ADONIS 000167i2910065E

A multinephron model of renal blood flow autoregulation by tubuloglomerular feedback and myogenic response A. H. 0 I E N a n d K.AUKLAND* Departments of Mathematics and * Physiology, University of Bergen, 5000 Bergen, Norway A. H. & AUKLAND,K. 1991. A multinephron model of renal blood flow autoregulation by tubuloglomerular feedback and myogenic response. Acia Ph>~szol Scand 143, 71-92. Received 11 November 1990, accepted 15 April 1991. ISSN 0001-6772. Department of Mathematics and Physiology, University of Bergen, 5000 Bergen, Norway.

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Tubuloglomerular feedback implies that a primary increase in arterial pressure, renal blood flow, glomerular filtration and increased flow rate in the distal tubule increase preglomerular resistance and thereby counteract the primary rise in glomerular filtration rate and renal blood flow. Tubuloglomerular feedback has therefore been assumed to play a role in renal autoregulation, i.e., the constancy of renal blood flow and glomerular filtration at varying arterial pressure. In evaluating this hypothesis, the numerous tubular and vascular mechanisms involved have called for mathematical models. Based on a single nephron model we have previously concluded that tubuloglomerular feedback can account for only a small part of blood flow autoregulation. We now present a more realistic multinephron model, consisting of one interlobular artery with an arbitrary number of evenly spaced afferent arterioles. Feedback from the distal tubule was simulated by letting glomerular blood flow exert a positive feedback on preglomerular resistance, in each case requiring compatibility with experimental openloop responses in the most superficial nephron. The coupling together of 10 nephrons per se impairs autoregulation of renal blood flow compared to that of a single nephron model, but this effect is more than outweighed by greater control resistance in deep arterioles. Some further improvement was obtained by letting the contractile response spread from each afferent arteriole to the nearest interlobular artery segment. Even better autoregulation was provided by spreading of full strength contraction also to the nearest upstream or downstream afferent arteriole, and spread ro both caused a renal blood flow autoregulation approaching experimental observations. However, when the spread effect was reduced to 25% of that in each stimulated afferent arteriole, more compatible with recent experimental observations, the autoregulation was greatly impaired. Some additional mechanism seems necessary, and we found that combined myogenic response in interlobular artery and tubuloglomerular feedback regulation of afferent arterioles can mimic experimental pressure-flow curves. Key words : glomerular filtration, kidney tubules, kidney circulation, kidney blood vessels, macula densa, mathematical modelling, rats.

Autoregulation of renal blood flow (RBF) implies that blood flow is maintained practically constant over a wide range of arterial pressure (AP). Correspondence to : Alf H. (dien, Dept. of Math, University of Bergen, Allegt 53/55, 5007 Bergen, Norway.

T h e r e is strong evidence that this is accomplished mainly by regulation of preglomerular vascular resistance, a n d two principally different mechanisms have been proposed as regulators: The tubuloglomerular feedback (TGF) mechanism assumes that flow rate i n the distal tubule regulates preglomerular resistance through some

