## Abstract

We developed a mathematical model of calcium (Ca^{2+}) transport along the rat nephron to investigate the factors that promote hypercalciuria. The model is an extension of the flat medullary model of Hervy and Thomas (*Am J Physiol Renal Physiol* 284: F65–F81, 2003). It explicitly represents all the nephron segments beyond the proximal tubules and distinguishes between superficial and deep nephrons. It solves dynamic conservation equations to determine NaCl, urea, and Ca^{2+} concentration profiles in tubules, vasa recta, and the interstitium. Calcium is known to be reabsorbed passively in the thick ascending limbs and actively in the distal convoluted (DCT) and connecting (CNT) tubules. Our model predicts that the passive diffusion of Ca^{2+} from the vasa recta and loops of Henle generates a significant axial Ca^{2+} concentration gradient in the medullary interstitium. In the base case, the urinary Ca^{2+} concentration and fractional excretion are predicted as 2.7 mM and 0.32%, respectively. Urinary Ca^{2+} excretion is found to be strongly modulated by water and NaCl reabsorption along the nephron. Our simulations also suggest that Ca^{2+} molar flow and concentration profiles differ significantly between superficial and deep nephrons, such that the latter deliver less Ca^{2+} to the collecting duct. Finally, our results suggest that the DCT and CNT can act to counteract upstream variations in Ca^{2+} transport but not always sufficiently to prevent hypercalciuria.

- calcium
- finite volume
- kidney
- mathematical model
- transport

calcium (ca^{2+}) is the most abundant cation in the body. It plays an essential role in cardiac, skeletal, and smooth muscle function, and extra- and intracellular Ca^{2+} concentrations ([Ca^{2+}]) must therefore be kept within a narrow range. Calcium homeostasis is maintained by the concerted action of the intestine, the parathyroid glands, and the kidneys. Preserving low Ca^{2+} concentrations in the urinary filtrate is especially important, since hypercalciuria is one of the major risk factors for the formation of kidney stones. Recent studies on Ca^{2+} handling by the kidney have focused on the molecular transporters and sensors of Ca^{2+} (2, 9), but our understanding of Ca^{2+} transport at the organ level remains limited. Whether there is a corticomedullary interstitial [Ca^{2+}] gradient has yet to be determined, and the contribution of each nephron segment to perturbations in the renal Ca^{2+} balance is not fully understood. To address these questions, we developed a mathematical model of renal Ca^{2+} transport across all nephron segments below the proximal tubules from the descending limbs of Henle to the inner medullary collecting ducts (IMCD). Our model is based on that of Hervy and Thomas (17), referred to below as the HT model, which describes the transport of water, NaCl, glucose, and lactate in the renal medulla. We modified the HT model to *1*) make it dynamic, *2*) explicitly represent the distal cortical segments, and *3*) incorporate Ca^{2+} transport. For this purpose, we developed a new finite-volume scheme that is robust and fast.

About two-thirds of the filtered load of calcium are passively reabsorbed in the proximal tubule and 20–25% in the loop of Henle. The renal excretion of Ca^{2+} is then fine tuned in the distal convoluted tubule (DCT) and the connecting tubule (CNT), which actively reabsorb ∼10% of the filtered load. Transepithelial Ca^{2+} exchanges are thought to be negligible in the collecting duct (9). Since our model does not represent the epithelial barrier at the cell membrane level, we described the active reabsorption of Ca^{2+} in the DCT-CNT as a saturable Michaelis-Menten relationship, dependent on intraluminal [Ca^{2+}], and we adjusted its parameters accordingly. Other Ca^{2+} transport parameter values were taken from experimental studies.

Our model predicts a significant interstitial [Ca^{2+}] gradient along the medullary axis, which stems mostly from passive Ca^{2+} reabsorption in the vasa recta and loops of Henle. Our results also suggest that the DCT and CNT are able to adjust to changes in Ca^{2+} transport upstream, but not always sufficiently to prevent hypercalciuria.

## MODEL DESCRIPTION

### Physical Representation

The model represents all tubules and vessels in the medulla, that is, descending (DL) and ascending (AL) limbs of Henle, collecting ducts (CD), and descending (DVR) and ascending (AVR) vasa recta, all of which are immersed in a common interstitium. In contrast with the HT model that specifies steady state boundary conditions at the inlet of the outer medullary collecting duct (OMCD), we explicitly represent the cortical distal tubules that connect the medullary thick ascending limb (mTAL) to the OMCD. The four cortical segments include the cortical thick ascending limb (cTAL), the distal convoluted tubule (DCT), the connecting tubule (CNT), and the cortical collecting duct (CCD).

We distinguish between superficial and deep nephrons. As depicted in Fig. 1, the superficial nephrons have short loops of Henle that do not penetrate into the inner medulla, whereas the deep nephrons have longer loops. The superficial nephrons also have a longer cTAL and a shorter CNT, relative to the deep nephrons. We assume that superficial and deep nephrons connect at the CCD entrance. Among the deep nephrons, we make a further distinction, between those that turn within the first millimeter of the inner medulla and those that reach below; as described below these two types have different hydraulic conductivity profiles. Within a given class of tubules or vasa recta, physical properties (e.g., permeability and radius) are identical at a given position but may vary with depth.

The renal medulla is divided into the outer medulla (OM) and the inner medulla (IM), as illustrated in Fig. 1. The outer medulla itself consists of an outer stripe (OS) and an inner stripe (IS). The inner medulla is also subdivided into an upper part (UIM) and a lower part (LIM). Whereas the cortical interstitium is taken to be isosmotic with plasma, solute concentrations in the renal medullary interstitium vary along the corticomedullary axis. The model explicitly represents three solutes, NaCl (denoted “NaCl”), urea (“U”), and Ca^{2+} (“Ca”).

### Number and Length of Vessels and Tubules

In the cortex, the DCT and CCD are taken to be 0.7- and 3.0-mm long, respectively. As noted above, the length of the cTAL and CNT varies between superficial and deep nephrons. These segments are taken to be, respectively, 2.5- and 1.0-mm long in superficial nehrons, and 0.5- and 3.0-mm long in deep nephrons. The combined length of the cTAL, DCT, and CNT (denoted *L*_{cTDC}) is therefore equal to 4.2 mm in both superficial and deep nephrons. We denote by *y*_{L} the spatial coordinate at the boundary between the DCT and the CNT in deep nephrons, by *y*_{S} that in superficial nephrons, and by *L*_{d} the total length of the cortical distal tubules (7.2 mm).

The total length of the rat medulla (*L*) is taken as 6 mm. The abscissas *x*_{OS/IS} (0.7 mm) and *x*_{OM/IM} (2 mm), respectively, represent the spatial coordinate at the boundary between the OS and IS and between the OM and IM. The long loops of Henle and vasa recta turn back at varying distances from the cortex, as depicted in Fig. 1. The total number of nephrons is taken as 30,000 (4). The numbers of vessels and tubules of type *j* (denoted *N*^{j}) are shown as a function of position in Fig. 2.

