Kidney International, Vol. 13 (1978), pp. 344360

Physicochemical aspects of urolithiasisBIRDWELL FINLAYSON

Division of Urology, Department of Surgery, University of Florida, College of Medicine, Gainesville, Florida

The following discussion centers on calcium oxa-late and calcium phosphate stones, but permits gen-eralization to other forms of stone. Although it con-stitutes a primer on the subject, it does attempt todeal with inconsistencies in our current meagerknowledge of the physical characteristics of uro-lithiasis. In addition, it describes concepts and ap-proaches that appear to be useful and that should beincorporated into urolithiasis research, to make fu-ture work in the field susceptible to analysis byconventional physical theory.

The known physicochemical features of uro-lithiasis are readily divided into four interrelated sub-jects: the driving force (supersaturation), nucleation,the growth of crystals and particles, and aggregation.

The chemical driving force, urinary supersaturation

Supersaturation of urine with respect to the saltsthat stones will or do consist of gives rise to thethermodynamic driving force for the formation ofstones. This driving force, expressed as free energy(AG), is given by

= RT ln (),where R is the gas constant, T is the temperature,and A1 and A0 are the activities of the unionized saltspecies in solution at any given condition and atequilibrium, respectively [1]. Activity (A) is relatedto concentration (C) through an activity coefficient(1) by

A = fC. (2)When urine is such that, for a given stone salt, A1/A0< 1, then G < 0, the urine is said to be undersatu-rated with respect to the stone salt, and any stonesthat are present can dissolve [2, 3]. As an example,treatment with allopurinol causes A1/A., to be lessthan 1 with respect to uric acid, and it is common for

00852538/78/00130344 503.40 1978, by the International Society of Nephrology.

uric acid stones to dissolve in this circumstance.When urine is such that, for a given stone salt, A1/A0= 1, then zG = 0, and the urine is said to besaturated. In this circumstance, old stones will notdissolve, and new ones will not form; but old stonescan grow, in the sense that aggregation of pre-exist-ing stones can occur. When, for a given stone salt,AIA0 > 1, then G > 0, and the urine is said to besupersaturated. In this circumstance, there is availa-ble free energy; if stones are present, they may grow,but if stone crystals are not present, then precipita-tion will not occur, unless A1/A0 exceeds an experi-mentally ill-defined limit called the "metastablelimit." Above the metastable limit, it is possible bothfor new stones to form and for old stones to grow(Fig. 1).

Inasmuch as the progress of stone disease is gov-erned by the available free energy, it is important tohave a quantitative measure of A1/A0; it makes itpossible to identify people who have an increasedlikelihood of stone disease and to monitor the effec-tiveness of the anti-stone therapies that operate byreducing A1/A0, such as magnesium oxide, hydro-

(1) chlorothiazide, and cellulose phosphate. In principle,there are a variety of methods for measuring A11A0,but during the last 10 years only three have receivedpersistent attention. Calculation of A1/A0 was pop-ularized by Robertson [41!. Pak and Chu [5] havedescribed a semi-empirical equilibration techniquethat capitalizes on the linear relations among theurinary concentrations of calcium, phosphate, andoxalate. In general, the relations used by Pak andChu are nonlinear; but in the range of change en-countered during calcium oxalate or calcium phos-phate precipitation in urine, the expected error of alinear assumption is less than 2%, as judged by theab initio calculations. Gill, Silvert, and Roma haveintroduced a radionuclide tracer into the Pakmethod, to simplify chemical quantitation [6]. With afirm grasp of the ab initio calculations, a theoreticalunderstanding of the methods of Pak and Gill willappear elementary.

344

Physicochemistry of urolithiasis 345

P P P

He 1.Region

ofI unstable

Isupersaturation

"-'--Regionof

metastablesupersaturation

Regionof

undersaturationI

Fig. 1. A mechanical analogy of chemical metastability. Thevertical bars represent concentrations of a precipitable salt. Theavailable thermal free energy is sufficient to cause fluctuations thattip the bars about the pivot point (P) through angular displacement(0). When the center of mass (C) is displaced (C') lateral to P, themechanical bar will fall over, which is equivalent to precipitation.In the region of undersaturation, the bar cannot be toppled byfluctuations equal toO. In the region of metastable supersaturation,catalytic surfaces can augment the fluctuations so that C' is dis-placed lateral to P. This case is equivalent to crystal growth orheterogeneous nucleation. In the region of unstable supersatura-tion, thermal concentration fluctuations, 0, are sufficient to placeC' lateral to P and cause the bar to fall over, which is analogous tospontaneous precipitation.

