For a total joint arthroplasty (TJA), its revision burden has been defined as the ratio of the number revised because of peri-prosthetic joint infection (PJI) to the total number of primary cases performed in a given period within a specific population  . The 2015 unweighted average burdens for total hip and knee arthroplasties between 6 arthroplasty registers are 0.97% and 1.03%, respectively  . In other words, the incidence of PJI is low; however, its impact is devastating. In fact, it is universally agreed that PJI is the most challenging complication of TJAs  . This is because of two reasons. First, a “gold standard” diagnostic test is lacking  . Second, even when diagnosed, it is difficult to treat/manage to the point of being intractable  . However, for patients who present with severe and persistent pain secondary to PJI diagnosis, the consensus is that the best treatment modality is two-stage exchange revision arthroplasty, which is a very painful and costly procedure (for example, in the United States, in 2010, annual hospital cost incurred for treating PJI cases was estimated to be between ~$769 million and ~$802 million, with it projected to rise to ~$1 billion by 2020)  . Thus, methods that aim to prevent PJI are an integral part of TJA practice. Arguably, the most widely used method is to anchor the joint replacement components in a bed of antibiotic-loaded poly (methyl methacrylate) bone cement (ALBC)  ; for example, each year, in the United States, over the period 2010-2015, ~80% of total hip arthroplasties were cemented using an ALBC  .
One of the shortcomings of the current generation of approved ALBC brands is that, upon polymerization, a high amount of heat is produced, which, some have postulated, may result in thermal necrosis of peri-prosthetic tissues  . Consequently, methods to reduce the maximum temperature reached during cement polymerization (Tmax) have been the subject of many studies  -  . In one recent study, two experimental ALBC formulations were prepared, one in which a phase change material (microencapsulated paraffin (MEPAR)) was added to the cement powder and another in which a chain-stopping agent (1-dodecyl mercaptan (DDM)) was added to the cement liquid  . It was found that each of these additives was effective in that Tmax was reduced by between 14% and 50% (with MEPAR additive) and between 31% and 47% (with DDM additive) compared to the corresponding value for the control ALBC  . However, in the study, although the cumulative gentamicin amount eluted obtained at different points during the test were determined, these results were only used to calculate CGE, which was defined as the cumulative mass of gentamicin eluted at the end of the test period (28 days) normalized by the product of the volume of phosphate buffered saline (PBS) solution used in the test and the initial mass of the test specimen  .
The purpose of the present study was to determine more information about the kinetics of elution of gentamicin from the two aforementioned experimental ALBC formulations; specifically, 1) determine the diffusion coefficient (Dgent), 2) determine the best-fit equation (from a collection of equations) for modeling the kinetics, and 3) use the results obtained in items 1) and 2) to provide guidelines for the selection of Tmax-reducing agents. There are very few literature reports that present diffusion coefficient for elution of an antibiotic from an ALBC and there are no reports that utilize the combination of diffusion coefficient and best-fit equations for antibiotic release to provide guidance on materials to be added to the powder and/or liquid of an ALBC to influence cement properties. Both of these aspects are covered in this present work, underscoring its novelty.
2. Materials and Methods
The control cement used was a commercially-available gentamicin-loaded ALBC brand (Table 1). The additive to the cement powder was MEPAR (Microtek Laboratories, Moraine, OH, USA) and the additive to the cement liquid was DDM (98% purity; Acros Organics, ThermoFisher Scientific, Pittsburgh, PA, USA) (Table 2). Details of methods used to blend MEPAR with the cement powder and to dissolve DDM in the cement liquid are given in the previous report  .
2.2. Specimen Preparation and Elution Test
Details of the methods used to obtain a homogenous cement dough and prepare the test specimens and the protocol used in performing the gentamicin elution test are given in the previous report  . After each measuring time-point (t) (1, 2, 5, 7, 10, 14, 21, and 28 d), a chemical derivatization method and a configurable microplate reader were used to determine the cumulative mass of gentamicin
Table 1. Composition of control cement brand  .
Table 2. Compositions of the cements in the study groups  .
eluted (in µg) after which it was normalized by the product of the volume of the PBS solution used (3 mL) and the initial mass of the test specimen (321 ± 38 mg) leading to a final result (in µg∙mL−1∙mg−1). For each cement group, 3 replicate specimens were used for each value of t. Further details of this computation are given in the previous report  .
