Inner medullary lactate production and accumulation: a vasa recta model
Catherine
Lloyd
Bioengineering Institute, University of Auckland
Model Status
This CellML version of the model has been checked in COR and PCEnv and it runs in PCEnv. The model will not run in COR because it does not contain any time derivatives - COR expects the units of the differential equations to be a function of time (not length as it is in this case). However, COR did allow for the units to be checked and they are consistent. The model runs in PCEnv but as yet it does not recreate the published results. This may be due to differences in the defined set of initial conditions. The model author has been contacted and we are currently receiving their help and advice.
Model Structure
S. Randall Thomas here investigates the possibility that recycling of lactate produced by anaerobic glycolysis in the inner medulla of the kidney can provide sufficient of a lactate gradient to contributes significantly to the urine concentrating mechanism.
Assuming (from other sources) that 20% of the glucose delivered to the inner medulla, Thomas uses a mathematical model of the inner medullary vasa recta, based on observed mass distribution and distribution of loops, to investigate a range of plausible values of lactose and glucose permeabilities, to see which values would allow a sufficient accumulation of a lactate gradient to be a significant contributor to urine concentrating ability, in different circumstances of blood flow and diuresis.
The complete original paper reference is cited below:
Inner medullary lactate production and accumulation: a vasa recta model, S. Randall Thomas, 2000,
American Journal of Physiology
, 279, F468-F481.
PubMed ID: 10966926
model diagram
Schematic representation of the steady-state model.
$\mathrm{N\_x}=\mathrm{N\_0}e^{-(\mathrm{ksh}x)}$
$\frac{d \mathrm{F\_DVR\_v}}{d x}=-(\mathrm{Jv}+\mathrm{ksh}\mathrm{F\_DVR\_v})$
$\frac{d \mathrm{F\_DVR\_GLU}}{d x}=-(\mathrm{JGLU}+\mathrm{ksh}\mathrm{F\_DVR\_GLU})$
$\frac{d \mathrm{F\_DVR\_LAC}}{d x}=-(\mathrm{JLAC}+\mathrm{ksh}\mathrm{F\_DVR\_LAC})$
$\frac{d \mathrm{F\_AVR\_v}}{d x}=\mathrm{Jv}+\mathrm{ksh}\mathrm{F\_DVR\_v}+\mathrm{J\_ABS\_V}$
$\frac{d \mathrm{F\_AVR\_GLU}}{d x}=\mathrm{JGLU}+\mathrm{ksh}\mathrm{F\_DVR\_GLU}-\mathrm{JGLY}$
$\frac{d \mathrm{F\_AVR\_LAC}}{d x}=\mathrm{JLAC}+\mathrm{ksh}\mathrm{F\_DVR\_LAC}+2.0\mathrm{JGLY}$
$\mathrm{JGLU}=\mathrm{N\_x}\mathrm{PGLU}(\mathrm{c\_DVR\_GLU}-\mathrm{c\_AVR\_GLU})-1.0\mathrm{Jv}\frac{\mathrm{c\_DVR\_GLU}+\mathrm{c\_AVR\_GLU}}{2.0}$
$\mathrm{JLAC}=\mathrm{N\_x}\mathrm{PLAC}(\mathrm{c\_DVR\_LAC}-\mathrm{c\_AVR\_LAC})-1.0\mathrm{Jv}\frac{\mathrm{c\_DVR\_LAC}+\mathrm{c\_AVR\_LAC}}{2.0}$
$\mathrm{c\_DVR\_GLU}=\frac{\mathrm{F\_DVR\_GLU}}{\mathrm{F\_DVR\_v}}$
$\mathrm{c\_AVR\_GLU}=\frac{\mathrm{F\_AVR\_GLU}}{\mathrm{F\_AVR\_v}}$
$\mathrm{c\_DVR\_LAC}=\frac{\mathrm{F\_DVR\_LAC}}{\mathrm{F\_DVR\_v}}$
$\mathrm{c\_AVR\_LAC}=\frac{\mathrm{F\_AVR\_LAC}}{\mathrm{F\_AVR\_v}}$
$\mathrm{JGLY}=\mathrm{N\_x}\frac{\mathrm{Vmax}\mathrm{c\_AVR\_GLU}}{\mathrm{Km}+\mathrm{c\_AVR\_GLU}}\mathrm{Vmax}=\frac{\mathrm{ksh}}{\mathrm{N\_0}(1.0-e^{-(\mathrm{ksh}L)})}\mathrm{GlyFract}\mathrm{F\_DVR\_G\_0}$
$\mathrm{J\_ABS\_V}=\mathrm{kv}\mathrm{N\_x}$
$\mathrm{kv}=\frac{\mathrm{ksh}}{\mathrm{N\_0}(1.0-e^{-(\mathrm{ksh}L)})}\mathrm{VolFract}\mathrm{F\_DVR\_V\_0}$
$\mathrm{Jv}=0.3\frac{\mathrm{F\_DVR\_v}}{\mathrm{N\_0}b}\mathrm{N\_x}\mathrm{F\_DVR\_V\_0}=3.75\mathrm{N\_0}\mathrm{x\_L}=\frac{x}{L}\mathrm{F\_DVR\_G\_0}=\mathrm{F\_DVR\_V\_0}\mathrm{c\_DVR\_GLU\_0}$
Species
Gene
number of descending vasa recta (DVR) at depth x in the inner medullary vasa recta model
volume (blood) flow in the ascending vasa recta
tubular outflow of volume (blood) in the ascending vasa recta from the long descending Henle's loop
concentration of lactate in the desceding vasa recta (?? mapping with c_DVR_GLU in the CellML model ??)
