A Kinetic Model for Hydroxyapatite Precipitation in Mineralizing

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A kinetic model for hydroxyapatite precipitation in mineralizing solutions Jiaojiao Yun, Brian Holmes, Alex Fok, and Yan Wang Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.7b01330 • Publication Date (Web): 19 Mar 2018 Downloaded from http://pubs.acs.org on March 20, 2018

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Crystal Growth & Design

Title page A kinetic model for hydroxyapatite precipitation in mineralizing solutions Jiaojiao Yun†, Brian Holmes‡, Alex Fok*,‡ ,Yan Wang*,† †

Department of Prosthodontics, Guanghua School of Stomatology & Hospital of Stomatology,

Guangdong Key Laboratory of Stomatology, Sun Yat-sen University, Guangzhou 510055, China ‡

Minnesota Dental Research Center for Biomaterials and Biomechanics, School of Dentistry,

University of Minnesota, Minneapolis, MN 55455, USA.

* Corresponding author at: 56 Lingyuan West Road, Guangzhou, Guangdong 510055, China. Tel.: +86 83802805; Fax: +86 20 83822807. E-mail address: [email protected] * Corresponding author at: 515 Delaware Street SE, Minneapolis, MN 55455, USA. Tel.: +1 612 625 0950; Fax: +1 612 626 1484. E-mail address: [email protected]

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ABSTRACT

Numerous studies have shown that there is an amorphous calcium phosphate (ACP) phase preceding the precipitation of crystalline hydroxyapatite (HA) in calcium phosphate solutions. It has also been shown that the addition of magnesium to the solutions has a stabilizing effect by inhibiting the transformation of ACP to HA. The stabilizing effect of Mg2 is attributed to the stronger bonds between water molecules and the magnesium ions adsorbed on the surface of the ACP particles, making it harder for them to dehydrate. However, the kinetics of the reactions between calcium and phosphate ions to form ACP and then HA crystals, and the effect of varying concentrations of Mg on the kinetics have not been studied theoretically in detail. In this study, we develop and validate a kinetic model for analyzing such reactions. The pertinent rate constants are derived by calibrating the model against temporal changes in Ca2 concentration reported by others. The predicted onset and growth of HA crystallization for solutions with different Mg concentrations are consistent with those measured. As it is capable of predicting the production of ACP and the subsequent transformation to HA under different assumed conditions, the kinetic model developed can help further our understanding of the mechanism of mineralization of calcium phosphate solutions.

Keywords: Kinetic model; Hydroxyapatite; Amorphous calcium phosphate; Mineralizing solution


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INTRODUCTION Due to their excellent biocompatibility, the mineralization of collagen scaffolds has been extensively studied as a way for repairing hard tissues. Hydroxyapatite (HA), the main inorganic component of natural bone and teeth with the composition of Ca5 PO4 3 OH,1-3 is produced when a calcium phosphate solution is used to mineralize the scaffolds. The initial solid phase that precipitates depends on the solution’s degree of supersaturation.4 Amorphous calcium phosphate (ACP) is a precursor phase that precedes the formation of crystalline hydroxyapatite in a highly supersaturated calcium phosphate solution.5,

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It is now generally agreed that ACP particles

consist of a random assembly of ion clusters, with a calcium-to-phosphate (Ca/P) molar ratio of about 3/2,5 compared to the ratio of 5/3 found in hydroxyapatite. Based on this Ca/P ratio, it is generally believed that the average cluster of ACP particles have a chemical composition of Ca9 PO4 6 .6-8 Further, there is 15−20% of water in the interstices between, and not within, the individual clusters.9 Due to its disordered structure, ACP is a highly unstable phase that reacts readily with body fluids, leading to rapid calcium phosphate precipitation.3 Some researchers have shown that ACP directly converts to HA:10, 11 the calcium and phosphate ions in ACP rearrange, accompanied by an exchange of ions with the surrounding solution, extending the order of ACP to longer distances to form HA embryos.12 Once the HA embryos reach a critical size, a continuous but slow increase of HA crystallinity will take place. Other researchers state that the mineral transformation takes place by ACP dissolution, followed by subsequent HA crystallization.13 Yet others found other intermediate phases during such a transformation process. For example, Tung et al.14 found the existence of octacalcium phosphate (OCP) (Ca8 PO4 6 H2 ·5H2 O, Ca/P=1.33) during the transformation.