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kind of signal elicited in the macula densa. T h e without transmission of contraction stimuli myogenic response mechanism, on the other between vascular segments, a change in resistance hand, assumes that increased vessel wall tension in one particular afferent arteriole (a.a.) would resulting from increased arterial blood pressure influence its blood flow and thereby the pressure elicits contraction of the vascular smooth muscle along the interlobular artery and thereby also the cells and thereb!- increased resistance. While feeding pressure for all other glomeruli. T h i s there is strong evidence for the existence of both effect would clearly also influence the total of these mechanisms, there is no agreement on m?-ogenic response in a ‘i.1.a. tree’. Because of the shortcomings of previous their relati\ e importance. Because of the large number of haemodynamic models we have now studied autoregulation of and tubular parameters involved, mathematical renal blood flow in a system of an arbitrary models have become an important tool in this number (n) of evenly spaced affcrent arterioles, area of research. I n most models of the renal and have in particular calculated the predictions circulation the preglomerular resistances have for n = 10 numerically. Both ‘descending myogenic autoregulation’ (0ien & Aukland 1983) been lumped together as one non-distributed resistance, usually assumed to be located in the and tubuloglomerular feedback have been exaffcrent arterioles (Huss i v 01. 1975, Jensen ei a / . plored, but the main emphasis has been placed 1081, 1,ush & FraS- 1984, Holstein-Rathlou 8i on the latter mechanism in which the potential I,c\ssao 1987). In our ohvn studies on renal effect of-a multinephron model would seem to be autoregulation we introduced a distributed pre- most pronounced. glonierular resistance (0ien & Aukland 1983), and more recently, Haberle & Davis used a G L O S S .4R Y model with 2 or 3 preglomerular resistances in series (Hgberle 1988). However, in all these a.a.(s), afferent arteriole(s) models it has been assumed that the pre- AP, arterial pressure glomerular resistance could be represented by /I,],spread effect parameter one unbranched tube, thereby excluding any c, subscript for ‘control’ interaction between the nephrons. This simplifi- ARBF, ABF, change in (renal) blood flow A.4P, change in arterial pressure cation may be misleading especialll- in the case of AGFR, change in glomerular filtration rate the most superficial nephrons in the rat kidney, APG, change in glomerular pressure because a sizeable part of their preglomerular ASFP, change in stop flow pressure resistance is located in the interlobular artery, e.a., efferent arteriole i.c. in a segment that gives off many afferent ,f; blood flow in afferent arterioles, when identical in all arterioles. (Kallskog et u / . 1976, Tonder & A,,control afferent arteriolar blood flow, identical in all Aukland 1979/80, Heyeraas & Aukland 1987). A, varied blood flow in arteriole i &, control and varied distal tubular flow %lore specifically, Sjoquist et al. (1981) pointed out that activation ofsmooth muscle cells induced F , blood flow in unspecified vascular segment by a tuhuloglomerular feedback signal from only F.4K, tubuloglomerular response sensitivity parameter one nephron might well spread to the nearest G, niJ-ogenic gain factor part of the interlobular artery, but probably not GFR, glomerular filtration rate to its entire length. h, subscript for ‘high’ On the other hand, in the w-hole kidney, a i.l.a., interlobular artery change in arterial pressure w-ould elicit a response 1, subscript for ‘low’ in all nephrons, and could well spread to the /i, length of afferent arteriole i entire interlobular artery, and might therefore Li, length of interlobular artery segment i g i w a stronger autoregulation than that expected n, number of afferent arterioles originating from the interlobular artery from a single nephron response. Since our own p‘,, p, control and varied pressure modelling of tubuloglomerular feedback (.4ukpa?, pa, control and varied arterial pressure land & 0 i e n 1987) was based on the open loop pet, p e , control and varied pressures at the end of the response of one single nephron, this objection interlobular artery also questions the conclusion that TGF is not by PG, glomerular pressure far strong enough to explain the experimentall! pgc,control glomerular capillary pressure, common for observed autoregulation. Furthermore, even all nephrons

Autoregulation in a multinephron model pp,,varied glomerular capillary pressure of nephron t p,,, p,, control and varied pressure at inlet of arteriole i Tic, r , , control and varied resistance of arteriole z R, resistance of unspecified segment R,?, R,, control and varied resistance of interlobular artery segment no ‘ I ’ RBF, renal blood flow pc, radius of unspecified segment in control state T, tubuloglomerular response function TGF, tubuloglomerular feedback VAR, parameter for TGF induced fractional resistance variation z , position variable along segments

G E N E R A L D E S C R I P T I O N OF THE MODEL The model consists of n a.a.s branching off from one interlobular artery (i.l.a)., as shown schematically in Figure 1. In the control state (subscript c), blood pressure is assumed to fall linearly along i.1.a. from arterial pressure (paJ at the entrance to p,, at the end. The pressure at the inlet to each a.a. is denoted by pic, i = 1,2, . . ., n, taken in downstream sequence. The corresponding glomeruli, tubuli, and efferent arterioles (e.a.s) are identified by the same numbers. T h e position on i.1.a. of the inlets to the a.a.s divides i.1.a. into n segments. The pressure difference between the inlets of any two i = 2,3, ..., n , consecutive a.a.s (p+,)c -jic), and pac-pl, are considered to be equal. Also control blood flow (f,) is assumed to be equal in all a.a.s. In accordance with these specifications one may consider i.1.a. in the control state as a series of n tubes, each of uniform inner radius, and also each a.a. as one uniform tube. In varied states the radii of the tubes may be altered by myogenic mechanisms or by T G F . Glomerular capillary pressures in the control state, pgo are all assumed to be equal. From each glomerulus the blood enters the e.a., each modelled as a uniform tube, all having equal and constant diameters and lengths both in control and varied states. The pressures at the end of the e.a.s are assumed to be equal to the ‘tissue pressure’, which is taken to be zero. The most extensive information on TGF has been obtained by open loop microperfusion studies in rats : after blocking the last accessible portion of the proximal tubule, downstream tubular flow is replaced by microinfusion of saline or artificial tubular fluid. The response to varying infusion rate, referred to here as ‘ distal