Ab initio calculations of A,1A0. Urine is a solutioncontaining a set of cations (C) and a set of anions (A).Some of the cations and anions will very rapidly(relaxation time, 1O see) undergo ion complexformation

Ck + A3 CAk,).

For each complex formed, the equilibrium is gov-erned by the mass action relation,

Kk,3 = [CAk,3]fk,j/[Ck][A3]fkfJ,

where Kk, is the stability constant for the (k,j)thcomplex, f is the activity coefficient for the n1charged species, and brackets indicate concentra-tion. If [TCk] is the sum of the concentrations of allspecies containing Ck, then conservation of massrequires that

[TCRI = [C3] +,ma

and IITA3] = [As] + [CAk,J]nk,3,J.

+ [CA,](Z)2, (8)k=j=1

with R, S, U, and V as the numbers of the speciesbeing summed. In practice, to make an ab initiocalculation, urine is analyzed for pH and total so-

(3) dium, potassium, calcium, magnesium, ammonium,sulfate, phosphate, citrate, oxalate, urate, and chlo-ride, and the calculation is made with Equations 58.There are far too many equations to attempt such acalculation for urine by hand. A number of computerprograms, however, have been written and can beobtained from their authors''. (The interestedreader will find a step-by-step guide through the abinitio calculation presented elsewhere [7]).

Working with urine at 25C, Robertson [4] showeda high degree of correlation between ab initio calcu-

G. Nancollas, Department of Chemistry, New York University(Sa) at Buffalo, Buffalo, New York.

b W.G. Robertson, MRC Unit, Leeds, Great Britain. J. Meyer, National Institute of Dental Research, Building 30,Room 211, Bethesda, Maryland 20014.d The Royal Institute of Technology, Department of Inorganic

(5b) Chemistry, Stockholm, Sweden. Ask for LETAGROP andHALTAFALL.

- Metastae

0

0

a,0.

a,>'aa,

Cl

-I-.

e

-tCII, IIC' I

I.._ ,ILi

In Equations 5a and Sb, m is the number of possiblecomplexes, and nk,,1 is the stoichiometric number ofthe 1th species in the (k,j)th complex. The stoichio-metric number is required for polynuclear com-plexes, such as CaaCa O42Equation 5a transforms to

[CE] = [TCE]/(1 + ,[CAk,3]nk,3,k). (6)

Equation Sb undergoes a similar transformation.Equations of the genre of Equation 4 can be substi-tuted into Equations 5a and 5b, giving a set of nonlin-ear simultaneous equations in [TCE], [TA3], [CE],and [A3], whose solution rapidly converges to self-consistency by iterative approximation. In my labo-ratory, it was found empirically that in most urinesthe activity coefficient, f, can be taken to be O.732,with Z being the electronic charge of the species inquestion. Alternatively, f at 38C can be calculatedwith

= exp( 1 .202Z2ft.Jw/( 1 +.Jw)) 0.285w)). (7)

Equation 7 is the Davies modification of the Guggen-heim approximation of the Debye-HUckel first-ordersolution of the Poisson-Boltzman equation for theenergy of the electrostatic field of an ion in ionicsolution [2], in which w is the ionic strength given by

