In silico design of magnesium implants: Macroscopic modelling

In silico design of magnesium implants: macroscopic modelling J.A. Sanz-Herrera aE. Reina-Romo aA.R. Boccaccini b aSchool of Engineering, University of Seville. Camino de los descubrimientos s/n, 41092 Seville, Spain. Tel.: +34 954 486079. Fax: +34 954 487295. E-mail: jsanz,err[email protected] bInstitute of Biomaterials, Department of Materials Science and Engineering, University of Erlangen-Nuremberg Abstract Magnesium-based biomedical implants offer many advantages versus traditional ones although some challenges are still present. In this context, mathematical modeling and computational simulation may be a useful and complementary tool to evaluate in silico the performance of magnesium biomaterials under different conditions. In this paper, a phenomenologically-based model to simulate magnesium corrosion is developed. The model describes the physico-chemical interactions and evolution of species present in this phenomenon. A set of 7 species is considered in the model, which allows to simulate hydrogen release, pH evolution, corrosion products formation as well as degradation of magnesium. The model is developed under the continuum media theory and is implemented in a finite element framework. In the results section, the effect of model parameters on outcomes is firstly explored. Second, model results are qualitative validated versus two examples of application found in the literature. Two main conclusions are derived from this work: (i) the model captures well the experimental trends and allows to analyze the main vari- Preprint submitted to Elsevier December 19, 2017 This is an Accepted Manuscript of an article published by Elsevier in Journal of the Mechanical Behavior of Biomedical Materials, Vol. 79, on March 2018 Cop. 2017 Elsevier. En idUS: Licencia Creative Commons CC BY-NC-ND ables present in magnesium corrosion, (ii) even though further validation is needed the model may be a useful standard in the design of degradable metal implants. Key words: Magnesium implants, Biodegradable metals, Tissue Engineering, Mathematical modeling, Computational simulation. 1 Introduction 1.1 Magnesium corrosion problem Magnesium offers great advantages versus traditional implants, such as ceramics, polymers or metals, fundamentally due to: (i) it is biodegradable and hence a second removal surgery is not necessary, (ii) it shows similar mechanical properties than bone tissue avoiding stress shielding and osseointegration related problems, (iii) it can be used as a load bearing implant or tissue engineering scaffold application (4; 5; 18). Despite these features are known since the early use as a biomedical material in 1878 (34), there has been limited clinical application of magnesium as an implant solution since then to 2000s, probably due to limited knowledge of corrosion phenomenon and uncertain behavior under circumstances of uncontrolled degradation (29; 30). In the last 15 years there has been an exponential trend in the research of magnesium-based biomaterials which has provided better knowledge about the related physicochemical characterization of magnesium degradation and has shed light about the response of magnesium at different circumstances and applications (19). In parallel, many biomedical applications have been conceived and tested using magnesium biomaterials, such as cardiovascular stents (12; 25), bony implants (32; 37) or tissue engineering scaffolds (13; 30; 35), to cite a few. 2 Magnesium biomaterial behavior and time evolution is an extremely complex problem. Briefly, once the implant is immersed within an aqueous ionic solution, e.g. simulated body fluid (SBF) or in vivo environment, several ionic reactions are activated. Released ions interact with those present in the aqueous medium yielding to layer precipitation of corrosion products in the surface of the biomaterial and consumption of magnesium, i.e. degradation (33). It is clear that magnesium corrosion does not affect to implant mass loss by itself but also to the performance of mechanical properties such as stiffness, strength, toughness or fatigue resistance which are fundamental quantities for orthopedic applications (6; 15; 16; 17; 33). Magnesium related problems are associated to uncontrolled degradation which derives in hydrogen release or pH amplification. On the one hand, hydrogen release gives place to the formation of a bubble which induces irregularities in implant-tissue interaction as well as the risk of explosion (21; 23). Indirectly, hydrogen release also induces a brittle mechanical behavior and decreases the fatigue resistance of the implant (6; 15; 16). On the other hand, pH alteration is an extremely dangerous issue in the human body although this risk is mitigated due to the buffer capacity in vivo (19). In order to circumvent magnesium corrosion associated problems referred above, several strategies are adopted, namely, alloying and coating. Alloying and coating allow to tune degradation rate and corrosion performance of magnesium-based implants, and hence to get controlled release of hydrogen gas and hydroxil ions (19). Most popular magnesium alloys are based on Mg-Al, Mg-Zn, Mg-Ca and Mg-Rare earth elements (19). Some studies also reported on an enhanced osseointegration and osteoblast proliferation in the presence of certain alloys (7; 36). On contrary, magnesium alloys may present secondary 3 effects such as cytotoxicity or difficulty to evacuate alloy products from the human body (19; 33). On the other hand, surface modification and coating are usually performed by means of micro-arc oxidation, plasma spraying, hydrothermal method, electrochemical deposition, sol-gel deposition, chemical conversion coating deposition techniques (see (19) for a review). Localized break of the coated protective layer may induce an accelerated and localized corrosion in the biomaterial (pitting corrosion) (24). 1.2 Simulation of magnesium corrosion There exists few models at the macroscale available to simulate degradation of magnesium biomaterials. Physically-based continuum models are presented in Grogan et al. (9; 10) with application to corrosion of metal stents. Recently, Bajger et al. (2) presented a similar model using a level-set strategy. However, these models consider a limited number of species and neglect the dynamics and evolution of secondary (although important) elements such as pH or corrosion products. A more detailed modeling of corrosion phenomena is necessary for the following reasons: (i) to properly describe the underlying physico-chemical features of corrosion phenomena and (ii) to get additional information and variables of analysis which may be compared by experimental outcomes. 1.3 Objectives and organization of the paper The objective of the paper is the derivation of a complete and general mathematical model and its computational implementation, to simulate the degrada- 4 tion of magnesium biomaterials taking into consideration the physico-chemical interactions and evolution of the main involved species. The paper is organized as follows: Firstly, the mathematical framework and model is introduced in the materials and methods section. Secondly, the results section shows an analysis of model parameters as well as qualitative validation of the model based on two examples found in the literature. Finally, a discussion of the results obtained from the model is established at the end of the paper. 2 Material and Methods 2.1 Species and evolution Magnesium degradation is the result of the following chemical processes: (i) dissolution of magnesium in contact with an aqueous medium by means of ionic reactions, (ii) corrosion products formation in a surface layer and (iii) eventual layer dissolution in the presence of certain substances in the medium such as chloride. This process can be described by the following set of reaction equations (33): Mg(s)+2·H2O(aq)Mg2+(aq)+H2(g)+2·OH−(aq) Ionic reactions (1) 5 Mg2+(aq)+2·OH−(aq)MgO2H2(s) Corrosion products precipitation (2) MgO2H2(s) + Cl−(aq)MgCl2(s)+2·OH−(aq) Layer dissolution (3) where aq,sand grefers to aqueous, solid and gas states of the species, see additionally Fig. 1. Eqs. (1) and (2) can be summarized in the following reaction: Mg(s)+2·H2O(aq)MgO2H2(s) + H2(g) Magnesium degradation (4) Reaction equations (3) and (4) are followed in the subsequent derivation of the theoretical framework. Therefore, 7 species are identified according to Fig. 1, namely: •[H2O] [mol/m3] aqueous species which represents water content present in the aqueous solution of the medium. It can diffuse into the porous materials of the corrosion products layer and magnesium biomaterial. •[Cl−] [mol/m3] aqueous species which represents chloride ions content present in the aqueous solution of the medium. It can diffuse into the porous materials of the corrosion products layer and magnesium biomaterial. •[H2] [mol/m3] gaseous species which represents hydrogen gas content produced as consequence of reaction (4). It can be encountered