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ACP plays an essential role in biomineralization. There are some stable biogenic ACP in mineralized tissues which act as a temporary storage of calcium and phosphorus for bone apatite.15 Moreover, ACP has better osteoconductivity and biodegradability than tricalcium phosphate and hydroxyapatite in vivo.16 Therefore, gaining an understanding of the mechanism and regulation of calcium mineral crystallization is important in biomineralization research. In the presence of various proteins,17 metal ions,18-20 and polyelectrolytes,21,

22

ACP may

persist for an appreciable period due to kinetic stabilization. Recently, the sodium salt with polyaspartic acid (polyAsp) has been extensively used to stabilize a highly hydrated ACP precursor,21 apparently facilitating its infiltration into the interstices of collagen fibrils in the socalled Polymer-Induced Liquid-Precursor (PILP) process. Compared to traditional methods that result in only extrafibrillar precipitation of HA clusters on the surface of collagen scaffolds, such as the soaking of collagen in a simulated body fluid (SBF), the PILP process produces homogeneous intra- and extrafibrillar minerals indensified collagen films, leading to a nanostructure and a woven microstructure analogous to those of woven bone. Among the metal ions, Mg2 , ubiquitous in many biological systems, inhibits the precipitation of calcium minerals such as calcium phosphates.18-20,

23, 24

Boskey and Posner18 studied the

kinetics of the conversion of ACP to HA at a constant pH of 8. They showed that increasing the magnesium content lengthened the induction period but did not alter the rate of HA precipitation once it began. They also found that the calcium ion concentration dropped significantly in the beginning as ACP was formed, and began to rise again as some of the ACP started to dissolve. With time the Ca2 concentration leveled off to a value associated with the "solubility" of ACP. It then began to decreaseagain as HA crystals appeared, and continued to decrease until all the ACP was converted. At high Mg/Ca ratios the initial ascent of Ca2 concentration never


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occurred, as the ACP was stabilized at these high Mg concentrations. With high Mg concentrations and reduced ACP solubility there was an increase in the rate of ACP formation with a concomitant decrease in ACP particle size. The amount of Mg incorporated into the amorphous solid was directly proportional to the Mg content of the precipitating solution. But Mg was not incorporated into the HA crystals and the final HA crystallite size was independent of Mg concentration. Boskey and Posner18 claimed that in the ACP-to-HA transformation the role of the magnesium was to decrease the Ca2 concentration of the mediating solution by reducing the "solubility" of the amorphous calcium phosphate. But Mg did not affect crystal growth, nor did it appear to poison the HA seeds which were added to the slurry in some cases. Ding et al.19 also studied the effect of Mg on the crystallization of HA from supersaturated calcium phosphate solutions by measuring the changes in pH, Ca2 , PO4 3 and Mg2 concentrations during the reaction. Unlike Boskey and Posner,18 they did not use a buffer to control the pH. Rather, the changes in pH were used as an indication of the crystallization process. Their results showed that the precipitation of HA from supersaturated solutions could be divided into five stages: s1, formation of ion clusters and ACP; s2, stabilization of ACP; s3, transformation from ACP to HA via dissolution and crystallization; s4, classical crystal growth of HA; s5, HA aging under a near equilibrium state. They found that the Mg2 adsorbed on the surface of ACP played the most important role in inhibiting HA crystallization, especially in the phase transformation from ACP to HA. Similar to Boskey and Posner,18 they also found that the amount of Mg incorporated into the ACP was directly proportional to the Mg content in the solution. Although the works mentioned above have provided some useful insights into the different stages of HA precipitation from calcium phosphate solutions and the role of metal ions in