2w = [Ck](Zh)2 +

346 Finlayson

lated and experimentally measured calcium concen-tration, [Ca2]. Robertson's early ab initio calcula-tions used an inappropriate stability constant forcalcium oxalate that necessarily gave an appreciableerror in the calculated activity of A1(CaC2O4). Butapproximately the same fractional error occurs in thecalculated value of A0(CaC2O4); thus, wheneverRobertson's calculations for calcium oxalate are pre-sented as A1/A0 (i.e., relative supersaturation), theestimates are reasonable from a theoretical point ofview. In recent calculations, Robertson et al hasused a calcium oxalate stability constant of 1,900 M'[8]. It is not clear whether his current program hasbeen altered for 38C. In my laboratory, using a 38Cprogram and a calcium oxalate stability constant of2,746 M1 [91, we are able to calculate the equilib-rium value of calcium oxalate precipitating from arti-ficial urine to within 20%. Ab initio calculations ofA1/A0 for wheweilite do not agree well with the re-suits of the semi-empirical methods to be discussedlater; however, if for no other reason, ab initio calcu-lations are useful because they provide an incisivetechnique for investigating the semi-empiricalmethods.

In an effort to reduce the number of chemicalanalyses needed for the ab initio calculation of uri-nary supersaturation, Marshall and Robertson haveempirically analyzed the results of their ab initiocalculations of urinary supersaturation and have de-vised nomograms for estimating supersaturation withconsiderably fewer chemical analyses [10]. Only thechemical determination of citrate and oxalate pose aproblem in practice. The nomogram approach ob-viates the citrate determination and access to acomputer.

Semi-empirical method of Pak and Chu. In themethod of Pak and Chu [5], the concentration ofdissolved stone salt is measured before and afterequilibration with solid stone salt. Relative super-saturation, A1/A0, for whewellite is calculated with

A1/A0 =

[TCa1][(TC204,1)[ (f1)8[TCa0] [TC2O4)0](f0)8.

Pak and Chu have devised an approximation to esti-mate f, and I. Detailed calculations, however, showthatf1 differs from f, by

Physicochemistry of urolithiasis 347

hibitors in urine. This speculation, which remainsuntested, arises from observations that crystals cov-ered with a film of inhibitor will not continue to growwhen A > A, > A0, in which A is a critical concen-tration and by empirical testing would appear to beA0 [13]. Ohata and Pak [14] have shown that ethane-1-hydroxyl- 1 ,2-diphosphonate, a crystal-growth, in-hibitor, gives an apparent A0 greater than the ther-modynamic A

348 Finlayson

Classical homogeneous nucleation. As with mostphysicochemical processes, we start with free-en-ergy considerations. The standard free-energychange (G) resulting from the formation of a spher-ical new phase can be written as

= + Fci,

in which I is the sphere diameter. If / is too small, thesurface-energy term prevails, and the new phase willdissolve. If I is large enough, the volume-energyterm prevails, and the new phase will either stay thesame size or grow. The critical value of 1(1*), neededfor a new particle to remain stable or grow, is givenby

1* = 4ciIzG. (11)

In principle, ci can be calculated, but in practice, itis usually experimentally measured. However,

= mkT in(S), (12)

in which m is the number of ions in the neutralmolecule, v is the molecular volume, k is the Boltz-mann constant, T is absolute temperature, and A1!A0is denoted as S to simplify the notation (Fig. 2).

0x

0)

Co

C)

0)0Cw

16Hcr32zG* = _________3(mkT lnS*)2

= 4)ciS/(lnS*)2 (14)

(10) with 4) defined by Equation 13. Because G* isequivalent to an activation energy, the rate of nuclea-tion (J) is written as

J = Fexp(_G*IkT),or J = Fexp(4)ciSIkT(lnS*)2).

The value of F is not known with certainty, but isusually taken to be 10 to 1082.

S', the metastable limit, can be experimentallymeasured in two ways. In the first, J is measured as afunction of S. Because it is virtually impossible toprepare solutions that are free of particulate matterthat acts as sites of heterogeneous nucleation at S >exp(4)(f(8)1cr3Z)), then

