Presentation of our (flat) model exploring a possible role for metabolically produced interstitial osmoles (lactate)

• Overview

• Tubule system

  • Nephrons
  • Collecting ducts and vasa recta

• System equations

  • Flow equations
  • Epithelial flux equations
  • Mass conservation

• Numerical methods

• Parameter values

• Simulation

Overview

As sketched out in the brief history above, no idea offered to date as an explanation of the steep osmotic gradient in the inner medulla has proved compatible with the lack of active transport in the inner medulla and the high urea permeability of the long descending limbs of Henle (LDL) in this region. As mentioned, revival of the idea that external (i.e., interstitial) osmoles may contribute (Kuhn & Ramel 1959 and Niesel & Röskenbleck 1963) received theoretical attention (Jen & Stephenson, 1994; Thomas & Wexler, 1995) but at the time there seemed to be no viable candidates for the IM osmole production.

More recently, it came to our attention that due to the well-established relative hypoxia of the IM (ref), anaerobic glycolysis (AG) furnishes a significant portion of the ATP for IM cells. We then realiized that since AG splits each glucose into two lactates, it is also a source of net osmoles. Furthermore, the sugar molecules are excellent candidates for the role of external osmoles, since they are excluded from the nephron and can thus exert their full osmotic effect across the highly water permeable long descending limbs of the IM. Experimental measurement of interstitial lactate levels would be in order but is difficult. An early study (Ruiz-Guinazu 1961) measured glucose and lactate in vasa recta fluid from in vivo micropuncture at the papillary tip of rat kidneys: they concluded there was no reason to follow up on this idea, since they found only marginally increased lactate (and decreased glucose) compared to arterial plasma levels. However, the long puncture times needed to accumulate sufficient sample volume compromised the IM gradient. A few later biochemical studies on IM slices using radioisotopes consistently showed at least a doubling of lactate content per gram of tissue towards the papillary tip. However, none of these studies was carried out in frankly antidiuretic animals.

We thus felt it worthwhile to explore quantitatively (i.e., in modelling studies) the possibility that IM lactate production might significantly affect the osmotic gradient by playing the role so long attributed to urea dumped from the IMCD (inner medullary collecting ducts) but rendered ineffective by the high measured urea permeability of the LDL.

We began with a "feasibility study", a very simple model of glucose and lactate flows and recycling in the IM vascular space (Thomas 2000) in order to establish rough limits. The results of that study encouraged following up by incorporating glycolytic conversion of glucose to lactate in a medullary model explicitly including the nephrons and collecting ducts, but rather than going straight to a 3D model, we first applied the idea in an extended "flat" model. As mentioned in the History section above, it has already been well established that such models, which assume a common interstitial bath for all structures (i.e., they cannot account for the lateral gradients allowed by 3D features such as the vascular bundles), are inadequate, at least when NaCl and urea are the only solutes included. We wanted to see how much of an osmotic gradient they could build if glycolytic lactate production is accounted for. The resulting model is the one presented here (Hervy and Thomas, 2003).

Briefly, this is a flat, lumped-tube model of the medulla representing the flows of water, NaCl, urea, glucose, and lactate in the long and short limbs of Henle, the collecting-ducts, and the vasa recta, in which the ascending vasa recta are assimilated with the interstitium. We also include KCl input to the collecting ducts. The main purpose was to see the predicted effect on the inner medullary osmotic gradient of converting an increasing proportion of the entering glucose into lactate within the extra-nephron (i.e., interstitial) space.

Tube system

The tubular distribution within the medulla is based on the anatomical data for the rat (Jamison & Kriz).

The nephrons.

As in the 3D WKM models, the net effect of the distal tubules (DT) is represented here by a set of boundary constraints instead of explicitly integrating the DT transport processes. Thus, flows and concentrations entering the outer medullary collecting ducts (OMCD) are calculated from flows exiting the ascending limbs of Henle, based on the constraints shown in the equations opposite, namely, isotonicity, fixed NaCl (and KCl) concentrations, fixed proportion of DT urea reapsorption (1-ufact), and conservation of glucose and lactate (assumed impermeable).

The descending and ascending long limbs of Henle (LDL and LAL) are all represented by a lumped structure with shunts at each inner medullary level representing the tubes that turn back, according to an exponential distribution given opposite this paragraph. The equation gives the number of tubes at depth x, where ksh = 1.213 mm-1 is the constant of exponential decrease of the number of tubes with depth, and xOMIM is the position of the outer/inner medullary border (in mm from the cortico-medullary border).

Collecting ducts and vasa recta

The number of vasa recta and collecting ducts also diminish exponentially within the IM according to the same equation. For vasa recta, ksh has the same value as for LDL & LAL, but for the IMCD ksh = 1.04 mm^-1.

System of equations

Flow equations

The differential equations for the flows of volume v and solutes j in the different tubes i are:

the unknowns are volume flow F_iv and solute concentrations c_ij.

Epithelial flux equations

Flux equations for the different solutes j in tubes i are:

Mass conservation

The equation for mass conservation at each level, following Stephenson et al 1974:

Numerical methods

The system is solved using a method developed by Stephenson et al. in 1974 and used by us previously in a cascading six-nephron flat model (Thomas 1991). The principle is to discretize the ODEs by chopping the space into a number of equal slices. Considering a tube i chopped into n slices and thus n+1 nodes k, the equations then become:

Adopting the midpoint method, the J's are calculated in the center of each slice n, i.e., for the arithmetic average of k and k-1 (at k-1/2).

Parameter values

As nearly as possible, permeabilities and other parameters are equal to those of our previous 3D models (Wang, Thomas, & Wexler 1998; Thomas 1998). The table below gives the values used here. L_p, P_u, P_s, P_g, and P_l are the permeabilities for water, urea, NaCl, glucose, and lactate, respectively. σ_u, σ_s, σ_g, and σ_l are the reflection coefficients for urea, NaCl, glucose, and lactate, respectively. Vm is the rate of active transport of NaCl from the MTAL.