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stabilizing ACP, a kinetic model that can predict changes in the concentrations of the different ions and molecules is still lacking. Such a model would be useful in addressing some of the uncertainties that still remain with the reaction. For example, it is still unclear whether ACP dissolves to supersaturate the solution so that nucleation of HA can occur, or ACP converts directly to HA, exchanging ions with the solution. Further, the stability of the ACP has been found to be crucial in inducing intrafibrillar mineralization of collagen. A kinetic model of the creation, stabilization and transformation of ACP will not only help to optimize the composition of the mineralizing solution, it may also help to test the validity of the different theories proposed for mineralizing biological tissues such as collagen. Several empirical equations have been proposed to describe the kinetics of HA crystal growth.25 These equations are based on driving forces calculated from disequilibrium, and are not based on experimentally derived reaction orders using traditional kinetic techniques. The aim of our study is therefore to develop and validate a kinetic model for studying the effects of stabilizing agents, such as Mg2 , on the stability of calcium phosphate solutions and the formation of HA. EXPERIMENTAL SECTION The overall reactions. Here, we develop a kinetic model based mainly on the experiments conducted by Boskey and Posner.18 It is assumed that when mixing calcium and phosphate solutions containing magnesium ions, ACP is formed immediately, but the reaction is reversible. Further, it is assumed that there are X Ca2 , Y PO4 3 and Z Mg2 in one ACP molecule. With time, the ACP molecules will transform into crystalline HA and, in the process, release some of the surplus ions back into the solution. Once formed, the HA crystals will grow in size by acquiring more calcium and phosphates ions from the solution. The above reactions can be summarized as follows:


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K1

XCa2 + YPO4 3 + ZMg2

ACP CaX PO4 Y MgZ

K 1

K2

ACP CaX PO4 Y MgZ HACa5 PO4 3 OH +X − 5Ca2 + Y − 3PO4 3 + ZMg2+ K3

HACa5 PO4 3 OH + 5nCa2 + 3nPO4 3

(1)

HA crystal growth

where K1, K−1, … are the rate constants and n is an integer. The kinetic models. Using the law of mass action, the rates of change of concentration of Ca2 , PO4 3 and Mg2 with respect to time, t, can be written as: dCa2 X Y Z = −XK1 Ca2 PO4 3 Mg2 + XK1 ACP dt 5

3

+X − 5K2 ACP − 5K3 Ca2 PO4 3 HAα dPO4 3 X Y Z = −YK1 Ca2 PO4 3 Mg2 + YK1 ACP dt 5

3

+Y − 3K2 ACP − 3K3 Ca2 PO4 3 HAα dMg2 X Y Z = −ZK1 Ca2 PO4 3 Mg2 dt +ZK1 ACP + ZK2 ACP

(2.1)

Note that the hydroxyl group and water molecules are not considered as pH is constant and their quantities in the solution are so large that they can be ignored in the formulation above. For simplicity, direct conversion of ACP to HA is assumed; no other intermediate phases are considered. And the rate of conversion is assumed to be proportional to the amount of ACP present. Further, it is assumed that no magnesium ions are incorporated into the HA crystals and that the rate of HA crystal growth is proportional to the surface area of the crystals. The exponent, α, associated with the concentration of HA in the crystal growth term depends on the shape of the HA crystals formed. For spherical and sheet-like crystals, α = 2/3 and 1, respectively.23


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Similarly, the rates of change of concentration of HA and ACP with respect to time can be written as: d[HA] 5 3 = K2 ACP + K3 Ca2 PO4 3 HAα dt d[ACP] X Y Z = K1 Ca2 PO4 3 Mg2 dt −K1 ACP − K2 ACP

(2.2)

The values of the rate constants (K) will depend on the concentration of magnesium in the solution. To facilitate computation, Eq. 2.1 and Eq. 2.2 are nondimensionalized. Thus, all the concentration terms are normalized by the initial calcium concentration, [Ca2 ]0; time t is normalized by t0=180 min, the duration in Boskey and Posner’s experiment for which Ca2 concentration was measured;18 and the rate constants are normalized by combinations of t0 and the initial calcium concentration taken to the appropriate power. The dimensionless kinetic equations can then be written as: dc⁄dτ = −Xk1 cX pY mZ + Xk1 a +X − 5k2 a − 5k3 c5 p3 hα dp⁄dτ = −Yk1 cX pY mZ + Yk1 a +Y − 3k2 a − 3k3 c5 p3 hα dm⁄dτ = −Zk1 cX pY mZ + Zk1 a + ZK2 a dh⁄dτ = k2 a + k3 c5 p3 hα da⁄dτ = k1 cX pY mZ − k1 a − k2 a

(3)

where [c], [p], [m], [h] and [a] are the normalized concentrations of Ca2 , PO4 3 , Mg2 , HA and ACP, respectively; k1, k−1, k2 and k3 are the normalized rate constants; and τ is dimensionless time. The relationships between the dimensionless and actual rate constants are shown in Table 1.


