Statistical Studies of the Physicochemical Analytic Results of a Series of Synthetic Calcium Hydroxyapatite Containing Carbonate and Sodium

Faouzia Rockh B. Hadj Yahia, Ismail Khattech

Faculty of Science, Chemistry Department, Applied Thermodynamics Laboratory, Tunis El Manar, Tunisia.

DOI: 10.4236/ajac.2014.55042                

Abstract

The objective of this study is to present a simple method of statistical calculation that allowed us to determine the relationship between the different data obtained from the characterization of the synthetic carbonated apatites containing sodium, in order to find the fundamental substitution mechanism(s) for incorporation of Na⁺ and CO₃^2- and to establish the general formula. For that, a series of hydroxyapatites containing carbonate and sodium (Na-CO₃HAps) has been obtained by the precipitation method. All the compounds were characterized by infrared spectra (IR), powder X-ray diffraction (PXRD) and elemental analysis. The statistical treatment of the experiment result allows us to determine the relationship between one variable and the change in the other and to found the fundamental substitution mechanism(s) for incorporation of Na⁺ and CO₃^2- . Analysis of variance (ANOVA) allows us to test the models proposed.

Keywords

Carbonated Calcium Hydroxyapatite Containing Sodium Na-CO₃HAps, Statistical Studies, Multiple Linear Regression, Analysis of Variance (ANOVA)

1. Introduction

Carbonated calcium hydroxyapatite containing sodium and/or potassium, magnesium etc. is the most important mineral compound in human dental, enamel and bone [1] [2] . We can also found this type of apatite in sedimen- tary phosphate apatite [3] . Consequently, these materials have necessitated studies of the effects of ions, such as Na⁺, Mg²⁺, on the physicochemical properties and especially the mode of Na⁺ incorporation in apatite structure. But, the complexity of biological and sedimentary phosphates apatite has returned these studies very difficultly.

For this reason, multiple techniques have been used to prepare these products, with wet chemical methods and solid-state reactions [4] -[14] . Carbonate incorporation in synthetic carbonated calcium hydroxyapatites “CO₃ HAp” has been classified as either type A or type B depending on the mode of substitution: for type A substitution and for type B substitution [10] [11] .

In the precipitation of “CO₃HAps”, different products are obtained if the calcium solution is added to the phosphate plus carbonate solution “direct” rather than vice versa “inverse” apatite [6] [8] [12] -[14] . All these products are B-Apatite, but it seems that the maximum possible carbonate content was greater in the “direct” apatite compared with “inverse” apatite [13] . Na-free B-type “CO₃HAps” has been prepared with a maximum content between 10 wt% at 14 wt%. In the presence of ions such as (Na⁺, K⁺ or Mg²⁺), we have a coupled substitution. With Na⁺ for Ca²⁺, B-type apatite “Na-CO₃HAps” contains up to 22 wt% [14] .

Based on some studies, such as EPR and IR spectroscopy, X-ray and neutron diffraction, chemical analyses, infrared and Raman spectroscopy, the crystallite size, chemical composition and physicochemical nature of synthetic apatite have been determinated. But there have been multiple studies, speculations and controversy about the mechanism(s) by which and alkali metal(s) are incorporated in the apatite lattice [15] -[20] . According to De Maeyer et al., the fundamental substitution mechanisms for the incorporation of Na⁺ and in hydroxyl apatite “HAp” can be described using six basic processes [21] .

1)

2)

3)

4)

5)

6)

where V^X is the vacancy on a regular apatite lattice site occupied by X.

Moreover, these authors propose a coupling of different fundamental mechanisms in a fixed proportion, lead- ing to the definition of apparently new mechanisms. The composition of B-type of Na-CO₃HAps is determined by the occurrence of one or more of these fundamental mechanisms. Thus, these authors suggest that the stoi- chiometry would be given by:

where a, b, c, d and e are the contribution per unit cell of basic substitutions (I to V) resulting in the fundamental substitution mechanisms for incorporation of Na⁺ and in the crystal lattice of hydroxyapatite “HAp”.

