96-45d5-a115-0ca4903aff08.jpg width=54.9099992752075 height=32.1099992752075  /> and. Thus, with 31 unknowns and 31 equations, the degree of freedom is zero. However, the equilibrium relation of gaseous carbon dioxide and aqueous carbon dioxide (24) demands a value for partial pressure in the gaseous phase. This equilibrium can be calculated by the following expression [7]:

, (33)

where is the fugacity coefficient of carbon dioxide in gas phase and is the partial pressure of the same component in gas phase. Also in (33), is the activity of carbon dioxide in liquid phase. The fugacity coefficient is evaluated by (33) from the reference [7]:

(34)

Table 1 presents the analysis of degrees of freedom for Situations (a) and (b), summarizing this information.

The nonlinear system representing Situation (a) or (b) is solved by a damped Newton method (in order to enhance convergence properties) [8].

4. Results and Discussion

In this section, we present some results concerning the effects of using buffers [Situation (a)] and the effect of carbon dioxide [Situation (b)] in simulated body fluids.

Table 1. Degrees of freedom analysis: Situations (a) and (b).

4.1. Situation (a)

We present some simulated results for a solution with the composition detailed in Table 2, for the TRIS and HEPES buffers. One can note that the concentrations shown in Table 2 do not correspond, for example, to the actual concentration of bicarbonate ion in solution, since we consider the carbonate/bicarbonate equilibrium in liquid phase. We also consider the same concentrations for both buffers in our simulated results. Equilibrium constants for the chemical reactions were presented in references [1,2] and [9-13]. There is an enormous quantity of works regarding to the measurements of formation constants for complexes in aqueous solutions involving calcium, magnesium, phosphates and sulfates at different temperatures and ionic forces. Therefore, in silico experiment previsions depend on the reliable values of these formation/dissociation constants.

Table 3 contains the equilibrium constants for the chemical reactions used in this work: the protonation/ deprotonation of buffers, and the complexation reactions of the calcium and magnesium ions. As far as we can see, HEPES buffer does not form complexes with calcium or magnesium when isolated (without ATP) and, therefore, these reactions were not considered in our results. Although out of scope of the present context,  

Table 2.Total composition of solution [4,18].

Table 3. Equilibrium constants for protonation/deprotonation and complexation reactions (298.15 K) [9].

there is information about the complex Ca2+-HEPESATP and HEPES that act as an interferent in the amperometric determination of ATP (adenosinetriphosphate) in clinical applications [14]. Besides, HEPES buffer can form complexes with copper(II) ions in aqueous solutions [15-17]. Complexation constants for TRIS buffer are available in [9].

Figures 1 and 2 show, respectively, the concentration profiles for Ca2+ (free) and phosphate ions, as a function of pH for the TRIS and HEPES buffers. Clearly, we can observe that the concentration of free calcium ions is higher for HEPES buffer, because TRIS forms complexes with the calcium reducing its content in solution. At low pH, we noted an increase of free calcium ions. This situation seems to be favourable to hydroxyapatite formation, but we must consider that, at lower pH, the hydroxyapatite is not stable. Other interesting concentration profiles are related to phosphates in solution. It can be noted that phosphate ion concentration is very low in comparison with total phosphate (1.0 mmol·L−1, according to Table 2), as indicated in Figure 2. On the other hand, almost all phosphate appears as a complex with calcium ions (), as shown in Figure 3. Just for a comparison with calcium ions, Figure 4 shows the concentration of as a function of pH. The concentrations of are not significant when compared tothe total magnesium concentration (1.5 mmol·L−1), but the concentrations of are relevant and can affect the driving force for hydroxyapatite precipitation. The high stability of this calcium phosphate complex in aqueous solutions was described by [10].

All the information presented and discussed previously can be condensed in other thermodynamic quantities, such as the Gibbs free energy variations for precipitation of calcium phosphates or supersaturations [4]. For instance, the Gibbs free energy variation between a supersaturated solution and the saturation condition (a driving force for the hydroxyapatite precipitation) is represented by [1]:

, (38)

where IAP is the product of ionic activities (calculated from the solution of the nonlinear algebraic system that represents chemical equilibria and material balances, for situations (a) and (b)). The quantity is the solubility constant product for hydroxyapatite at 298.15 K. The quantity R is the universal gas constant and T is the system temperature (Kelvin). A similar equation can be obtained by octacalcium phosphate.

Figures 5 and 6 present the supersaturations of hydroxyapatite (HAP) and octacalcium phosphate (OCP) precipitations, respectively. An analysis of Figure 5 indicates that buffering with HEPES produces higher values for supersaturations, thereby, enhancing the driving

Figure 1. Concentration profiles for Ca2+ as a function of pH for the TRIS and HEPES buffers.

Figure 2. Concentration profiles for phosphate as a function of pH, for the TRIS and HEPES buffers.

Figure 3. Concentration profiles for as a function of pH, for the TRIS and HEPES buffers.