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From the literature,6-8 X and Y were taken to be 9 and 6, respectively, and Z was based on the Mg/Ca ratios in ACP given by Boskey and Posner.18 With an initial normalized concentration of 1 for both Ca2 and PO4 3 , the normalized rate constants were determined by calibrating the numerical model against the measured calcium ion concentration for different initial normalized concentrations of Mg2 , i.e. 0.00, 0.004, 0.04 and 0.20.18 Excel’s nonlinear regression was used for this purpose. The predicted concentration curves for the other ions or molecules were then compared with other available experimental results to validate the model. Table 1. Relationships between actual and normalized rate constants Normalized rate constants k1: K1t0[Ca2 ]0(X+Y+Z−1) k−1: K−1t0 k2: K2t0 k3: K3t0[Ca2 ]07+α

RESULTS AND DISCUSSION Effect of Mg concentration on the different stages of HA production. Figure 1a shows the changes in the normalized calcium ion concentration for the different initial Mg/Ca molar ratios as calculated by using Eq. 3 and α = 1 after calibration. As can be seen, they compare well with the measured values (extracted from Ref. 18 using Image J (https://imagej.net)) shown in Figure 1b, reproducing clearly the five stages of reaction leading to the precipitation of HA (Figure 1c). The sharp decrease in Ca2 concentration at Stage 1 is due to the rapid formation of ACP, which consumes Ca2 (and PO4 3 ), upon mixing of the two starting solutions. The concentration of Ca2 then rises almost immediately again as some of the ACP begins to dissolve, releasing some


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calcium and phosphate ions back into the solution before reaching a quasi-steady state (Stage 2). At Stage 3, the concentration of Ca2 begins to fall again as ACP transforms into HA. It is suggested that the calcium and phosphate ions in ACP rearrange, accompanied by an exchange of ions with the solution, extending the order of ACP to longer distances to form HA embryos. As ACP disappears, the crystal growth of HA dominates the reaction at Stage 4, which consumes Ca2 (and PO4 3 ) in the solution quickly. At the final stage, Stage 5, as the amount of Ca2 (and PO4 3 ) diminishes, the HA crystals reach equilibrium with their surrounding solution. Tables 2 and 3 summarize the values determined for the normalized and actual rate constants, respectively, and the stoichiometric coefficient Z taken from Ref. 18 for each initial Mg/Ca molar ratio. The actual rate constants are further plotted in Figure 2 to help illustrate their dependence on the initial Mg concentration. K1 increases with increasing Mg concentration, while K−1 and K2 reduce. K3, which controls the rate of HA crystal growth, is assumed to stay constant, as suggested by Boskey and Posner.18 They also found that ‘With higher Mg concentrations … there is a resultant increase in the rate of ACP precipitation …’ As K1 controls the rate of ACP production, it is necessary to have K1 increase with increasing Mg concentration. Figure 3a shows more closely the initial reduction in Ca2 concentration, as a result of ACP production, with different initial Mg concentrations. K−1 controls the rate of dissolution of ACP, and K2 the rate of conversion of ACP to HA. Experiments showed that the presence of Mg ions retarded these two processes (Figures 1 and 3). That is why both of their rate constants reduce with Mg concentration. The reduction is more significant in K2, i.e. the rate of conversion of ACP to HA, than in K−1, i.e. the dissolution of ACP. This is consistent with the observation that the main effect of Mg ions seems to be the stabilization of ACP, thus lengthening the induction period for HA precipitation.18


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Figure 1. Changes in normalized Ca2 concentration with time for different initial Mg concentrations: (a) Predicted values after calibration and (b) Measured values.18 (c) The different stages of reaction for an initial Mg/Ca ratio of 0.004.


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Table 2. Normalized rate constants as functions of initial magnesium concentration Normalized

Initial molar ratio of Mg/Ca

rate constants

0.00

0.004

0.04

0.20

k1

22

30

90

2100

k−1

38

10

7

0.0001

k2

1

0.01

0.0005

0.0001

k3

180

180

180

180

Z

0.000

0.030

0.333

2.34

Note: Z is calculated from the ACP Mg/Ca ratios given in Ref. 19 with X = 9

Table 3. Actual rate constants as functions of initial magnesium concentration Actual

Initial molar ratio of Mg/Ca

rate constants

0.00

0.004

0.04

0.20

K1

1.10×1039

1.83×1039

4.01×1040

5.00×1047

((L/mol)14.0/min)

((L/mol)14.0/min)

((L/mol)14.3/min)

((L/mol)16.3/min)

K−1 (/min)

0.211

0.056

0.039

5.56×10−7

K2(/min)

0.006

5.56×10−5

2.78×10−6

5.56×10−7

K3((L/mol)8/min)

6.78×1022

6.78×1022

6.78×1022

6.78×1022

Z

0.000

0.030

0.333

2.34


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Figure 2. Actual rate constants as functions of initial Mg concentration (see Table 3 for units)

Figure 3. Effect of different initial Mg concentrations on the first two stages of HA production: (a) Stage 1 – ACP production. (b) Stage 2 – induction period with stable ACP.


