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Influence of a variable differential function on the stone-growth-related urinary depletion effect.

Recently, we showed that urine flowing past growing stone material on its way through the urinary tract becomes systematically depleted of its lithogenic substances (1). This effect can be large enough to be clinically relevant (2). However, only the altered urine can be assessed by urine collection, which leads to a urinalysis that does not sufficiently reflect the patient's true metabolic status. The excessively low concentrations and excretion values of the lithogenic urinary constituents may lead to "optimistic" misinterpretation of the patient's actual health status and overestimation of treatment success.

Apart from geometric and physiologic simplifications, the previously used model assumes that the differential volume function of the kidneys amounts to 50%. Consequently, both the stone-free and the stone-forming kidney contribute equally to the total urine volume. However, even in healthy individuals, the differential function varies by [+ or -] 6%.

Parenchymal defects, such as tumors, renal arteriopathy, and posttraumatic failures, or morphologic abnormalities, such as pyeloureteral junction stenosis and vesico-ureteral-renal reflux, can lead to a dramatic decrease in renal function, causing a reduction in the glomerular filtration rate. Patients suffering from these diseases often present with calcium nephrolithiasis and proteinuria, occasionally with progression to nephrocalcinosis, urinary concentrating defects, and renal insufficiency caused by tubular atrophy, intestinal fibrosis, and glomerulosclerosis.

The process of kidney stone formation follows fundamental physical principles such as mass conservation. In a recently introduced model (1), we evaluated the influence of in vivo-growing uroliths on urinary composition. The material from which the stones are formed originates completely from the urine streaming through the kidneys; therefore, the urinary concentrations of the lithogenic components, such as calcium and oxalate, in the excreted urine must be depleted by the mass deposited in the stone(s). As long as no method exists to measure those concentrations in vivo, mathematical models describing the depletion process are invaluable for quantitatively estimating the effect. These models can be used to "correct" measured concentration values of lithogenic urinary constituents.

In the example calculations below, we refer to the mineral phase calcium oxalate monohydrate (COM), which is currently the most common stone type formed in individuals in developed countries (3).

When the differential function of a kidney is 50% and only one kidney is affected by urolithiasis, the stone-growth-related depletion effect can be estimated according to Eq. 1 (1):

[c.sub.i] = [1/2(1 + [V/[tau]] / v[phi]c) + [square root of 1/4[(1 + [V/[tau]] / v[phi]c).sup.2] + 2 [V/[tau]] / v[phi]c]c (1)

with [c.sub.i] as the proximal (with respect to the stone) concentration of, e.g., oxalate, and c as the distal concentration (i.e., the measured concentration); [c.sub.i] is the concentration that maintains stone formation ([c.sub.i] [greater than equal to] c). The term V/T is the mean growth rate of the stone ([mm.sup.3]/day) within the observation period between t = 0 and t = [tau], and V denotes the total stone volume that has been formed within the period [tau]. The variable [phi] is the mean urinary flow rate, and v represents the volume of a stone composed of 1 mol of substance.

The former model (1) assumes that urine excretion is equally distributed between the stone-bearing and stonefree kidneys. Because this is not necessarily the case in totally stone-free individuals, our model can be refined by considering the distribution of excretion between the kidneys. As the depletion effect takes place only in the stone-forming kidney, an asymmetric differential function influences the correction of measured values.

For quantitative determination of the differential-function-dependent depletion effect, we have to enhance our formalism of measured concentration c and initial concentration [c.sub.i] by introducing the exposure fraction (y) for urine to stone(s), with 0 [less than or equal to] [gamma] [less than or equal to] 1. [gamma] denotes the volume fraction of a, e.g., 24-h urine that is flowing past the stone(s); therefore, (1 - [gamma]) is the volume fraction formed by the stone-free kidney, or the urine fraction excreted without contact to stone(s).

Assuming identical flow rates of (P through both kidneys, Eq. 14 of Laube et al. (1) becomes:


By transforming and resolving this equation in a way analogous to that described in detail in Laube et al. (1), we find for the initial concentration [c.sub.i]:


The above equation (Eq. 3) is an important improvement of our former model because it offers a more general approach. For [gamma] = 1, all urine is streaming through the stone-bearing kidney, and Eq. 2 reproduces case 2 of our former model (1). Case 2' from our former model can be directly deduced for [gamma] = 0.5 (1). In that case, only one-half of the urine is produced by the stone-forming kidney. For the theoretical situation of [gamma] = 0, all urine originates from the stone-free kidney; therefore, the measured concentration c equals [c.sub.i].

However, if the value of the stone formation rate (V/[tau]) is positive, a lower limit for[gamma]exists, which is established by the positiveness of the numerator in Eq. 2; this assures the existence of the numerical solution of Eq. 3.

The interpretation of [gamma] can be further generalized. The model permits its interpretation not only as being the volume fraction formed by the stone-bearing kidney, but also allows it to be regarded as the urine fraction that actually passes the growing stone. Thus, the case when not all of the urine produced by the stone-bearing kidney comes in contact with the stone is also covered by the improved depletion model.

With this generalized interpretation, not only can the differential function be mapped to our model, but also all limitations through partial exposure of stone(s) to lithogenic substances. Thus, the introduction of [gamma] reflects not only the case of stone forming in both kidneys with different volume fractions, but also the reasonable assumption that only a fraction of the urine formed by the stone-bearing kidney takes part in stone formation. Kavanagh (4) estimated that fraction to be 1/6.