The aim of the present study is to present a statistical method which allows us to determine the relationship between the obtained values and experimental conditions and to estimate the change in one variable from the given increase or decrease in another. The analysis of variance (ANOVA) allows us to test the mathematical model. Finally, the statistical method allows us to find the fundamental substitution mechanism(s) for incorpora- tion of Na⁺ and. So, the stoichiometry of the solid can be described as a function of the contributions of each mechanism to the composition of the unit cell.

2. Experimental Procedure

2.1. Preparation of Na-CO₃HAps

The preparation method is described in detail elsewhere [12] [13] , In brief, The precipitated Na-CO₃HAps were prepared by dripping calcium solution 0.03 M (Ca(NO₃)₂×4H₂O) into a phosphate solution 0.008 M of (Na₂HPO₄×12H₂O) which contains also different concentrations of (Na₂CO₃) such as the molar ratio R:

. So that, the concentrations of and [Na⁺] were varied in the hydrolysis solution:,. Then, the concentration ratio

After hydrolysis which took 3 h, the precipitates were filtered, washed abundantly with hot distilled water (95˚C), dried for 12 h at 70˚C and then heated at 400˚C in air for 24 h in order to eliminate inter- and intra- crystalline water.

2.2. Characterization

2.2.1. Infrared Spectroscopy and X-Ray Diffraction

Na-CO₃HAps samples are characterized by IR absorption spectroscopy and X-ray diffraction. In infra-Red, we use pellets of absorption. They are prepared using the usual KBr disk technique. It consists in mixing 1 mg of powder of a sample of Na-CO₃HAp with 300 mg KBr then pressed at 6 psi. The pellets so prepared were then scanned on a Shimadzu IR spectrometer in the range (4000 - 400) cm⁻¹.

Powder X-ray diffraction analysis for the Na-CO₃HAps samples were carried out using an X-ray diffractome- ter MRD with a generator (40 kV and 40 mA). After indexation of the full pattern, cell parameters were refined using the program “WINCELL”

2.2.2. Chemical Analysis and Density Measurements

The dehydrated samples were subjected to a chemical analysis. The phosphorus content of precipitates is deter- mined by colorimetry after complexation with vanado-molybdate [22] . The sodium and calcium content are ob- tained by atomic absorption spectroscopy and the carbonate content was determined by coulometry method (re- lease of CO₂ by dissolution in acid). The hydroxide content % OH was calculated on the basis of electro neutral- ity. Density of the solids was measured by a flotation method [13] .

3. Results

3.1. Physical Analysis

The infra-Red spectrums presented in Figure 1 show that they are typical of apatite containing B-type carbonate [13] . The assignment of absorption bands was made according to the studies [12] [13] . IR analyses of the sam- ples show the effect of increasing and Na⁺ on the spectral properties of apatites. However, increasing and Na⁺ contents caused increasing of the intensities of the absorption bands at 1420 - 1460 cm^‒1 (n₃ C‒O) and 872 cm^‒1 (n₂ C‒O) and decreasing in the resolution of (n₃ P‒OH) absorption band at 1032 cm⁻¹, The shoulder observed at 740 cm^‒1 is attributed to the n_L mode of OH^‒ close neighbors to Na⁺ ions [13] .

Figure 1. Absorption Spectra of Na-CO₃Haps obtained from solutions with: (A) R = 5, (B) 15 AND (C) 25.

In Figure 2 we presented the X-ray diffraction patterns. The peaks are relatively sharp and well resolved and can all attributed to the hexagonal crystal form of hydroxyapatite, but some shifts of peaks position can be ob- served, reflecting a change in unit cell dimensions due to incorporation of and Na⁺. This allows us to consider our samples as pure well-crystallized phases of apatite type. The results of dimensions “a” and “c” and the volume of the hexagonal apatitic unit cell obtained from X-Ray analysis as well as the densities of the com- pounds allow determining the molar weight (Mw). The results are grouped together in Table 1.

