Amendment history:Errata (January 1941)

FORMULAE FOR AFFERENT AND EFFERENTARTERIOLAR RESISTANCE IN THE HUMANKIDNEY: AN APPLICATION TO THE EFFECTSOF SPINAL ANESTHESIA

Harold Lamport

J Clin Invest. 1941;20(5):535-544. https://doi.org/10.1172/JCI101246.

Research Article

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FORMULAEFOR AFFERENTAND EFFERENTARTERIOLARRESISTANCE IN THE HUMANKIDNEY: AN APPLICATION

TO THE EFFECTS OF SPINAL ANESTHESIA

By HAROLDLAMPORT

(From the Department of Neurology, College of Physicians and Surgeons,Columbia University and The Neurological Institute, New York City)

(Received for publication May 21, 1941)

Glomerular dynamics are steadily becomingmore coinprehensible as the trenchant methodsfor studying renal physiology developed byHomer Smith and his coworkers (1, 2) are furtherextended. Their recent analysis of the problem,using the current conception of the glomerulus asa pure ultra-filter operated by the hydrostaticpressure of the blood, is based on various pre-sumptions-some of them explicit and supportedby evidence, others tacit or not supported byevidence (2).

An examination of these basic questions leadsto formulae, different in certain respects fromthose developed by these workers, which can beapplied clinically to determine the actual afferentand efferent arteriolar resistance of human kid-neys. We shall first present these formulae,with an example of a clinical application, beforeproceeding much further with their detailedderivation.

Fundamentally, the essence of a quantitativedescription of resistance, whether applied to flowof fluid or electricity, is the representation ofresistance in terms of flow and pressure, the cus-tomarily measured quantities. In electricity,Ohm's law defines resistance as the ratio of thepotential drop, in volts, along the resistor, tothe electric current flow in it, in amperes. Inhydraulics, Poiseuille, who was brought to theproblem by his interest in the resistance of theblood vessels to the circulation of the blood, hassimilarly defined resistance as the ratio of thefall in pressure from one end to the other of acontinuous wetted tube, to the rate of flowthrough it, after being multiplied by the viscosityof the fluid. Thus:

Resistance = drop in pressurerate of flow X viscosity

It is, of course, applicable in the simple formonly if certain conditions are fulfilled. For ex-

ample, it is presumed that the flow into theresistance is the same as the outflow: there areno leaks. In the discussion of the derivation ofour formulae, the matter of the applicabilityand modification of Poiseuille's law and thevalidity of the choice of the points along thekidney's vascular system between which resist-ance is to be measured will be of importance.Approximations for the changes in viscosity ofthe blood will also be offered and supported.For this purpose, empirical formulae are to bederived which will relate the viscosity of plasmaand whole blood to the serum protein contentand hematocrit.

By itself, Poiseuille's law may not be directlyapplied to renal afferent and efferent arteriolarresistance. One must add the hypothesis, wellsupported by Homer Smith et al. (2), and whichrepresents currently accepted views on the kid-ney's function, that in the glomerular capillariesthe pressure of the blood filters out non-proteinfluid into Bowman's capsule until finally the op-posing osmotic force of the blood proteins therebyconcentrated rises to equal it. This fluid thenbecomes the raw material from which the kid-ney's tubules elaborate urine. In the glomeru-lus, the filtrate is osmotically in equilibrium,across the capillary membrane, with the blood asit starts to enter the efferent arteriole.

This hypothesis of osmotic equilibrium, likePoiseuille's law, is subject to quantitation.Knowing the degree of increased concentrationof the blood proteins produced by glomerularfiltration, we can estimate from an empiricalequation the resulting increased osmotic pressureand, therefore, the blood pressure in the terminalportion of the glomerular capillaries. In thisway, we obtain one of the pressure measurementsneeded to apply Poiseuille's law to the kidney.

The rates of flow of blood and of glomerular535

HAROLDLAMPORT

filtration are determined, clinically, by means ofthe diodrast and inulin clearances (1).

PART I

FormulaeThe final formulae developed require definitions of cer-

tain symbols. Weshall give them here briefly; they aremore exactly defined in Part II.