#### Vessels.

The number of long descending vasa recta (*N*_{L}^{DVR}) is taken to decrease exponentially in the IM

The number of short descending vasa recta (*N*_{S}^{DVRL}) is taken to be constant in the OS and to linearly decrease in the IS

Note that in the model representation, the number of AVR (or AL) at a given level is equal to that of DVR (or DL) at that level.

#### Loops of Henle.

The number of long descending limbs (*N*^{LDL}) is given by

The number of short descending limbs (*N*^{SDL}) is taken to be constant in the OM

#### Cortical distal tubules.

We distinguish between the cTAL, DCT, and CNT of long-looped (superscript “dL”) and short-looped (superscript “dS”) nephrons. In both deep and superficial nephrons, the number of cTAL and DCT remains constant, and the number of CNT is taken to decrease exponentially as CNT merge on their way towards CCD. Thus we have
where *N*^{dL}(0) is the number of long loops at the corticomedullary (CM) junction, i.e., *N*^{LDL}(0). Similarly
where *N*^{dS}(0) is the number of short loops at the CM junction, i.e., *N*^{SDL}(0).

We assume that deep and superficial nephrons join at the CCD entrance. The number of CCD (*N*^{CCD}) is taken to decrease exponentially

#### Medullary CDs.

The number of medullary collecting ducts (*N*^{mCD}) is constant in the OM and decreases in the IM due to coalescence; we assume the following distribution

The constants *k*^{DVR}, *k*^{LDL}, and *k*^{mCD} are computed so that the number of tubules is ≈75 at *x* = *L* (16). We also make the assumption that the parameter *k*^{d}, which characterizes the exponential decrease of tubules in the cortex, is the same for deep and superficial nephrons; *k*^{d} is adjusted so that the number of CD leaving the cortex is equal to that entering the medulla (that is, 5,000). With these assumptions, *k*^{DVR} = *k*^{LDL} = 1.22 mm^{−1}; *k*^{mCD} = 0.70 mm^{−1}; and *k*^{d} = 0.40 mm^{−1}.

### Physical Variables

The unknowns to be determined are defined for all time *t* > 0, and for all *x* ∈ [0, *L*] for the following compartments: DVR, AVR, long descending limbs (LDL), long ascending limbs (LAL), medullary CD, and the interstitium; for all *x* ∈ [0, *L*/3] for short descending limbs (SDL) and short ascending limbs (SAL); and for all *y* ∈ [0, *L*_{d}] for cortical distal tubules. Let the superscript *j* denote a given type of tube, and the subscript *i* a given solute (*i* = NaCl, U, Ca).

*F*^{j}(*t*, *x*) (in nl/min) is the water flow in one representative tube of type *j* at depth *x* and time *t*. It represents the volume of water passing through the area centered on *x* in 1 min. Thus the variable *N*^{j}(*x*)*F*^{j}(*t*, *x*) represents the total water flow in all tubes *j* at depth *x* and at time *t*. Note that in ALs and AVR, *F*^{j} is a negative number.

C_{i}^{j}(*t*, *x*) (in mmol/l) is the concentration of solute *i* in one representative tube *j*. The product *F*^{j}(*t*, *x*)C_{i}^{j}(*t*, *x*) (in pmol/min) represents the number of pmoles of solute *i* passing through the area centered on *x* in 1 min.

*J*_{i}^{V}(*t*, *x*) (in min^{−1}·mm^{−1}) is the flux of water into a representative tube *j*. It represents the signed surface of water entering one tube *j* across the wall at depth *x* in 1 min.

*J*_{i}^{j}(*t*, *x*) (in pmol·min^{−1}·mm^{−1}) is the flux of solute *i* into a representative tube *j*. It represents the number of pmoles of *i* entering one tube *j* across the wall at depth *x* in 1 min. The fluxes are taken to be positive if directed into the tubular or vascular lumen, and negative otherwise.

### Model Equations

#### Conservation equations.

The variables are related by conservation equations, which are formally derived in appendix a. For DVR, AVR, LDL, LAL, SDL, and SAL, and for each solute *i*, the equations are written as
(1)where *r*^{j} is the radius of tube *j*, and *j*′ = DVR if *j* = DVR or AVR, *j*′ = LDL if *j* = LDL or LAL, and *j′* = SDL if *j* = SDL or SAL.

For the cortical distal tubules and medullary CD we have
(2)An exception to *Eq. 1* must be made. In plasma, a significant fraction of calcium is bound to proteins such as albumin. Elsewhere (i.e., in the interstitial and tubular fluids), protein concentration is taken to be negligible. If Ca.P denotes protein-bound calcium, the conservation equation for the total amount of calcium in DVR is expressed as
(3)

We assume that the fraction of calcium that is protein-bound is fixed and equal to 50% (41), so that C_{Ca.P}^{DVR} = C_{Ca}^{DVR} everywhere. The previous equation is then rewritten as
(4)A similar equation describes the conservation of calcium in AVR.

We complete the conservation equations with two closure conditions. Ignoring the elasticity of the vessels, tubules, renal pelvis and that of the renal capsule, we assume interstitial, tubule and vessel volumes to be constant (they do not accumulate water). Hence, we have (5)Solute conservation in the medullary interstitium (denoted “int”) is written as (6)

In the cortical interstitium, assumed to be isosmotic with plasma, the concentrations of NaCl, urea, and calcium are taken as 139, 10, and 1.2 mM respectively.

#### Flux equations.

The water flux is given by
(7)where *L*_{p}^{j} is the water conductivity of tube *j*, σ_{i}^{j} is the reflection coefficient of tube *j* to solute *i*, and δ is taken as 1 in tubules and 0 in DVR. Note, firstly, that the water conductivity of AVR is so large that the corresponding water flux is calculated based on water conservation instead (see *Eq. 5*). Secondly, *Eq. 7* implicitly assumes that hydrostatic pressures are negligible relative to osmotic pressures. Lastly, the mechanisms underlying the formation of the axial osmolality gradient in the IM remain to be fully elucidated, as discussed below. To reproduce the observed gradient, we follow the approach of Thomas and Wexler (35) and add an external osmotic driving force in the IM interstitium. Thus *E* is the concentration of external osmolytes in the interstitium, taken as 0 in the OM and 75 mM in the IM.

Solute transport is driven by electrodiffusion, convection, and active transport. The flux of uncharged solutes (NaCl and urea) is written as
(8)with
(9)The first term represents diffusive transport; *P*_{i}^{j} is the permeability of tube *j* to solute *i*. The second term, which represents convective transport, is formulated differently than in other kidney models (17, 26), where it is expressed as:
(10)To insure that our system is well-posed and positive, we use instead an upwind term.^{1} The third term represents active transport, with Michaelis-Menten kinetics. For a charged solute such as Ca^{2+}, the flux is written as
with
(12)and
(13)

Solute conservation implies that
(14)This last equation, which closes the system, expresses the fact that the net flux of solute into the interstitium (*J*_{i}^{int}) is the opposite of the net flux from the interstitium into surrounding tubules and vessels.