N = E F exp(4)(f(O)1&'Z))dt, (36)fl

and N = Q' exp(4)(f(6)1cr8Z(t = 0))) (37)

with Q = F1/(+4)cr3(dZ!dt)f(O)!) and (dQ1/dt) 0.A transformation of the data in Figure 4 to a plot ofln(N) vs. Z(t = 0) shown in Figure 6 strongly sug-gests that the assumptions giving rise to Equation 37are valid and that n = 2 (i.e., the experimentalsystem has only one class of heterogeneous nuclei).Much of the information suggested by Figure 6 canbe given intuitively simple interpretations. For ex-ample, since f(O)1 = 1, the slope of the steepest limbof the curve is 4)o. 4) is a combination of knownphysical constants; and we calculate, from Figure 6,that a for calcium oxalate is 69 erg/cm2, which is inexcellent agreement with the reported value of 67erg/cm2 [20]. Since the slope of the other limb of thecurve in Figure 6 is 4)af(O)2, and 4)a2 is known,f(O)2 can be estimated. Furthermore, the intercept ofthe lower limb at Z = 0 is a count of the number ofheterogeneous nuclei.

The important points to be gained from Figures 4,5, and especially 6 are: 1) Homogeneous nucleationof calcium oxalate in urine is most improbable. Thekidney is incapable of creating sufficient supersatura-tion. It also follows that current methods of estimat-ing apparent formation products measure either thecatalytic efficiency of heterogeneous nuclei or analteration in the liquid-solid interfacial energy of pre-cipitating salts. 2) The catalytic efficiency (1 f(O)1)of heterogeneous nuclei in urine is experimentallymeasurable. Nucleation inhibitors in urine can act byaltering either the calcium oxalate liquid-solid inter-face (a) or the catalytic efficiency of heterogeneousnuclei (f(O)1). Appropriate use of Equation 36 wouldpermit evaluation of both effects. 3) The number ofnuclei in urine can be measured. The point of view

developed in this section is a means of removing theambiguity associated with existing observations offormation products.

I hope that the preceding discussion and Figures46 make it apparent that the concept of a uniqueformation product does not naturally derive from atheoretical foundation. A criterion for selecting aformation product is arbitrarily imposed by workersto provide a practical means of comparing the ten-dency to precipitate spontaneously in various urinesamples.

Seeded crystals. To study seeded-crystal growth,seed crystals are added to a supersaturated solution,and the reaction is monitored. Nancollas and Gard-ner [43] and Marshall and Nancollas [46] have ex-ploited the seeded-crystal growth system for whew-elite and brushite. By experimental design, thesurface-area change in the whewellite system was

00

20

15

a,C,

'aa00C'a2 10C-J

5

Ln(relative supersaturationL2 x 102

Fig. 6. A transjbrmation of dolt: in Figure 4: The nuturul logo-rithm ofthefiualpurticle couceutrution vs. (ln(relative supersutur-ution))2. The presence or absence of allopurinol is suppressed.The circles are experimental data. The solid line is a least squaresfit of Equation 37 with n = 2.

4 6 S 10

Physicochemistiy of urolithiasis 355

/ M\ 2/3=ovS,

6.0

32 my of calcium oxalate5.0

'7 4.0

(J-)

3.0

2.0

1.0

Time, mm

small, and, using Equation 32, they observed that n= 2. However, in the analysis of the brushite experi-ments, the reaction variable, W, in Equation 32 wasthe reacting ion concentration product. Again, it wasfound that n = 2. In seeded crystal growth experi-ments, Meyer and Smith [47] measured the linear-growth-rate constant for whewellite with Equation32 and with n = 2 at 0.2 j.tm!min at urine concentra-tions of calcium and oxalate. (The method for trans-posing the rate constants in Equations 32 and 33 ispresented elsewhere [48].) Because of growth inhibi-tors in urine, the growth rate is expected to be muchsmaller in urine than the value found in uninhibitedsimple solutions.

In our laboratory, working with seeded whewellitecrystals, we have been unable to fit our data toEquation 32 with the reaction variable W being activ-ity or concentration of either reactant, in contrastwith the experience of Meyer and Smith [47] andNancollas and Gardner [431. Our systems, however,examine a larger range of supersaturation, a largersurface-area variation, and a larger extent of reactionthan those of the other workers. We found that if A1!A0 is the reaction variable in Equation 32, good fitsare obtained (Fig. 7), but the calculated surface-normalized rate constants have ratios approximatingthe ratios of the initial reacting surface. Therefore, itappears that an equation like Equation 33 should beused in the analysis. Because of the difficulty inintegrating Equation 33, workers have used the dif-ferential form of the equation or tables of numeri-cally integrated values. If a linear approximation ismade of the b213 term in Equation 33, the error is

356 Finlayson

in which = (V 0)2. Equation 48 is readily inte-grated to

Ks1t = [ (-J) ln(0 VS) + [(1)

1 (02V)f l+ln(S_1)]++ constant,= G(S) + constant.

