-
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
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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 +
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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
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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
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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
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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.
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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
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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.
-
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