Let RA- resistance of afferent arterioles in mm. Hg percc. per minute.

R-- resistance of efferent arterioles in mm. Hg percc. per minute.

R = RA + R,, total resistance of renal arteriolesin mm. Hg per cc. per minute.

I - inulin clearance in cc. per minute.D = diodrast clearance in cc. per minute.

IF 5 I , the filtration fraction.

S - grams of protein in 100 cc. serum.H. - hematocrit, as a fraction of 1.

H-I1-f H.k a constant dependent on the hematocrit.

For all, except grossly abnormal hematocrits,k - 0.47.

PM mean of systolic and diastolic brachial bloodpressures in mm. Hg of the recumbent subject.

Where- 2.34S and P 2.34S

1 - 0.05425 1 - 0.0542S -F

R =PM- Po, -40HD

R (1 - kF)(Po. - Po + 10)HD

or- (1 - 0.47F)(Po, - Po + 10)

H'DFor normal subjects, such as those reported by Smith and

his coworkers (3), the formulae can be simplified. Usingthe average hematocrit of 43 per cent 1; serum protein as7 grams per 100 cc., with an osmotic pressure of the corre-sponding plasma of 26.4 mm. Hg, the formulae become

RA P-PoM-- 40; R, - (1- 0.47F)(Po. -16.4)

1.755D 1.755D

Clinical applicationWeshall now proceed to exemplify the use of

these formulae in clinical material. There hasbeen considerable discussion as to the part man'srenal nerves play in controlling blood flowthrough the kidneys. On the assumption thatspinal anesthesia abolishes any possible neuro-

I Dr. Homer Smith has very kindly supplied this datum.

genic control of the kidneys, Smith and hiscollaborators (3) have examined the changesproduced in diodrast and inulin clearances andin blood pressure in a series of normal, basal,supine subjects. They have reached the con-clusion that the kidneys are " not dependent upontonic activity of the central nervous system."

Examining their cases, as reported, we omit,as does Smith, any correction factor for diodrastclearance. Weuse the set of simpler formulaeabove, as individual hematocrit and serum pro-tein are not recorded. The results of this appli-cation are given in Table I.

The average fall in renal arteriolar resistanceis found to be 32 per cent with a probable errorof 4.2 per cent. Although so large a change maybe due solely to autonomous activity of the de-nervated kidneys, it has not been demonstratedthat abolition of central nervous influence is notinvolved. An analysis of data on blood pressureand cardiac output before and during spinalanesthesia in 11 cases, using the Poiseuille lawformula with 20 mm. correction subtracted frommean blood pressure,2 shows no consistent changein total body vascular resistance (4, 5). Conse-quently, a 32 per cent fall in renal arteriolarresistance would appear significant.

The afferent arterioles always dilated; theaverage is 70 per cent fall in resistance.3 Theafferent arteriolar resistance always fell morethan the efferent arteriolar resistance. The ef-ferent arteriolar resistance fell in 5 of 7 cases.In the last case, with a 50 per cent increase inefferent arteriolar resistance (if there is notechnical failure) one would suspect either auton-omous constriction of the efferent arteriole or aresponse to an increase in circulating adrenalinelicited by the fall of mean blood pressure to 75mm. Hg. (Normally, in the unanesthetizedsubject both arterioles appear to be active (6).)

' See Part II, The adual applicaion of Poiscuilk's law.'In 2 cases under spinal anesthesia with extremely

abnormal blood pressures of 64 and 75 mm. Hg, the formulafor afferent resistance breaks down, becoming negative.This would indicate that in this circumstance intracapsularand intrarenal pressures are less than the values we haveused as normal. A modification of these quantities sothat the intracapsular urinary perfusion pressure remains10 mm. Hg would leave efferent arteriolar resistance un-changed, while increasing afferent arteriolar resistance and,to a much lesser extent, total arteriolar resistance.