A plot of G(S) against r for a seeded-growth experi-ment is shown in Figure 8. Plots of zS against C arelinear for both our experimental system and for fourrandom urine samples that we checked by ab initiocalculation. This validates the first approximationthat was used (i.e., Equation 41) and substantiatesthe estimated maximal error. Thus, from Figure 10 itappears that we have a growth law valid over reason-able ranges of concentration and surface-area changein a convenient integrated form. Of course, Equation49 requires additional experimental verification.

Seeded-crystal growth studies have been donewith hydroxyapatite crystals [491. Depending on theconcentration, however, one or more phases other

(00

4.0

3.0

2.0

1.0

than hydroxyapatite can be growing simultaneously,and the growth curves, even in simple solutions, arecomplex and difficult to quantitatively analyze. Cal-cium phosphate precipitates adsorb a variety of in-hibitors known to be present in urine, and the actualgrowth rate of calcium phosphate in urine is a matterof conjecture.

Continuous crystallizers. A continuous crystallizer(49) is a well-mixed compartment that continuously re-

ceives a supersaturated solution from an inlet andcontinuously or intermittently discharges its contentsthrough an outlet. (The urinary tract can be consid-ered as a series of continuous crystallizers, i.e., col-lecting duct, renal pelvis, and urinary bladder [501.)If the volume of the crystallizer and the concentra-tion at the inlet are constant, the crystallizer dynam-ics are described by n = n0exp(x/ar), in which n isthe concentration density of particles of size x, a isthe growth rate, and r is the system volume dividedby the flow rate [51]. Inasmuch as (an0) is the nuclea-tion rate, the system simultaneously gives informa-tion about crystal growth and nucleation rates. Wehave measured a whewellite growth rate of 0.79 mImm with a calculated A1/A0 of 32. With a similarinput, Miller, Randolph, and Drach observed that thegrowth rate of weddellite was less than 1 m!min[52, 531.

The measurement of crystal growth rate in solu-tions has been dealt with at some length for severalreasons. The most important is that the growth rategives an upper bound on the time required to form astone, and a firm grasp of growth rate permits us tostart speculating about what is and what is not possi-ble with regard to mechanisms in stone disease. Inaddition, with a clear understanding of how stonecrystals grow, it will be possible to increase thesophistication of the in vitro tests done on urine tomeasure the tendency to grow stones and the effec-tiveness of antistone therapy.

Crystal growth in gels. Stone-salt crystals havebeen grown in gel systems [54, 551. It is quite difficultto measure growth rate as a function of concentra-tion in these systems because the analysis requires acomplex diffusion calculation. However, growth ingel systems offers the best opportunity, so far, ofgrowing large crystals (>100 m in diameter) ofstone salts. Gel systems typically yield crystals s Imm.

Aggregation

Urinary stones and crystalluria particles are oftendescribed as polycrystalline aggregates. Robertson etal [81 are using aggregation inhibition as a factor intheir evaluation of the stone-forming potential of

5.0

32 mg of calcium

0 5 10 15 20 25 30 35 40 45 50 55 60Time, rn/ri

Fig. 8. A plot of G(S) vs. time for seeded calcium oxalate crystalgrowth. See text, Equation 49, for definition of G(S). Conditionsare the same as in Figure 7.

Physicochemistry of urolithiasis 357

urine. Although it is generally agreed that aggrega-tion is important in urolithiasis, very little work hasbeen done on the details of stone-salt aggregation.The following is an outline of the problem as itpertains to urolithiasis.