#### Boundary conditions.

Water and solute flows are specified at the inlet of DVR and of long and short DL for all times (15)

In addition, by continuity between adjacent tubules and vessels, we have The last two equations express the fact that the total water and molar flows are conserved when superficial and deep nephrons merge at the CCD inlet.

### Model Parameters

Model parameters are estimated based on measurements in rats. Water and solute permeability values are summarized in Table 1. Based on recent reconstructions of renal anatomy (29), the SDL are taken to be impermeable to water in the IS. The water permeability of the LDL that turn at a distance <1 mm below the OM-IM junction is zero in the IM and that of the remaining LDL is zero along the last 60% of their IM length.

At a given axial position *x*, the average hydraulic conductivity of the latter class of LDL (denoted LDL2 immediately below) is calculated as follows. We denote by *L*_{p}(*x*, *y*) the conductivity at *x* of a tubule turning at depth *y*. It is piecewise constant over the first 40% of the tubule IM length, and zero below that
where *L*_{p}^{o}(*x*) is a constant that differs in the OS, IS, and IM, and whose values are given in Table 1. We then compute the average hydraulic conductivity at position *x* as
It can be shown that
(16)

Also given in Table 1 are radius values for each set of tubules and vessels. As for size of the interstitium, we calculate its equivalent radius *r*^{int} (see *Eq. 6*) by assuming that at a given depth *x*, the cross-sectional area of the interstitium is 40% that of tubules (25). In other words
(17)

The permeability of the rat cTAL to Ca^{2+} is taken as 5 × 10^{−5} cm/s, i.e., an intermediate value between the value of 8.4 × 10^{−5} cm/s measured in our laboratory (30) and a lower value of 1 × 10^{−5} cm/s estimated from the study of Mandon et al. (24). That study suggested that the permeability of the rat mTAL to Ca^{2+} is 10 times lower than that of the cTAL; the former is thus taken as 0.5 × 10^{−5} cm/s here. To our knowledge, the permeability of the tAL to Ca^{2+} has not been measured in rats. In rabbits, it is similar to that of the mTAL (33). Thus we take it as 0.5 × 10^{−5} cm/s in the present study.

The rat DL was found to transport Ca^{2+} (18), but permeability values could not be inferred from the latter study. Based on measurements in rabbits (31, 32), the DL permeability to Ca^{2+} is taken as 0.7 × 10^{−5} cm/s. The Ca^{2+} permeability of the DCT and CNT (0.1 × 10^{−5} cm/s) and that of the CD (0.01 × 10^{−5} cm/s) are estimated based on a compilation of values from several studies (33).

We did not find any measurements of the permeability of vasa recta to Ca^{2+}. The latter is chosen as 1×10^{−5} cm/s, and the impact of this parameter value is assessed below.

Shown in Table 2 are active transport parameters for NaCl and Ca^{2+}. The NaCl parameter values are comparable to those of the HT model; they are slightly modified so that ∼25% of the calcium filtered load is reabsorbed in the ALs of superficial nephrons. The Ca^{2+} parameters are chosen so that the fractional Ca^{2+} reabsorption in the DCT and CNT is about 10%.

### Numerical Method

To approach numerically the solution to this model, we developed a finite-volume scheme that is described in appendix b. The scheme is robust in that it converges towards the steady-state solution independently of our choice of initial conditions. It is implemented using Matlab on a computer equipped with an Intel Core I5 CPU M 460 @ 2.53 GHz × 4 processor with 3.5 GB of RAM. Steady state is typically reached within 15 min of CPU time.

## RESULTS

We first examined Ca^{2+} flow and concentration profiles along the nephron under baseline conditions. We then simulated scenarios under which Ca^{2+} handling by specific segments is altered to investigate how the kidney responds overall to such changes.

### Base Case Osmolality Profiles

Base case steady-state profiles of NaCl concentration, urea concentration, and total osmolality in medullary segments are similar to those of the HT model and are not shown here. The osmolality of the tubular fluid at the CD outlet is predicted as 1,440 mosmol/kgH_{2}O (with an external osmole concentration of 75 mM in the IM). Shown in Fig. 3 are tubular fluid osmolality profiles in cortical distal segments. Note that these profiles differ significantly between superficial and deep nephrons, since the latter have a shorter cTAL and a longer CNT. We assumed that the water permeability of the DCT and CNT is about five times lower in deep nephrons than in superficial ones (Table 1), so as to avoid luminal fluid depletion in the former.

The fluid exiting the mTAL is significantly hypo-osmotic relative to plasma, following the massive reabsorption of NaCl that occurs in this water-impermeable segment. In the cTAL, which is similarly water impermeable, tubular fluid osmolality continues to decrease but less steeply.

Water reabsorption resumes in the DCT, but the hydraulic conductivity of this segment is too small to counteract the effects of active sodium reabsorption; tubular fluid osmolality is therefore predicted to decrease in the DCT. It begins to increase in the CNT, the water permeability of which is significantly greater than that of the DCT. Osmotic equilibration is predicted to occur very early in the CCD. Once the osmolality of the CCD fluid has reached that of the surrounding interstitium, it remains constant.

### Base Case [Ca^{2+}] Profiles

[Ca^{2+}] profiles at steady state are shown in Fig. 4, *A* (medulla) and *B* (cortex). In the longest DLs, the predicted [Ca^{2+}] increases severalfold between the corticomedullary junction and the midpoint of the medulla, because of massive water reabsorption. Assuming that the permeability of DL to Ca^{2+} is not negligible, it allows for some passive diffusion of Ca^{2+} into the interstitium. The latter is too small to counterbalance the effect of water abstraction in the upper regions of the medulla. However, in the deep medulla, where DL are impermeable to water, luminal [Ca^{2+}] diminishes slightly as a result of diffusion (Fig. 4*A*). In shorter DLs, luminal [Ca^{2+}] similarly increases rapidly along the water-permeable portion of the tubule, and decreases slightly along the water-impermeable portion (not shown).

Luminal [Ca^{2+}] continues to decrease along (water-impermeable) ALs, because of passive Ca^{2+} reabsorption. In the thick portion of the limb, the lumen-positive transepithelial voltage gradient (Δ*V*_{TE} = +15 mV) greatly increases the driving force for Ca^{2+} transport. Overall, the luminal [Ca^{2+}] in the AL is predicted to decrease by 35% between the papillary tip and the CM junction and then more abruptly in the cTAL because this segment is significantly more permeable to Ca^{2+}. Note that [Ca^{2+}] decreases more in the cTAL of superficial nephrons (Fig. 4*B*), since the latter is taken to be five times longer than that of deep nephrons.