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ARTERIOLAR RESISTANCE IN THE KIDNEY

TABLE I

Effects of spinal anesthesia on renal arteriolar rcsistance*

Clearances__________ ~~~~~~~~~~~~RA/RJFARAIRA ARB/R.F &RIR

PG RA RzJ R Ratio Change Change ChangeHD PM Glo- Resist- Resist- Total of in in in

Subject Condition Inulin D Renal Mean merular ance ance arte- afferent afferent efferent totalnumber (Glo- Dio- blood blood intra- of of riolar to arte- arte- arte-

mer- drast flow pressure capillary afferent efferent resist- efferent riolar riolar riolarular (Plasma pressure arterioks arterioes ance resist- resist- resist- resist-

filtra- flow) ance ance ance ancetion)

X iOVXi X 1Of0X0 X lcMXprc per mcc. p Hr mm. Hg mm. Hg mm. Hg | p

minute minute minute mm gm. zpr" pe ".pr" e ej e tPrcnminute minute minute

18 Control 81 421 739 100 57.4 30.5 25.9 56.4 1.1818 Spinal 84 455 798 90 56.9 16.5 23.3 39.8 0.71 -46 -10 -2913 Control 139 607 1065 100 60.9 17.9 20.5 38.4 0.8713 Spinal 128 655 1150 80 57.7 2.0 16.8 18.8 0.12 -89 -18 -5111 Control 128 893 1567 94 53.7 13.0 10.3 23.3 1.2611 Spinal 116 910 1597 80 52.6 4.6 9.5 14.1 0.48 -65 -8 -3920 Control 176 812 1425 100 59.7 14.2 14.7 28.9 0.9720 Spinal 155 771 1353 83 58.2 3.5 14.6 18.1 0.24 -75 -1 -3717 Control 152 648 1137 95 61.5 11.9 19.7 31.6 0.6017 Spinal 102 596 1046 64 55.7 17.0 17.0 0.00 -100 -14 -4612 Control 99 468 821 110 59.7 37.5 25.0 62.5 1.5012 Spinal 80 426 747 100 57.1 30.6 25.2 55.8 1.21 -18 +1 -1110 Control 146 897 1574 86 55.1 6.9 11.0 17.9 0.6310 Spinal 128 664 1165 75 57.5 16.5 16.5 0.00 -100 +50 -8

* See footnote 3 in text.

If we compute the average of the ratios tomean blood pressure of the blood pressure in theterminal portion of the glomerular capillaries(PG = Po, + 20, see Part II) in the cases duringanesthesia (renal denervation), we find a ratio of0.70. If the 2 cases with abnormally low bloodpressures are omitted, the ratio is 0.66. Wintonand his collaborators report an average value of0.66 (7) in the dog's isolated (denervated) kidney.

PART II

FORMULAEFOR GLOMERULO-DYNAMICS

Glomerular equilibriumSmith el al. (2) first consider the relationship

between the forces acting in the glomerulus.They reach the conclusion that, even under thecondition of greatest renal blood flow, equili-brium exists between the difference in hydro-static pressures on both sides of the glomerularmembrane and the osmotic pressure of the con-centrated blood before it leaves the glomerulus.

The change in hematocrit during glomerularfiltration

Using the methods of Van Slyke (8), it can beshown that even during maximal glomerularhemoconcentration, when the plasma proteins

are concentrated 33 per cent (2), the resultingchange in pH of the plasma (neglecting the buf-fering action of the red cells) is less than 0.006,which is less than the change occurring in thelungs. The hematocrit, with so small a shift,changes less than 1 per cent (9, 10). Conse-quently, in considering the effect of glomerularfiltration on the red cells, the change in pH canreasonably be neglected.

Weshall turn to the effect of the increase inplasma protein concentration on the red cells.While this increase is quite significant with re-spect to the glomerular membrane, which ispermeable to all the crystalloid ions, it becomesinsignificant so far as the red cell is concerned.

Recent studies using radioactive salts (11)have confirmed the previous finding that, in man,the red cell is, at least for a few hours, imperme-able to sodium and potassium. The determina-tion of the forces acting on the red cell membranemust, therefore, include the oncotic action of theimpermeable crystalloid ions (sodium and potas-sium), as well as the protein, on both sides of thered cell surface.4 On this basis, a rough compu-

'We are very much indebted to Dr. Homer W. Smithfor calling this fact and its implication to our attention.