When particles are about one centimeter in diame-ter or larger, gravitational forces tend to be greaterthan adhesional forces. But as the size of particlesdiminishes, the effect of adhesion relative to gravita-tion rapidly becomes dominant. For particles aboutone micrometer in diameter, adhesional forces areabout a millions times greater than gravitationalforces [56]. Thus, in dealing with fine-particle pro-cesses, adhesion must be taken into consideration.In dealing with crystalluria, it appears to be neces-sary to consider both particle-to-particle and parti-cle-to-membrane adherence [48]. In addition, pre-liminary measurement of stone density has shownthat stones have densities approaching the density ofstone crystals [571very much higher than would beexpected if stones formed purely by close-packedaggregation. Therefore, if aggregation is significant inurolithiasis, densification of the aggregate must alsooccur.

There are six basic mechanisms by which aggre-gates are held together [561. In order of increasingenergy, they are electrostatic attraction, van derWaal forces, liquid bridge, capillarity, viscousbinder, and solid bridge. Because there is total im-mersion, liquid bridges and capillarity are not ex-pected to play a large role in crystalluria particleinteraction or in urolithiasis. Because of the zetapotential on particles immersed in urine, the electro-static forces, if significant at all, will be repulsive. Itis expected, on the basis of protein-adsorption iso-therms [581, that each calcium oxalate particle inurine is coated 75% or more with a monomolecularlayer of protein that may act as a viscous binder.Solid bridges can occur only after particle-to-particleapposition due to other adhesive forces. Therefore,we write in a qualitative way for particles in urine,

force of adhesion = van der Waal electrostatic + viscous binding (50)

For two spheres of equal size,

van der Waal = hor/16]ITat, (51)

in which hw is a tabulated function, r is the radius,and a is the separation distance;

electrostatic = Hij2rl2a,

in which and , have their customary electrostatic

meaning of electric permitivity, and Ja is the surfacecontact potential; and

viscous binding M(8 RTIn(k))h(a), (53)

in which M is moles of binder, 0 is the referenceenergy, R is the universal gas constant, T is theabsolute temperature, k is the reciprocal of the con-centration at which half surface saturation by theviscous binder occurs, and h(a) is a Heaviside unitfunction = 1 for a 20A.

The elements of Equation 50 are susceptible toindividual investigation. Measurements of the affin-ity of viscous binders, e.g., proteins, for calciumoxalate surfaces have been reported [581. Thesemeasurements indicate that if the protein content ofurine is 10 mgldl, the surface of calcium oxalateparticles will be covered more than 50% with ad-sorbed protein. Relating this observation to aggrega-tion will require study of the effect of protein onaggregation kinetics. The electrostatic contributionof adhesion energy can be evaluated by study of thezeta potential of stone-salt precipitates. Figure 9schematically shows the origin of zeta potential, andFigure 10 shows the effect of some urinary anions onzeta potential and the ease with which surface ad-sorption of the anions is demonstrated. The ability ofour ab initio ion-equilibrium program to compute atwo-phase equilibrium, given the total componentsof a system, makes it much easier to interpret zeta-

0-D

0

+

+

+

+

+++

+

+

Solid+

+

+

+

(+1

Electricpotential

(I

0

00

09 00 0

Zeta potential

(A) (B)

0

Zeta potential

Fig. 9. Zeta potential as an indicator for chemical adsorption ofions. Whewellite is normally positively charged. A) In the absenceof chemisorbed ions, a diffuse double layer of anions () existsand the zeta potential is positive. B) In the presence of chemi-sorbed anions 0 , the net charge on the surface becomes negative

52 and the counter ions are cations. The zeta potential is ameasure of the electrical potential between the layer of chemi-sorbed ions and the diffuse double layer of counter ions. The largerthe extent of chemisorption, the more negative the zeta potential.

358 Fin/ayson

E

0C!3

N

Fig. 10. Effect of various ionic species on the Zeta potential ofwheivel/ite. The reversal of zeta potential by increasing amounts ofpyrophosphate, citrate, and EHDP indicates the strong adsorbabil-ity of these ions. This effect may relate to the mechanisms ofinhibition of polyvalent anions in urine. (Unpublished work byCURRERI, ONODA, and FINLAY50N.)

potential experiments. From the zeta potential, wecan compute surface potential (iji) with the Gouy-Chapman equation [591. Because of the technicaldifficulty of measuring zeta potentials at ionicstrength greater than 0.05, zeta potentials have yet tobe measured in urine-like solutions. We anticipatedoing it by a short extrapolative process. Van derWaal calculations for stone-salt particles have notbeen made.