In the DCT, even though passive diffusion favors Ca^{2+} secretion (since Δ*V*_{TE} < 0 and [Ca^{2+}] is lower in the lumen than in the interstitium), active transport mechanisms predominate and [Ca^{2+}] decreases steeply in deep and superficial nephrons. Calcium continues to be vigorously reabsorbed along the CNT. In deep nephrons, where the CNT is taken to be 3-mm long, luminal [Ca^{2+}] is predicted to decrease be <0.1 mM.

In the model configuration, the superficial and deep nephrons meet at the CCD entrance. Solute concentrations at the CCD inlet are therefore a weighted average between the CNT outlet concentrations of deep and superficial nephrons. Beyond the CCD inlet, transepithelial Ca^{2+} exchanges are small. The predicted [Ca^{2+}] increase along the medullary CD (Fig. 4*A*) is predominantly due to water abstraction.

We assumed that the vasa recta are permeable to Ca^{2+}. The model predicts that [Ca^{2+}] increases steadily along DVR, even though there is some passive Ca^{2+} diffusion into the interstitium, because of water reabsorption. In contrast with DLs, DVR are water-permeable over their full length, so that vascular [Ca^{2+}] rises along the entire vessel (Fig. 4*A*). Conversely, the predicted [Ca^{2+}] decrease in AVR is predominantly driven by water uptake.

### Base Case Ca^{2+} Molar Flow

The molar flow rate of Ca^{2+} at steady state is displayed in Fig. 5. Note that the molar flow per tubule can vary solely as a result of variations in tubule number, that is, even when there is no transepithelial solute transport. Such effects are particularly manifest in CDs, which exchange very little Ca^{2+} with the interstitium; yet, because the model represents 16,475 CDs that converge to 75 CDs in the IM, the Ca^{2+} molar flow per tubule increases steeply along the CCD and the IMCD. We refer to this phenomenon hereafter as the “coalescing” effect.

As shown in Fig. 5*A*, the Ca^{2+} load diminishes from 12 to ∼10 pmol·min^{−1}·tubule^{−1} along the longest DL, and continues to diminish slightly along the tAL. This decrease results primarily from passive diffusion into the interstitium in the OM and upper IM. In the mTAL, the driving force for passive Ca^{2+} reabsorption is enhanced by the lumen-positive voltage gradient. As a result, the Ca^{2+} load per mTAL decreases significantly. The rate of passive Ca^{2+} reabsorption is further augmented along cTALs, since they are severalfold more permeable to Ca^{2+} than mTALs.

In deep nephrons, the load of Ca^{2+} entering the DCT is predicted as ∼4 pmol·min^{−1}·tubule^{−1}. Following active reabsorption, it becomes almost negligible at the CNT outlet. In superficial nephrons, the molar flow of Ca^{2+} at the DCT entrance is ∼4.5 pmol·min^{−1}·tubule^{−1}. It decreases less along the DCT and CNT (relative to long nephrons), because the latter segment is much shorter. Along the CD, a very small amount of Ca^{2+} is secreted into the tubular lumen, as a result of the lumen-negative transepithelial voltage and the large interstitial-to-lumen [Ca^{2+}] gradient in the medulla. Thus the total Ca^{2+} molar flow rate increases slightly. The Ca^{2+} load per tubule rises very steeply, however, owing to the coalescing effect.

Base case results are summarized in Table 3. The concentration of Ca^{2+} at the CD outlet (C_{out}) is predicted to be 2.7 mM and the Ca^{2+} molar flow at that point represents 0.32% of the filtered load. These model predictions are in good agreement with experimental data, as discussed below.

### Sensitivity Analysis

Most parameter values were taken from experimental studies, except the following: the permeability of vasa recta to Ca^{2+}, which has not been measured to our knowledge, and the macroscopic parameters that describe the rate of active Ca^{2+} transport in the DCT-CNT. Other uncertain values include transepithelial voltage differences and the IM osmotic force. Table 3 illustrates the sensitivity of model predictions to these parameters.

#### Vasa recta permeability to calcium.

The baseline vasa recta permeability to Ca^{2+} was chosen as 1.0 × 10^{−5} cm/s. We found that a twofold increase or decrease in this value has a negligible effect on urinary Ca^{2+} excretion but a significant impact on interstitial [Ca^{2+}] in the medulla (Table 3). Indeed, vasa recta are a major determinant of the axial interstitial [Ca^{2+}] gradient, as described below.

#### Active transport.

Our model represents the epithelial layer as a single barrier, and we used Michaelis-Menten equations to describe the active transport of Ca^{2+}. The corresponding parameters, that is, the maximal rate of active Ca^{2+} reabsorption in the DCT-CNT (*V*_{m,Ca}^{d}) and the [Ca^{2+}] at which the rate is half its maximal value (*K*_{m,Ca}^{d}) were fitted so that the DCT-CNT would reabsorb ∼10% of the filtered load of Ca^{2+}. Not surprisingly, model results are very sensitive to *V*_{m,Ca}^{d} and *K*_{m,Ca}^{d}. A twofold increase in *V*_{m,Ca}^{d} would reduce C_{out} by a factor of 3. Conversely, a twofold decrease in *V*_{m,Ca}^{d} would raise C_{out} by a factor of 4.5. A twofold increase in *K*_{m,Ca}^{d} would have comparable effects on Ca^{2+} transport as a twofold decrease in *V*_{m,Ca}^{d} and vice-versa (Table 3). Note that *V*_{m,Ca}^{d} and *K*_{m,Ca}^{d} have little impact, however, on interstitial [Ca^{2+}] in the medulla.

#### Transepithelial voltage.

The transepithelial potential difference (Δ*V*_{TE}) greatly enhances the paracellular reabsorption of Ca^{2+} in the TAL, where it is lumen-positive (9). Thus as shown in Table 3, a twofold increase or decrease in the TAL Δ*V*_{TE} is predicted to respectively reduce or augment C_{out} by ∼25%. Average interstitial [Ca^{2+}] values are then predicted to vary by ∼10% in the OM, and <3% in the IM. Beyond the cTAL, Δ*V*_{TE} is lumen negative (9) and thus promotes Ca^{2+} secretion. Hence, in the DCT, CNT, and CD, a twofold increase (or decrease) in Δ*V*_{TE} is predicted to enhance (or lower) C_{out}. The latter varies by 20% at most, and medullary interstitial [Ca^{2+}] remain unaffected. Note that the greater the Ca^{2+} permeability of a given segment, the greater the effects of Δ*V*_{TE} variations in this segment on C_{out}.

#### External osmoles.

The [Ca^{2+}] at the CD outlet is affected by the IM osmolality gradient, which depends here on the imposed concentration of external osmoles in the IM (*E*). As shown in Table 3, a twofold increase in *E* would raise C_{out} by 25% and the average interstitial [Ca^{2+}] in the IM by 40%. Conversely, a two-fold decrease in *E* would lower C_{out} by 50% and the average interstitial [Ca^{2+}] in the IM by 15%. In both cases, however, the fractional Ca^{2+} excretion would not vary significantly.

