639, 1978. [102] R. E. Oliver, D. R. Roy, and R. L. Jamison, “Urinary concentration in the papillary collecting duct of the rat,” J Clin Invest, vol. 69, pp. 157–164, 1982. [103] W. Sch¨utz and J. Schnermann, “Pelvic urine composition as a determinant of inner medullary solute concentration and urine osmolality,” Pfl¨ugers Arch, vol. 334, pp. 154–166, 1972. [104] T. M. Dwyer and B. Schmidt-Nielsen, “The renal pelvis: machinery that concentrates urine in the papilla,” News Physiol Sci, vol. 18, pp. 1–6, 2003. [105] H. E. Layton, “Concentrating urine in the inner medulla of the kidney,” Comments Theor Biol, vol. 1, pp. 179–196, 1989. [106] M. A. Knepper, G. M. Saidel, V. C. Hascall, and T. M. Dwyer, “Concentration of solutes in the renal inner medulla: interstitial hyaluronan as a mechano-osmotic transducer,” Am J Physiol Renal Physiol, vol. 284, pp. F433–F446, 2003. [107] J. M. Sands and H. E. Layton, “Urine concentrating mechanism and its regulation,” in The Kidney: Physiology and Pathology (3d ed.), D. W. Seldin and G. Giebisch, Eds. Philadelphia: Lippincott Williams & Wilkins, 2000, pp. 1175–1216. [108] C. S. Peskin, “The immersed boundary method,” Acta Numerica, vol. 11, pp. 479–517, 2002. [109] F. Bayle, L. Eloy, M. M. Trinh-Trang-Tan, J. P. Grunfeld, and L. Bankir, “Papillary plasma flow in rats. I. Relation to urine osmolality in normal and Brattleboro rats with hereditary diabetes insipidus,” Pfl¨ugers Archiv - European Journal of Physiology, vol. 394, no. 3, pp. 211–6, 1982. [110] V. Dzodic, S. Hervy, D. Fritsch, H. Khalfallah, M. Thereau, and S. R. Thomas, “Web-based tools for quantitative renal physiology,” Cell. Molec. Biol., vol. 50, no. 7, pp. 795–800, 2004. [111] A. J. Lonie, C. Stevens, and P. Harris, “Computer modelling of kidney function,” in Experimental Biology 2004, Washington D.C., 2004, p. A107. [112] A. Lonie, C. Stevens, P. Harris, and S. R. Thomas, “Abstract 4086: Visualization tools for the renal physiome,” in IUPS/Experimental Biology, San Diego, 2005. [113] S. R. Thomas, H. E. Layton, A. Layton, P. J. Harris, A. Lonie, and L. Moore, “Towards a web resource for quantitative renal physiology,” J. Physiol., vol. 565P, p. 98P, 2005.

S. Randall Thomas S. Randall Thomas received his B.A. in Biology from Swarthmore College in 1973, a Ph.D. in Physiology from Medical College of Virginia, Richmond, in 1977, and the HDR (Habilitation a` Diriger des Recherches) from Univ. Paris 5 in 1990. After postdocs at the French Atomic Energy Commission at Saclay (1979) and in the Dept. of Physiology at Univ. Texas Med. Center, Houston (1980-81), he obtained a permanent position with the National Center for Scientific Research (CNRS), France, in 1982. Until December 2004, he worked in INSERM Unit 467 at Necker Medical School, Paris. He is presently a Director of Research with the CNRS and head of the physiological modeling team in the LaMI (Laboratoire de M`ethodes Informatiques), UMR 8042 CNRS, in Evry, near Paris. His research centers on mathematical modeling of integrated transport systems in renal and epithelial physiology and on the development of web-based interfaces to such models and to related databases. He is on the editorial boards of Nephron Physiology and Synthetic and Systems Biology and is secretary of the French Society for Mathematical Biology (SFBT).

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Anita Layton Anita Layton received her M.Sc. and Ph.D. in Computer Science from the University of Toronto (Ontario, Canada) in 1996 and 2001, respectively. Afterwards, she was a postdoctoral fellow at the National Center for Atmospheric Research (Boulder, CO) and at the Department of Mathematics of the University of North Carolina (Chapel Hill, NC). She is now an Assistant Research Professor at the Department of Mathematics of Duke University (Durham, NC). Her research interests include numerical methods for combustion and multi-scale models, and of course modeling of renal physiology.

Harold Layton Harold Layton received an A.B. in mathematics from Asbury College in 1979, an M.S. in physics from the University of Kentucky in 1980, and a Ph.D. in mathematics from Duke University in 1986. From 1986 to 1988, he was a Visiting Member and NSF Postdoctoral Fellow at the Courant Institute of Mathematical Sciences of New York University. He returned to Duke University in 1988 as assistant professor of mathematics and was subsequently promoted to associate professor (1995) and to professor (2001). Layton’s research has been devoted to using mathematical methods and models to elucidate renal hemodynamics and the urine concentrating mechanism. He has regularly taught courses in scientific computing and mathematical biology. Layton and Alan M. Weinstein (of the Weill Medical College of Cornell University) were coeditors of Membrane Transport and Renal Physiology (Springer, 2002), a workshop proceedings volume that can serve as an introduction to renal modeling.

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Leon Moore Leon C. Moore received the B.S degree in Electrical Engineering from Brigham Young University, and the Ph.D. degree in Biomedical Engineering from the University of Southern California. He spent two years as a NATO Postdoctoral Fellow at the Physiological Institute of the University of Munich where he studied renal physiology. In 1978, he joined the Faculty of the State University of New York at Stony Brook where he is currently Professor of Physiology & Biophysics and Biomedical Engineering. His research interests include the regulation of renal hemodynamics, mathematical modeling, and the mechanism of tear production by the lacrimal gland.