It is appropriate to be skeptical about the impor-tance of aggregation in urolithiasis or crystalluria.Robertson et a! [81 have advanced the notion thataggregation inhibitors are important in urolithiasis. Ifaggregation occurs as a significant step in stone dis-ease, it might be expected to occur in a mannersomewhat like a Smoluchowski agglomeration [601,in which case,

N/N0 = 1/(1 + (t,'T)),

in which N/N0 is the fraction of particles remainingper unit volume after time (t) and T is the time for NIN0 = 1/2, r > 107/N0 if the unit of N0 is particles!milliliter and the unit of r is seconds [20]. Even if thel0 particles/ml in crystalluria reported by Robertson[32] is in error by two orders of magnitude, theexpected aggregation would be so slow that onecould not expect appreciable aggregation by a Smo-luchowski process. This kinetic consideration, plus

the small difference between the density of stonesand the density of crystals, raises some doubt aboutthe role of aggregation in urinary stone disease, andit is hoped that this important issue will receive moreattention in the future.

Although lesions such as Randall's Plaques andencrusting cystitis require crystals to adhere to tis-sue, the energy of adherence has not been measured.

Inhibition of crystal growth and aggregation

A variety of molecules that occur in urine inhibitthe crystal growth and aggregation of whewellite andapatite in simple solutions, e.g., pyrophosphate [61],nucleoside triphosphate [62], heparin, citrate, andEHDP [63]. As predicted by theory, zeta-potentialmeasurement is a good screening process to look forsurface-active urinary stone inhibitors. Whewellitezeta-potential perturbation by pyrophosphate be-haves as expected (Fig. 10). Citric acid also shows,by zeta-potential perturbation, significant surface ad-sorption on whewellite (Fig. 10). Meyer and Smith[61] looked at inhibition of whewellite-seededgrowth by citrate. By analysis of their rate-constantdata, they concluded that citrate concentration forhalf-surface coverage of whewellite is 16 tM. Thisvalue has been confirmed in our laboratory by mea-suring adsorption isotherms. Meyer and Smith [61]concluded that the effect of citrate surface inhibitionwas small compared with complexing in solution.This may be incorrect for two reasons: Meyer andSmith did not account for the possibility that citratecauses the equilibrium concentration to be A, in-stead of A0, and a Langmuir plot of their rate-con-stant data gives a negative intercept. Furthermore,the data in Figures 9 and 10 show a 39% inhibition ofthe growth rate by 50 /LM citrate.

One of the major problems in evaluating urinaryinhibitors is to know how they behave in urine.Current practice is to add an aliquot of urine to aseeded-growth system and observe its effect on crys-tal growth and aggregation [63]. This method doesnot necessarily indicate how the inhibitor works inundiluted urine. There can be profound dilutional

f54 effects. For example, the 100-fold dilution used byRobertson's group [64] will obscure the inhibitoryeffect of citrate and pyrophosphate that is expectedon the basis of adsorption isotherm observations.The key to predicting inhibitor effects is the concen-tration necessary for half-surface coverage (this con-centration is equivalent to a thermodynamic affinity).Another problem in evaluating the effect of inhibitorsin urine is that the competitive effects of variousurinary inhibitors are unknown. Nevertheless, Rob-ertson et al [8] have exploited the dilution approach

5 4 3

Log concentration, M

Physicochemistry of urolithiasis 359

Acknowledgments

This work was supported by NIH grants AM-13023 and AM-20586. Figure 9 was prepared by Dr.George Y. Onoda.

Reprint requests to Dr. Birdwell Finlayson, Division of Urology,Department of Surgery, University of Florida, College of Medi-cine, Gainesville, Florida 32610, U.S.A.