tubular inflow of volume (blood) in the descending vasa recta from inner/outer medulla border
concentration of glucose in the ascending vasa recta
concentration of glucose entering into the inner medullary descending vasa recta
flux of volume (blood) along the descending vasa recta from the inner medulla border
tubular inflow of volume (blood) in the descending vasa recta from inner/outer medulla border
distance from outer medulla (OM) or inner medulla (IM) border
glycolytic glucose flow from DVR to AVR
flux of glycolytic glucose consumption from all DVR to AVR treating the capillary walls as a single barrier (no distinction here between transcellular and paracellular transport)
total length of the inner medulla (depth at the papillary tip)
tubular inflow of volume (blood) in the descending vasa recta from inner/outer medulla border
glocose flow from DVR to AVR
flux of glucose from all DVR to AVR treating the capillary walls as a single barrier (no distinction here between transcellular and paracellular transport)
lactate flow from DVR to AVR
flux of lactate from all DVR to AVR treating the capillary walls as a single barrier (no distinction here between transcellular and paracellular transport)
flux of volume (blood) along the descending vasa recta from the inner medulla border
flux of glycolytic glucose consumption from all DVR to AVR treating the capillary walls as a single barrier (no distinction here between transcellular and paracellular transport)
number of entering descending vasa recta in the inner medullary vasa recta model
tubular inflow of glucose in the descending vasa recta from inner/outer medulla border
total baseline glucose delivery
tubular inflow of volume (blood) in the descending vasa recta from inner/outer medulla border
flux of lactate from all DVR to AVR treating the capillary walls as a single barrier (no distinction here between transcellular and paracellular transport)
Factor for describing the exponential decrease of vasa recta with depth (Coefficient for exponential decay of N_x)
number of entering descending vasa recta in the inner medullary vasa recta model
Inner medullary lactate production and accumulation: a vasa recta model
flux of volume (blood) along the descending vasa recta from the inner medulla border
number of entering descending vasa recta in the inner medullary vasa recta model
flux of glucose from all DVR to AVR treating the capillary walls as a single barrier (no distinction here between transcellular and paracellular transport)
fractions of total glucose consumption (Glycolytic rate parameter)
fraction of total baseline glucose delivery
number of descending vasa recta (DVR) at depth x in the inner medullary vasa recta model
reflection coefficient of solute glucose
tubular inflow of lactate in the descending vsa recta from inner/outer medulla border
glocose flow in the ascending vasa rceta
tubular outflow of glucose in the ascending vasa recta from the long descending Henle's loop
concentration of glucose in the ascending vasa recta
tubular outflow of glucose in the ascending vasa recta from the long descending Henle's loop
number of descending vasa recta (DVR) at depth x in the inner medullary vasa recta model
distance from outer medulla (OM) or inner medulla (IM) border
compartment2
compartment3
distance from outer medulla (OM) or inner medulla (IM) border
tubular inflow of lactate in the descending vsa recta from inner/outer medulla border
volume (blood) flow into the ascending vasa recta
flux of volume (blood) reabsorption into the ascending vasa recta from long descending Henle?s loop (LDL) and inner medullary collecting duct (IMCD)
concentration of lactate in the desceding vasa recta (?? mapping with c_DVR_GLU in the CellML model ??)