### Interstitial [Ca^{2+}] Gradient

At steady state, Ca^{2+} diffuses from the loops of Henle and DVR into the interstitium, from which it is carried into AVR by convection. Our model thus predicts a significant interstitial [Ca^{2+}] gradient along the medullary axis. If the permeability of all vessels and tubules to Ca^{2+} were zero, the gradient would vanish (results not shown). To assess the relative contribution of each structure, we computed the average interstitial [Ca^{2+}] in the OM and IM following a twofold increase in the Ca^{2+} permeability of selected tubules and vessels. As illustrated in Fig. 6, interstitial [Ca^{2+}] are determined mostly by the Ca^{2+} permeability of DVR and by that of AVR and limbs to a lesser extent. Our model predicts that DVR have a large impact on the interstitial [Ca^{2+}] gradient because they reabsorb water along their entire length and therefore concentrate Ca^{2+} to a higher extent than DL. Thus if their permeability to Ca^{2+} is on the order of 1 × 10^{−5} cm/s, as we have assumed, they strongly affect interstitial [Ca^{2+}] values.

As displayed in Fig. 4*A*, the predicted interstitial [Ca^{2+}] profile exhibits a sharp spike at the OM-IM boundary. This spike stems from abrupt changes in several parameter values at that point (especially in the transepithelial voltage difference), in combination with the manner in which the interstitial [Ca^{2+}] (C_{Ca}^{int}) is computed. As opposed to other variables, C_{Ca}^{int} is determined by an equation that does not contain a spatial derivative, which would act to smooth variations; as a result, the interstitial C_{Ca}^{int} profile echoes spatial discontinuities in model parameters.

### Effects of Local Perturbations

We then sought to investigate the specific contribution of each nephron segment in maintaining the renal Ca^{2+} balance. For this purpose, we simulated alterations in the Ca^{2+} transport properties of tubules and examined their impact on urinary Ca^{2+} excretion. Corresponding results are summarized in Tables 4 and 5 and in Fig. 7. Note that we did not account for the effects of hormonal regulation (such as the role of the parathyroid hormone) and feedback mechanisms (e.g., involving Ca^{2+} sensing receptors) in these simulations.

#### Calcium handling along the nephron.

The base case assumes that the filtered load of calcium equals 36 pmol·min^{−1}·tubule^{−1}, two-thirds of which are reabsorbed in the proximal tubule. Thus 12 pmol/min are delivered to each DL. If the latter amount is increased by a factor of 2, the fractional Ca^{2+} load at the mTAL entrance is predicted to remain approximately unchanged (Table 5), which means that in absolute terms, the DL and thin AL reabsorb twice as much calcium as in the base case. Downstream, the mTAL, cTAL, DCT, and CNT also reabsorb about the same proportion of the Ca^{2+} load. Hence, the fractional Ca^{2+} load at the CCD entrance is only slightly higher than in the base case (0.31 vs. 0.27%), and so is the fractional excretion of Ca^{2+} (0.37 vs. 0.32%). This means, however, that the urinary [Ca^{2+}], as reflected by C_{out}, is more than twofold greater than that in the base case (Table 4).

#### Calcium handling by the TAL.

The mTAL is generally thought to reabsorb insignificant amounts of Ca^{2+} relative to the cTAL (9). Based on flux measurements (24), we assumed that the mTAL is 10 times less permeable to Ca^{2+} than the cTAL. We found that this is sufficient to mediate some passive Ca^{2+} reabsorption in the mTAL (∼5% of the filtered load in the base case), given the large, lumen-positive Δ*V*_{TE}. When the mTAL permeability to Ca^{2+} was set to zero, the cTAL, DCT, and CNT each acted to compensate for the absence of Ca^{2+} reabsorption in the mTAL, and C_{out} was very close to its base case value (Tables 4 and 5).

When the cTAL permeability to Ca^{2+} was set to zero, the DCT and CNT also increased the active reabsorption of Ca^{2+} but not enough to fully offset the absence of reabsorption in the cTAL. In superficial nephrons, whose CNT is relatively short, the Ca^{2+} load at the CNT outlet was more than two times higher than in the base case. As a consequence, C_{out} was predicted to be >5 mM (Table 4). When the entire TAL was made impermeable to Ca^{2+}, the predicted hypercalciuria was even more severe. Even though the DCT and CNT together reabsorbed two to three times more Ca^{2+} than in the base case (Table 5), this did not fully compensate for the absence of reabsorption upstream, and the fractional excretion of Ca^{2+} was ∼2.5 times higher than in the base case.

#### Calcium handling by the DCT-CNT.

The renal excretion of Ca^{2+} is fine tuned in the DCT and CNT, wherein Ca^{2+} reabsorption is an active process that is mediated apically by transient receptor potential vanilloid 5 (TRPV5) channels, and basolaterally by type 1 Na^{+}-Ca^{2+} exchangers and plasma membrane Ca^{2+} pumps. Assuming that the passive permeability of the rat DCT and CNT to Ca^{2+} is 0.1 × 10^{−5} cm/s (33), the model predicts that the backflux of Ca^{2+} into these segments is not negligible, since the lumen-negative transepithelial voltage gradient favors Ca^{2+} secretion into the lumen. To assess the importance of this backflux, we set Δ*V*_{TE} to zero in the DCT and CNT. As shown in Table 4, the fractional Ca^{2+} load delivered to the CCD then decreased from 0.27 to 0.20% and C_{out} from 2.7 to 2.1 mM. When passive diffusion was fully abolished in the DCT and CNT (i.e., when the Ca^{2+} permeability of these segments was set to zero), the load of Ca^{2+} delivered to the CCD and C_{out} decreased slightly more (Table 4). These results suggest that if the Ca^{2+} permeability of the DCT and CNT is indeed on the order of 0.1 × 10^{−5} cm/s, then the back-flux of Ca^{2+} into the DCT and CNT lumen is not negligible.

We next simulated the full inhibition of active Ca^{2+} transport in the DCT and CNT, by setting *V*_{m,Ca}^{d} to zero. Owing to water abstraction, the luminal [Ca^{2+}] then increased continuously downstream from the cTAL, and the model predicted a massive increase in urinary Ca^{2+} excretion in the absence of compensating mechanisms along the CD or upstream (Table 5).

#### Calcium handling by the CD.

The luminal [Ca^{2+}] increases by a factor of ∼10 along the CD, primarily as a result of water reabsorption. In the base case, Ca^{2+} exchanges between the CD and the interstitium are predicted to be relatively minor. As shown in Table 4, setting the Ca^{2+} permeability of the CCD or medullary CD to zero slightly decreases urinary Ca^{2+} excretion, because it abolishes the small amount of Ca^{2+} secretion in that segment.

