Scandinavian Journal of Clinical & Laboratory Investigation, 2014; 74: 611–619
ORIGINAL ARTICLE
Evaluation of non-polynomial equations for one-compartment correction of slope-intercept GFR: Theoretical prediction and experimental measurement Natalie C. Finch1, Reidun Heiene2, Nicholas J. Bird3, Jonathan Elliott4 & A. Michael Peters5 1School
of Veterinary Sciences, University of Bristol, UK, 2Norwegian School of Veterinary Science, Norway, Hospital, Cambridge, UK, 4Royal Veterinary College, London, UK, and 5Departments of Nuclear Medicine, Royal Sussex County Hospital, Brighton, and Harley Street Clinic, London, UK 3Addenbrooke’s
Abstract Background. Polynomial equations for one-compartment correction of slope-intercept glomerular filtration rate (GFR) will underestimate values at high clearance rates. Non-polynomial correction equations that are independent of patient size and renal function would be advantageous and may have cross-species use. Materials and methods. The study explored the theoretical basis of firstly the Jodal and Brochner-Mortensen one-compartment correction equation, replacing plasma volume with extracellular fluid volume, and secondly an equation described by Peters. One-compartment correction factors (a which is related to plasma volume and v which is related to extracellular fluid volume) which avoided the need for scaling to body size were developed. Both factors were determined from the biexponential clearance curve of the markers iohexol and 51Cr-EDTA in humans and iohexol in cats and dogs. Relationships between a and v and filtration function and body size were then determined using data from humans, cats and dogs to assess their validity and compare this with theoretical predictions. Results. In all species, v was higher than a, as theoretically predicted. Both were significantly higher in humans than cats and dogs, ruling out cross-species use. Significant relationships were present between v and measures of filtration function in humans, but were weak with respect to a. Neither a nor v showed significant relationships with filtration function in animals or with body size in any species. Conclusions. a and v (which are factors independent of body size) can be used interchangeably for correcting slope-intercept clearance. However values of both for humans are higher compared to cats and dogs. Therefore a single cross-species factor cannot be used. Key Words: Kidney function tests, glomerular filtration rate, cats, dogs, humans Abbreviations: BSA, body surface area; BW, body weight; Cl, clearance; 51Cr-EDTA, Cr-51-ethylenediaminetetra-acetic acid; D, dose; ECFV, extracellular fluid volume; GFR, glomerular filtration rate; HPLC, high performance liquid chromatography; MRT, mean residence time; PV, plasma volume.
Introduction Glomerular filtration rate (GFR) can be estimated in a clinical context from the plasma clearance of a filtration marker, such as Cr-51-ethylenediaminetetraacetic acid (51Cr-EDTA) or iohexol. When blood sampling is started shortly following marker administration (multi-sample clearance), the clearance curve is traditionally resolved into two exponentials with respective zero-time intercepts A and B, and corresponding rate constants a1 and a2.
The first exponential (A, a1) represents redistribution of marker throughout its distribution volume, which for most filtration markers is essentially the extracellular fluid volume (ECFV). Modification of this technique by collection of a limited number of blood samples (generally 3) from 2 hours postmarker administration is known as the slope-intercept method. As a result of ignoring the first exponential (a1), clearance determined using the slope-intercept method requires a correction for marker lost in urine
Correspondence: Natalie C. Finch, currently at School of Veterinary Science, University of Bristol, Bristol, UK. Tel: 44117 9289420. Fax: 44117 9811277. E-mail: [email protected] (Received 4 August 2013; accepted 25 May 2014) ISSN 0036-5513 print/ISSN 1502-7686 online © 2014 Informa Healthcare DOI: 10.3109/00365513.2014.928941
612 N. C. Finch et al. during the redistribution phase (one-compartment correction). The most widely used correction formula for slope-intercept clearance in human patients was originally described by Brochner-Mortensen [1]. Similar correction formulae have since also been developed for dogs [2] and cats [3]. The effects of these one-compartment correction equations are shown in theoretical curves in Figure 1. However, these corrections are generally second order polynomials which, as quadratic functions, have a maximum value to the parabolic curve. This can result in underestimation of clearances at high rates of clearance. In human patients, the maximum uncorrected and corrected clearances when using the BrochnerMortensen equation will be 407 and 201 mL/ min/1.73 m2, respectively [4]. In feline patients, the maximum uncorrected and corrected clearances when using the cat correction formula can be calculated to be 8.35 and 4.33 mL/min/kg, respectively. In canine patients, the maximum uncorrected and corrected clearances when using the dog correction formula will be 7.53 and 3.98 mL/min/kg, respectively. Clearance rates of such magnitude are not often encountered in human patients. However, hyperthyroidism is very prevalent amongst the senior cat population and clearance rates in excess of 4.33 mL/min/kg have been reported [5,6]. The value at which underestimation becomes important has not been established for human or veterinary patients, but the potential inaccuracies of these polynomial correction formulae at high rates of clearance have led to recent interest in the development of new equations that may avoid the problem [4,7]. An equation for correcting slope-intercept clearance that is not quadratic, that provides a reliable estimate of multi-sample clearance and that could be used across species, across patients of different body size, across patients with varying GFR, and could
also be applied to different filtration markers is desirable. This study aims to explore such equations in humans, cats and dogs. The Jodal and Brochner-Mortensen equation The recently described Jodal and BrochnerMortensen correction equation [7] is as follows: Cl Cl1 / (1 f.Cl1)
(1)
where Cl is clearance, Cl1 is slope-intercept clearance prior to one-compartment correction and f is a correction factor. Note that this correction equation is applied to unscaled GFR, so f is required to account for body size. Jodal and Brochner-Mortensen [7] introduced the factor f as: f aPV / PV
(2)
where PV is plasma volume and aPV is ‘missing area’. The area is ‘missing’ in that the area under the full plasma concentration curve is larger than the area under the one-compartment curve; aPV is this difference scaled to initial plasma concentration (i.e. divided by D/PV where D is injected dose of filtration marker). This scaling gives aPV the unit of time. Jodal and Brochner-Mortensen [7] went on to show that aPV is relatively constant between children, women and men, indicating that f does not depend on renal function, only upon the size of the person. [4] More generally, f can be expressed as [7]: f ([1 / Cl] [1 / Cl1])
(3)
which follows from isolating f in Equation 1. Equation 2 can be rearranged as:
Figure 1. Theoretical equations showing the effect of polynomial correction factors (a) Brochner-Mortensen correction formula and (b) cat (bold line) and dog (dashed line) correction formulae. The curves were constructed using theoretical data. The regression line for corrected clearance against multi-sample clearance was plotted.
