CRYSTAL GROWTH & DESIGN
Kinetics of Growth of Columnar Triclinic Calcium Pyrophosphate Dihydrate Crystals Christoffersen,#
Margaret R. Jørgen Christoffersen*,#
Tonci
Balic-Zunic,‡
Søren
Pehrson,§
2001 VOL. 1, NO. 6 463-466
and
Department of Medical Biochemistry and Genetics, Biochemistry Laboratory A, The Panum Institute, University of Copenhagen, Blegdamsvej 3, DK-2200 Copenhagen N, Denmark, Geological Institute, University of Copenhagen, Østervoldgade 10, DK-1350 Copenhagen K, Denmark, and Haldor Topsøe A/S, Nymøllevej 55, DK-2800 Lyngby, Denmark Received August 15, 2001
ABSTRACT: Rates of growth of triclinic calcium pyrophosphate dihydrate microcrystals of columnar shape are reported for pH 4.5, 5.0, 5.5, and 6.5. These rates can be explained by the polynuclear growth mechanism in which nuclei of critical size form, grow laterally, and intergrow while spreading over the surface. The surface energy is found to be 75 ( 5 mJ/m2. The frequency for attachment of calcium ions to the crystal surface is found to be (2 ( 1) × 105 s-1, close to the theoretical value for a calcium ion to partially dehydrate and simultaneously make a diffusion jump into a kink site, 1.6 × 105 s-1. Introduction Many studies have reported the presence of monoclinic and triclinic calcium pyrophosphate dihydrate (m-CPPD and t-CPPD) in articular cartilage and synovial fluid of patients suffering from a type of arthritis called CPPD deposition disease.1-4 This is an agerelated problem with cartilage calcification identified in about 15% of the population of 80-89 years of age and up to 30-50% of the population over 90 years of age. No way is known of halting the progressive deposition of crystals or removing those already deposited in cartilage. In periods of joint pain and disability, CPPD crystals are also found in the synovial fluid and partially phagocytosed crystals are observed in polymorphonuclear leukocytes. In this acute stage of CPPD crystalinduced synovitis, the disease is referred to as pseudogout. CPPD deposition has also been reported to occur following joint trauma or surgery and in patients suffering from metabolic diseases such as hyperparathyroidism. An understanding of the formation, growth, and dissolution processes of CPPD is of importance for designing of methods for prevention and treatment of this disease. We have reported in vitro studies of the rate of dissolution of t-CPPD.5 The polynuclear mechanism, in which small holes or “nuclei” of a critical size spread and intergrow over the crystal surface, with a lateral rate depending mainly on the calcium ion concentration, could explain the rates observed. The results led to a value of about 30 mJ/m2 for the surface energy. We have also reported6 the rates of formation and growth of a calcium pyrophosphate tetrahydrate (CPPT) phase, which is often formed as a precursor to the more stable t-CPPD. The CPPT phase, previously * Corresponding author: Short address: IMBG/A, Panum Inst., Blegdamsvej 3, DK-2200 Copenhagen N, Denmark. Telephone (+45) 35327740, fax (+45) 35327741, e-mail:
[email protected]. # The Panum Institute, University of Copenhagen. ‡ Geological Institute, University of Copenhagen. § Haldor Topsøe A/S.
Figure 1. SEM micrograph of t-CPPD stock crystals. Bar: 20 µm.