It should be noted that several studies have found significant rates of Ca^{2+} transport in the IMCD (3, 23), possibly via active mechanisms. Magaldi et al. (23) measured a Ca^{2+} flux in the IMCD in the absence of significant chemical and electric potential gradients and showed that Ca^{2+} reabsorption was Na^{+} dependent. Based on their results, we estimated the Ca^{2+} flux to be ∼1 pmol·min^{−1}·mm^{−1} tubule in the IMCD, which is lower than measured values in the CNT (8). We thus performed simulations in which the maximum rate of active Ca^{2+} transport (i.e., *V*_{m,Ca}) in the IMCD was ∼10% (*case 1*) or 1% (*case 2*) that in the DCT and CNT. All else being equal, the predicted C_{out} was 0.28 mM in *case 1* and 2.17 mM in *case 2*. To examine whether active Ca^{2+} transport in the IMCD could act to compensate for dysfunctions upstream, we then eliminated the reabsorption of Ca^{2+} in the DCT and CNT. Under these conditions, C_{out} was computed as 71.6 mM in *case 1* and 87.6 mM in *case 2*. These results suggest that if Ca^{2+} reabsorption can indeed be induced in the IMCD, it hardly counterbalances Ca^{2+} transport defects upstream.

## DISCUSSION

### Calcium Transport in Deep vs. Superficial Nephrons

In comparing our results with experimental measurements of renal Ca^{2+} transport, we must bear in mind the distinction between superficial and deep nephrons. Indeed, our model predicts significantly different flow and concentration profiles in these two types of nephrons. In regards to Ca^{2+} transport specifically, our results suggest that not only the amount of Ca^{2+} that is delivered to the mTAL but also the fractional load of Ca^{2+} that is reabsorbed along downstream segments, vary between deep and superficial nephrons. According to our simulations, 27% of the filtered Ca^{2+} load reaches the cTAL in deep nephrons, vs. 32% in superficial nephrons. The model also predicts that the TAL and DCT-CNT each reabsorb 17 and 9% of the load in deep nephrons, vs. 20 and 12% in superficial nephrons. Thus the Ca^{2+} molar flow and concentration profiles differ significantly in the distal segments emanating from long and short loops, up until the point where they merge (Figs. 4 and 5). Whether this is indeed the case in vivo remains to be determined experimentally.

It should be noted that our predictions generally agree well with the experimental studies reviewed in references (12, 14): according to the latter, ∼20% of the filtered load is reabsorbed in the TAL, and 10 to 15% in the DCT and CNT. In addition, the fractional excretion of Ca^{2+} is reported to be <1% under normal conditions (3, 28); the value predicted by our model (0.32%) falls within the experimental range. We also compared our results with measured values of tubular [Ca^{2+}]. Our model predicts that [Ca^{2+}] ranges between 3.5 and 5 mM at the loop bend in deep nephrons, which is somewhat higher than the reported value of 3 mM (18). In the distal tubule, [Ca^{2+}] is predicted to drop sharply (Fig. 3), and the computed range of values encompasses the experimental finding of 0.5 mM (21). There may be, however, some discrepancy between model predictions and reported values for the base (0.82 mM) and tip (0.77 mM) of the CD (36). Nevertheless, the predicted [Ca^{2+}] in the urine is well within the experimental range (1).

### Calcium Reabsorption in the TAL

It is generally thought that Ca^{2+} reabsorption is negligible in the mTAL under basal conditions (9). Interestingly, one study showed that the presence of bicarbonate enhances the mTAL Ca^{2+} flux (42), whereas most perfusion studies are performed in the absence of HCO_{3}^{−}. Based on flux measurements, we assumed here that the mTAL permeability to Ca^{2+} is one tenth that of the cTAL. Despite the 1:10 permeability ratio, the mTALs are predicted to reabsorb almost as much Ca^{2+} as the cTALs (7% of the filtered load, vs. 10%) in deep nephrons. This is because *1*) the longest mTAL is taken to be four times longer than the cTAL in deep nephrons, and *2*) the transepithelial [Ca^{2+}] gradient is more favorable to reabsorption in the medulla than in the cortex (see Fig. 4). However, we also found that in the absence of Ca^{2+} reabsorption in the mTAL, the cTAL reabsorbs a significantly higher fraction of the filtered load. Since the DCT and CNT also do so, the [Ca^{2+}] at the CD outlet is not significantly affected (Table 4). These results suggest that Ca^{2+} transport dysfunctions in the mTAL may not, per se, significantly affect urinary Ca^{2+} excretion.

As noted above, our model does not take into consideration the role of hormones and calcium-sensing receptors in regulating renal Ca^{2+} transport. In the TAL, activation of the (basolateral) calcium-sensing receptor CaSR directly affects transepithelial Ca^{2+} reabsorption, but the underlying signaling pathway remains partly unknown (13). Some studies, but not others, suggest that CaSR activation also indirectly affects Ca^{2+} reabsorption, by modulating the rate of Na^{+} transport across the Na-K-2Cl cotransporter NKCC2 (13). Both the qualitative and quantitative effects of CaSR activation, not only in the TAL but also in the other nephron segments that express the receptor, have yet to be fully characterized.

### Role of the DCT and CNT

Our model predicts that the DCT and CNT are able to adjust to changes in Ca^{2+} transport upstream, but not always finely enough to prevent hypercalciuria. For instance, these two segments cannot fully counterbalance the abolition of Ca^{2+} reabsorption along the entire TAL: under these conditions, the model predicts a twofold increase in urinary Ca^{2+} excretion. It is difficult to compare these predictions with experimental data, since maneuvers that abolish the passive reabsorption of Ca^{2+} in the TAL (such as the administration of furosemide) typically affect the transport of NaCl as well, thereby altering the interstitial osmolality gradient and changing the driving force for Ca^{2+} exchange across other nephron segments (see below).

If Ca^{2+} exchanges across the CD epithelium are negligible, the [Ca^{2+}] in the CD fluid increases ∼10-fold from the CM junction to the papillary tip, due to massive water reabsorption along that segment. Thus, when Ca^{2+} reabsorption in the DCT and CNT is abolished, the model predicts a dramatic increase in the urinary [Ca^{2+}]. It is likely that in vivo, in the event of Ca^{2+} transport defects in the DCT and CNT, crosstalk with the proximal tubule enhances Ca^{2+} reabsorption therein so as to reduce Ca^{2+} delivery to the loop of Henle; such cross talk between proximal and distal tubules has been shown to occur following the administration of thiazide (27). CaSR-dependent mechanisms in the TAL and elsewhere may also come into play. Additionally, there may be compensating effects downstream of the DCT and CNT. As described above, several investigators have measured significant rates of Ca^{2+} transport along the IMCD (3, 23). However, since the underlying mechanisms have yet to be understood, whether the CD could act to limit hypercalciuria remains uncertain.

