Technical note on the determination of degradation rates of biodegradable magnesium implants

Magnesium‐based alloys are emerging as a capable alternative to traditional materials for bone implant applications. The implant's degradation rate is the main indicator of their performance; however, different formulas have been reported to determine it based on three‐dimensional imaging. In this technical note, we are presenting the deviation in the degradation rate determined by different equations for two sets of data and the implications for the comparison of different studies.


| INTRODUCTION
Biodegradable magnesium (Mg)-based implants have received increasing research interest in the past 15-20 years, as researchers look for alternatives to traditional implant materials for the temporary support of bone healing. As part of the development of new alloys, the degradation rate of the implants is a key measure of quality and suitability, with too high degradation rates leading to alkalization of the local environment and potential cell death. Multiple methods have been proposed for the determination of the degradation rate, [1,2] the most prominent of which being hydrogen gas evolution and mass loss measurements.
Based on mass loss measurements, the degradation rate DR i given in mm/year at time t i is determined as [3] m m k ρA t DR = ( − ) .
With ρ the material density, A 0 the initial surface, m 0 the initial mass of the sample, and m i the mass following degradation over time t i and removal of the degradation products using chromic acid treatment. k is a constant used to convert the unit of the degradation rate into mm/year and will be omitted in the following.
Volume loss is the key measure to determine degradation rates of Mg alloys based on three-dimensional (3D) in vivo (micro) computed tomography (µCT) and can similarly be employed for other tomographic imaging techniques. [4] However, different approaches exist on how to exactly determine the degradation rate. Specifically, as µCT imaging enables the longitudinal assessment of the degradation in vivo or in situ, the degradation rate can be determined both with respect to the initial time point or the preceding time point, that is and researchers may use the mean A ̅ i of the changed surface area A i at time t i and the initial surface area A 0 [5] or refer only to the initial surface area A 0 [3] for the calculation of the degradation rate: While each formula has merit, these differences in calculating the degradation rate call into question whether a comparison of degradation rates from different studies is prudent without differentiating between the formula used for the calculation.
In this work, we are presenting a comparison of the degradation rates calculated using Equations (2)-(5) for the in situ degradation data from µCT images shown in Zeller-Plumhoff et al. [6] Additionally, the difference between Equations (2) and (4) is assessed using highresolution ex vivo µCT data published by Krüger et al. [7] 2 | METHODS, RESULTS, AND DISCUSSION To calculate the degradation rate according to Equations (2)-(5) for the data of a Mg-2wt.%Ag pin degraded in alpha Modified Eagle's Medium during in situ µCT imaging published in Zeller-Plumhoff et al.; [6] the volume V t ( ) and surface A t ( ) of the pin at time t were determined based on the mean radius r t ( ) of the pin such that V t πr t h ( ) = ( ) 2 and A t πr t h ( ) = 2 ( ) . For the µCT data published in Krüger et al., [7] three datasets F I G U R E 1 (a) 3D rendering of degraded Mg-2Ag pin after 5 days reprinted with permission from Zeller-Plumhoff et al. [6] (b) Comparison of degradation rate based on Equations (2)-(5) for the pin. (c) Relative difference in degradation for different equations for the pin data. (d) 3D rendering of degraded Mg-5Gd screw overlayer with original after 12 weeks of healing as published in Krüger et al. [7] (e) Comparison of degradation rate based on Equations (2) and (4)  of Mg-5wt.%Gd screws implanted in rat tibia per healing time point of 4, 8, and 12 weeks were selected randomly. V t ( ) was determined based on the number of voxels of the residual screw and A t ( ) was determined using a custom Matlab R2021b (The Mathworks Inc.) script counting the number of voxel faces in contact between the residual metal and the surrounding. For the latter data set only Equations (2) and (4) were compared, as the data stems from different samples, and the implementation of Equations (3) or (5) would need to be adjusted further. The values for V t ( ) and A t ( ) for all time points t and for both data sets are presented in the supplementary information. Figure 1a,d shows 3D renderings of the degraded specimen of the respective studies under investigation. Figure 1b shows the comparison of degradation rates for the data in Zeller-Plumhoff et al. [6] and Figure 1e that for Krüger et al. [7] Note that the values of the degradation rate differ from those published in Zeller-Plumhoff et al., [6] due to a previously incorrect implementation of the formula. It is visually apparent that the use of Equations (2) and (4), or (3) and (5), yields similar kinetics, respectively. Negative degradation rates at the early time points during in situ degradation shown in Figure 1a are due to the uncertainty in the correct segmentation of the residual metal when the degradation layer thickness measures only 1-2 pixels. The higher degradation rates observed for the in situ study compared to the literature can be attributed to the flow of the medium and potential contamination of the medium, which generally leads to accelerated corrosion. [8] It is apparent that in particular at late time points, the difference in Equations (2) and (4), or (3) and (5), for both data sets increases. Figure 1c,f show the relative differences in degradation rate determined based on Equations (2) and (4), and in the case of the in situ degradation also for Equations (3) and (5). From the graphs, it is clear that the degradation rates can vary strongly, with the maximum difference being as high as 16% for one sample after 12 weeks of degradation in vivo. In cases of more complex geometries, the use of different equations can lead to even higher differences in degradation rate. [9] Consequently, variations in degradation rates based on the use of a different formula may be larger than those determined using the same formula for different alloys, which could bias any such comparison between different studies. This is the case for the screw data from Krüger et al. [7] at 8 weeks of healing, where the difference in degradation rate between Equations (2) and (4) in the current work is higher (mean difference in degradation rate 0.03) than that between Mg-5Gd and Mg-10Gd in the published study (mean difference in degradation rate 0.02). Moreover, to minimize the error for a comparison of volume loss and mass loss experiments Equation (4) should be used.

| CONCLUSION
The differences in the degradation rate determined depending on Equations (2)- (5) shown in this technical note have important applications for the comparison of published degradation rates. This is valid in particular for the in vivo evaluation, where low image contrast in clinical CT images may obscure the accurate determination of A .
i [10] Therefore, an evaluation with respect to A 0 (Equation 4) in accordance with the ASTM standard [3] may generally be preferred. Additionally, it is pivotal to include the single values for A i and V i for all time points t i in the supplementary information of every publication, so that a comparison of degradation rates between different experiments and measurement methods can be performed.

DATA AVAILABILITY STATEMENT
Data that supports the findings of this study are available in the supplementary material of this article.