Correcting slope-intercept GFR 613 aPV f.PV PV([1/Cl] [1/Cl1])
(4)
(6)
ECFV([1/Cl] [1/Cl1]) (7) Both aPV and a have the unit of time. Since ECFV PV, it follows from Equations 4 and 6 that a aPV. In summary, Equation 1 can be used with f aPV/PV. However, as shown by Peters [8], it is also possible to use f a/ECFV. There was good agreement between the two approaches. In a human population, Jodal and Brochner-Mortensen [7] showed that PV in mL could be estimated as 1336 BSA1.34, and their final (rounded) result was f 0.0032 BSA1.3, where BSA is body surface area. Peters [8] showed that ECFV in mL could be estimated as 6080 BSA1.34 and found f 0.00308 BSA1.34. Using f a/ECFV, Equation 1 can be written as: Cl Cl1 / (1 a.[Cl1/ECFV])
(8)
Note that the term [Cl1/ECFV] has units of a rate constant (min1) so a has units of time. Peters’ equation [9] It was previously suggested by Peters [9] that a2 can be used instead of Cl1 in a correction equation of similar form to Equation 1, as follows. For a bi-exponential plasma concentration-time curve: Cl
d a / α1 B / α 2
(9)
where D is the dose of the filtration marker administered. For slope-intercept clearance, in which the initial redistribution exponential is ignored, the equation becomes: Cl1
d B / α2
(10)
Therefore, Cl1 a / α1 B / α 2 Cl B / α2
Cl Cl1 / (1 v.a2)
(5)
Jodal and Brochner-Mortensen [7] argued on theoretical grounds that PV is the most appropriate marker distribution volume to use in Equations 2 and 3. However, a more recent study showed that any marker distribution volume that is proportional to PV, such as ECFV, can also be used [8]. So, rather than PV and aPV, it is possible to substitute with ECFV and a corresponding constant a. With this terminology Equation 4 can be replaced with: a f.ECFV
which can be rearranged to give:
(11)
v A / (B.a1)
where
(12) (13)
a2 has units of 1/time, so v, like a, has units of time. Note the similarity between Equations 8 and 12. Relationship between a and v From inspection of Equations 8 and 12, it is clear that the terms a.(Cl1/ECFV) and v.a2 must be equal, so, n Cl1 (14) a α 2 ECFV When calculating clearance using the slope-intercept technique, d Cl1 (15) (B / α 2 ) which can be written as: Cl1 a2.Vd
(16)
where Vd D/B which is generally referred to as the ‘virtual volume of distribution’. So combining Equations 14 and 16. Vd n a ECFV
(17)
Vd exceeds ECFV by a factor that depends on a2 [10], and the relation of a2 with v/a is the same as its relationship with Vd/ECFV, namely non-linear in which v/a approaches unity as a2 approaches zero, when a v. Relationships between a, v and parameters of the bi-exponential curve Similar to aPV which is the missing area under the curve when scaling to PV, a is the missing area under the curve when scaling to ECFV. Substituting for Cl and Cl1 in Equation 7 a ECFV
(a / α1 B / α 2 B / α 2 ) d
ECFV A/(D.a1)
(18) (19)
ECFV can be determined from multi-sample clearance following as ECFV Cl MRT
(20)
where MRT is mean residence of tracer in its distribution volume (ECFV) before filtration and determined following as MRT
a / α12 B / α 22 a / α1 B / α 2
(21)
614 N. C. Finch et al. Therefore, a
a
d α1 B
2
α2
a α1 B a α1 B
2
α2 a (22) α2 d.α1
a α1 2 B α2 2 a 2 α1 ( a α1 B α2 )
(23)
So it can be seen that a, like v, is a function only of the parameters of the bi-exponential plasma clearance curve, not injected dose of marker (D) or body measures (such as BSA or BW). Equation 23 cannot be further reduced to determine the dependence of a on a2. However, assuming that the value of a2 is small compared with the value of a1, following then a ≈ A / (B.a1)
(24)
When a2 is zero, Equation 24 is therefore identical to Equation 13, and a v. By inserting typical values of a1 into Equation 23 and assuming A B, then a non-linear inverse relationship is seen between a and a2. In contrast, it can be seen from Equation 13 that v is not directly dependent on a2. However, in practice, A, B and a1 are all influenced by a2, so v may be indirectly dependent on a2, and a may change its theoretical relation with a2. The aim of the present study was to explore these theoretical principles and to determine the values of a and v for one-compartment correction of slopeintercept GFR. Ideally, notwithstanding the above considerations, these factors should be independent of filtration function and body size and, in order to be used across veterinary and human medicine, also of species. Therefore, relationships between a and v and renal function and body size in humans, cats and dogs were tested. Methods Patients Humans. Multi-sample clearance (Cl) was performed in 20 healthy volunteers and 60 patient volunteers referred for routine, clinically indicated measurement of GFR. All volunteers gave informed consent and the study was approved by the local research ethics committee. Patients were given simultaneous injections of 51Cr-EDTA and iohexol into opposite arms. Following administration, bilateral exactly timed venous blood samples were collected at approximately 20, 40, 60, 120, 180 and 240 min and assayed for marker injected contralaterally, 51CrEDTA by gamma counting and iohexol by X-ray fluorescence, as previously described [11]. The healthy volunteers were studied on several occasions as part of repeatability and fasting versus non-fasting studies [11]. Therefore, there were a total of 110 clearance
measurements available for analysis. In 6 studies, the clearance curves could not be resolved into two exponentials and were therefore, excluded leaving 104 studies for analysis. Feline patients. Multi-sample clearance was performed in 16 healthy client owned cats as described in a previous study [3]. Briefly, a bolus dose of iohexol (Omnipaque™) was administered via an intravenous catheter placed in the right cephalic vein. Exactly timed blood samples (8–9) were collected between 5 and 300 min following administration. Iohexol was analyzed at a reference laboratory (Epsom and St Helier NHS trust) using high performance liquid chromatography (HPLC). Canine patients. Multi-sample clearance was performed in 43 client-owned dogs with varying renal function at the Norwegian School of Veterinary Science as described in a previous study [12]. Briefly, a bolus dose of iohexol (Omnipaque™) was administered via an intravenous catheter placed in the right cephalic vein. Four exactly timed blood samples were collected between 5 and 240 min following administration (two from the initial redistribution phase and two from the elimination phase). Iohexol was analyzed using HPLC as described [12]. In three dogs, the curves could not be resolved into two exponentials, leaving 40 clearance measurements available for analysis. Data analysis Values of a and v were determined using Equations 23 and 13, respectively. Body size was assessed in human patients using BSA estimated from height and weight using the formula of Haycock [13] and in feline and canine patients using body weight (BW; kg). Statistical methods Parametric testing was performed in all statistical analyses. Comparisons between species were performed using the independent samples t-test. A Bonferroni correction was applied for multiple testing. Relationships were explored using linear regression analysis and by determining the coefficient of determination (R2). Correlations were assessed using Pearson’s correlation coefficient (r). Results are presented as mean SD. Significance was set at p 0.05. Results Parameters of biexponential plasma clearance curves In humans A/B, a1 and a2 were not significantly different between 51Cr-EDTA and iohexol. There was no significant difference between species in A/B but a1 and a2 were both significantly higher in cats and dogs compared to humans (p 0.001). In all
