Presentation of our (flat) model exploring a possible role for metabolically produced interstitial osmoles (lactate)"."
";
$str .= par('
');
$str .= par('Overview
');
$str .= par(' 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.');
$str .= par('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. ');
$str .= par('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.');
$str .= par('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).');
$str .= par('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.');
$str .= par('Tube system
');
$str .= par('The tubular distribution within the medulla is based on the anatomical data for the rat (Jamison & Kriz).');
$str .= par('The nephrons.
');
$str .= tab(tr(td(
par('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).',"paratd").
td('
')
)));
$str .= tab(tr(td(
par('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).',"paratd").
td('
')
)));
$str .= par('Collecting ducts and vasa recta
');
$str .= par('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.');
$str .= par('System of equations
');
$str .= par('Flow equations
');
$str .= par('The differential equations for the flows of volume v and solutes j in the different tubes i are:');
$str .= par('
');
$str .= par('the unknowns are volume flow Fiv and solute concentrations cij.');
$str .= par('Epithelial flux equations
');
$str .= par('Flux equations for the different solutes j in tubes i are:');
$str .= par('
');
$str .= par('Mass conservation
');
$str .= par('The equation for mass conservation at each level, following Stephenson et al 1974:');
$str .= par('
');
$str .= par('Numerical methods
');
$str .= par('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:');
$str .= par('
');
$str .= par('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).');
$str .= par('Parameter values
');
$str .= par('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. Lp, Pu, Ps, Pg, and Pl are the permeabilities
for water, urea, NaCl, glucose, and lactate, respectively. σu, σs, σg, and σl are the reflection coefficients for urea, NaCl, glucose, and lactate, respectively. Vm is the rate of active transport of NaCl from the MTAL.');
$str .= '
| |
Membrane Parameters |
| Tube |
Region |
Radius |
Lp |
Pu |
Ps |
Pg |
Pl |
σu |
σs |
σg |
σl |
Vm |
| |
|
µm |
10-6 mm/(s.mosm/l) |
10-4mm/s |
10-4mm/s |
10-4mm/s |
10-4mm/s |
|
|
|
|
pmol/mm2-sec |
| ldl: |
os |
10 |
66.6 |
2 |
20 |
0 |
0 |
1 |
0.9 |
1 |
1 |
0 |
| |
is |
10 |
62.5 |
2 |
20 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
| |
iu |
10 |
58.3 |
12 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
| |
il |
10 |
58.3 |
12 |
0.5 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
| |
|
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
|
| lal: |
os |
10 |
0 |
4.5 |
2 |
0 |
0 |
1 |
1 |
1 |
1 |
0.247 |
| |
is |
10 |
0 |
4.5 |
2 |
0 |
0 |
1 |
1 |
1 |
1 |
0.247 |
| |
iu |
10 |
0 |
23 |
80 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
| |
il |
10 |
0 |
23 |
80 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
| |
|
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
|
| cd: |
os |
15 |
10 |
0.5 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
0.02 |
| |
is |
15 |
5.33 |
0.5 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
0.02 |
| |
iu |
15 |
2.67 |
1 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
0.02 |
| |
il |
15 |
3.33 |
70 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
0.02 |
| |
|
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
|
| dv: |
os |
9 |
66.6 |
360 |
80 |
0.78 |
39 |
0.5 |
0.5 |
0.5 |
0.5 |
0 |
| |
is |
9 |
25 |
360 |
80 |
0.78 |
39 |
0.5 |
0.5 |
0.5 |
0.5 |
0 |
| |
iu |
9 |
33.3 |
120 |
80 |
0.78 |
39 |
0.5 |
0.5 |
0.5 |
0.5 |
0 |
| |
il |
9 |
33.3 |
120 |
80 |
0.78 |
39 |
0.5 |
0.5 |
0.5 |
0.5 |
0 |
| |
|
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
|
| av: |
os |
9.5 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| |
is |
9.5 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| |
iu |
9.5 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| |
il |
9.5 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| |
|
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
|
| sdl: |
os |
11 |
58.3 |
8.5 |
2.3 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
| |
is |
11 |
50 |
8.5 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
| |
iu |
11 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| |
il |
11 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| |
|
0 |
|
0 |
0 |
0 |
0 |
|
|
|
|
|
| sal: |
os |
10 |
0 |
4.5 |
2 |
0 |
0 |
1 |
1 |
1 |
1 |
0.247 |
| |
is |
10 |
0 |
4.5 |
2 |
0 |
0 |
1 |
1 |
1 |
1 |
0.247 |
| |
iu |
10 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
| |
il |
10 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
';
$page_content = $ma.$str;
// echo main_page($page_content,"home");
toppage("Model Description",$page_content);
endpage();
?>