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HAROLDLAMPORT

tation shows that plasma proteins account foronly 1 per cent of the oncotic pressure of theplasma with respect to the red cell. Since thecrystalloid ions permeate the glomerular mem-brane, they are not concentrated in the glomeruliand an increase of 33 per cent in plasma proteinwill produce a change of less than Y per cent inthe oncotic pressure acting on the red cells.Such a change is, of course, trivial so that we arejustified in concluding that little, if any, waterleaves the red cells during glomerular filtrationand that their absolute volume remains fixed.A formula based on this premise is easily derivedand is presented later when the change in vis-cosity of the blood during its passage throughthe kidney is considered.

Osmotic pressure of concentrated bloodSince the red cells are not osmotically active in

the ultrafiltration occurring in the glomerulus,they can be excluded while this problem isexamined.

To apply the results of in vitro determinationof the osmotic pressure of blood to the kidney,one must be sure that the glomerular membraneacts like the artificial membranes in use. Thereappears to be little difference among the artificialmembranes (12) and a suggested electrical chargeon the glomerular membrane (13) seems unlikelyand unimportant (14, 15). The observations onhuman transudates by Loeb, Atchley, andPalmer (16) under in vitro conditions showed thesimilarity between the living and artificial semi-permeable membrane and it therefore appearsvery likely that the results of laboratory deter-minations of osmotic pressure can be applied toglomerular ultra-filtration.

Excellent osmotic pressure measurements ofblood have been made with an instrument in-vented by Hepp (10, 17, 18). It has not, as yet,been applied to human blood or blood moreconcentrated 'than normal-the range we areconcerned with-so that, in determining the rela-tionship between osmotic pressure and proteinconcentration in human plasma, we shall use therecent results of Adair and his coworkers (19).They have studied the osmotic pressure of freshand redissolved dehydrated human serum up to,but not beyond, normal concentration and havegiven an empirical formula. At the maximum

glomerular hemoconcentration of 33 per cent, theosmotic pressure is found to be increased 65per cent, so that the assumption by Smith et al.(2) of proportionality between osmotic pressureand protein concentration can lead to consider-able error (20 per cent).

The Adairs' and Greaves' (19) formula, modi-

fied for 37.50 C., is: Po' 2.34S' + 0.1,wher'~Oisthe smoIc - 0.0543S'+01

where PO, is the osmotic pressure in mm. of Hgof one source of normal human plasma and S' isthe concentration of protein in grams per 100 cc.solution of the corresponding serum. The addednumber, 0.1, is derived from Hepp's work oncow's blood. The osmotic pressure of plasma isabout 13 mm. H20 less than that of the corre-sponding serum (17). But the osmotic pressureat pH 7.4 (arterial blood) is about 14W mm. ofH20 greater than at pH 6.9, the value studied byAdair. The difference, 1 mm. H20, is 0.1 mm.Hg. Since this is less than Wper cent of Po,it will be neglected. Wethen have:

(1) 2.34 S'Pop = 1 - 0.0542 S'

This formula is to be used to obtain the osmoticpressure of plasma at any degree of hemo-concentration.

Intracapsular and intrarenal pressureIn the glomerulus, let us call the hydrostatic

pressure of the blood, when it reaches osmoticequilibrium with its filtrate, PF, expressed inmm. Hg, and correspondingly let Pc be theintracapsular pressure, the pressure againstwhich the filtrate is formed. Smith et al's (2)description of equilibrium in the terminal capil-laries of the glomerulus gives the equality:

PG = Po'+ Pc-

Is the intracapsular pressure the same as in-trarenal pressure, which, in the dog, has beenestimated to average 10 mm. Hg and to be inde-pendent of arterial pressure (20, 21, 22)? Smithand his coworkers, in their analysis of the bloodflow in and out of the glomerulus, assume thatPC = PR. In such a circumstance, glomerularfiltrate might conceivably be perfused throughthe tubules up to the loop of Henle where it be-