3.2. Chemical Results

The compositions of samples in Weight % determined by chemical analysis are displayed in Table 2. The results of quantitative analysis of phosphorus, calcium and sodium were determined with standards deviations 0.17; 0.03 and 0.09 respectively and the amount of CO₃ was determined on relative uncertainty 2%.

The results of the chemical and physical analysis (Table 1 and Table 2) allowed us to calculate the number of each ion X per unit cell, n_x according to:

, (1)

where M (r×V_cell×N) is the molar mass, V_cell is the unit cell volume, M_x is the atomic or ionic mass of X and N Avogadro’s constant. The amount of OH ions was calculated taking into account the electroneutrality of each compound. Table 3 gathers the calculation results.

4. Discussion

As in the case of the reference [23] , the last column of Table 3 shows that the sum of the number of phosphate and carbonate per unit cell is approximately equals to 6 with experimental error as well as the XRD and IR spectra clearly demonstrate that the samples “Na-CO₃HAps” of this study are pure B-type carbonated apatites, this indicates that substitutes for on a 1:1 basis which agrees with the fundamental substitution mechanisms for B-type [21] [23] -[25] . On the basis of the observation, the real number of each ion per unit cell can be obtained using the following equation:

Figure 2. XRD patterns of synthetic apatites obtained from solutions with: (A) R = 5, (B) 15, (C)25.

Table 1. Values of lattice parameters “a”, “c” and volume molar; density and weight (Mw) of the prepared samples.

Table 2. Chemical composition (weight %) and total mass balance S% of Na- CO₃HAps prepared from solutions with.

(2)

Similarly, we have calculated the number of vacancies on Ca²⁺ and OH⁻ lattice sites respectively:

(3)

and

(4)

The results of these calculations are summarized in Table 4. The errors in Table 4 were estimated by means of error propagation theory.

Table 3. Unit cell composition of Na-CO₃AHps calculated on the basis of the chemical composition and by means of Equa- tion (1).

Table 4. Unit cell compositions of Na-CO₃Aaps calculated on the basis of the chemical composition and by means of fol- lowing Equation (2).

Comparing our experiment data with the data available in the literature [24] -[26] , we can found that the presents results are in agreement with those of reference [25] , though, this series was prepared by hydrolysis of monetite in solution with varied and alkali metal Na⁺ concentration.

So, it may be said in the present study, that the fundamental substitution mechanisms (I, III and IV) could ac- count for the incorporation of and Na⁺ in the HAp lattice. If a, c and d are the contributions of mechan- isms I, III and IV respectively, thus,

(5)

(6)

and

(7)

and the generic formula has the following expression:

5. Statistical Studies

The objective of this study is to construct a mathematical model which allows us to estimate the change in Y the dependent variable from a given increase or decrease in X the independent variable and to determine which me- chanism(s) is related with experimental conditions.

The mathematical model is expressed as:

(8)

where e_i is the random variable drawn from N(0, s²), b₀ and b_j are the estimated regression coefficients. The model assumes that their deviation ε from the line is normally distributed with means 0 and constant variances s².

Least square method [26] allows calculating the regression and correlation coefficients, the variance of the b parameters and to test the null hypothesis H₀: b_j = 0 and their significances level. The analysis of variance for the linear regression or the F test allows us to be confident that at least one of X-variable contributes to the regression. The theoretical basis of these calculations is given in references [12] [26] .

5.1. Multiple Linear Regression of Y = c/a Crystallographic Parameters Ratio on and X₂ = % Na⁺ the % Weightions in the Hexagonal Apatite Lattice

In attempts to disentangle and to measure the effects of the insertion of and Na⁺ ions on the hexagonal apatite lattice dimensions, we have constructed a multiple linear regression on two X-variables where, and X₂ = %Na⁺ (data Table 2) and Y is the estimate ratio of the hexagonal lattice dimensions (da- ta Table 1). The results of the calculation values of the different correlations coefficients, the estimated values of b₀, b₁ and b₂, the variances and t-test of b₁ and b₂ are summarized in Table 5(a). The analysis of the variance for the linear regression or the F-test is given in Table 5(b).