### Interstitial [Ca^{2+}] Gradient

Our results suggest that a significant interstitial [Ca^{2+}] gradient develops along the medullary axis if the vasa recta and limbs of Henle are not fully impermeable to Ca^{2+}. In the base case, the average interstitial [Ca^{2+}] is predicted as 1.2 mM in the OM and 3.9 mM in the IM (Fig. 6). As described above, we estimated the thin limb permeability to Ca^{2+} in rats by extrapolating rabbit data, but there may be significant interspecies differences. The resulting values are small compared with the Ca^{2+} permeability of the proximal tubules and cTALs, yet they are sufficient to allow some Ca^{2+} to passively diffuse from the thin DLs and ALs to the interstitium. A substantial amount of Ca^{2+} is predicted to diffuse from the DVR as well, if their permeability to Ca^{2+}, which has not been measured to our knowledge, is similar to that of thin limbs. Since predicted interstitial [Ca^{2+}] profiles are exquisitely sensitive to the Ca^{2+} permeability of DVR, as illustrated in Table 3 and Fig. 6, experimental measurements of this parameter would be very valuable.

The interstitial [Ca^{2+}] increases overall along the medullary axis because as water is reabsorbed from DVR and DLs, the luminal [Ca^{2+}] in these vessels and tubules increases, thereby raising that in the interstitium as well. Axial interstitial concentration gradients can be maintained because axial transport is negligible in the medullary interstitium: interstitial cells are stacked like the rungs of a ladder and their orientation impedes axial diffusion (22). Likewise, we assumed no exchanges between the cortical and medullary interstitium, and the model therefore predicts a sharp discontinuity in the interstitial [Ca^{2+}] at the CM junction. It is likely, however, that the change is more gradual in vivo.

### Interstitial Osmolality Gradient

Our results suggest that the contribution of Ca^{2+} to the axial osmolality gradient is generally negligible relative to that of NaCl and urea. As described above, the driving force behind the urinary concentrating mechanism in the IM remains to be entirely understood. None of the hypotheses that have been proposed in the last decades has been fully validated (10). A recent model of the urinary concentrating mechanism that accounts for preferential interactions among tubules and vessels and for new permeability data was able to predict a substantial urea gradient in the IM, but failed to reproduce the observed increase in sodium concentration along the IM axis (20), which suggests that additional explanations for concentrating NaCl are needed. Reasoning that the IM osmolality gradient could be important for Ca^{2+} transport, we generated this gradient by adding an external osmolyte (*Eq. 7*), a hypothesis initially proposed by Hargitay and Kuhn (15), revived by Jen and Stephenson (19), and subsequently probed by Thomas (34) and Thomas and Wexler (35). The HT model demonstrated that in principle, the accumulation of lactate (via anaerobic glycolysis) in the IM could drive water out of DLs and thereby amplify the IM osmolality gradient. This hypothesis has yet to be confirmed, and we did not include the transport of glucose and lactate in the current model. Our results indicate that the IM osmolality gradient has indeed a significant impact on Ca^{2+} transport in the kidney. In the absence of such a gradient, the luminal fluid remains much more dilute, and there is less passive reabsorption of Ca^{2+} along long loops of Henle (Table 5). Since the DCT and CNT downstream compensate for this reduction by actively reabsorbing more Ca^{2+}, the fractional Ca^{2+} load delivered to the CCD is very similar to that in the base case (0.26 vs. 0.27%). Nevertheless, as water abstraction in the CD is greatly diminished, [Ca^{2+}] in the CD fluid at the papillary tip is about one-fourth its base-case value (Table 4).

### New Features of Mathematical Model

Most models of solute transport in the medulla do not represent the cortical distal tubules, and instead specify steady-state boundary conditions at the entrance to the medullary CD. We explicitly included the cTAL, DCT, CNT, and CCD in our model because the DCT and the CNT are the nephron segments where Ca^{2+} reabsorption is exquisitely regulated. Without their specific inclusion, we would not have been able to investigate the role of these segments in maintaining the Ca^{2+} balance. In addition, adding these segments was necessary to build a model with arbitrary (transient) initial conditions. Even though we focused on steady-state results in this study (partly because adding an external osmolyte constrains the transient behavior of the model), a dynamic strategy to obtain steady-state solutions has several advantages. The finite-volume scheme that we developed is robust, in that it converges to the steady-state solution no matter what the initial conditions are. It is also relatively fast: reaching steady state from an arbitrary initial state usually requires <15 min of CPU time.

In summary, we have developed the first detailed model of calcium transport in the rat kidney. Our results suggest that Ca^{2+} reabsorption profiles differ between superficial and deep nephrons. Our model also predicts the existence of an axial [Ca^{2+}] gradient in the medullary interstitium.

## GRANTS

Funding for this study was provided by the program EMERGENCE (EME 0918) of the Université Pierre et Marie Curie (Paris Univ. 6).

## DISCLOSURES

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

## AUTHOR CONTRIBUTIONS

Author contributions: M.T., N.S., B.P., S.R.T., and A.E. conception and design of research; M.T. and N.S. performed experiments; M.T. analyzed data; M.T., S.R.T., and A.E. interpreted results of experiments; M.T. prepared figures; M.T. and A.E. drafted manuscript; M.T., S.R.T., and A.E. edited and revised manuscript; M.T., N.S., B.P., S.R.T., and A.E. approved final version of manuscript.

## ACKNOWLEDGMENTS

We thank P. Houillier for helpful discussions.

## Appendix A:

### FORMAL DERIVATION OF CONSERVATION EQUATIONS

We derive the equations for DVR and AVR (denoted by D and A). We assume that *F*^{D} ≥ 0 and that *F*^{A} ≤ 0. Analogous equations apply to descending and ascending tubules. As in the HT model, the vasa recta and tubules of a given class are grouped into lumped “tubes” (i.e., vessels or tubules), with virtual shunts at each depth to represent the fact that a number of tubes turn back at every depth. A discussion on the error introduced by the shunted-tube approach can be found in reference (37).

#### Volume Conservation

The difference in the total water flow in DVR between depths *x* and *x* + *dx* equals
(18)Conservation of water implies that this difference equals the amount of water entering DVR across vessel walls between *x* and *x* + *dx*, minus the amount of water that is shunted to AVR. The latter corresponds to the water flow at the end of the vessels that turn back between *x* and *x* + *dx*. The sum of these two terms is written as
(19)Note that
−*dN*^{D}/*dx*(*x*)
represents the number of DVR that end at depth *x*. Setting (*Eq. 18*) equal to *Eq. 19*, we obtain
(20)The differential form of this equation is
(21)Following the same reasoning, the water conservation equation for AVR is written as
(22)

#### Mass Conservation

The total amount of solute in the DVR domain comprised between depths *x* and *x* + *dx* at time *t* is

Variations in *M* between times *t* and *t* + *dt* are due to:

1. The net amount of solute entering the domain axially between *t* and *t* + *dt*

2. The net amount of solute entering the domain laterally (across the vessel walls) between *t* and *t*

3. The amount of solute that is shunted to AVR at the end of the DVR turning back between *x* and *x* + *dx*
Assembling all these terms, we obtain
This equation is rewritten as
The differential form of this equation is

Note that there are no shunt terms for cortical distal tubules and CDs, hence the form of *Eq. 2*.