Figure 4. Concentration profiles for as a function of pH, for the TRIS and HEPES buffers.

Figure 5. Supersaturation in hydroxyapatite precipitation, as a function of pH, for the TRIS and HEPES buffers.

Figure 6. Supersaturation in octacalcium phosphate precipitation, as a function of pH, for the TRIS and HEPES buffers.

force for calcium phosphates precipitations. As expected—and also verified by in vitro and in vivo conditions—high pHs promote an increase in hydroxyapatite stability (verified by the increase in supersaturation); on the other hand, the octacalcium phosphate supersaturation decreases in the same scenario [19].

4.2. Situation (b)

For this problem, present in in vitro as well as in vivo situations, we used the same solution described in Table 1, but only with TRIS buffer. Clearly, in this aqueous system, a diminishing of the pH as a consequence of an increase of carbon dioxide partial pressure is observed. The partial pressures used in this work are compatible with the in vivo values. The equilibrium constant for Equation (24) was obtained from reference [7].

The calcium free concentration is important in the driving force for the phosphate precipitations (for hydroxyapatite, as previously discussed) and is presented in Figure 7. As expected, the calculations indicate that an increase of carbon dioxide partial pressure promotes an increase of free calcium concentration. Conversely, low pH values are not favourable for hydroxyapatite formation. Thus, there is a “trade-off” between these two quantities (pH and free calcium concentration).

Figures 8 and 9 show, respectively, the concentrations of carbonate and bicarbonate ions as functions of the partial pressure of carbon dioxide. As expected, an increase of corresponds to a lower pH and, thus, carbonate concentrations tend to zero. Moreover, the concentration of bicarbonate ion presents a maximum value close to the bar.

Figure 7. Free calcium concentration as a function of carbon dioxide partial pressure (bar).

Figure 8. Carbonate concentration as a function of carbon dioxide partial pressure (bar).

Figure 9. Bicarbonate concentration as a function of carbon dioxide partial pressure (bar).

Figure 10. Gibbs free energies of hydroxyapatite precipitation, as a function of carbon dioxide partial pressure(bar).

Finally, Figure 10 presents the Gibbs free energy variations for hydroxyapatite precipitation, as functions of carbon dioxide partial pressures. As previously discussed, we noted a “trade-off” between the two most important quantities for hydroxyapatite formation: free calcium concentration and pH. However, the increase in free calcium availability—as indicated in Figure 7—is not capable of compensating for a lower pH due to the high carbon dioxide partial pressures. Indeed, we observed an increase in Gibbs free energies as consequences of high. This situation represents a less favourable condition for hydroxyapatite formation in the considered system. For higher pH (lower carbon dioxide partial pressures), we noted a significant increase in the thermodynamic driving force for hydroxyapatite precipitation. These results are fully compatible with experimental ones (see, for instance, [19]), which indicate a pH stability range for hydroxyapatite close to 9.5 - 12.0. This finding is only possible to achieve by numerical simulations due to the coupling of dozens of non-linear equations and chemical species present in simulated body fluids. Therefore, due to the typical (and high) non-linearity of equilibrium equations, the use of SBF with high concentration of chemicals generally does not give the expected proportionality results.

5. Conclusion

In this work, we investigated the influence of buffer types and carbon dioxide partial pressure (in the gaseous phase) in the chemical equilibrium of simulated body fluids. In addition, the driving force of hydroxyapatite formation, a calcium phosphate of great biomedical interest, was evaluated. We also presented a computational framework oriented to the solution of multielectrolyte chemical equilibria in this context, with different degrees of freedom and specifications. The main results indicated that: 1) HEPES buffer promotes a more favourable condition for hydroxyapatite/ octacalcium phosphate precipitation than TRIS buffer, since the first does not form complexes with calcium. These findings can be observed in supersaturation profiles for both buffers; 2) the considerations of a specified partial pressure of carbon dioxide (in gaseous phase) and other chemical equilibrium relations in aqueous phase have produced Gibbs free energy profiles of hydroxyapatite precipitation. At low partial pressures of carbon dioxide (high pH), hydroxyapatite is thermodynamically stable—compatible with experimental observations. An increase incarbon dioxide partial pressure reduces the driving force for hydroxyapatite precipitation. As an important remark, it is always advisable that equivalent in silico evaluation be performed before experimental works to know the effect of each chemical in the solution. Thus, in silico experiments are extremely useful tools for the design of simulated body fluids for different purposes.

6. Acknowledgements

G. M. Platt acknowledges the financial support of Fundação Carlos Chagas de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) and Coordenação de Aperfeiçoamento de Pessoal Superior (CAPES). I. N. Bastos, M. C. Andrade and G. D. Soares acknowledge the financial support of CNPq, CAPES and FAPERJ Brazilian agencies. G. M. Platt, I. N. Bastos and M. C. Andrade acknowledge UERJ for their ProCiencia Grants.

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