dissolved in the aqueous solution of the medium. It can diffuse into the porous materials of 6 the corrosion products layer and magnesium biomaterial. •[OH−] [mol/m3] aqueous species which represents hydroxide ions content produced as consequence of reaction (3). It can be encountered dissolved in the aqueous solution of the medium. It can diffuse into the porous materials of the corrosion products layer and magnesium biomaterial. •[Mg] [mol/m3] solidus species which represents magnesium biomaterial. •[MgO2H2] [mol/m3] solidus species which represents overall magnesium hydroxide produced as consequence of the balance between precipitation in reaction (4) and dissolution in reaction (3). •[MgCl2] [mol/m3] solidus species which represents magnesium chloride produced as consequence of precipitation in reaction (3). The evolution of the kinetics of the involved species proceeds as follows. Both magnesium and magnesium hydroxide dissolution are considered to follow a first order kinetics as in other models (2; 27), such that: ˙ [Mg] = −k(1) d·[Mg] (5) ˙ [MgO2H2]diss =−k(2) d·[MgO2H2] (6) where k(1) dand k(2) dare the kinetic constants involved in eqs. (5) and (6), respectively (remark on the minus sign which denotes consumption). Then, having into consideration the stoichometry of reactions (3) and (4) yields, ˙ [H2O]=2·˙ [Mg] (7) 7 ˙ [MgO2H2]prec =−˙ [Mg] (8) ˙ [H2] = −˙ [Mg] (9) ˙ [Cl−] = ˙ [MgO2H2]diss (10) ˙ [MgCl2] = ˙ [MgO2H2]diss (11) ˙ [OH−] = ˙ [MgO2H2]diss (12) Finally, [MgO2H2]=[MgO2H2]diss + [MgO2H2]prec. 2.2 Model The model is established in the biomaterial domain Ω(x, t), where xis the vector position of a material point of the domain and trepresents time. This domain includes both magnesium implant bulk volume and surface layer. Aqueous species, represented by its density ρaq(x, t), can diffuse within the biomaterial. Therefore, in a control volume of the domain the net balance is stated as follows, ˙ρaq(x, t) = ˙ρd aq(x, t) + ˙ρr aq(x, t) (13) 8 Figure 1. Sketch of the phenomenon of magnesium implants degradation. Species involved at different domains and description of physical processes. Species in red denote newly formed products during the process. where ˙ρd aq and ˙ρr aq are the diffusive and reactive rates, respectively. On the one hand, ˙ρd aq is assumed as a fickean diffusion, ˙ρd aq(x, t) = −∇ · (−D∇ρaq) (14) On the other hand, ˙ρaq(x, t) = ˙ρH2O+ ˙ρH2+ ˙ρOH−+ ˙ρCl−(15) 9 Figure 3. Magnesium mass loss, corrosion rate, hydrogen release and pH evolution quantities: Parametric analysis. (a) D= 1 ·10−4(mm2/day), (b) D= 2 ·10−4 (mm2/day) and (c) D= 4 ·10−4(mm2/day). 16 Figure 4. Mass loss chart (data in %) at the edge region of the magnesium specimen (see Fig. 2) at the end of the analysis (25 days). Parametric matrix of analyzed values for Dand kd. 3.2 Example of application 1 In this example of application, the in vitro corrosion test presented in Seitz et al. (28) is reproduced and simulated. Then, neodymium-containing magnesium alloy MgNd2 specimens having a length of 25 mm and diameter of 2.5 mm were immersed in 700 mL of SBF solution (see details of the composition of the solution in (28)). A 2D axisymmetric model was used. Mass loss and corrosion rate were recorded along 25 days of immersion (28). The results of the simulation of this experimental test is compared in Fig. 17 Figure 5. Corrosion products (ρMgO2H2+ρMgCl2) chart (data in g/L) at the edge region of the magnesium specimen (see Fig. 2) at the end of the analysis (25 days). Parametric matrix of analyzed values for Dand kd. 6 in terms of mass loss and corrosion rate. These quantities were computed from the definition in eqs. (20) and (21). The model parameters were fitted to D= 1.6·10−4(mm2/day) and kd= 8.4·10−2(1/day). 3.3 Example of application 2 In order to explore the availability of the presented model for in vivo applications, where higher corrosion rates are found, the animal test presented in Li et al. (20) is reproduced and simulated. In this test, magnesium screw alloys (Mg-Zn-Zr) were implanted in adult Japanese white rabbits (see Fig. 7). The 18 Figure 6. Mass loss (left) and corrosion rate (right) simulation of the experimental test presented in (28). evolution and degradation of the geometry of the screws was recorded along the implantation time (6 months) by means of micro-computed tomography (20). On the other hand, the geometry of the screw was approximated and modeled by using a 2D axisymmetric model as a first approach. Due to the lack of data and complexity to get