Factor for describing the exponential decrease of vasa recta with depth (Coefficient for exponential decay of N_x)
S. Randall Thomas
number of descending vasa recta (DVR) at depth x in the inner medullary vasa recta model
Factor for describing the exponential decrease of vasa recta with depth (Coefficient for exponential decay of N_x)
concentration of lactate in the ascending vasa recta
tubular inflow of lactate in the descending vsa recta from inner/outer medulla border
Factor for describing the exponential decrease of vasa recta with depth (Coefficient for exponential decay of N_x)
Thomas vasa recta model
tubular inflow of volume (blood) in the descending vasa recta from inner/outer medulla border
Factor for describing the exponential decrease of vasa recta with depth (Coefficient for exponential decay of N_x)
permeation velocity of lactate
lactate permeability across DVR
vasa recta model
glucose
Lactate
tubular inflow of volume (blood) in the descending vasa recta from inner/outer medulla border
total length of the inner medulla (depth at the papillary tip)
tubular inflow of glucose in the descending vasa recta from inner/outer medulla border
lactate flow in the ascending vasa rceta
tubular outlow of lactate in the ascending vasa recta from the long descending Henle's loop
distance from outer medulla (OM) or inner medulla (IM) border
Factor for describing the exponential decrease of vasa recta with depth (Coefficient for exponential decay of N_x)
flux of volume (blood) along the descending vasa recta from the inner medulla border
reflection coefficient of solute lactate
tubular outlow of lactate in the ascending vasa recta from the long descending Henle's loop
distance from outer medulla (OM) or inner medulla (IM) border
distance from outer medulla (OM) or inner medulla (IM) border
concentration of lactate in the ascending vasa recta
tubular inflow of glucose in the descending vasa recta from inner/outer medulla border
flux of volume (blood) reabsorption into the ascending vasa recta from long descending Henle?s loop (LDL) and inner medullary collecting duct (IMCD)
Factor for describing the exponential decrease of vasa recta with depth (Coefficient for exponential decay of N_x)
fraction of total inner medullary volume (blood) absorption
flux of lactate from all DVR to AVR treating the capillary walls as a single barrier (no distinction here between transcellular and paracellular transport)
total length of the inner medulla (depth at the papillary tip)
concentration of lactate in the desceding vasa recta
Factor for describing the exponential decrease of vasa recta with depth (Coefficient for exponential decay of N_x)
concentration of lactate in the desceding vasa recta (?? mapping with c_DVR_GLU in the CellML model ??)
number of entering descending vasa recta in the inner medullary vasa recta model
tubular outflow of volume (blood) in the ascending vasa recta from the long descending Henle's loop
number of descending vasa recta (DVR) at depth x in the inner medullary vasa recta model
flux of glucose from all DVR to AVR treating the capillary walls as a single barrier (no distinction here between transcellular and paracellular transport)
Since anaerobic glycolysis yields two lactates for each glucose consumed and since it is reported to be a major source of ATP for inner medullary (IM) cell maintenance, it is a likely source of "external" IM osmoles. It has long been known that such an osmole source could theoretically contribute to the "single-effect" of the urine concentrating mechanism, but there was previously no suggestion of a plausible source. I used numerical simulation to estimate axial gradients of lactate and glucose that might be accumulated by countercurrent recycling in IM vasa recta (IMVR). Based on measurements in other tissues, anaerobic glycolysis (assumed to be independent of diuretic state) was estimated to consume approximately 20% of the glucose delivered to the IM. IM tissue mass and axial distribution of loops and vasa recta were according to reported values for rat and other rodents. Lactate (P(LAC)) and glucose (P(GLU)) permeabilities were varied over a range of plausible values. The model results suggest that P(LAC) of 100 x 10(-5) cm/s (similar to measured permeabilities for other small solutes) is sufficiently high to ensure efficient lactate recycling. By contrast, it was necessary in the model to reduce P(GLU) to a small fraction of this value (1/25th) to avoid papillary glucose depletion by countercurrent shunting. The results predict that IM lactate production could suffice to build a significant steady-state axial lactate gradient in the IM interstitium. Other modeling studies (Jen JF and Stephenson JL. Bull Math Biol 56: 491-514, 1994; and Thomas SR and Wexler AS. Am J Physiol Renal Fluid Electrolyte Physiol 269: F159-F171, 1995) have shown that 20-100 mosmol/kgH(2)O of unspecified external, interstitial, osmolytes could greatly improve IM concentrating ability. The present study gives several plausible scenarios consistent with accumulation of metabolically produced lactate osmoles, although only to the lower end of this range. For example, if 20% of entering glucose is consumed, the model predicts that papillary lactate would attain about 15 mM assuming vasa recta outflow is increased 30% by fluid absorbed from the nephrons and collecting ducts and that this lactate gradient would double if IM blood flow were reduced by one-half, as may occur in antidiuresis. Several experimental tests of the hypothesis are indicated.
flux of glycolytic glucose consumption from all DVR to AVR treating the capillary walls as a single barrier (no distinction here between transcellular and paracellular transport)
distance from outer medulla (OM) or inner medulla (IM) border
flux of volume (blood) along the descending vasa recta from the inner medulla border
tubular inflow of volume (blood) in the descending vasa recta from inner/outer medulla border
Factor for describing the exponential decrease of vasa recta with depth (Coefficient for exponential decay of N_x)
total baseline glucose delivery
number of descending vasa recta (DVR) at depth x in the inner medullary vasa recta model
tubular outflow of volume (blood) in the ascending vasa recta from the long descending Henle's loop
distance from outer medulla (OM) or inner medulla (IM) border
Factor for describing the exponential decrease of vasa recta with depth (Coefficient for exponential decay of N_x)
permeation velocity of glucose
glucose permeation across DVR
concentration of glucose in the ascending vasa recta