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The predicted curves of HA precipitation for the different Mg-stabilized solutions are consistent with those measured (Figure 4). This validates the kinetic model as the HA precipitation curves were not used for its calibration. The predicted amount of HA increases slowly in the beginning, then rapidly before plateauing to its final value. Increasing the initial concentration of Mg lengthens the induction of HA, but has less effect on the rate of HA formation. This is consistent with experiments which showed that, although a higher concentration of Mg2 could delay the induction of HA, it did not affect its growth rate, as the shape of the remainder of the curve did not change.18

Figure 4. Effect of different initial Mg concentrations on the precipitation of HA: (a) Prediction relative to initial Ca concentration. (b) Measurements relative to final HA content.18 In addition to the amount of Ca2 and HA present in the solution, the numerical model can also predict the amount of PO4 3 and ACP as functions of time. Figure 5a shows the predicted changes in the quantities of these ions or molecules in a solution with an initial Mg/Ca molar ratio of 0.004. The five stages of reaction leading to the precipitation of HA can again clearly be seen. The changes in the concentration of PO4 3 with time follow closely those of Ca2 . The


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difference in their concentrations depends on the composition of the predominant solid phase of calcium phosphate, i.e. ACP or HA, that is present in the solution at the time. The predictions for PO4 3 concentration compare well with the trends reported by Ding et al.;19 see Figure 5b. The initial sharp drop in Ca2 and PO4 3 concentrations is mirrored in an equally sharp peak predicted in ACP production (Stage 1). Some of the ACP produced quickly dissolves as it reaches quasi equilibrium with its constituent ions in a reversible reaction (Stage 2). This is followed by the start of its transformation to HA (Stage 3) and then rapid reduction in its quantity, as the rate of HA production increases (Stage 4), before disappearing altogether from the solution (Stage 5).

Figure 5. (a) Numerical solutions for changes in the amount of Ca2 , PO4 3 , ACP and HA in a calcium phosphate solution containing Mg. The initial Mg/Ca molar ratio is 0.004. The mineralization process can be divided into five stages as marked by the dashed lines: Stage 1) formation of ACP; Stage 2) stable ACP period; Stage 3) transformation of ACP to HA; Stage 4) crystal growth of HA; and Stage 5) ripening of HA. (b) Measured Ca2 and PO4 3 concentrations in a solution with an initial Mg/Ca molar ratio of 0.2.19


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Figure 6a shows the effect of Mg concentration on the production and stabilization of ACP as predicted by the model. The higher the Mg concentration, the more ACP is produced and the longer it remains stable. Figure 6b shows the corresponding effect on the changes in PO4 3 concentration with time. As can be seen, the changes are similar to those in Ca2 concentration as depicted in Figure1, i.e. with increasing concentration of Mg, the PO4 3 concentration is stabilized for a longer time following the initial ACP formation.

Figure 6. Predicted changes in the concentrations of ACP and PO4 3 in solutions containing different initial amounts of Mg: (a) relative ACP content. (b) normalized concentration of PO4 3 . The predicted changes in Mg2 concentration with time for different initial concentrations are shown in Figure 7a. As can be seen, there is a sharp drop in concentration right at the beginning of reaction. The changes in Mg2 concentration are larger in solutions with higher initial Mg concentrations. It means that more Mg ions are incorporated into the ACP as the initial Mg concentration increases, which agrees with previous findings.18, 19 There is a small rebound in Mg2 concentration during Stage 1 as some of the ACP dissolves, releasing some Mg2 back to the solution before reaching its quasi-steady state. As the ACP begins to transform into HA,


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more Mg ions are returned to the solution. These results are consistent with those reported by Ding et al.19 (Figure 7b).