Correcting slope-intercept GFR 615
Table I. Values of the biexponential plasma concentration versus time curves obtained in humans, cats and dogs. Humans (n 104) Mean SD Iohexol A/B a1 (min1) a2 (min1)
1.076 0.57 0.039 0.022 0.0051 0.0018
Cats (n 16) Mean SD
Dogs (n 40) Mean SD
0.944 0.370 0.950 0.408 0.084 0.040* 0.095 0.034* 0.01 0.002* 0.010 0.002*
51Cr-EDTA
A/B a1 (min1) a2 (min1)
0.898 0.33 0.036 0.011 0.0052 0.0018
*Significantly different to human patients (p 0.005)
three species, A/B was close to one. See Table I for values of biexponential plasma clearance curve. Values of a and v Mean v was higher than a in all 3 species (see Table II). Neither a nor v was significantly different between cats and dogs but both were significantly lower compared with humans (p 0.001). Relations of a and v with filtration function Based on 51Cr-EDTA in humans, there were weak positive relationships between a and multi-sample GFR (R2 0.052, p 0.021) and between v and multi-sample GFR (R2 0.144, p 0.001; Figure 2a). The regression lines for a and v diverged but the intercepts (i.e. at zero GFR) were similar (15.0 versus 16.5 min) in accordance with the theoretical prediction that a v for zero GFR. Similar relations were seen when filtration function was expressed as a2, when the respective intercepts were 20.3 and 19.8 min (data not shown). Similar results were obtained using iohexol (data not shown). Similar results were also obtained when a and v based on one marker were regressed on filtration function measured with the other marker in humans (data not shown). In cats and dogs, however, neither a nor v showed significant relationships with filtration function (Figures 2b and 2c). In humans and dogs, v/a showed a positive relationship with filtration function (humans R2 0.382, p 0.001, dogs R2 0.126, p 0.025) whereas in cats the relationship was not significant (R2 0.000, p 0.943; Figure 3). Relations of a and v with body size In humans, BSA showed no significant relation with either a or v based on either 51Cr-EDTA (Figure 4a) or iohexol (data not shown). Likewise, neither a nor v showed any significant relation with BW in cats (Figure 4b) or dogs (Figure 4c).
Table II. Mean SD values for the correction factors a and v in humans, cats and dogs. The values of a and v were significantly higher in humans (p 0.001). A single mean value could not be applied for use across species.
a a v v
(51Cr-EDTA) (min) (iohexol) (min) (51Cr-EDTA) (min) (iohexol) (min)
Humans (n 104)
Cats (n 16)
Dogs (n 40)
20.5 6.4 22.3 8.4 27.0 11.3 31.0 15.9
ND 10.6 5.5 N/D 14.2 9.2
ND 8.9 4.2 N/D 11.5 6.8
Discussion The theoretical curves constructed in Figure 1 illustrate the one-compartment correction by the original Brochner-Mortensen study [1] and those reported for cats [3] and dogs [2]. These polynomial corrections will, however, underestimate high GFR values. The purpose of the present study was to explore the theoretical basis of non-quadratic one-compartment correction equations and identify a unique factor to correct slope-intercept clearance that could be used across patients of varying renal function, body size and different species. Correction factors that are independent of body size and renal function are of interest in developing a single correction factor for use with slope-intercept clearance in both human and veterinary patients. In veterinary patients, BSA is not used for scaling GFR and correction equations developed in humans are not applicable [3]. The present paper has first considered the nonpolynomial correction equation by Jodal and Brochner-Mortensen [7]: Cl Cl1 / (1 f.Cl1) (Equation 1) where f ([1/Cl] [1 / Cl1]) (Equation 3) As shown by Peters [8], the factor f can also be expressed by scaling to ECFV giving the following variant of the Jodal and Brochner-Mortensen equation: Cl Cl1 / (1 a.[Cl1/ECFV]) (Equation 8) The two studies gave very similar expression for the factor f, but these expressions were established only for humans and based on BSA. For cats and dogs, the f factors may differ from humans, and scaling is usually made to BW rather than BSA. Second, an alternative equation by Peters [9] (referred to as the Peters equation) has been considered: Cl Cl1 / (1 v.a2) (Equation 12) where a2 is the rate constant of the final exponential of the plasma clearance curve, and v is a constant. Since Equation 12 does not involve body measures (such as BW and BSA), it is more easily applied to non-human species. The aim of the present study was
616 N. C. Finch et al.
Figure 2. Relationship of the correction factors a and v with multi-sample clearance in humans, cats and dogs. (a) a and v in human patients determined from clearance data using the filtration marker 51Cr-EDTA. Unfilled circles represent a and dashed line is corresponding regression line (R2 0.052, p 0.021). Filled circles represent v and bold line is corresponding regression line (R2 0.144, p 0.001). (b) a and v in cats determined from clearance data using the filtration marker iohexol. Unfilled circles represent a and dashed line is corresponding regression line (R2 0.041, p 0.452). Filled circles represent v and bold line is corresponding regression line (R2 0.015, p 0.649). (c) a and v in dogs determined from clearance data using the filtration marker iohexol. Unfilled circles represent a and dashed line is corresponding regression line (R2 0.060, p 0.128). Filled circles represent v and bold line is corresponding regression line (R2 0.069, p 0.101).
specifically to examine whether v was speciesdependent or not. To allow inter-species comparison of the parameters of Equation 1, this equation was used in the form with ECFV (Equation 8), ECFV was expressed from bi-exponential expression of the plasma clearance curve, and finally an expression for a was derived (Equation 23). Like v, these values for a could be compared between species because Equation 23 is independent of body size. Finally, the properties of a and v were compared. In humans, apv is the numerator in the constant 0.0032 described by Jodal and Brochner-Mortensen
[7] and likewise a is the numerator in the constant 0.00308 described in a previous study by Peters [8]. The denominator is an estimate of PV in the Jodal and Brochner-Mortensen [7] equation and of ECFV in the Peters [8] equation, both based on BSA. Analogous equations based on body weight still require development in small animals because BSA is not conventionally used as a scaling variable in animals. Values of a and v As theoretically predicted, v was higher than a in all three species. The values of a and v were respectively
Figure 3. Relationship between v/a and a2 in humans (black filled circles), cats (grey filled circles) and dogs (unfilled circles). In humans and dogs, v/a showed a positive relationship with filtration function (humans R2 0.382, p 0.001, dogs R2 0.126, p 0.025) whereas in cats the relationship was not significant (R2 0.000, p 0.943).