2.3. Calculation of Diffusion Coefficient
The applicable governing equation for the elution of gentamicin out of the test specimen (for release ratio ≤ 0.8) was taken to be that for diffusion of a drug out of a long, cylindrical specimen; thus, it is  :
where Mt is the cumulative amount of released gentamicin at a given time-point, M∞ is the equilibrium amount of released gentamicin (defined as the value of Mt at which the Mt-versus-t plot flattened), Dgent is the gentamicin diffusion coefficient, t is time, l is the length of the test specimen (0.062 m), and a is the mean radius of the specimen (0.00374 m). With the aid of a commercially-available software package (Wolfram Mathematica, version 11; Citrix StoreFront, Citrix Corp., Fort Lauderdale, FL, USA), the Mt/M¥-versus-t results and Equation (1) were used to compute Dgent.
2.4. Model Equations
For each of the study groups, four kinetics equations that have been widely used to gain insight into the mechanism(s) involved in the release of an antibiotic from ALBC specimens were fitted to the Mt/M¥-versus-t results. These equations, in order, are those given by Korsmeyer et al.  , Lindner and Lippold  , Frutos et al.  , and Hesaraki et al.  ; namely
The physical meanings of the coefficients (k1, n1, b, k2, n2, A, B, n3, C, k3, τ and d) in Equations (2)-(5) are given in Table 3, and their values were determined using a nonlinear least squares method contained in a commercially-available software package (OriginPro 8.6; OriginLab Corp, Northampton, MA, USA).
2.5. Statistical Analysis
The Dgent results are presented as mean ± population standard deviation and the mean values are reported for the coefficients in the model equations. For Dgent and each of the aforementioned coefficients, intergroup comparison was carried out using the Kruskal-Wallis test, with Bonferroni correction (SAS Version 11.5; SAS Institute, Inc., Cary, NC, USA). Significance was denoted when p < 0.05.
The gentamicin elution test results are presented in Figure 1  . The computed
Table 3. Physical meanings of coefficients in Equations (2)-(5).
Figure 1. Summary of the gentamicin elution test results  .
values for Dgent are given in Table 4 and the best-fit values of the coefficients in the model equations in Equations (2)-(5) are given in Table 5, from which it is seen that the fits were excellent (as an example, see Figure 2).
The difference between Dgent of an experimental cement formulation and that of the control cement was not significant (p = 0.06). The difference in Dgent among the experimental cement formulations was not significant (p = 0.109). Only one coefficient in one of the model equations displayed a consistent trend in going from the control cement to the experimental cement formulations; that is, coefficient n2 in Equation (3). It was found that n2 of an experimental cement formulation was greater than the corresponding value for the control cement but not significantly so (p = 0.534). The difference in n2 among the experimental cement formulations was not significant (p = 0.625).
Table 4. Summary of computed values of Dgent.
Table 5. Summary of best-fit mean values of parameters: fits between release kinetics models and experimental results.
*Adjusted R2 = coefficient of multiple determination, adjusted for the degrees of freedom of the equation. Control: no additive to powder or liquid; 5MEPAR: 5 wt./wt.% MEPAR in cement powder; 15MEPAR: 15 wt./wt.% MEPAR in cement powder; 25MEPAR: 25 wt./wt.% MEPAR in cement powder; 1DDM: 1 vol./vol.% DDM in cement liquid; 2DDM: 2 vol./vol.% DDM in cement liquid; 3DDM: 3 vol./vol.% DDM in cement liquid.
Figure 2. Sample fit between experimental results and Equation (3): specimen E2 (specimen #2 in the 5MEPAR group; for compositional details of the group, see Table 2).
Many TJAs are grouted in the contiguous bone in an ALBC bed as a way of reducing the likelihood of PJI, which is the most challenging complication of TRJs (primary cases), or of treating a patient from whom an infected TJA has been removed (2-stage revision cases)  .
There is a sizeable body of literature on research into methods of addressing one of the shortcomings of ALBCs, namely, high Tmax  -  . In one such study, the relative influence of addition of a phase change material to the cement powder or of a chain-stopping agent to the cement liquid was investigated  . As part of that study, Tmax and other cement properties were determined, among which was cumulative amount of gentamicin eluted over the test period of 28 days. No other index of gentamicin elution kinetics was determined. The purpose of the present study was to determine some of these indices, specifically, the coefficient for the diffusion of gentamicin (Dgent) and values of coefficients in four equations that have been postulated to be relevant to modeling the kinetics of elution of an antibiotic from an ALBC.