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ARTERIOLAR RESISTANCE IN THE KIDNEY

comes osmotically concentrated (1), in part bydiffusion from a high to a low concentration ofwater. But it is doubtful that this would be atall a potent factor, and certainly its effect wouldbe reversed during the secretion of hypotonicurine found at times in all mammals (1). An-other possibility is the suction effect of the ab-sorption of about 99 per cent of the filtrate (1)by the time the urine has reached the first partof the distal tubule. Its result would be to makea suction pump out of the proximal tubules, loopof Henle, and part of the distal tubules, theenergy coming from the tubular epithelium. Anobjection to this notion is that the pump wouldhave no valve; it would similarly suck urinebackwards, up the collecting tubules, from thepelvis of the kidney. Neither of these hypo-theses offers a force to move the concentratedurine through the last part of the distal and thecollecting tubules. That the arterial pulsationsof renal tissue milk urine towards the calicesbecause the resistance of the remote retrogradebranches of the system is greater than that ofthe central trunks seems a minor likelihood, in-capable of effectively explaining the perfusion ofurine through distal and collecting tubules. Weare forced to the conclusion that PC > PR; intra-capsular pressure is greater than interstitial renalpressure.

O'Connor (23) calculated the glomerular fil-trate perfusion pressure in the rabbit (PC - PR)to be a minimum of 10 mm. Hg if the bore of thetubules were lO,u. In view of the fact that"histological sections would indicate a ratherhigher value" for the lumen (24), and becausewe have noted other factors helping in perfusion,we should choose this minimum value for humanfiltrate perfusion pressure: PC- PR = 10. WithWinton's average for interstitial pressure (PR= 10 mm. Hg), we have: Pc = 20 mm. Hg; andPG = Po, + 20.

Poiseuile's lawIdeally, the resistance of a fixed tubular

system to the flow of liquid is a constant definedby the ratio of the pressure difference used toperfuse it to the product of the rate of flow there-by produced and the relative viscosity of thefluid with respect to some standard. Inflow andoutflow at the points of pressure measurement

must be equal for the definition of resistance toapply.

Resistance Perfusion pressurerate of flow X relative viscosity

Experimental in vivo confirmation of this rela-tionship, somewhat modified for the kidney, iscrude (21). Perfusion experiments have shownthat, while it applies with modification (a con-stant is subtracted from pressure) to the isolatedhind-limb of the dog (25), it does not apply tothe isolated dog's kidney (26). However, theisolated dog's kidney behaves quite differently, incertain respects, from the kidney in the anes-thetized or unanesthetized dog. Its urine ishypotonic; its inulin clearance (glomerular filtra-tion rate) is quite considerably lower (more than50 per cent); its creatinine clearance is not uni-formly proportional to inulin clearance (27).While, in man, inulin clearance is approximatelyindependent of renal blood flow (2), in the iso-lated or anesthetized dog's kidney, this is notthe case (27). The mechanism of dilutiondiuresis also appears to differ in the isolated andanesthetized kidney (28).

The presumption attending the application ofPoiseuille's law to an isolated organ is that itsresistance remains constant while the flow andthe perfusion pressure which cause it vary. Ifthe isolated organ's arterioles respond auton-omously to the change in pressure, and thedenervated anesthetized kidney, unlike theextremities, appears to do so (29, 30), this pre-sumption would be false. It does not seem un-reasonable, therefore, to apply Poiseuille's lawto man's intact kidney. Smith and his co-workers have done so, without the modificationfound necessary in the hind-limb (25), analyzingseparately afferent and efferent arteriolar resist-ance. Weshall proceed with this analysis.

Resistance of afferent arterioksSince the blood undergoes no change in vis-

cosity before the glomerulus, no correction forviscosity is needed if only the afferent arteriolarresistance referred to normal blood is measured.However, the two points of pressure referencechosen by Smith el al. are average arterialpressure and hydrostatic blood pressure in theglomerulus when osmotic equilibrium with ultra-

539

HAROLDLAMPORT

filtrate is reached, which must be near the efferentarteriole.

Wecan consider afferent resistance defined inthis way (RA) to be composed of true afferentarteriolar resistance, ending at Bowman's cap-sule (RA) and the resistance of the glomerularcapillaries (RG) to perfusion (Rp) and filtration(Rp). Thus, we would have:

RA= RsA+ RG.