similar between cats and dogs but were both significantly higher in humans, ruling out a cross-species correction factor. The value of v originally determined in human patients using 51Cr-EDTA was 15.4 min [9] but it was later modified to 25.0 min [14] on the basis of further work. The mean value determined in the present study is 27.0 min. This is significantly higher compared with v in feline (10.6 min) and canine (8.9 min) patients. This is probably explained by more rapid mixing of marker throughout the distribution volume in these smaller species, resulting in higher values of a1 (Table I). Relationships between a and v and filtration function a and v showed no significant relationships with GFR in either cats or dogs, possibly because of relatively small patient numbers. It may also be explained by the small range of GFR values in the cats and dogs of which the majority had normal or near normal renal function compared to the range of GFR values in the humans. In humans, however, v and to a lesser extent a, correlated positively with measures of filtration function, although the relationships were weak. As a2 does not feature in Equation 13 (defining v), a positive relationship between filtration function and v would not be expected. We believe that the theoretical relationships that a and v, respectively, show or do not show with filtration function are modified by the effects of variations in a2 on A, B and a1. The intercept values, however, remained very similar between a and v, as predicted theoretically. Relations of a and v with body size Neither a nor v showed any relation with body size. Redistribution rates are almost certainly influenced
Correcting slope-intercept GFR 617 by body size, as shown by differences in a1 between small animals and humans, and would therefore be expected to impact on a and v (Table I). Thus, clear differences in both a and v between animals and humans have been demonstrated. a and v would therefore be expected to correlate with body size. That fact that they do not is presumably because the variation in body size within an individual species is not wide enough. However, Jodal and BrochnerMortensen [7] found that apv (based on plasma volume) was not significantly different between children and adults. As v is related to a1 (see Equation 13), and it is demonstrated that A/B does not differ between species (see Table I), the differences in a1 may explain inter-species differences in v. Indeed, as can be seen from table I, a1 is higher in cats and dogs compared to humans. This suggests initial mixing and redistribution of marker is faster in these species. Furthermore a2 is higher in cats and dogs compared to humans. These findings may suggest the form and function of microcirculation in cats and dogs facilitates more rapid transfer than in humans. It has previously been shown in human patients that the plasma clearance of filtration markers thought to distribute within the ECFV comprises three rather than only two exponentials [15,16]. The first phase reflects exchange between plasma and interstitial fluid, the second phase reflects exchange between the compartments within the interstitial fluid and the third phase reflects filtration. Plasma samples from 20 min following injection allows fitting to two exponentials. This gives a very small bias, which could have been compensated by earlier plasma sampling and more exponentials, however, this was not the topic of the present study. It is unknown whether the same is true for cats and dogs as such studies have not been performed. If the same were to hold true for these species then this may have artificially elevated a1. Relation of a and v with filtration marker in humans It was not possible to perform measurements using in the feline or canine patients due to lack of nuclear medicine facilities. Human values of a and v respectively based on 51Cr-EDTA and iohexol, were not significantly different and their relations with filtration function and body size were similar. 51Cr-EDTA
Conclusion In this study we have explored the theoretical basis of equations that are not second order polynomials for correcting slope-intercept GFR in human, feline and canine patients and exposed the close similarity between the Jodal and Brochner-Mortensen and Peters equations. For the Jodal and BrochnerMortensen equation (Cl Cl1 / (1 a.[Cl1/ECFV])), an estimate of ECFV would be required in the term
618 N. C. Finch et al.
Figure 4. Relationship of the correction factors a and v and body size in humans, cats and dogs. (a) a and v in human patients determined from clearance data using the filtration marker 51Cr-EDTA. Unfilled circles represent a and dashed line is corresponding regression line (R2 0.018, p 0.173). Filled circles represent v and bold line is corresponding regression line (R2 0.008, p 0.355). (b) a and v in cats determined from clearance data using the filtration marker iohexol. Unfilled circles represent a and dashed line is corresponding regression line (R2 0.180, p 0.102). Filled circles represent v and bold line is corresponding regression line (R2 0.167, p 0.116). (c) a and v in dogs determined from clearance data using the filtration marker iohexol. Unfilled circles represent a and dashed line is corresponding regression line (R2 0.034, p 0.255). Filled circles represent v and bold line is corresponding regression line (R2 0.017, p 0.421).
Cl1/ECFV to calculate corrected clearance. In the Peters equation clearance is calculated as Cl1 / (1 v.a2) (i.e. Cl1/ECFV is replaced by a2). Cl1/ECFV and a2 are both closely related to the turnover rate of ECFV by glomerular filtration (i.e. GFR per unit ECFV). The question is then posed as to which of the two equations is more useful in clinical practice. The Jodal and Brochner-Mortensen equation has the advantage that a is less influenced by filtration function than v but has the disadvantage that it requires prediction of ECFV or PV to calculate correction clearance. ECFV can be predicted from BSA in
humans but not for cats and dogs. In humans, even though BSA is raised to the index of 1.34, which addresses the non-linearity between BSA and fluid volumes such as ECFV, Peters et al. [17] have recently questioned the use of single, gender-independent, formulae for estimating BSA (in contrast to the gender-specific formulae for lean body mass, total body water, ECFV and red cell and plasma volumes). The use of a in cats and dogs would require the use of alternative formulae based on weight to predict ECFV. These have been developed in cats (Finch et al., under review) but not dogs. Both factors a and
Correcting slope-intercept GFR 619
v are independent of body size and therefore are attractive for cross-species use but unfortunately turn out to be markedly different between humans, cats and dogs. Therefore a cross-species value is not available. Further studies are required to test a and v in separate populations and to assess agreement with other methods of estimating GFR. Declaration of interest: The authors report no conflict of interest. The authors alone are responsible for the content and writing of the paper.