Figure 7. (a) Predicted changes in Mg2 concentration with time for different initial Mg/Ca molar ratios. (b) Measured changes in Mg2 concentration with time (initial Mg/Ca ratio was 0.2).19 The quantity of Mg2 incorporated in the ACP (Mg/Ca) given by Boskey and Posner18 was about 0.037 and 0.26 in solutions with an initial Mg/Ca molar ratio of 0.04 and 0.20, respectively (Table 5). Ding et al.19 also studied the amount of Mg in the ACP; however, as already mentioned, they did not control the pH of the mineralizing solutions and a drop in pH was observed when ACP (or HA) was produced. The amount of Mg2 they found incorporated in the ACP with an initial Mg/Ca molar ratio of 0.2 was about 0.015, which was less than that found by Boskey and Posner,18 who used a buffer to maintain the solutions’ pH at 8. As can be seen from another paper,26 the pH of the mediating solution has an effect on the transformation kinetics. An increase in pH leads to an increase in both the induction time and the amount of time spent in the proliferation period of the reaction, probably due to an increase in Mg uptake by the ACP.


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Using the ACP’s Mg/Ca molar ratios given by Boskey and Posner,18 we then proceeded to estimate the size of the ACP molecules using the ionic radii of the constituent ions listed in Table 4. For simplicity, the hydroxyl groups and water molecules in the complex are ignored. The particle size calculated is for 1 Mg ion. The quotient between this and the actual ACP particle size gives the number of Mg ions in each ACP particle. The results are tabulated in Table 5 and plotted in Figure 8 for different initial Mg concentrations. We can see more clearly from these calculations the increase in the number of Mg ions in each ACP particle as the initial Mg/Ca ratio increases, but this number plateaus to about 40,000 as the size of the ACP particles reduces. The reduction in ACP particle size by the addition of Mg was also reported by others.12 Note the close resemblance between the ACP particle-size curves (Figure 8a) and those of the rate constants that control the dissolution and transformation of ACP (Figure 2). This strongly suggests that Mg ions limit the size of the ACP particles and their reactivity at the same time by surrounding them with water molecules. Despite the relatively small number of parameters used in our kinetic model for the reactions of calcium phosphate solutions, it is capable of reproducing the rather complicated changes of Ca2 concentration with time. The predicted curves for HA precipitation and Mg2+ concentration also resemble those obtained experimentally. These show that the model is reasonable in capturing the various stages of HA precipitation. The model is particularly useful in quantifying the enhancing or inhibitory effect of Mg2 on the rate of each stage of reaction. It also allows sensitivity studies to be performed to investigate the effect of changes in the other parameters or assumptions made on the process.


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Table 4. Ionic radius27 and volume of ions that constitute the ACP Ionic radius Volume of ions Ions (pm)

(pm3)

Ca

100

4.19×106

PO4

238

5.65×107

Mg

72

1.56×106

Table 5. Estimated number of Mg ions in each ACP particle formed in the presence of different concentrations of Mg.18 Mg/Ca molar ratio Mg/Ca Estimated volume of preparative molar ratio of ACP per Mg solution of ACP ion (pm3)

Measured particle size of ACP (pm3)

Estimated number of Mg ions in each ACP

0.00

0.000

-

1.95×1014

0

0.004

0.003

1.26×1010

1.37×1014

10,935

0.04

0.037

1.13×109

5.30×1013

46,843

0.20

0.260

1.64×108

6.62×1012

40,311


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Figure 8. (a) Estimated (per Mg ion) and measured particle sizes of ACP18 for different initial Mg/Ca molar ratios. (b) Estimated number of Mg ions in each ACP particle for different initial Mg/Ca molar ratios. Sensitivity studies. In this section, the kinetic model developed is used to perform some sensitivity studies on the assumptions made. First, we consider the effect of the shape of the HA particles assumed on the numerical solutions. HA can be produced in various morphologies, such as spheres, sheets, needles, wires and bamboo leaves, by controlling the synthesis conditions. In this study, the growth of HA particles is assumed to be proportional to their surface area. Depending on the form of HA particles assumed, different volume-to-surface exponents (α) need to be used in association with the volume of HA in the kinetic eqs (2) and (3). For spherical particles, the volume-to-surface exponent is 2/3. For sheet-like particles, it is 1, which is the value used in the earlier calculations, in accordance with the observations made by Ding et al.19 It was found that the numerical solution often became unstable when α

A Kinetic Model for Hydroxyapatite Precipitation in Mineralizing

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