To calculate values of [c.sub.i] in dependence of c, [phi], and V/[tau], the following reasonable values for the variables [tau], V, [phi], v, and c are chosen: growth rate (V/[tau]) =1, 2, 5, 10, or 20 [mm.sup.3]/day. These growth rates result after a period of 1 year in uroliths indicated by radii of 4.43, 5.59, 7.58, 9.55, and 12.03 mm, respectively. The molar volume of COM (v) = M/[delta] = 65.82 [cm.sup.3]/mol, where M = 146.1 g/mol and 8 = 2.22 g/[cm.sup.3] as the molar weight and the density of COM, respectively. The mean urine flow rate ([[phi]) during period T amounts to 1500 cm/day, a typical value observed in stone-forming persons. The measured urinary oxalate concentration (c) is set to be 0.37 mmol/L. The latter value is calculated from [phi] and the established limit value of the oxalate excretion of 0.56 mmol/day, which is applied to distinguish normooxaluric urines from "(mild) hyperoxaluric" ones (5, 6).

To gain an overview of the effect of exposure fraction ([gamma]) on the extent of the urinary depletion effect caused by in vivo-growing uroliths, [gamma] is varied nearly within the entire theoretical value range, i.e., from 0.01 to 1. However, only in rare cases will a patient's differential function fall below 0.25 or exceed 0.75.

The top panel in Fig. 1 illustrates the importance of the presented refinement to the depletion model with a fixed [gamma] = 0.5 (1). This graph shows the dependence of the ratio of the results of the refined and the former model ([gamma] = 0.5) on V/[tau]. As expected, all functions show for [gamma] = 0.5 a ratio indicated by the value 1. The slopes of the functions increase with the value of V/[tau]. Within the range of normal fluctuation for the differential function, i.e., 0.44 [less than or equal to] [gamma] [less than or equal to] 0.56, the refined model discloses relative deviations of up to 4%. From a clinical point of view, these deviations are of minor relevance. However, for a more asymmetric differential function, the deviations get much larger; at [gamma] <0.3 or [gamma]>0.8, the deviations in the results obtained with the refined model compared with those obtained with the basic model increase, at stone formation rates >15 [mm.sup.3]/ day, an order of magnitude, which is of increasing relevance.

The bottom panel in Fig. 1 displays the dependency of the degree of underestimation of a measured urinary oxalate concentration of 0.37 mmol/L on the values of [gamma] and V/[tau]. It is obvious that the degree of underestimation increases strongly, in particular for high rates of stone formation, with both, [gamma] and V/[tau].

Our results document the importance of the step-by-step generalization of theoretical concepts. Consideration of a variable differential kidney function demonstrates how stone-forming processes depend on often-neglected medical variables.

The potential difference between urinary concentrations of lithogenic components distal to in vivo-growing stone material and the measured concentrations can be on the order of tens of percentages. Lack of awareness of this fact may lead to misinterpretation of a patient's health status and, consequently, to an unfavorable treatment strategy. Thus, the depletion effect should be considered when interpreting data from the clinical metabolic work-up of stone-bearing patients.


The extended depletion model presented above allows for estimation of the extent of the depletion effect attributable to stone growth itself and takes into account the individual variability of the differential volume function of a patient's kidneys. Moreover, it confirms that the usual practice of neglecting variation in the differential function within normal values (0.44 [less than or equal to] [gamma] [less than or equal to] 0.56) is appropriate in the context of stone formation.


(1.) Laube N, Pullmann M, Hergarten S, Hesse A. Influence of urinary stones on the composition of a 24-hour urine sample. Clin Chem 2003;49:281-5.

(2.) Laube N, Pullmann M, Hergarten S, Schmidt M, Hesse A. The alteration of urine composition due to stone material present in the urinary tract. Eur Urol 2003;44:595-9.

(3.) Hesse A, Brandle E, Wilbert D, Kbhrmann KU, Alken P. Study on the prevalence and incidence of urolithiasis in Germany comparing the years 1979 vs. 2000. Eur Urol 2003;44:709-13.

(4.) Kavanagh JP. Enlargement of a lower pole calcium oxalate stone: a theoretical examination of the role of crystal nucleation, growth, and aggregation. J End urol 1999;13:605-10.

(5.) Sutton RAL, Walker VR. Enteric and metabolic hyperoxaluria. Miner Electrolyte Metab 1994;20:352-60.

(6.) Robertson WG, Hughes H. Importance of mild hyperoxaluria in the pathogenesis of urolithiasis-new evidence from studies in the Arabian Peninsula. Scan Microsc 1993;7:391-402.

DOI: 10.1373/clinchem.2004.035972

Michael Pullmann, [1] * Stefan Hergarten, [1] and Norbert Laube [2]

[1] Geodynamics and Physics of the Lithosphere, and [2] Division of Experimental Urology, Department of Urology, University of Bonn, Bonn, Germany; * address correspondence to this author at: Geodynamics and Physics of the Lithosphere, University of Bonn, Nussallee 8, D-53115 Bonn, Germany; fax 49-228-73-2508, e-mail
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Title Annotation:Technical Briefs
Author:Pullmann, Michael; Hergarten, Stefan; Laube, Norbert
Publication:Clinical Chemistry
Date:Sep 1, 2004
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