[8]
[9]
[10]
[11]
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Analytically correct relationship between total and one-pool clearance. Scand J Clin Lab Invest 2009;69:305–13. Peters AM. Re-evaluation of the new Jodal-BrochnerMortensen equation for one-pool correction of slopeintercept measurement of glomerular filtration rate. Nuc Med Commun 2011;32:375–80. Peters AM, Henderson BL, Lui D, Blunkett M, Cosgriff PS, Myers MJ. Appropriate corrections to glomerular filtration rate and volume of distribution based on the bolus injection and single-compartment technique. Physiol Meas 1999;20:313–27. Peters AM. The kinetic basis of glomerular filtration rate measurement and new concepts of indexation to body size. Eur J Nucl Med Mol I 2004;31:137–49. Bird NJ, Peters C, Michell AR, Peters AM. Reproducibilities and responses to food intake of GFR measured with chromium-51-EDTA and iohexol simultaneously and independently in normal subjects. Nephrol Dial Transpl 2008; 23:1902–9. Heiene R, Eliassen KA, Risoen U, Neal LA, Cowgill LD. Glomerular filtration rate in dogs as estimated via plasma clearance of inulin and iohexol and use of limited-sample methods. Am J Vet Res 2010;71:1100–7. Haycock GB, Schwartz GJ, Wisotsky DH. Geometric method for measuring body surface area: a height-weight formula validated in infants, children and adults. J Pediatr 1978;93:62–6. Bird NJ, Henderson BL, Lui D, Ballinger JR, Peters AM. Indexing glomerular filtration rate to suit children. J Nucl Med 2003;44:1037–43. Cousins C, Gunasekera RD, Mubashar M, Mohammadtaghi S, Strong R, Myers MJ, Peters AM. Comparative microvascular inulin and Tc-99m DTPA exchange. Clin Sci 1997;93:471–7. Cousins C, Skehan SJ, Rolph SM, Flaxman ME, Ballinger JR, Bird NJ, Barber RW, Peters AM. Comparative microvascular exchange kinetics of [77Br]bromide and 99mTc-DTPA in humans. Eur J Nucl Med Mol I 2002; 29:655–62. Peters AM, Perry L, Hooker CA, Howard B, Neilly MD, Seshadri N, Sobnack R, Irwin A, Snelling H, Gruning T, Patel NH, Lawson RS, Shabo G, Williams N, Dave S, Barnfield MC. Extracellular fluid volume and glomerular filtration rate in 1878 healthy potential renal transplant donors: effect of age, gender, obesity and scaling. Nephrol Dial Transpl 2012;27:1429–37.
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ORIGINAL ARTICLE
Evaluation of non-polynomial equations for one-compartment correction of slope-intercept GFR: Theoretical prediction and experimental measurement Natalie C. Finch1, Reidun Heiene2, Nicholas J. Bird3, Jonathan Elliott4 & A. Michael Peters5 1School
of Veterinary Sciences, University of Bristol, UK, 2Norwegian School of Veterinary Science, Norway, Hospital, Cambridge, UK, 4Royal Veterinary College, London, UK, and 5Departments of Nuclear Medicine, Royal Sussex County Hospital, Brighton, and Harley Street Clinic, London, UK 3Addenbrooke’s
Abstract Background. Polynomial equations for one-compartment correction of slope-intercept glomerular filtration rate (GFR) will underestimate values at high clearance rates. Non-polynomial correction equations that are independent of patient size and renal function would be advantageous and may have cross-species use. Materials and methods. The study explored the theoretical basis of firstly the Jodal and Brochner-Mortensen one-compartment correction equation, replacing plasma volume with extracellular fluid volume, and secondly an equation described by Peters. One-compartment correction factors (a which is related to plasma volume and v which is related to extracellular fluid volume) which avoided the need for scaling to body size were developed. Both factors were determined from the biexponential clearance curve of the markers iohexol and 51Cr-EDTA in humans and iohexol in cats and dogs. Relationships between a and v and filtration function and body size were then determined using data from humans, cats and dogs to assess their validity and compare this with theoretical predictions. Results. In all species, v was higher than a, as theoretically predicted. Both were significantly higher in humans than cats and dogs, ruling out cross-species use. Significant relationships were present between v and measures of filtration function in humans, but were weak with respect to a. Neither a nor v showed significant relationships with filtration function in animals or with body size in any species. Conclusions. a and v (which are factors independent of body size) can be used interchangeably for correcting slope-intercept clearance. However values of both for humans are higher compared to cats and dogs. Therefore a single cross-species factor cannot be used. Key Words: Kidney function tests, glomerular filtration rate, cats, dogs, humans Abbreviations: BSA, body surface area; BW, body weight; Cl, clearance; 51Cr-EDTA, Cr-51-ethylenediaminetetra-acetic acid; D, dose; ECFV, extracellular fluid volume; GFR, glomerular filtration rate; HPLC, high performance liquid chromatography; MRT, mean residence time; PV, plasma volume.
Introduction Glomerular filtration rate (GFR) can be estimated in a clinical context from the plasma clearance of a filtration marker, such as Cr-51-ethylenediaminetetraacetic acid (51Cr-EDTA) or iohexol. When blood sampling is started shortly following marker administration (multi-sample clearance), the clearance curve is traditionally resolved into two exponentials with respective zero-time intercepts A and B, and corresponding rate constants a1 and a2.
The first exponential (A, a1) represents redistribution of marker throughout its distribution volume, which for most filtration markers is essentially the extracellular fluid volume (ECFV). Modification of this technique by collection of a limited number of blood samples (generally 3) from 2 hours postmarker administration is known as the slope-intercept method. As a result of ignoring the first exponential (a1), clearance determined using the slope-intercept method requires a correction for marker lost in urine
Correspondence: Natalie C. Finch, currently at School of Veterinary Science, University of Bristol, Bristol, UK. Tel: 44117 9289420. Fax: 44117 9811277. E-mail: [email protected] (Received 4 August 2013; accepted 25 May 2014) ISSN 0036-5513 print/ISSN 1502-7686 online © 2014 Informa Healthcare DOI: 10.3109/00365513.2014.928941
612 N. C. Finch et al. during the redistribution phase (one-compartment correction). The most widely used correction formula for slope-intercept clearance in human patients was originally described by Brochner-Mortensen [1]. Similar correction formulae have since also been developed for dogs [2] and cats [3]. The effects of these one-compartment correction equations are shown in theoretical curves in Figure 1. However, these corrections are generally second order polynomials which, as quadratic functions, have a maximum value to the parabolic curve. This can result in underestimation of clearances at high rates of clearance. In human patients, the maximum uncorrected and corrected clearances when using the BrochnerMortensen equation will be 407 and 201 mL/ min/1.73 m2, respectively [4]. In feline patients, the maximum uncorrected and corrected clearances when using the cat correction formula can be calculated to be 8.35 and 4.33 mL/min/kg, respectively. In canine patients, the maximum uncorrected and corrected clearances when using the dog correction formula will be 7.53 and 3.98 mL/min/kg, respectively. Clearance rates of such magnitude are not often encountered in human patients. However, hyperthyroidism is very prevalent amongst the senior cat population and clearance rates in excess of 4.33 mL/min/kg have been reported [5,6]. The value at which underestimation becomes important has not been established for human or veterinary patients, but the potential inaccuracies of these polynomial correction formulae at high rates of clearance have led to recent interest in the development of new equations that may avoid the problem [4,7]. An equation for correcting slope-intercept clearance that is not quadratic, that provides a reliable estimate of multi-sample clearance and that could be used across species, across patients of different body size, across patients with varying GFR, and could
also be applied to different filtration markers is desirable. This study aims to explore such equations in humans, cats and dogs. The Jodal and Brochner-Mortensen equation The recently described Jodal and BrochnerMortensen correction equation [7] is as follows: Cl Cl1 / (1 f.Cl1)
(1)
where Cl is clearance, Cl1 is slope-intercept clearance prior to one-compartment correction and f is a correction factor. Note that this correction equation is applied to unscaled GFR, so f is required to account for body size. Jodal and Brochner-Mortensen [7] introduced the factor f as: f aPV / PV
(2)
where PV is plasma volume and aPV is ‘missing area’. The area is ‘missing’ in that the area under the full plasma concentration curve is larger than the area under the one-compartment curve; aPV is this difference scaled to initial plasma concentration (i.e. divided by D/PV where D is injected dose of filtration marker). This scaling gives aPV the unit of time. Jodal and Brochner-Mortensen [7] went on to show that aPV is relatively constant between children, women and men, indicating that f does not depend on renal function, only upon the size of the person. [4] More generally, f can be expressed as [7]: f ([1 / Cl] [1 / Cl1])
(3)
which follows from isolating f in Equation 1. Equation 2 can be rearranged as:
Figure 1. Theoretical equations showing the effect of polynomial correction factors (a) Brochner-Mortensen correction formula and (b) cat (bold line) and dog (dashed line) correction formulae. The curves were constructed using theoretical data. The regression line for corrected clearance against multi-sample clearance was plotted.