Since the filtration and perfusion resistances ofthe capillaries are parallel, they are computedin the usual way:

1 _1 1

It is found that, at the mean basal plasma filtra-tion fraction of 19 per cent (2), where the effectof changes in viscosity of perfused blood andfiltered plasma (they are opposite in directionand tend to cancel) are neglected, the result ofcomputing RG with the above formula, usingPoiseuille's law, is less than 11 per cent differentfrom the result when we compute RGas:

fall in pressure in glomerular capillariesRa = rate of flow of blood into glomerulus

While an error of 11 per cent in RG may seemconsiderable, it is relatively small in its effecton RA, as defined above to include RG, since RGis small compared with RA; the pressure drop inthe glomerulus itself is small compared withthat in the afferent arteriole.5

By the time the blood in the glomerular capil-laries reaches the efferent end, its volume hasbeen diminished by filtration and its viscosityhas increased due to the increase in the hemato-crit and the concentration of plasma protein.In Poiseuille's law, the denominator of the ex-pression for resistance is the product of flow andviscosity. Since flow decreases and viscosityincreases, the effects of the two errors in neglect-ing these changes are in opposite directions.Later, in the consideration of the efferent arterio-lar resistance, it will be shown that these errorsalmost cancel each other. It is for this reason,as well as because RG is small compared with

' The change of kinetic into potential energy is negligible(31).

RA, that we introduce little error in computingRA up to the equilibrium point in the glomerularcapillaries, near the efferent arterioles, withoutallowing separately for filtration resistance, re-duced flow, and increased viscosity. Smith andhis associates likewise make the same choice,including the glomerulus in afferent arteriolarresistance without special adjustment (2).

Resistance of efferent arteriolesThe problem of the efferent arteriole is differ-

ent in that in it the blood viscosity is increasedthroughout the major fall of pressure, up to thepoint where the glomerular filtrate has been re-absorbed in the peritubular capillaries. Smithand his coworkers make no allowance for theincrease in viscosity since they consider itnegligible (2).

Effect of change in viscosityLet us compute the importance of this change

to ascertain how large a factor it is.It has been shown that the viscosity of blood

depends on the hematocrit, but that it is alsoproportional to the viscosity of the plasma(32, 33). The maximum plasma protein con-centration observed by Smith and his associatesis 33 per cent (2), so we are interested in therange of plasma viscosity from 7 grams per cent(taken as normal) to 9.3 grams per cent. Whilethere have been many attempts to offer generalformulae relating the concentration of solute toviscosity, none has been completely successful.However, in our range, a simple linear relation-ship is an excellent empirical one (34, 35). Sucha formula, based on the data for serum viscosityin the proper range (35), has been derived by themethod of least squares, modified with an in-crease of 222 per cent at normal hemoconcen-tration to obtain the viscosity of the correspond-ing plasma (36, 37). Where V' is relative vis-cosity of the corresponding plasma with referenceto water as unity, and S' is defined as previouslyfor osmotic pressure, we obtain:

(2) V' = 0.60 + 0.204S'.

Wemust now compute the change in hemato-crit caused by glomerular filtration.

Let H. = hematocrit of blood before ultrafiltration.H.' = hematocrit of blood after ultrafiltration.

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ARTERIOLAR RESISTANCE IN THE KIDNEY

D = diodrast clearance of plasma in cc. perminute corrected for incomplete renal extrac-tion and residue in red cells (2) (rate of flowof plasma through kidney).

I = inulin clearance in cc. per minute (2) (rate ofglomerular filtration by the kidney).

IF = D-, filtration fraction.

The new hematocrit is found:

_ unchanged volume of RBCvolume of RBC+ initial volume of plasma

degree of increased plasma concentration

Or,

He + (1 - He)(I - F)

On this basis, with F at its maximum value of0.33, and taking a high initial hematocrit of60 per cent, we find that H0' is 69 per cent.Consequently, we are interested in a range ofhematocrit from about 35 to 70 per cent. Theeffect of hematocrit on the viscosity of blood inthe dog has been carefully studied by Whittakerand Winton (25). Using their curve (p. 358) inthis range, we have obtained an excellent fit withthe following empirical equation:

U' = VI 027 + 0.983 )

Here U' is the viscosity of whole blood of hemato-crit HX' and plasma viscosity V', while U and Vare the values for unconcentrated normal blood(Hc' = Hc; V' = V).