The multiple linear regression (Table 5(a)) indicates that the ratio of the hexagonal lattice dimensions of Na-CO₃Aps “c/a” vary linearly with the carbonate and sodium content according to:

(9)

The analysis of variance (Table 5(b)/ANOVA) shows further that F-test = 100.5 is higher than criterion F(5%; 2.9) = 4.26. This allowed us to rejet H₀: b₁ = b₂ = 0 and affirm that at least one of the predictors is linearly asso- ciated to the response.

T-test of of the estimated regression coefficients b₁ and b₂ has shown that b₂ and b₁ are significant at 70% level.

5.2. Multiple Linear Regression of the Molar Ratio in Na-CO₃HAps Solid on the Concentration of Na⁺: X_1,i and the Concentration of: X_2,i in Aqueous Solution

To understand the influence of the experimental conditions on the composition of these apatites (data Table 4), we conducted a multiple linear regression [22] on two X-variables where, X₁ and X₂ are the Na⁺ and ions

(a) (b)

Table 5. calculation for fitting a multiple linear regression analysis of the estimated Y = c/a the ratio of the lattice parameters of Na-CO₃HApson carbonate and sodium contents X₁ = wt% CO₃, X₂ = wt% Na. (a) Calculation of regression, correlation coefficients and variances, (b) F-test: Analysis of Variance.

concentrations in solution and Y is the estimates molar ratios of in the solids. The values of the different correlations coefficients, the estimated values of b₀, b₁ and b₂, their variances and the F-test or the test of the variance are summarized in Table 6 and Table 7.

The multiple linear regression (Table 6(a)) indicates that the molar CO₃/P ratio of the Na-CO₃HAps vary linearly with the concentrations of carbonate and sodium in solution according to:

(10)

The result of the F-test (Table 6(b)) F = 635.5 and the individual t-test of the b’s Table 6 b for b₁, t = −7.82. For b₂, t = 8.26 confirm that the molar CO₃/P ratio of Na-CO₃HAps solid is significantly depends on both con- centrations of carbonate and sodium [Na⁺] = X₂ in aqueous solution. However, constantly we have, thus the expression in brackets, hardly varied, thus,

CO₃/P molar ratio is constant. This shows that for concentration ratio in solution constant the

increase on molar in solid is equal to the decrease on molar. Thus, the molar ratio in the solid is not significantly influenced by the increase of Na⁺ and ions in solution. This result is in a good agree- ment with the results in literature [24] .

The multiple linear regression (Table 7(a)) indicates that the molar Na/P ratio of Na-CO₃HAps vary linearly with the carbonate content according to:

. (11)

An analysis of variance (Table 7(b)/ANOVA), further shows that F-test = 43.5 is higher than criterion F(1%, 2.10) = 7.56, this allowed us to rejet H₀: b₁ = b₂ = 0 and affirm that at least one of the predictors is linearly as- sociated to the response. T-test of the regression coefficients b₁ and b₂ has shown for b₁, t = −0.45 signifiant at 30% level. For b₂, t = 1.67 signifiant at 80% level.

5.3. Multiple Linear Regression of the Estimated Molar Content of Calcium on and the Molar Contents of Sodium and Carbonate in the Solid

The examination of Table 4 showed too that n_Ca molar content of the solid decreases when and n_Na incor- porate increase. To determine the relationship between these three variables given in Table 4, a multiple linear regression analysis was undertaken. The results are summarized in Table 8.