## Appendix B:

### NUMERICAL SCHEME

We developed a new finite-volume scheme (based on the theory in Ref. 5), combined with a splitting method, to solve the dynamic model equations. We recently published a simplified version of this scheme (11) that did not account for the varying length of tubules and vessels. For simplicity, the scheme is written for the blood vessels (DVR and AVR), but it can easily be expanded to the renal tubules. The AVR appears specifically in this minimal scheme because its water flux is calculated differently than that of other structures. We also present the scheme with one solute (*I* = 1) but it is easily generalized to more solutes, the only differences lying in the computation of water fluxes (which then contain one more term).

For the initial condition, we define the cell average as
(23)We discretize the parameters *P* as
(24)We use a mesh size Δ*x* = *L*/*N* where *N* is the number of cells *Q*_{k} = (*x*_{k−1/2}, *x*_{k+1/2}), and C_{k}^{α,n} (with α = D, A, int) represents the value of the concentration in cell *k* after *n* time steps. The boundary conditions at the DVR inlet yield C_{0}^{D,n} = C^{DVR}(0, *t*); *F*_{0}^{D,n} = *F*^{DVR}(0, *t*). By continuity we have C_{N+1}^{A,n} = C_{N}^{D,n} and *F*_{N+1}^{A,n} = −*F*_{N}^{D,n}; in addition, by symmetry, we have C_{N+1}^{D,n} = C_{N}^{A,n} and *F*_{N+1}^{D,n} = −*F*_{N}^{A,n}. At the AVR outlet, we impose (and similarly for all ascending structures in the complete scheme) C_{0}^{A,n} = C_{1}^{A,n} and *F*_{0}^{A,n} = *F*_{1}^{A,n}, and we do so for every open tube outlet in the complete scheme. In contrast with our previous scheme (11), the current one does not account for flow reversal, as it would require inverting a nonlinear operator at each step and would result in much longer computational times. Thus we assume that *F*^{DVR} ≥ 0, *F*^{AVR} ≤ 0, *F*^{LDL} ≥ 0, *F*^{LAL} ≤ 0, *F*^{SDL} ≥ 0, as expected under most physiological conditions. For the two-tube case, we have
(25)

We note that the equations for DVR can be written as (26)

At the beginning of each time step, all concentrations are assumed to be known. In each cell, first transmural water fluxes then water flows are calculated based on the known concentrations. We then update the optimal time step Δ*t* given the CFL condition (7). Finally, the concentration in each cell is updated using a splitting method in two steps: first, a provisional concentration is obtained using the axial convection part of *Eq. 8*; it is then updated to account for the transmural solute flux.

##### First step.

We compute the discrete water fluxes and flows as (27) (28)For the CD, we have (29)

##### Second step.

The CFL condition is a stability condition of the scheme which binds Δ*t* and Δ*x* (7). Our stability criterion is that the concentration values need to remain positive. For this condition to be satisfied, we compute Δ*t* such that
(30)

##### Third step.

To compute the provisional concentration, we write the scheme under a conservative form (i.e., in which the conserved quantity appears explicitly). The term that accounts for the dependence of *N* on *x* is included in the first step of the splitting process:
(31)
(32)For the CD, we have
(33)

##### Fourth step.

To compute the definitive value of the concentration, we use a semi-implicit scheme (i.e., only the active transport term is written in an implicit manner) so as to avoid limitations on the CFL condition. Thus there is no need to invert a system of equations, and we write (34) with (35)

Then we compute (36) (37)

This numerical scheme is convergent, has conservation properties, and preserves positivity under the CFL condition (*Eq. 30*). To demonstrate the conservation property, we rewrite and rearrange *Eq. 28* to obtain the following system

(38)Adding these two equations demonstrates that (39)This property represents water conservation.

Using a similar approach, it can be shown that the scheme satisfies solute conservation at steady state (for large *t* values): under stationary conditions, the quantity ∑_{j = D,A} *N*_{k}^{j}*F*_{k}^{j}C_{k}^{j} does not depend on *k*. This property represents mass conservation at the discrete level and can be used as a criterion to determine when the system has reached steady state.

Our scheme converges adequately when a spatial discretization of *N* = 300 is used in the axial (*x*) direction, and *N* = 200 in the radial direction (for the cortical distal tubules). Further increasing *N* does not change the numerical results by more than 1 × 10^{−3} in relative terms.

## Footnotes

↵1 For a discussion of the consequences of this assumption, see Thomas SR and Mikulecky DC. Transcapillary solute exchange. A comparison of the Kedem-Katchalsky convection-diffusion equations with the rigorous nonlinear equations for this special case.

*Microvasc Res*, 15: 207–220, 1978.

## Glossary

- AL
- Ascending limb of Henle's loop
- AVR
- Ascending vas rectum
- C
_{i}^{j} - Concentration of solute
*i*in vessel or tubule*j* - CCD
- Cortical collecting duct
- CD
- Collecting duct
- CM
- Corticomedullary
- CNT
- Connecting tubule
- cTAL
- Cortical thick ascending limb
- DCT
- Distal convoluted tubule
- DL
- Descending limb of Henle's loop
- DT
- Cortical distal tubule
- DVR
- Descending vas rectum
*F*- Water flow
- IM
- Inner medulla
- IS
- Inner stripe of outer medulla
*K*_{m,i}^{j}- Michaelis-Menten constant for active transport of solute
*i*in vessel or tubule*j* - L
- Total length of medulla
*L*_{cTDC}- Combined length of the cTAL, DCT, and CNT
*L*_{d}- Total length of cortical distal tubules
- LIM
- Lower part of inner medulla
*L*_{P}^{j}- Permeability of vessel or tubule
*j*to water - mTAL
- Medullary thick ascending limb
*N*^{j}- Number of vessels or tubules of type
*j* - OM
- Outer medulla
- OS
- Outer stripe of outer medulla
*P*_{i}^{j}- Permeability of vessel or tubule
*j*to solute*i* *r*^{j}- Radius of vessel or tubule
*j* - SAL
- Short ascending limb
- SDL
- Short descending limb
- tAL
- Thin ascending limb
- TAL
- Thick ascending limb
- UIM
- Upper part of inner medulla
*V*_{j}^{m,i}- Maximal rate of active transport of solute
*i*in tubule*j* *x*_{OM/IM}- Spatial coordinate at the OM-IM boundary
*x*_{OS/IS}- Spatial coordinate at the OS-IS boundary
- Δ
*V*_{TE}^{j} - Electric potential difference between the lumen of tubule
*j*and the interstitium - σ
_{i}^{j} - Reflection coefficient of vessel or tubule
*j*to solute*i*

- Copyright © 2013 the American Physiological Society