Formulae have been given for all the quantitiesdetermining U' so that we can compute the rela-tive change caused by glomerular filtration. Weuse for the normal hematocrit (He) Smith's aver-age value of 43 per cent.'

The maximum increase in blood viscosityoccurs when the filtration fraction is 33 per cent.In this case, with relative viscosity entering intothe denominator of Poiseuille's law, efferentarteriolar resistance would be estimated at themost 46 per cent too high, if the increase inblood viscosity caused by glomerular filtration isneglected. But if the loss of glomerular filtrateis also disregarded in estimating efferent arterio-lar flow, along with the increase in viscosity, theresult is at the worst 118 per cent of the moreexact value, an error of 18 per cent. We see,

then, that if an approximation is to be made, itis wiser to neglect both the reduction in flow andthe increase in viscosity of the blood in theefferent arteriole, rather than either one alone,since they tend to compensate each other. Atmean basal plasma filtration fraction (19 percent), these errors are considerably reduced-23WX per cent and 10 per cent, respectively.This illustrates in numerical form our previousstatement that it is proper to neglect the oppos-ing effects on resistance of changes in viscosityand blood flow occurring in the glomerulus whendefining afferent arteriolar resistance. Strik-ingly enough, we see that the error introduced bythis approximation for the resistance of theefferent arteriole is nearly proportional to thefiltration fraction. Thus 0.47F can be appliedto correct our simplified formula for efferentresistance where the hematocrit is properly takenas the average, 43 per cent. Other factors of Fcan be used for other values of the hematocrit,if need should arise. It may be noted that themore precise expression involving viscosity andreduced blood flow depends only on clinicallymeasurable quantities. Wehave:

Efferent arteriolar resistancepressure difference X (1 - 0.47F)whole blood flow through kidney

The definition of efferent arteriolar resistanceThe next step in the definition of efferent

arteriolar resistance, having considered thechanged viscosity and the reduced flow in thisarteriole, is to consider what the perfusion pres-sure is. Certainly the equilibrium pressure atthe efferent end of the glomerular capillaries,used as the terminal one in the definition ofafferent resistance, is the correct initial one andwe shall follow Smith et al. (2) in using it, al-though computing its value, as above, somewhatdifferently. The ideal terminal pressure wouldbe the one at the arterial end of the peritubularcapillaries, before significant change in the rateof blood flow due to reabsorption of the glomeru-lar filtrate has occurred. But this pressure is notknown and is certainly a variable one, dependingon rate of flow and other factors, so that a fixedestimate of its size is unwarranted. As the blood

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HAROLDLAMPORT

passes through the capillaries to the venules,6 itsvolume is restored to 99 per cent of its originalvalue by the reabsorbed glomerular filtrate andits viscosity falls back to normal. This is roughlythe reverse of the situation in the glomeruli. Bychoosing venous capillary pressure as our ter-minal pressure in the definition of efferent ar-teriolar pressure, we introduce an error, but onceagain the effects of falling viscosity and risingflow are in opposite directions; the fraction in-volved of the total efferent arteriolar resistancedefined is small, and so the net effect can beneglected. Homer Smith and his coworkers (2)have similarly chosen for their terminal perfusionpressure the venous end of the capillaries wherehydrostatic blood pressure is very likely to beapproximately equal to the sum of intrarenaland normal osmotic pressures. Weshall, there-fore, chose PR + PO, where PO is normal bloodosmotic pressure in mm. Hg, for the terminalpressure in our definition of efferent arteriolarresistance.

Arterial pressureIn measuring the perfusion pressure of the

afferent arteriole, we should naturally choosethe pressure in the renal artery. Since this ispulsatile, the best approximation would be itsintegrated mean value. This value is not theaverage of systolic and diastolic blood pressurebut diastolic blood pressure plus 44 per cent of thepulse pressure (38). The fall in this more precisemean, as obtained from subclavian and femoralarteries in man, is usually about 2 mm. of mer-cury. The average of systolic and diastolicpressure is about 3 mm. Hg too high (38). It istherefore clear that the average of systolic anddiastolic blood pressure taken at the arm is agood enough measure of mean renal arterypressure.