Correcting slope-intercept GFR 613 aPV f.PV PV([1/Cl] [1/Cl1])
(4)
(6)
ECFV([1/Cl] [1/Cl1]) (7) Both aPV and a have the unit of time. Since ECFV PV, it follows from Equations 4 and 6 that a aPV. In summary, Equation 1 can be used with f aPV/PV. However, as shown by Peters [8], it is also possible to use f a/ECFV. There was good agreement between the two approaches. In a human population, Jodal and Brochner-Mortensen [7] showed that PV in mL could be estimated as 1336 BSA1.34, and their final (rounded) result was f 0.0032 BSA1.3, where BSA is body surface area. Peters [8] showed that ECFV in mL could be estimated as 6080 BSA1.34 and found f 0.00308 BSA1.34. Using f a/ECFV, Equation 1 can be written as: Cl Cl1 / (1 a.[Cl1/ECFV])
(8)
Note that the term [Cl1/ECFV] has units of a rate constant (min1) so a has units of time. Peters’ equation [9] It was previously suggested by Peters [9] that a2 can be used instead of Cl1 in a correction equation of similar form to Equation 1, as follows. For a bi-exponential plasma concentration-time curve: Cl
d a / α1 B / α 2
(9)
where D is the dose of the filtration marker administered. For slope-intercept clearance, in which the initial redistribution exponential is ignored, the equation becomes: Cl1
d B / α2
(10)
Therefore, Cl1 a / α1 B / α 2 Cl B / α2
Cl Cl1 / (1 v.a2)
(5)
Jodal and Brochner-Mortensen [7] argued on theoretical grounds that PV is the most appropriate marker distribution volume to use in Equations 2 and 3. However, a more recent study showed that any marker distribution volume that is proportional to PV, such as ECFV, can also be used [8]. So, rather than PV and aPV, it is possible to substitute with ECFV and a corresponding constant a. With this terminology Equation 4 can be replaced with: a f.ECFV
which can be rearranged to give:
(11)
v A / (B.a1)
where
(12) (13)
a2 has units of 1/time, so v, like a, has units of time. Note the similarity between Equations 8 and 12. Relationship between a and v From inspection of Equations 8 and 12, it is clear that the terms a.(Cl1/ECFV) and v.a2 must be equal, so, n Cl1 (14) a α 2 ECFV When calculating clearance using the slope-intercept technique, d Cl1 (15) (B / α 2 ) which can be written as: Cl1 a2.Vd
(16)
where Vd D/B which is generally referred to as the ‘virtual volume of distribution’. So combining Equations 14 and 16. Vd n a ECFV
(17)
Vd exceeds ECFV by a factor that depends on a2 [10], and the relation of a2 with v/a is the same as its relationship with Vd/ECFV, namely non-linear in which v/a approaches unity as a2 approaches zero, when a v. Relationships between a, v and parameters of the bi-exponential curve Similar to aPV which is the missing area under the curve when scaling to PV, a is the missing area under the curve when scaling to ECFV. Substituting for Cl and Cl1 in Equation 7 a ECFV
(a / α1 B / α 2 B / α 2 ) d
ECFV A/(D.a1)
(18) (19)
ECFV can be determined from multi-sample clearance following as ECFV Cl MRT
(20)
where MRT is mean residence of tracer in its distribution volume (ECFV) before filtration and determined following as MRT
a / α12 B / α 22 a / α1 B / α 2
(21)
614 N. C. Finch et al. Therefore, a
a
d α1 B
2
α2
a α1 B a α1 B
2
α2 a (22) α2 d.α1
a α1 2 B α2 2 a 2 α1 ( a α1 B α2 )
(23)
So it can be seen that a, like v, is a function only of the parameters of the bi-exponential plasma clearance curve, not injected dose of marker (D) or body measures (such as BSA or BW). Equation 23 cannot be further reduced to determine the dependence of a on a2. However, assuming that the value of a2 is small compared with the value of a1, following then a ≈ A / (B.a1)
(24)
When a2 is zero, Equation 24 is therefore identical to Equation 13, and a v. By inserting typical values of a1 into Equation 23 and assuming A B, then a non-linear inverse relationship is seen between a and a2. In contrast, it can be seen from Equation 13 that v is not directly dependent on a2. However, in practice, A, B and a1 are all influenced by a2, so v may be indirectly dependent on a2, and a may change its theoretical relation with a2. The aim of the present study was to explore these theoretical principles and to determine the values of a and v for one-compartment correction of slopeintercept GFR. Ideally, notwithstanding the above considerations, these factors should be independent of filtration function and body size and, in order to be used across veterinary and human medicine, also of species. Therefore, relationships between a and v and renal function and body size in humans, cats and dogs were tested. Methods Patients Humans. Multi-sample clearance (Cl) was performed in 20 healthy volunteers and 60 patient volunteers referred for routine, clinically indicated measurement of GFR. All volunteers gave informed consent and the study was approved by the local research ethics committee. Patients were given simultaneous injections of 51Cr-EDTA and iohexol into opposite arms. Following administration, bilateral exactly timed venous blood samples were collected at approximately 20, 40, 60, 120, 180 and 240 min and assayed for marker injected contralaterally, 51CrEDTA by gamma counting and iohexol by X-ray fluorescence, as previously described [11]. The healthy volunteers were studied on several occasions as part of repeatability and fasting versus non-fasting studies [11]. Therefore, there were a total of 110 clearance
measurements available for analysis. In 6 studies, the clearance curves could not be resolved into two exponentials and were therefore, excluded leaving 104 studies for analysis. Feline patients. Multi-sample clearance was performed in 16 healthy client owned cats as described in a previous study [3]. Briefly, a bolus dose of iohexol (Omnipaque™) was administered via an intravenous catheter placed in the right cephalic vein. Exactly timed blood samples (8–9) were collected between 5 and 300 min following administration. Iohexol was analyzed at a reference laboratory (Epsom and St Helier NHS trust) using high performance liquid chromatography (HPLC). Canine patients. Multi-sample clearance was performed in 43 client-owned dogs with varying renal function at the Norwegian School of Veterinary Science as described in a previous study [12]. Briefly, a bolus dose of iohexol (Omnipaque™) was administered via an intravenous catheter placed in the right cephalic vein. Four exactly timed blood samples were collected between 5 and 240 min following administration (two from the initial redistribution phase and two from the elimination phase). Iohexol was analyzed using HPLC as described [12]. In three dogs, the curves could not be resolved