References

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2. DAVIES CW: Ion Association. London, Butterworths, 19623. ROBINSON RA, STOKES RH: Electrolyte Solutions. London,

Butterworths, 19594. ROBERTSON WG: Measurement of ionized calcium in biologi-

cal fluids. Clin Chim Acta 24:149157, 19695. PAK CYC, CHU S: A simple technique for the determination

of urinary state of supersaturation for brushite. Invest Urol11:211215, 1972

6. GILL WB, SILVERT MA, ROMA MS: Supersaturation levelsand crystallization rates of calcium oxalate from urines ofnormal humans and stone formers determined by a '4C-oxalatetechnique. Invest Urol 12:203209, 1974

7. FINLAYSON B: Calcium stones: Some physical and clinicalaspects, Chapter 10, in Calcium Metabolism in Renal Failureand Nephrolithiasis, edited by DAVID DS, New York, JohnWiley, 1977

8. ROBERTSON WG, PEACOCK M, MARSHALL RW, MARSHALLDH, NORDIN BEC: Saturation-inhibition index as a measureof the risk of calcium oxalate stone formation in the urinarytract. N Engl J Med 294:249252, 1976

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10. MARSHALL RW, ROBERTSONWG: Nomogram for the estima-tion of the saturation of urine with calcium oxalate, calciumphosphate, magnesium ammonium phosphate, uric acid, so-dium acid urate, ammonium acid urate, and cystine. ClinChim Acta 72:253260, 1976

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13. SEARS GW: Influence of adsorbed films on crystal growthkinetics. J Phys Chem 25:154159, 1958

14. OHATA M, PAK CYC: The effect of diphosphonate on calciumphosphate crystallization in urine in vitro. Kidney mt 4:401406, 1973

15. PAK CYC, OHATA M, HOLT K: Effect of diphosphonate oncrystallization of calcium oxalate in vitro. Kidney Jut 7:154160, 1975

16. ERWIN D, ROBERTS JA, SLEDGE G, BENNETT Di, FINLAY-SON B: Estimating urine supersaturation: A comparison of theresult of two methods of evaluating changes induced by drink-ing milk, in Urolithiasis Research, edited by FLEISH H, ROB-ERTSON WG, SMITH LH, VAHLENSIECK W, New York,Plenum, 1976, pp. 261264

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18. BURDETTE WG, THOMAS WC, FINLAYSON B: Urinary super-saturation with calcium oxalate before and during orthophos-phate therapy. J Urol 115:418422, 1976

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to look at inhibition of calcium oxalate crystalgrowth and aggregation (termed "crystallization" bythe Robertson group) by aliquots of urine in seededsupersaturated calcium oxalate solutions. They havefound that stone-formers have less of a heparin-likeinhibitor molecule in their urine than normal. A clus-ter analysis of the two variates, supersaturation andinhibitor concentration, readily separates stone-for-mers from non-stone-formers and has led them toformulate a saturation-inhibition index for classifyingthe predisposition of urine to form calcium oxalatestones. These workers have further observed thatthe effective concentration of heparin-like inhibitorin urine is a function of the total concentration of theinhibitor and the concentration of urate in urine [641.

There can be no doubt that crystal growth andaggregation inhibitors are present in urine; however,their role in the pathogenesis of stone disease is farfrom clear. Several causes of doubt about the role ofaggregation in stone formation have already beencited. Another possibility is that the differences inurinary inhibitor concentration between stone-for-mers and normal subjects are a result of solutiondepletion by adsorption on crystalluria particles. Ac-cordingly, inhibitor concentration becomes a correc-tion on the estimate of supersaturation. Thishypothesis could explain the apparent success of thesaturation-inhibition index as well as the more ob-vious hypothesis of urinary inhibitors directly reduc-ing crystallization activity in stone disease.

A final comment on the future of urolithiasis research

The last decade has seen a resurgence of interest inthe physicochemical features of urolithiasis. Duringthese years, techniques have been developed forevaluating ion equilibrium in complex urine-like so-lutions. The ability to calculate complex equilibriahas put us in range of making penetrating studies ofnucleation, crystal growth, and aggregation. Beyondthese studies, we need to develop a valid understand-ing of the supersaturation and inhibitor-concentra-tion profile along the nephron. We will then be ableto start building a comprehensive kinetic picture ofwhat can happen in urine as it moves through theurinary passages.

360 Finlayson

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