referred to as orthorhombic, was found7 to have a monoclinic structure and is referred to as m-CPPTβ. This phase shows an interesting phenomenon in formation and growth around pH 7. The rate of growth is increasingly inhibited with increasing supersaturation. Induction times for spontaneous precipitation are longer at pH 7 than at pH 5 for a given supersaturation. We suggested that this “autoinhibition” was due to chelation of Ca2+ and P2O74- in the crystal surface blocking normal growth and formation processes. This hypothesis was supported by the results of a determination of the crystal structure.7 The aim of the present work is to study the rate of growth of t-CPPD as a function of pH, to interpret the experimental results in terms of molecular events taking place in the crystal surface and thereby contribute to the understanding of the mechanism of growth of these crystals. Materials and Methods The triclinic CPPD crystals used in this work and shown in Figure 1 were from the same stock as those for which the dissolution rate was reported.5 The crystals have a slightly
10.1021/cg015547j CCC: $20.00 © 2001 American Chemical Society Published on Web 10/10/2001
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Table 1. Rates of growth of t-CPPD at 10% Growth (m/m0 ) 1.1) in solutions with Ca/PP ) 2.0 pH
m0 (mg)
V0 (mL)
CCa (mM)
CCa,s (mM)
S
dnCa/ dnOH
log(J/m0) (mol/g‚s)
φ
ln(J/m0Hgp(C)) (s-1)
4.5 4.5 4.5 4.5 4.5 5.0 5.0 5.0 5.0 5.0 5.0 5.5 5.5 5.5 5.5 5.5 6.5 6.5 6.5 6.5 6.5
11 11 11 11 11 12.8 15.5 19.7 12.9 14.5 12.6 10.1 11.0 10.5 11.2 10.2 10.8 12.0 10.5 10.0 10.4
918 913 909 906 904 907 905 915 930 879 898 898 897 896 895 894 883 943 875 869 862
2.64 2.12 1.70 1.49 1.29 1.62 1.40 1.06 1.04 0.97 0.76 0.65 0.542 0.432 0.366 0.323 0.141 0.139 0.118 0.101 0.080
0.396 0.387 0.379 0.375 0.371 0.168 0.166 0.162 0.162 0.162 0.159 0.0727 0.0718 0.0709 0.0703 0.0698 0.0171 0.0171 0.0169 0.0169 0.0167
4.87 4.17 3.56 3.22 2.88 6.89 6.20 5.05 4.98 4.73 3.91 6.65 5.80 4.88 4.28 3.88 6.03 5.97 5.24 4.68 3.88
1.03 1.02 1.02 1.01 1.00 1.10 1.09 1.08 1.07 1.07 1.06 1.18 1.17 1.15 1.13 1.12 1.55 1.54 1.52 1.49 1.46
-6.26 -6.52 -6.79 -6.99 -7.20 -6.31 -6.51 -6.75 -6.82 -6.81 -7.04 -6.69 -6.95 -7.12 -7.33 -7.39 -7.23 -7.29 -7.45 -7.65 -7.91
1.75 1.76 1.77 1.77 1.77 1.62 1.66 1.71 1.72 1.73 1.75 1.65 1.69 1.73 1.75 1.76 1.74 1.75 1.76 1.76 1.77
10.05 9.69 9.32 9.02 8.73 10.05 9.75 9.51 9.37 9.47 9.23 9.81 9.41 9.28 8.99 9.00 9.65 9.53 9.34 9.06 8.73
elongated parallelepiped (“columnar”) habit similar to some of the crystals observed in synovial fluid of patients suffering from CPPD deposition disease.1-4 The longest dimension of the stock crystals is of the order of 10 µm. New BET measurements of the specific surface area lead to the value of 0.6 m2/g used here. (The value 1 m2/g was used previously.5) All kinetic experiments were made at 37.0 ( 0.1 °C. Rates of growth were measured by the rate of addition of KOH to keep pH constant for pH values in the range 4.5-6.5. Experiments were made in the same way as previously described.6 Initial supersaturations were in the range 3 < S0 < 7. (Detailed definitions of symbols are given in the List of Symbols). These supersaturated solutions are unstable. An induction time of 4 h was observed at pH 7 for a solution with composition corresponding to S0 ) 7 for t-CPPD.6 Longer induction times were observed for lower supersaturations. Induction times at pH 5 were found to be shorter than at pH 7. In the present work, the rates at the highest supersaturations and lowest pH were measured approximately 15 min after preparation of the supersaturated solutions. There was no sign of spontaneous precipitation (S-shaped titration curves). The rates were always monotonically decreasing at the point where they were measured. Samples of reaction mixtures were also checked microscopically for extra precipitation.
Results and Discussion Experiments were analyzed as described,6 using the same values of constants, in particular the value of pKs ) 18.35 for the solubility product for t-CPPD. The rate of growth, J ) dnCPPD/dt, can be expressed by
J ) kJm0F(m/m0)g(C)
(1)
in which kJ is a rate constant proportional to the initial specific surface area, m0 the initial mass of crystals, F(m/m0) is a function representing the change in the morphology/surface area of the crystals, and g(C) is a function of the solution composition.8 The rate was calculated in each experiment from the rate of addition of KOH at the point where the crystals had grown 10%, and F(m/m0) was assumed equal to 1. In Table 1 the pH, initial mass, m0, and the initial volume, V0, are given in columns 1-3. The initial calcium concentration was approximately 0.01 mM higher than the concentration of calcium ions, CCa, after 10% growth, given in column 4. The pyrophosphate
Figure 2. Logarithmic plot of J/m0 in molCPPD/s‚g as a function of S-1. Squares: pH ) 4.5, circles: pH ) 5.0, triangles: pH ) 5.5, diamonds: pH ) 6.5. The slopes of the lines are 3.0, 2.4, 2.4, and 2.8, respectively.