into two exponentials, leaving 40 clearance measurements available for analysis. Data analysis Values of a and v were determined using Equations 23 and 13, respectively. Body size was assessed in human patients using BSA estimated from height and weight using the formula of Haycock [13] and in feline and canine patients using body weight (BW; kg). Statistical methods Parametric testing was performed in all statistical analyses. Comparisons between species were performed using the independent samples t-test. A Bonferroni correction was applied for multiple testing. Relationships were explored using linear regression analysis and by determining the coefficient of determination (R2). Correlations were assessed using Pearson’s correlation coefficient (r). Results are presented as mean SD. Significance was set at p 0.05. Results Parameters of biexponential plasma clearance curves In humans A/B, a1 and a2 were not significantly different between 51Cr-EDTA and iohexol. There was no significant difference between species in A/B but a1 and a2 were both significantly higher in cats and dogs compared to humans (p 0.001). In all
Correcting slope-intercept GFR 615
Table I. Values of the biexponential plasma concentration versus time curves obtained in humans, cats and dogs. Humans (n 104) Mean SD Iohexol A/B a1 (min1) a2 (min1)
1.076 0.57 0.039 0.022 0.0051 0.0018
Cats (n 16) Mean SD
Dogs (n 40) Mean SD
0.944 0.370 0.950 0.408 0.084 0.040* 0.095 0.034* 0.01 0.002* 0.010 0.002*
51Cr-EDTA
A/B a1 (min1) a2 (min1)
0.898 0.33 0.036 0.011 0.0052 0.0018
*Significantly different to human patients (p 0.005)
three species, A/B was close to one. See Table I for values of biexponential plasma clearance curve. Values of a and v Mean v was higher than a in all 3 species (see Table II). Neither a nor v was significantly different between cats and dogs but both were significantly lower compared with humans (p 0.001). Relations of a and v with filtration function Based on 51Cr-EDTA in humans, there were weak positive relationships between a and multi-sample GFR (R2 0.052, p 0.021) and between v and multi-sample GFR (R2 0.144, p 0.001; Figure 2a). The regression lines for a and v diverged but the intercepts (i.e. at zero GFR) were similar (15.0 versus 16.5 min) in accordance with the theoretical prediction that a v for zero GFR. Similar relations were seen when filtration function was expressed as a2, when the respective intercepts were 20.3 and 19.8 min (data not shown). Similar results were obtained using iohexol (data not shown). Similar results were also obtained when a and v based on one marker were regressed on filtration function measured with the other marker in humans (data not shown). In cats and dogs, however, neither a nor v showed significant relationships with filtration function (Figures 2b and 2c). In humans and dogs, v/a showed a positive relationship with filtration function (humans R2 0.382, p 0.001, dogs R2 0.126, p 0.025) whereas in cats the relationship was not significant (R2 0.000, p 0.943; Figure 3). Relations of a and v with body size In humans, BSA showed no significant relation with either a or v based on either 51Cr-EDTA (Figure 4a) or iohexol (data not shown). Likewise, neither a nor v showed any significant relation with BW in cats (Figure 4b) or dogs (Figure 4c).
Table II. Mean SD values for the correction factors a and v in humans, cats and dogs. The values of a and v were significantly higher in humans (p 0.001). A single mean value could not be applied for use across species.
a a v v
(51Cr-EDTA) (min) (iohexol) (min) (51Cr-EDTA) (min) (iohexol) (min)
Humans (n 104)
Cats (n 16)
Dogs (n 40)
20.5 6.4 22.3 8.4 27.0 11.3 31.0 15.9
ND 10.6 5.5 N/D 14.2 9.2
ND 8.9 4.2 N/D 11.5 6.8
Discussion The theoretical curves constructed in Figure 1 illustrate the one-compartment correction by the original Brochner-Mortensen study [1] and those reported for cats [3] and dogs [2]. These polynomial corrections will, however, underestimate high GFR values. The purpose of the present study was to explore the theoretical basis of non-quadratic one-compartment correction equations and identify a unique factor to correct slope-intercept clearance that could be used across patients of varying renal function, body size and different species. Correction factors that are independent of body size and renal function are of interest in developing a single correction factor for use with slope-intercept clearance in both human and veterinary patients. In veterinary patients, BSA is not used for scaling GFR and correction equations developed in humans are not applicable [3]. The present paper has first considered the nonpolynomial correction equation by Jodal and Brochner-Mortensen [7]: Cl Cl1 / (1 f.Cl1) (Equation 1) where f ([1/Cl] [1 / Cl1]) (Equation 3) As shown by Peters [8], the factor f can also be expressed by scaling to ECFV giving the following variant of the Jodal and Brochner-Mortensen equation: Cl Cl1 / (1 a.[Cl1/ECFV]) (Equation 8) The two studies gave very similar expression for the factor f, but these expressions were established only for humans and based on BSA. For cats and dogs, the f factors may differ from humans, and scaling is usually made to BW rather than BSA. Second, an alternative equation by Peters [9] (referred to as the Peters equation) has been considered: Cl Cl1 / (1 v.a2) (Equation 12) where a2 is the rate constant of the final exponential of the plasma clearance curve, and v is a constant. Since Equation 12 does not involve body measures (such as BW and BSA), it is more easily applied to non-human species. The aim of the present study was
616 N. C. Finch et al.
Figure 2. Relationship of the correction factors a and v with multi-sample clearance in humans, cats and dogs. (a) a and v in human patients determined from clearance data using the filtration marker 51Cr-EDTA. Unfilled circles represent a and dashed line is corresponding regression line (R2 0.052, p 0.021). Filled circles represent v and bold line is corresponding regression line (R2 0.144, p 0.001). (b) a and v in cats determined from clearance data using the filtration marker iohexol. Unfilled circles represent a and dashed line is corresponding regression line (R2 0.041, p 0.452). Filled circles represent v and bold line is corresponding regression line (R2 0.015, p 0.649). (c) a and v in dogs determined from clearance data using the filtration marker iohexol. Unfilled circles represent a and dashed line is corresponding regression line (R2 0.060, p 0.128). Filled circles represent v and bold line is corresponding regression line (R2 0.069, p 0.101).