The actual application of Poiseuille's lawPreviously, we have mentioned Whittaker and

Winton's (25) finding that Poiseuille's law appliesto the hind-limb of the dog perfused with blood,but with modification. They find that theperfusion pressure plotted against the flow is astraight line with an intercept at about 20 mm. of

6 The cortical tissues are mainly nourished by glomerularfiltrate on its way to the capillaries.

mercury. That is to say, Poiseuille's law forblood is really:

Resistance = Perfusion pressure - 20Rate of flow

This phenomenon appears to be an inherent oneof blood itself (25). There is further confirma-tion of it in the intact animal (39). In applyingPoiseuille's law to the afferent arteriole, we shalltherefore use this 20 mm. Hg correction. Butwe shall not use it in the efferent arteriole whereblood flows from capillaries back to capillaries,not from artery to capillaries, as in the afferentarteriole and in Whittaker and Winton's ex-periments.

We are now ready to utilize Smith and hiscoworkers' (2) analysis of renal dynamics asmodified above.

Let PM = Average of svstolic and diastolic blood pres-sure measured at the arm in mm. Hg.

D = Diodrast clearance of plasma in cc. perminute, etc., as noted previously.

I = Inulin clearance of plasma in cc. per minute,as noted previously.

HC = Hematocrit, as a fraction of 1.1 whole blood volume

H = 1 - H. plasma volume

HD = Total blood flow in cc. per minute.

F = D plasma filtration fraction.

RA = Afferent arteriolar resistance, as defined.RE = Efferent arteriolar resistance, as defined.R = RA + RE, total arteriolar resistance, as

defined.

The resistance units will be mm. Hg per cc. per minutewith the viscosity of the unconcentrated blood take asunity.

Other symbols have already been defined.

The perfusion pressure for the definition ofafferent arteriolar resistance is the differencebetween mean renal artery pressure and theequilibrium blood pressure in Bowman's capsule.Thus:

PM-Pa-20R,,= HD(3)

The perfusion pressure for RE is PG- (PO + PR); the flow is HD - I. Wehave:

(4)PG PO PR

(HDD-I) U

542

ARTERIOLAR RESISTANCE IN THE KIDNEY

Substituting for PG its equivalent, PC + Pot,we write:

(5) Po' - Po + 10RB= ~~U'

(HD-I) U

Wehave justified the substitution, where k is a

constant dependent on the hematocrit,

U' HDI 1-kF

Accordingly,

(6) RR= (1- kF)(Po -Po + 10)HD

PM- Po, -40RA= HD(7)

The formula for Po, has already been given(see Equation (1)), and Po is found from thisformula when for S' we substitute S, the clini-cally observed value for unconcentrated serum

protein. The degree of hemoconcentration being

we have S' = 1F

Substituting the formula for S' in that for PO,,we find that:

(8) Po' ~~2.34S

(8) P0' = 1- 0.0542S -F

Po is the value of Po, when F is set equal tozero.

Equations (6) and (7) for RR and RA are

therefore seen to be completely determined, withthe aid of Equation (8), by the clinically deter-minable entities: serum protein, hematocrit,inulin and diodrast clearances, and blood pres-

sure.7SUMMARY

The application of Poiseuille's law to the kid-ney has been discussed and formulae have beendeveloped to measure clinically, in man, afferentand efferent arteriolar resistance. A practicalapplication to available clinical data on the renaleffect of spinal anesthesia (denervation) in nor-

mal man has also been offered. At present,while normal man may lack tonic centralnervous control of renal blood flow, this does not

7 The method of computing k has already been givenunder Effect of change in viscosity. Usually, k = 0.47is adequate, as most hematocrits are near 43 per cent.

appear to have been demonstrated. The evi-dence also does not preclude autonomous controlby the kidney of its blood supply.

Incidentally, empirical formulae for the vis-cosity of human plasma and whole blood havebeen derived from observations in the literature.

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