information about the in vivo fluid environment, a chloride concentration of 5 g/L was considered in the model, as in the previous example. This parameter has no influence on the specific analyzed variables in this case, as discussed in the next section. The model parameters were fitted for this example to D= 1.6·10−4(mm2/day) and kd= 3.4·10−1(1/day). Results regarding mass loss and corrosion rate are presented in Fig. 7. As in the previous examples, these quantities were computed from their definition in eqs. (20) and (21). Finally, the evolution of the degradation of the screw magnesium geometry is presented in Fig. 8 both for simulation results and experimental comparison. 19 Figure 7. Left: In vivo magnesium screw implantation for degradation experiment (20). Right: Mass loss and corrosion rate simulation of the experimental test presented in (20). Figure 8. In vivo magnesium screw degradation map along time (6 months) (20) and comparison with computer simulation. 20 4 Discussion The analysis of the results presented in the previous section allows to discuss several interesting points. Firstly, the parametric analysis in Fig. 3 was established in the range of compact (low porosity) metals and slow corrosion rate. At this range, evolution of analyzed variables i.e. magnesium mass loss, corrosion rate, hydrogen release and pH fall within the order of magnitude and trend of experimental measurements for these quantities (1; 17; 20; 22) to cite a few. Although the model was validated with two specific applications, a proper fitting of model parameters may reproduce any of the reported experimental observations. In the range of analyzed values for the diffusion coefficient and kinetic constant, it is seen in Fig. 3 that a higher kinetic rate yields to a higher consumption of magnesium (and hence faster corrosion rate) as well as an increase in the release of ionic substances. However, in the range of analyzed values, diffusion coefficient has a minor effect on model outcomes. This conclusion is achieved just analyzing the impact of parameter Don curves shown in Fig. 3. Moreover, Figs. 4 and 5 show that the corrosion activity is produced in a close layer region near to the surface of the implant. The order of magnitude of this layer is 100 µm (see Figs. 4 and 5). Consequently, aqueous solution diffusion within the implant may be neglected for compact (low porosity) magnesium as a first approach. However, the presented model in this paper is general and allows to simulate porous metals (scaffolds) in different applications such as tissue engineering problems. On the other hand, the established model allows the analysis of pH evolution 21 as a model variable. For the analyzed cases, pH is dependent on hydroxide release (as a consequence of reaction equations) and aqueous solution composition (chloride and initial pH). Secondary reactions which involve the hydroxide ions were not considered in the model. Moreover, some magnesium coatings biomaterials include oxide release (11) which should be considered in the formulation in order to properly reproduce the pH evolution. The effect of the aqueous solution in the model is only related to pH evolution according to the main considered reaction equations in (1), (2) and (3). The analysis of the results in the different examples of application allows to conclude that the model captures well the observed experimental trends of magnesium degradation, both in vitro (section 3.2) and in vivo (section 3.3). The effect of in vitro versus in vivo is considered phenomenologically in the model through the kinetic constant kd. Therefore, different effects such as fluid circulation or fluid environment composition are considered in this parameter under the meaning of a catalytic effect in the chemical reactions. Consequently, the effect of different solutions on magnesium corrosion may be indirectly incorporated in the model through kd. The fitted values and results of in vitro and in vivo scenarios give a ratio of 4 times faster for degradation in vivo. This value is within the order of magnitude of reported works (19; 20). It allows to conclude that the model is valid both for slow and fast degradation rate regimes. Future work of the preliminary development and analysis presented in this paper includes, firstly, a thorough experimental validation of the proposed model. Secondly, the model allow to couple the corrosion modeling to the mechanical behavior of magnesium-based implants. 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