1. Introduction
Hydrogen potential (pH) is a parameter of prime relevance in aqueous liquid phases, either with in vivo or in vitro situations. Therefore, species that are able to control their pH values deserve special attention in these systems. In the context of fluids that simulate those of body solutions, two chemicals are very important: carbon dioxide and buffers. In spite of this situation, few studies focus on the thermodynamic aspects of these chemicals in the precipitation of calcium phosphates.
Carbon dioxide partial pressure affects the pH of aqueous solutions and can promote an increase/decrease of carbonate/bicarbonate content in the liquid phase. Indeed, its partial pressure plays a key role in in vivo pH regulation.
The use of buffers is advisable in order to maintain the pH of aqueous systems within narrow ranges and, thus, several chemical reactions can take place under controlled physico-chemical conditions. However, besides protonation/deprotonation reactions of buffers, in some cases, several complexation reactions can occur, mainly with alkaline-earth ions. In these cases, the availability of specific ions is diminished, influencing all chemical equilibrium reactions, as well as the driving force to the precipitation of phosphate phases when these ions take part. In the specific case of calcium phosphates of biomedical interest, the quantity of free calcium ions in an aqueous system is extremely important because this parameter affects the stoichiometry of solid phases (such as hydroxyapatite). For example, using BISTRIS buffer, we noted a drastic depletion of calcium ion concentrations in simulated body fluids [1]. In the same context, Nakon and Krishnamoorthy [2] showed that, among 20 buffers known as “Good’s buffers”, three of them showed complexation reactions with metal ions, interfering particularly in protein analysis. In this paper, we analyze the depletion of calcium ions in the presence of TRIS buffer, comparing the results with HEPES buffer. Besides, the determination of the carbon dioxide effects on calcium phosphate precipitation was performed.
To evaluate the buffers and carbon dioxide in simulated body fluids, we choose a thermodynamic analysis, also known as in silico experiments, due to the scarcity of works that use a similar theoretical approach.
2. Models
2.1. Thermodynamic Models
In this subsection, we present the thermodynamic models employed in the in silico simulation of chemical equilibria of SBF fluids (essentially, multielectrolyte-diluted aqueous solutions).
Activity coefficients for charged species (
) were calculated using long-range interactions. In this sense, we used the extended Debye-Hückel model [1] as follows:
, (1)
where 
is the ionic radius, 
is the charge of chemical species i, I is the ionic force of medium and
, where 
is the dielectric constant of water at 298.15 K and T is the absolute temperature (Kelvin). Also in (1), we considered the product 
as suggested by Glinkina et al. [3] (BatesGuggenhein convention for chloride ions). This assumption is valid for ionic forces below 0.2 mol·L−1, as is the case for SBF solutions [4]. In this context, the ionic force is calculated by
, where 
is the molar concentration (expressed in mol·L−1) of ion i.
2.2. Chemical Equilibria and Material Balances
The numerical determination of concentrations—as well as activities and Gibbs free energy variations between supersaturated and saturated solutions—is conducted by a solution of a nonlinear algebraic system. This system contains chemical equilibrium relationships and material balances. We consider the following equilibrium reactions in aqueous solution.
a) Water formation/dissociation
(2)
b) Protonation/deprotonation of phosphates
(3)
(4)
(5)
c) Calcium ion equilibria
(6)
(7)
(8)
(9)
(10)
(11)
d) Carbonate/bicarbonate equilibrium
(12)
e) Protonation/deprotonation of sulphates
(13)
(14)
f) Magnesium ion equilibria
(15)
(16)
(17)
(18)
(19)
(20)
g) Protonation and complexation reactions of calcium and magnesium ions with buffers
(21)
(22)
(23)
h) Reactions of carbon dioxide in aqueous medium
(24)
(25)
(26)
In these equations, L represents the buffer (TRIS or HEPES, in the present work), 
is the calciumbuffer complex and 
is the magnesium-buffer complex.
Besides chemical equilibrium equations, some material balances must be included. These balances are responsible for keeping the chemical content constant. In the following expression, the subscript TOT means the total quantity of a given element.
i) Material balance for phosphorous
(27)
j) Material balance for inorganic carbon
(28a)
(28b)
k) Material balance for calcium
(29)
l) Material balance for magnesium
(30)
m) Material balance for sulfur
(31)
n) Material balance for buffer (TRIS or HEPES)
(32)
3. Degrees of Freedom Analysis
The system of nonlinear algebraic equations must be solved in order to access the concentrations of all chemical species (and, obviously, Gibbs free energies and supersaturations in the precipitation of calcium phosphates). The numerical resolution of equations depends on the necessary specification of the degrees of freedom in each approach. On the one hand, we consider a pH specification (in order to compare the effect of buffer type); on the other hand, a carbon dioxide partial pressure specification (to assess the resulting pH). Thus, the analysis of degrees of freedom for the nonlinear system is detailed for each case, here identified as Situation (a) (pH specification, buffer effect) and Situation (b) (carbon dioxide partial pressure specification, pH result). Situation (a) and (b) formulations were proposed by [1,5] and [6], respectively, for biomaterials applications; but, here, we present a unified approach for both situations and more detailed results/explanations concerning calcium phosphate precipitation.
3.1. Situation (a) (pH Specification, Buffer Effect)
The nonlinear system is formed by Equations (2)-(23), and by (27)-(32) (considering (28a)). Thus, we have 28 algebraic equations. The unknowns of this problem are:
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
and
. Since we have 29 undetermined values for concentrations, one specification is necessary. If the pH value is available, the nonlinear system can be solved, and all concentrations are subsequently calculated. Table 1 shows degrees of freedom analysis for both cases.
3.2. Situation (b) (Carbon Dioxide Partial Pressure Specification)
In this case, we must consider Equations (2)-(26) and (27)-(32) (using (28b)). Thus, this case considers 31 equations, with the same unknowns of case (a) plus 
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,