From Table 8, the multiple linear regression analysis shows that the relationship between these quantities is given by equation:

(a) (b)

Table 6. Calculation for fitting a multiple linear regression of the estimated molar ratio in Na-CO₃HAps solid on X₁ = [Na⁺] and the concentrations in so- lution. (a) Calculation of regression and correlation coefficients and variances, (b) Values of F-test: Analysis of Variance.

(a) (b)

Table 7. Calculation for fitting a multiple linear regression analysis of the estimated Y = n_Na/n_P molar ratio in Na-CO₃HAps solid on X₁ = [Na⁺] and the concentrations in solution. (a) Calculation of regression and correlation coefficients and variances, (b) Values of F-test: Analysis of Variance.

(12)

From the intercept of the following equation it can seen that, within experimental error, a carbonate-free apa- tite contains 10 Ca²⁺ ions per unit cell. The expression in brackets shows that the number of Ca²⁺ ions in the unit cell decreases with increasing of Na⁺ and numbers for about one Ca²⁺ ion.

The analysis of variance (Table 8(b)), further shows that F-test = 379.4 is higher than criterion F(5%, 2.9) = 4.26, this allowed us to rejet H₀: b₁ = b₂ = 0 and affirm that at least one of the predictors is linearly associated to the response. T-tests of the regression coefficients b₁ and b₂ show that the value of: b₁ is signifiant at 80% level and for b₂ is signifiant at 90% level.

5.4. Multiple Linear Regression of Y = a, c or d the Contributions of Mechanisms I, III and IV on X₁ = [Na⁺] and the Concentrations in Solution

For determine which mechanism(s) is related with experimental conditions, we have undertaken a statistical analysis [26] of the values of a, c and d the contribution of the mechanisms I, III and IV (Table 9) calculated according to the following Equations (5)-(7) as a function of Na⁺ and concentrations in aqueous solution. Results of these calculations are summarized in Tables 10-12.

(a) (b)

Table 8. Calculation for fitting a multiple linear regression of the estimated the molar number of calcium as a function of carbonate and sodium contents and in the solid Na-CO₃Haps (a) Calculation of regression, correlation coefficients and variances, (b) F-test: Analysis of Variance.

Table 9. Values of (a, c and d) the estimated contribution of the mechanism I, II and IV.

From Table 10, the multiple linear regression analysis shows that the relationship between these quantities is given by equation:

(12)

An analysis of variance (Table 10(b)/ANOVA), further shows that F-test = 25.36 is higher than criterion F (5%, 2.9) = 4.26, this allowed us to rejet: is linearly H₀: b₁ = b₂ = 0 and affirm that at least one of the predictors associated to the response. The individuals t-test of the regression show that for b₁, t = 0.79 signifiant at 60% level and for b₂, t = −0.70 signifiant at 50% level.

(a) (b)

Table 10. Calculation for fitting a multiple linear regression of the estimated the contribution of the mechanism I on the concentration of Na⁺: X_1,i and the concentration of: X_2,i in aqueous solution, (a) Calculation of regression and correlation coefficients and variances, (b) Values of F-test: Analysis of Variance.

(a) (b)

Table 11. Calculation for fitting a multiple linear regression of the estimated the contribution of the mechanism III on the concentration of Na⁺: X_1,i and the concentration of: X_2,i in aqueous solution, (a) Calculation of regression and correlation coefficients and variances, (b) Values of F-test: Analysis of Variance.

From Table 11, the multiple linear regression analysis shows that the relationship between these quantities is given by equation:

(13)

An analysis of variance (Table 11(b)/ANOVA), further shows that F-test = 163.55 is higher than criterion F(5%, 2.9) = 4.26, this allowed us to rejet H₀: b₁ = b₂ = 0 and affirm that at least one of the predictors is linearly associated to the response. The individual t-tests on the regression coefficients b₁ and b₂ and, Hence, for b₁, t = 10, for b₂, t = −9.8 t(b₁) and t(b₂) are signifiants at P > 99.9% level.