The literature on diffusion coefficient for elution of an antibiotic from ALBCs is very sparse, with the only reports being those by Shen et al.  , Salehi et al.  , and Shen et al.  . The values, computed by Shen et al. for gentamicin eluting from an approved brand (SmartSet GHV) and an experimental formulation (Simplex P (an approved plain brand) to which 3.4 wt./wt.% gentamicin was mixed with the cement powder) were 4.9 × 10−15 m2∙s−1 and 1.2 × 10−15 m2∙s−1, respectively   . These values are not on the same order as the present Dgent result for the control cement, which may be a consequence of a difference in the method of computation used. Salehi et al.  determined the diffusion coefficient of daptomycin (Ddapt) from an experimental cement formulation (1.36 g of daptomycin per 40 g of cement powder) to be (37.50 ± 2.10) × 10−12 m2∙s−1. While the present value of Dgent for the control cement ((26.5 ± 7.6) × 10−12) is comparable to Ddapt, a difference in antibiotic incorporation method (that is, method used to mix/blend the antibiotic into the control cement powder) in the two studies should be noted; namely, proprietary (in the present study) and via a manual mixing device (in the study by Salehi et al.  ). Various curing and cured properties of a commercially-available gentamicin-loaded bone cement (1.69 g of gentamicin per 40 g of cement powder; proprietary antibiotic incorporation method group (Cement A)) and two experimental formulations, each with the same powder as in Cement A but gentamicin incorporation method being mixing in an open bowl (Cement B) or mixing in a cement powder mixer (Cement C) have been reported  . It was found that while the difference in gentamicin elution rates (into PBS, at 37˚C) between Cement A and Cement B specimens was not significant, elution rate from Cement C specimens was significant higher than from either Cement A or Cement B specimens  . These results point to the likelihood that antibiotic incorporation method would exert a significant influence on Dgent.
Lack of significant difference in both Dgent and n2 between experimental cement formulations, on the one hand, and the control cement, on the other, suggests that there is no significant difference in gentamicin elution mechanism between these two groups   . In other words, addition of either MEPAR or DDM to the control cement powder and liquid, respectively, does not interfere with or alter the gentamicin elution mechanism of the control cement   . It is worth noting that for both control cement and experimental cement formulations, n2 < 0.5, consistent with the postulate of Fickian diffusion of gentamicin through pores in the matrix of the cement  . Thus, among the possible gentamicin elution mechanisms are diffusion of the gentamicin through the solid part of the cement matrix or through channels of cracks and voids in the cement matrix   . Furthermore, lack of significant difference in n2 between experimental cement formulations, on the one hand, and the control cement, on the other, indicates similarity in porosity among the specimens from these two groups.
Taken together, the trends in the Dgent and n2 provide a guide with regard to selection of additives to incorporate in an ALBC from the perspective of reducing Tmax without simultaneously affecting the kinetics of diffusion of the antibiotic. An appropriate additive is one that does not significantly affect the porosity of the cement specimen.
We recognize two limitations of our study. First, the gentamicin elution tests were conducted in a medium (1X PBS solution, at 37˚C) whose ionic composition is similar to that of synovial fluid (the medium to which a joint and a TJR is exposed in vivo) but differs from it in terms of other constituents, such as proteins and cells and, additionally, the test conditions do not account for key in vivo features, such as the response of the host  . Second, the study was only on one approved ALBC brand, and, as such, generality of the applicability of the results to other approved ALBC brands cannot be claimed.
Based on the results obtained on one gentamicin-loaded PMMA bone cement brand, we conclude that:
· Addition of either a phase change material to the cement powder or a chain-stopping chemical to the cement liquid does not have a significant influence on the coefficient of diffusion of the gentamicin.
· A consistent trend was found in one coefficient in one of the four gentamicin elution kinetics equations tested for fit to the experimental elution test results, with this trend suggesting that 1) gentamicin elution mechanism in the control cement is not different from that in an experimental cement formulation and 2) the experimental cement formulation and control cement specimens have similarity porosity.
· Taken together, the present results provide guidance in selecting an additive to an ALBC that will simultaneously significantly reduce Tmax but have an insignificant influence on antibiotic elution kinetics.
The authors thank Jiyang Chen (Department of Physics and Materials Science, The University of Memphis) and Liang Zhang (Department of Electrical Engineering and Computer Engineering, The University of Memphis) for their contribution to the computation of the diffusion coefficient. We also thank Bentham Science Publishers for permission to use Table 1, Table 2, and Figure 1, which were reproduced from the article by McKee et al. (2018).
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