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

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 3 CO − 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 3 CO − . Analysis of variance (ANOVA) allows us to test the models proposed.


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- Ca PO CO Na 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, leading to the definition of apparently new mechanisms.The composition of B-type of Na-CO 3 HAps is determined by the occurrence of one or more of these fundamental mechanisms.Thus, these authors suggest that the stoichiometry would be given by: CO − 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 incorporation of Na + and 2 3 CO − .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.

Preparation of Na-CO 3 HAps
The preparation method is described in detail elsewhere [12] [13], In brief, The precipitated Na-CO 3 HAps were prepared by dripping calcium solution 0.03 M (Ca(NO 3 ) 2 ⋅4H 2 O) into a phosphate solution 0.008 M of (Na 2 HPO 4 ⋅12H 2 O) which contains also different concentrations of (Na 2 CO 3 ) such as the molar ratio R: Na CO 2.
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 intracrystalline water.

Infrared Spectroscopy and X-Ray Diffraction
Na-CO 3 HAps 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-CO 3 HAp 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-CO 3 HAps samples were carried out using an X-ray diffractometer MRD with a generator (40 kV and 40 mA).After indexation of the full pattern, cell parameters were refined using the program "WINCELL"

Chemical Analysis and Density Measurements
The dehydrated samples were subjected to a chemical analysis.The phosphorus content of precipitates is determined by colorimetry after complexation with vanado-molybdate [22].The sodium and calcium content are obtained by atomic absorption spectroscopy and the carbonate content was determined by coulometry method (release of CO 2 by dissolution in acid).The hydroxide content % OH was calculated on the basis of electro neutrality.Density of the solids was measured by a flotation method [13].

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 samples show the effect of increasing PO − (ν 3 P-OH) absorption band at 1032 cm −1 , The shoulder observed at 740 cm -1 is attributed to the ν L mode of OH -close neighbors to Na + ions [13].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 observed, reflecting a change in unit cell dimensions due to incorporation of 2 3 CO − 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 compounds allow determining the molar weight (Mw).The results are grouped together in Table 1.

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 CO 3 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: %X 100 where M (ρ⋅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.

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 CO − [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: Similarly, we have calculated the number of vacancies on Ca 2+ and OH − lattice sites respectively: The results of these calculations are summarized in Table 4.The errors in Table 4 were estimated by means of error propagation theory.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 2 3 CO − and alkali metal Na + concentration.So, it may be said in the present study, that the fundamental substitution mechanisms (I, III and IV) could account for the incorporation of and the generic formula has the following expression:

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 mechanism(s) is related with experimental conditions.
The mathematical model is expressed as: where ε i is the random variable drawn from N(0, σ 2 ), β 0 and β 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 σ 2 .Least square method [26] allows calculating the regression and correlation coefficients, the variance of the β parameters and to test the null hypothesis H 0 : β 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].CO − and Na + ions on the hexagonal apatite lattice dimensions, we have constructed a multiple linear regression on two X-variables where,  2) and Y is the estimate ratio of the hexagonal lattice dimensions (data Table 1).The results of the calculation values of the different correlations coefficients, the estimated values of β 0 , β 1 and β 2 , the variances and t-test of β 1 and β 2 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-CO 3 Aps "c/a" vary linearly with the carbonate and sodium content according to: 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 H 0 : β 1 = β 2 = 0 and affirm that at least one of the predictors is linearly associated to the response.
T-test of of the estimated regression coefficients β 1 and β 2 has shown that β 2 and β 1 are significant at 70% level.

CO − : 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, X 1 and X 2 are the Na + and The result of the F-test ( CO − in solid is equal to the decrease on molar 3 4 PO − .Thus, the molar ratio in the solid is not significantly influenced by the increase of Na + and 2 3 CO − 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-CO 3 HAps vary linearly with the carbonate content according to: 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 H 0 : β 1 = β 2 = 0 and affirm that at least one of the predictors is linearly associated to the response.T-test of the regression coefficients β 1 and β 2 has shown for β 1 , t = −0.45signifiant at 30% level.For β 2 , t = 1.67 signifiant at 80% level.

= 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 3 CO n and n Na incorporate 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: From the intercept of the following equation it can seen that, within experimental error, a carbonate-free apatite ( ) shows that the number of Ca 2+ ions in the unit cell decreases with increasing of Na + and 2 3 CO − numbers for about one Ca 2+ 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 H 0 : β 1 = β 2 = 0 and affirm that at least one of the predictors is linearly associated to the response.T-tests of the regression coefficients b 1 and b 2 show that the value of: β 1 is signifiant at 80% level and for β 2 is signifiant at 90% level.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 2 3

Multiple Linear Regression of
CO − concentrations in aqueous solution.Results of these calculations are summarized in Tables 10-12.From Table 10, the multiple linear regression analysis shows that the relationship between these quantities is given by equation: 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 H 0 : β 1 = β 2 = 0 and affirm that at least one of the predictors associated to the response.The individuals t-test of the regression show that for β 1 , t = 0.79 signifiant at 60% level and for β 2 , t = −0.70signifiant at 50% level.
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 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 H 0 : β 1 = β 2 = 0 and affirm that at least one of the predictors is linearly associated to the response.The individual t-tests on the regression coefficients β 1 and β 2 and, Hence, for β 1 , t = 10, for β 2 , t = −9.8t(β 1 ) and t(β 2 ) 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: 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 2 3 CO − in HAp.