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tubular flow rate’ (also called loop perfusion rate), is then measured as a change in blood flow, glomerular capillary pressure or filtration, or as a change in proximal tubular stop flow pressure in the same nephron. Since each numerical solution for 10 complete nephrons would have taken too much time on the most powerful computer available to us, the tubular system was not modelled in the present study. Instead, the tubuloglomerular feedback was modelled as a function of glomerular blood flow, or glomerular pressure, with parameters that gave preglomerular resistance changes in response to changes in distal tubular flow rate similar to those obtained in our previous complete nephron study (Aukland & Oien 1987), which again had been modelled to comply with open loop studies in rats. To be more specific, the TGF parameters in each model were adjusted in ‘distal perfusion simulations ’ first such that the response to changes in flow rate in the distal tubule of one superficial nephron became compatible with open loop studies. These parameters were then used in closed loop mode in corresponding autoregulation simulations, where blood flow response in all a.a.s to varying arterial pressure was found. Separate solutions were obtained for TGF induced resistance changes located in the afferent arterioles and with the effect spreading to the whole, or part of the i.l.a., and to neighbouring afferent arterioles. The effect of myogenic response was also studied, alone, as well as in combination with TGF.

MATHEMATICAL DESCRIPTION

OF T H E MODEL Control state. Quantities in control state are designated by subscript c. For the pressure at the inlet of the arteriole i, i = 1, ..., n, we have Pic = ~ a c - ( ’ / n ) ( ~ a c - ~ w ) , where p e r = ~ n is r the pressure at the end of i.1.a. and at the inlet of the nth arteriole (Fig. 1). Let R,, = (pat -p,?)/ (nL),where f, is the control blood flow in each a.a., and R, = -piJ/((n 1 - 4A.), i = 2 , . .., n, denote the resistances of the n segments along i.1.a. in the control state. For the pressure profile along the first i.1.a. segment we 0 G 2 d L,. have P&) = ~,,-(nf,)R,,(z/~J, L, is the length of the segment. For the pressure profile along the ith i.1.a. segment, i = 2 , . . . ,n, we likewise have p,(n) = fi(i-l)c - (n i-I - t) f,

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R E G U L , q T I O N O F BLOOD FLOW .IT i-..IRYING A R T E R I A I , PRESSURE .Ilyogen/crtsponse in [ifia.a.s und the mhole length of' i./.iz. T h e pressure along the i.1.a. segments and the a.a. segments are evaluated following the ila f same lines as the model of %en & Aukland Fig. 1. The architecture of the interlobular arter!(i.1.a.) 'tree' I I = number of afferent arterioles (a,%.) (1983). W e assume a myogenic gain factor G = 1 on all segments implying that tangential wall branching off from i,l.a., .I..%. = arcuate arter!-, p;,= arterial pressure, p,.= pressure at the end of tension, i.e. the product of transmural pressure i.l.a., p,, i = 1, 2, ..., n, = pressures a t inlets of the and inner vessel radius, is kept constant. T h e a.6.s. p,,, i = I , 2, ..., r l , = glornerular pressures, .(, pressure uriation along an arbitrary tube segi = 1,2, . .. N , = a . ~ hlood . flon-s. ment follows from the Hagen-Poiseuille law on each infiwitrsiwziil segment, i.e. from

Ric(c/L,), 0 6 :. < Z,i, the length of the ith segment. 1,et r,( = ( p , , -pK,.)/'/i,, / = 1,.... n, dcnotc the resistance of the ith a.a. T h e n the prcssure profile along the ith a.a. becomes p, ( z ) I= p , , --/,' r,((z/I,), 0 6 2 ,< I,, the length of the rth arteriole. Because all glomeruli :Ire assumed to h a w the same pressure. pg,, in control state n-c simpl!- have

J,,

,),

= r,,

+(I?+lb/)R,