specifically to examine whether v was speciesdependent or not. To allow inter-species comparison of the parameters of Equation 1, this equation was used in the form with ECFV (Equation 8), ECFV was expressed from bi-exponential expression of the plasma clearance curve, and finally an expression for a was derived (Equation 23). Like v, these values for a could be compared between species because Equation 23 is independent of body size. Finally, the properties of a and v were compared. In humans, apv is the numerator in the constant 0.0032 described by Jodal and Brochner-Mortensen
[7] and likewise a is the numerator in the constant 0.00308 described in a previous study by Peters [8]. The denominator is an estimate of PV in the Jodal and Brochner-Mortensen [7] equation and of ECFV in the Peters [8] equation, both based on BSA. Analogous equations based on body weight still require development in small animals because BSA is not conventionally used as a scaling variable in animals. Values of a and v As theoretically predicted, v was higher than a in all three species. The values of a and v were respectively
Figure 3. Relationship between v/a and a2 in humans (black filled circles), cats (grey filled circles) and dogs (unfilled circles). In humans and dogs, v/a showed a positive relationship with filtration function (humans R2 0.382, p 0.001, dogs R2 0.126, p 0.025) whereas in cats the relationship was not significant (R2 0.000, p 0.943).
similar between cats and dogs but were both significantly higher in humans, ruling out a cross-species correction factor. The value of v originally determined in human patients using 51Cr-EDTA was 15.4 min [9] but it was later modified to 25.0 min [14] on the basis of further work. The mean value determined in the present study is 27.0 min. This is significantly higher compared with v in feline (10.6 min) and canine (8.9 min) patients. This is probably explained by more rapid mixing of marker throughout the distribution volume in these smaller species, resulting in higher values of a1 (Table I). Relationships between a and v and filtration function a and v showed no significant relationships with GFR in either cats or dogs, possibly because of relatively small patient numbers. It may also be explained by the small range of GFR values in the cats and dogs of which the majority had normal or near normal renal function compared to the range of GFR values in the humans. In humans, however, v and to a lesser extent a, correlated positively with measures of filtration function, although the relationships were weak. As a2 does not feature in Equation 13 (defining v), a positive relationship between filtration function and v would not be expected. We believe that the theoretical relationships that a and v, respectively, show or do not show with filtration function are modified by the effects of variations in a2 on A, B and a1. The intercept values, however, remained very similar between a and v, as predicted theoretically. Relations of a and v with body size Neither a nor v showed any relation with body size. Redistribution rates are almost certainly influenced
Correcting slope-intercept GFR 617 by body size, as shown by differences in a1 between small animals and humans, and would therefore be expected to impact on a and v (Table I). Thus, clear differences in both a and v between animals and humans have been demonstrated. a and v would therefore be expected to correlate with body size. That fact that they do not is presumably because the variation in body size within an individual species is not wide enough. However, Jodal and BrochnerMortensen [7] found that apv (based on plasma volume) was not significantly different between children and adults. As v is related to a1 (see Equation 13), and it is demonstrated that A/B does not differ between species (see Table I), the differences in a1 may explain inter-species differences in v. Indeed, as can be seen from table I, a1 is higher in cats and dogs compared to humans. This suggests initial mixing and redistribution of marker is faster in these species. Furthermore a2 is higher in cats and dogs compared to humans. These findings may suggest the form and function of microcirculation in cats and dogs facilitates more rapid transfer than in humans. It has previously been shown in human patients that the plasma clearance of filtration markers thought to distribute within the ECFV comprises three rather than only two exponentials [15,16]. The first phase reflects exchange between plasma and interstitial fluid, the second phase reflects exchange between the compartments within the interstitial fluid and the third phase reflects filtration. Plasma samples from 20 min following injection allows fitting to two exponentials. This gives a very small bias, which could have been compensated by earlier plasma sampling and more exponentials, however, this was not the topic of the present study. It is unknown whether the same is true for cats and dogs as such studies have not been performed. If the same were to hold true for these species then this may have artificially elevated a1. Relation of a and v with filtration marker in humans It was not possible to perform measurements using in the feline or canine patients due to lack of nuclear medicine facilities. Human values of a and v respectively based on 51Cr-EDTA and iohexol, were not significantly different and their relations with filtration function and body size were similar. 51Cr-EDTA
Conclusion In this study we have explored the theoretical basis of equations that are not second order polynomials for correcting slope-intercept GFR in human, feline and canine patients and exposed the close similarity between the Jodal and Brochner-Mortensen and Peters equations. For the Jodal and BrochnerMortensen equation (Cl Cl1 / (1 a.[Cl1/ECFV])), an estimate of ECFV would be required in the term
618 N. C. Finch et al.
Figure 4. Relationship of the correction factors a and v and body size in humans, cats and dogs. (a) a and v in human patients determined from clearance data using the filtration marker 51Cr-EDTA. Unfilled circles represent a and dashed line is corresponding regression line (R2 0.018, p 0.173). Filled circles represent v and bold line is corresponding regression line (R2 0.008, p 0.355). (b) a and v in cats determined from clearance data using the filtration marker iohexol. Unfilled circles represent a and dashed line is corresponding regression line (R2 0.180, p 0.102). Filled circles represent v and bold line is corresponding regression line (R2 0.167, p 0.116). (c) a and v in dogs determined from clearance data using the filtration marker iohexol. Unfilled circles represent a and dashed line is corresponding regression line (R2 0.034, p 0.255). Filled circles represent v and bold line is corresponding regression line (R2 0.017, p 0.421).
Cl1/ECFV to calculate corrected clearance. In the Peters equation clearance is calculated as Cl1 / (1 v.a2) (i.e. Cl1/ECFV is replaced by a2). Cl1/ECFV and a2 are both closely related to the turnover rate of ECFV by glomerular filtration (i.e. GFR per unit ECFV). The question is then posed as to which of the two equations is more useful in clinical practice. The Jodal and Brochner-Mortensen equation has the advantage that a is less influenced by filtration function than v but has the disadvantage that it requires prediction of ECFV or PV to calculate correction clearance. ECFV can be predicted from BSA in
humans but not for cats and dogs. In humans, even though BSA is raised to the index of 1.34, which addresses the non-linearity between BSA and fluid volumes such as ECFV, Peters et al. [17] have recently questioned the use of single, gender-independent, formulae for estimating BSA (in contrast to the gender-specific formulae for lean body mass, total body water, ECFV and red cell and plasma volumes). The use of a in cats and dogs would require the use of alternative formulae based on weight to predict ECFV. These have been developed in cats (Finch et al., under review) but not dogs. Both factors a and
Correcting slope-intercept GFR 619
v are independent of body size and therefore are attractive for cross-species use but unfortunately turn out to be markedly different between humans, cats and dogs. Therefore a cross-species value is not available. Further studies are required to test a and v in separate populations and to assess agreement with other methods of estimating GFR. Declaration of interest: The authors report no conflict of interest. The authors alone are responsible for the content and writing of the paper.
[8]
[9]
[10]
[11]
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