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 X 1 = a, X 2 = c, X 3 = d.Least square allows calculating the regression and correlation coefficients.The sample regression (prediction equation) is: , where 0 1 2 3 , , , β β β β 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: 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: 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 H 0 : β 1 = β 2 = β 3 = 0 and affirm that at least one of the predictors is linearly associated to the response.
for testing the regression coefficients β 1 , β 2 and β 3 and discovering with variable(s) is related to estimate Y = n Ca , we conduct on the one hand individual t-tests on the β's, Hence, for β 1 , t = −5.28,for β 2 , t = −28.3and for β 3 , t = −1.2t(β 1 ) and t(β 2 ) are signifiants at P > 99.9% level.Thus, the molar n Ca of Na-CO 3 Aps solid is significantly depends on both a and c contributions of mechanisms I and III.The general formula can be written as follows: Also, structural study for two samples obtained under comparable conditions [28] has been investigated extensively by physicochemical analysis and by Rietveld method refinements.The results of unit cell content calculated from the occupations of the atomic sites and the data of chemical composition ( )  These results show that we cannot consider unique and well defined substitution mechanisms resulting in apatites, 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.

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 stoichiometry of the apatite and especially of the mechanisms by which 2 3 CO − and Na + are incorporated in the lat- tice must be known. 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 2 3

2 3 CO 3 CO 3 CO
− and Na + on the spectral properties of apatites.However, increasing 2 − and Na + contents caused increasing of the intensities of the 2 − absorption bands at 1420 -1460 cm -1 (ν 3 C-O) and 872 cm -1 (ν 2 C-O) and decreasing in the resolution of3 4

3 CO − substitutes for 3 4 PO
to 6 with experimental error as well as the XRD and IR spectra clearly demonstrate that the samples "Na-CO 3 HAps" of this study are pure B-type carbonated apatites, this indicates that 2 − on a 1:1 basis which agrees with the fundamental substitution mechanisms for B-type 2 3

2 3 CO
− and Na + in the HAp lattice.If a, c and d are the contributions of mechan- isms I, III and IV respectively, thus,

X 2 =
% Na + the % Weightions in the Hexagonal Apatite LatticeIn attempts to disentangle and to measure the effects of the insertion of 2 3

X 2 =
%Na + (data Table Concentration of Na + : X 1,i and the Concentration of2 3 2+ ions per unit cell.The expression in brackets ( )

1 =
Y = a, c or d the Contributions of Mechanisms I, III and IV on X

2 3 CO
− : X 2,i in aqueous solution, (a) Calculation of regression and correlation coefficients and variances, (b) Values of F-test: Analysis of Variance.

(
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).

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 Σ% of Na-CO 3 HAps prepared from solutions with

Table 3 .
Unit cell composition of Na-CO 3 AHps calculated on the basis of the chemical composition and by means of Equation (1).

Table 4 .
Unit cell compositions of Na-CO 3 Aaps calculated on the basis of the chemical composition and by means of following Equation (2).

Table 5 .
calculation for fitting a multiple linear regression analysis of the estimated Y = c/a the ratio of the lattice parameters of Na-CO 3 HApson carbonate and sodium contents X 1 = wt% CO 3 , X 2 = wt% Na.(a) Calculation of regression, correlation coefficients and variances, (b) F-test: Analysis of Variance.
of the different correlations coefficients, the estimated values of β 0 , β 1 and β 2 , their variances and the F-test or the test of the variance are summarized in Table6 and

Table 7 .
The multiple linear regression (Table6(a)) indicates that the molar CO 3 /P ratio of the Na-CO 3 HAps vary linearly with the concentrations of carbonate and sodium in solution according to:

Table 6
For β 2 , t = 8.26 confirm that the molar CO 3 /P ratio of Na-CO 3 HAps solid is significantly depends on both con- (b)) F = 635.5 and the individual t-test of the β's Table 6 b for β 1 , t = −7.82.

Table 6 .
Calculation for fitting a multiple linear regression of the estimated

Table 7 .
Calculation for fitting a multiple linear regression analysis of the estimated Y = n Na /n P molar ratio in Na-CO 3 HAps solid on X 1 = [Na + ] and

Table 8 .
Calculation for fitting a multiple linear regression of the estimated

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

Table 11 .
Calculation for fitting a multiple linear regression of the estimated

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

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

Table 13 .
Calculation for fitting a multiple linear regression of the estimated Y = n Ca =Y in Na-CO 3 HAps on the contribution of mechanism I a: X 1i , contribution of mechanism III c: X 2i contribution of mechanism IV d: X 3i .