concentration was in all cases a factor of 2 lower than the calcium concentration. The concentration of calcium ions in a saturated solution under the conditions of the experiment, CCa,s, the supersaturation, S, at 10% growth, and the conversion factor, dnCa/dnOH, all calculated using the ion speciation program Ionics,9 are given in columns 5-7. The logarithm of J/m0 in the units molCPPD/s‚g is given in column 8. A logarithmic plot of J/m0 as a function of S-1 is shown in Figure 2 for the values of pH 4.5, 5.0, 5.5, and 6.5. The slopes of the lines in Figure 2 are 3.0, 2.4, 2.4, and 2.8, respectively. It was found5 that the dissolution process is not purely limited by the polynuclear surface process but is also slightly influenced by the transport of substances away from the crystal surface. The rates of the growth process are lower and the bulk concentration is higher than for the dissolution process. A calculation of the type5 showed that there is no significant difference between the concentration at the crystal surface calculated from the diffusion equation and the bulk concentration. The growth process is therefore purely surface-controlled,
Kinetics of Growth of Triclinic Ca2P2O7‚2H2O Crystals
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most likely either by a spiral mechanism10 or the polynuclear mechanism.8,11 The slopes of the lines in Figure 2 are not in good agreement with that expected for the spiral mechanism,10 i.e., 1.9:
g(C) ) S1/2 ln S(S - 1) ≈ (S - 1)1.9
(2)
For the polynuclear mechanism, nuclei, small islands, are formed by fluctuations in the surface. To be stable or to grow larger, the nuclei have to have a critical size depending of the degree of saturation. The nuclei intergrow with one another. For this mechanism g(C) is given11 by
g(C) ) gp(C)e-R/3β
(3)
in which β ) ln S and
R ) π(σd2/kT)2
(4)
where σ is the surface energy, d is the mean ion diameter in the crystal (3.36 × 10-10 m), k is the Boltzmann constant, and T is the temperature in Kelvin. The preexponential concentration term, gp(C), is given by
gp(C) ) φ-1/3β1/6a1/3(CCa - CCa,s)2/3
(5)
in which φ is an error function integral calculated numerically; see the List of Symbols. In eqs 3-5, formation of nuclei is described in terms of mean ion activity, and the lateral growth rate of nuclei is described in terms of the concentration of calcium ions. The reason for this choice is that cations, in contrast to anions, have to dehydrate or hydrate (at least partly) when entering or leaving the solid phase. According to the polynuclear theory of growth, the rate constant kJ is proportional to the frequency for a calcium ion to partially dehydrate and simultaneously make a diffusion jump into a kink site, νin,
kJ ) Hνin
(6)
The proportionality constant, H, is a function of the specific surface area. νin and the corresponding frequency for detachment of a calcium ion, νout, are connected through the equilibrium condition:
νout ) 2d3NACCa,sνin
(7)
In Table 1, the value of the error function, φ, which appears in the polynuclear expression is given in column 9 and the natural logarithm of the polynuclear expression J/m0Hgp(C) is in column 10. In Figure 3 ln(J/m0Hgp(C)) is plotted against -1/β. The slopes of the lines in Figure 3, R/3, and the intercepts with -1/β ) 0, ln νin, lead to the values of σ, νin, and νout given in Table 2. The ranges of x*, the number of building units (ions) in a critical unit, given by the formula:
x* ) π
( ) σd2 kTβ
2
(8)
are also given in Table 2. For a constant value of S the raw data plot, Figure 2, shows a strong pH dependence. This is in contrast to
Figure 3. Plot of the results for growth of t-CPPD according to the polynuclear mechanism. Units of J/m0Hgp(C): s-1. Squares: pH ) 4.5, circles: pH ) 5.0, triangles: pH ) 5.5, diamonds: pH ) 6.5. Values of the surface energy, the rate constant, νin, and the number of building units in a critical nucleus obtained from these plots are given in Table 2. Table 2. Results of Plots of the Rates (Figure 3) According to the Polynuclear Mechanism pH
R/3
10-5 νin (s-1)
σ (mJ/m2)
νout (s-1)
x*
4.5 5.0 5.5 6.5
4.2 3.6 3.7 4.9
3.1 1.3 1.1 2.2
76 70 71 82
5.4 1.0 0.36 0.17
5-11 3-6 3-6 5-8
the polynuclear plot in Figure 3, in which only a minor pH dependence is seen. The main reason for this is that the lateral growth rate for the polynuclear mechanism is expressed as a function of (CCa,s - CCa) ∝ (CPP,s CPP).11 The surface energy is 75 ( 5 mJ/m2 and νin is (2 ( 1) × 105 s-1, a value close to that expected if the rate determining step for the lateral growth rate is partial dehydration of a calcium ion combined with a diffusion jump into a kink site, 1.6 × 105 s-1.10 The surface energy found here for growth of t-CPPD is about twice the value obtained for dissolution.5 Similar difference in surface energy found for growth and dissolution processes was also observed for calcium hydroxyapatite and fluorapatite.11 This effect was explained by partial protonation of anions in the surface weakening bonds to calcium ions and facilitating surface nucleation of the dissolution process in contrast to the growth process. Whereas the frequency or first order rate constant for ion integration is practically independent of pH in the pH interval investigated, the frequency for detachment of surface ions, νout, increases with decreasing pH, and thus also increasing with surface protonation. To elucidate the kinetic behavior of CPPD crystals in vivo, in vitro formation, growth, and dissolution processes need to be described in terms of molecular events on the crystal surfaces. Once such information is available, the interaction between CPPD crystals and relevant body fluids may possibly be understood in terms of molecular events. This description is necessary for understanding the in vivo processes involved and thereby for predicting possible methods for prevention and treatment of CPPD deposition disease. Acknowledgment. We are most grateful for the technical assistance of Ms. Julita Kuzimska and Ms.