From Table 12, the multiple linear regression analysis shows that the relationship between these quantities is given by equation:

(14)

An analysis of variance (Table 12(b)/ANOVA), further shows that F-test = 0.10 is lower than criterion F(5%, 2.9) = 4.26, this test affirm that mechanism IV is unrelated to experiment conditions. The result of the present study provide that mechanisms I and III are the main in incorporation of Na⁺ and in HAp.

5.5. Multiple Linear Regression Analysis for the Determination of Formula of Unit Cell

The determination of unit cell has been realized by a multiple linear regression between the variables Y = n_Ca and X₁ = a, X₂ = c, X₃ = d. Least square allows calculating the regression and correlation coefficients. The sample regression (prediction equation) is:

, where are the estimated regression coefficients. These have been calculated from the values of correlation coefficients, variance and covariance according to the method of Scherrer [27] . The results of calculations are given in Table 13. To testing the utility of the model, we conduct the F-test according to:

, (15)

where n is sample size, m is number of parameters and is degree of freedom.

From Table 13, the multiple linear regression analysis shows that the relationship between these quantities is given by equation:

(16)

(a) (b)

Table 12. Calculation for fitting a multiple linear regression of the estimated Y = d = n_Na ? c the contribution of the mechanism IV on the concentration of Na⁺: X_1,i and the concentration of: X_2,i in aqueous solution, (a) Calculation of regression and correlation coefficients and variances, (b) Values of F-test: Analysis of Variance.

Table 13. Calculation for fitting a multiple linear regression of the estimated Y = n_Ca =Y in Na-CO₃HAps on the contribution of mechanism I a: X_1i, contribution of mechanism III c: X_2i contribution of mechanism IV d: X_3i.

An analysis of variance (ANOVA), further shows that F-test = 2018.9 is higher than criterion F(5%, 2.9) = 4.26, this allowed us to rejet H₀: b₁ = b₂ = b₃ = 0 and affirm that at least one of the predictors is linearly asso- ciated to the response.

for testing the regression coefficients b₁, b₂ and b₃ and discovering with variable(s) is related to estimate Y = n_Ca, we conduct on the one hand individual t-tests on the b’s, Hence, for b₁, t = −5.28, for b₂, t = −28.3 and for b₃, t = −1.2 t(b₁) and t(b₂) are signifiants at P > 99.9% level. Thus, the molar n_Ca of Na-CO₃Aps solid is signifi- cantly depends on both a and c contributions of mechanisms I and III. The general formula can be written as fol- lows:

Also, structural study for two samples obtained under comparable conditions [28] has been investigated ex- tensively by physicochemical analysis and by Rietveld method refinements. The results of unit cell content cal- culated from the occupations of the atomic sites and the data of chemical composition may be represented by the following ideal substitution scheme and corresponding solid solution :

(The sum of mechanism II and III)

, with x =1.5 (I) and 2.4 (II).

These results show that we cannot consider unique and well defined substitution mechanisms resulting in apa- tites, especially for homogeneous precipitation methods in aqueous solutions, because the lack of control of the reaction parameters as well as incomplete analyses of the solids could result in erroneous interpretations of the substitution mechanism.

On the other hand El Feki et al. [28] confirm that no vacancies of OH⁻ are observed by Rietveld refinements. But, small fraction of vacancies is undetectable by this method and was ignored in the structure refinements.

6. Conclusion

The different statistical analyses present in this review mainly focused on original and new approaches of the knowledge of the substitutions mechanisms. However, in biological calcifications, part of lattice ions of Hap are substituted to considerable extent ions. Consequently, these substitutions have an important influence on several processes (the growth, the dissolution, the mineralization and the demineralization processes. In order to derive the fundamental thermodynamic properties of the solid which determine the course of these processes, the stoi- chiometry of the apatite and especially of the mechanisms by which and Na⁺ are incorporated in the lat- tice must be known.

Conflicts of Interest

The authors declare no conflicts of interest.

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