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Mette Kjær Schou, The Panum Institute. We acknowledge with thanks financial support from the Danish Medical Research Council (J.C., M.R.C.) and the Vera and Carl Johan Michaelsen Foundation (J.C.). List of Symbols Asp ) specific surface area of crystals, Asp,0 ) 0.6 m2/g for t-CPPD here a ) mean ionic activity, a ) IP1/3, for CPPD as ) mean ionic activity in a saturated solution, as ) Ks1/3, for CPPD CX ) stoichiometric (total) concentration of component X CX,s ) saturation value of the stoichiometric concentration of X d ) mean ionic diameter in crystal, d(CPPD) ) 3.36 × 10-10 m dnCa/dnOH ) differential relation between the amount of Ca ions consumed and the amount of OH ions required to maintain a constant pH at a given solution composition F(m/mo) ) term expressing the influence that the relative change in the mass of the crystals has on the geometry of the crystals and thereby on the overall rate of growth or dissolution g(C) ) term expressing the influence of solution composition on the rate of growth or dissolution gp(C) ) preexponential concentration term, polynuclear mechanism H ) constant, H ) kJ/νin ) 2dπ1/2Asp0/31/3ν, H ) 1 × 10-10 m3/g for the CPPD crystals used here IP ) ionic product J ) overall rate of growth or dissolution, J ) kJm0F(m/ m0)g(C) Ks ) ionic product for solubility product, pKs(CPPD) ) 18.35 used here k ) Boltzmann constant kJ ) overall rate constant m ) mass of crystals at time t NA ) Avogadro constant n ) amount of substance (mol)
Christoffersen et al. PP ) pyrophosphate S ) supersaturation ratio, (IP/Ks)1/3 for CPPD; S ) a/as T ) absolute temperature t ) time x* ) number of ions in critical nucleus, x* ) π(σd2/kTβ)2 R ) π(σd2/kT)2 β ) ln S ν ) number of particles in formula unit, for Ca2P2O7‚2H2O, ν)5 νin, νout ) frequency for a calcium ion to enter or leave a growth site F ) density, for CPPD F ) 2.56 × 106 g/m3 σ ) Gibbs surface energy per unit area, surface tension φ ) integral: φ ) xπ/2 erf[(1 - xx*)xβ;∞] subscript 0 ) value at t ) 0
References (1) Pritzker, K. P. H. J. Am. Geriatr. Soc. 1980, 28, 439-445. (2) Beutler, A.; Rothfuss, S.; Clayburne, G.; Sieck, M.; Schumacher, H. R. Arthritis Rheum. 1993, 36, 704-715. (3) McCarty, D. J. In Arthritis and Allied Conditions; Koopman W. J., Ed.; Williams and Wilkins: Baltimore, 1997, pp 81102. (4) Ryan, L. M.; McCarty, D. J. In Arthritis and Allied Conditions; Koopman W. J., Ed.; Williams and Wilkins: Baltimore, 1997, pp 2103-2125. (5) Christoffersen, M. R.; Seierby, N.; Balic-Zunic, T.; Christoffersen, J. J. Cryst. Growth 1999, 203, 234-243. (6) Christoffersen, M. R.; Balic-Zunic, T.; Pehrson, S.; Christoffersen, J. J. Cryst. Growth 2000, 212, 500-506. (7) Balic-Zunic, T.; Christoffersen, M. R.; Christoffersen, J. Acta Crystallogr. 2000, B56, 953-958. (8) Christoffersen, J. J. Cryst. Growth 1980, 49, 29-44. (9) Berland, Y.; Olmer, M.; Grandvuillemin, M.; Madsen, H. E. L.; Boistelle, R. J. Cryst. Growth 1988, 87, 494-506. (10) Christoffersen, J.; Christoffersen, M. R. J. Cryst. Growth 1988, 87, 41-50. (11) Christoffersen, J.; Christoffersen, M. R.; Johansen, T. J. Cryst. Growth 1996, 163, 304-310.
CG015547J
