Figures
Abstract
Numerous experimental fracture healing studies are performed on rats, in which different experimental, mechanical parameters are applied, thereby prohibiting direct comparison between each other. Numerical fracture healing simulation models are able to predict courses of fracture healing and offer support for pre-planning animal experiments and for post-hoc comparison between outcomes of different in vivo studies. The aims of this study are to adapt a pre-existing fracture healing simulation algorithm for sheep and humans to the rat, to corroborate it using the data of numerous different rat experiments, and to provide healing predictions for future rat experiments. First, material properties of different tissue types involved were adjusted by comparing experimentally measured callus stiffness to respective simulated values obtained in three finite element (FE) models. This yielded values for Young’s moduli of cortical bone, woven bone, cartilage, and connective tissue of 15,750 MPa, 1,000 MPa, 5 MPa, and 1 MPa, respectively. Next, thresholds in the underlying mechanoregulatory tissue differentiation rules were calibrated by modifying model parameters so that predicted fracture callus stiffness matched experimental data from a study that used rigid and flexible fixators. This resulted in strain thresholds at higher magnitudes than in models for sheep and humans. The resulting numerical model was then used to simulate numerous fracture healing scenarios from literature, showing a considerable mismatch in only 6 of 21 cases. Based on this corroborated model, a fit curve function was derived which predicts the increase of callus stiffness dependent on bodyweight, fixation stiffness, and fracture gap size. By mathematically predicting the time course of the healing process prior to the animal studies, the data presented in this work provides support for planning new fracture healing experiments in rats. Furthermore, it allows one to transfer and compare new in vivo findings to previously performed studies with differing mechanical parameters.
Citation: Wehner T, Steiner M, Ignatius A, Claes L (2014) Prediction of the Time Course of Callus Stiffness as a Function of Mechanical Parameters in Experimental Rat Fracture Healing Studies - A Numerical Study. PLoS ONE 9(12): e115695. https://doi.org/10.1371/journal.pone.0115695
Editor: Christof Markus Aegerter, University of Zurich, Switzerland
Received: August 13, 2014; Accepted: November 26, 2014; Published: December 22, 2014
Copyright: © 2014 Wehner et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper and its Supporting Information files.
Funding: This work was funded by the German Research Foundation (WE 4859/1-1), www.dfg.de. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
The rat is of particular importance for experimental studies in fracture healing. The outcome of the healing process is often evaluated via determination of the callus stiffness by ex vivo mechanical testing at certain healing time points [1], [2], [3], [4], [5]. Mechanical and biological factors influence the manner in which fracture healing progresses [6], [7], [8]. The most important mechanical factor is the interfragmentary movement (IFM) [6]; IFM determines the tissue strains at the fracture site in relation to the fracture gap size and regulates the mechanically induced tissue differentiation [9], [10]. The IFM depends on the fixation stability and the musculoskeletal loads which are primarily dependent on the bodyweight of the animals [11]. Therefore, experimental studies in rats are not directly comparable with each other when different experimental parameters (bodyweight, fixation stiffness, fracture gap size) are used.
Over the last decade, several numerical models for simulating the fracture healing process were developed [12], [13], [14], [15], [16], [17], [18]. Those models, which take the experimental parameters mentioned above into consideration, have the potential to simulate the healing process prior to an in vivo investigation helping to refine and reduce the number of animals in experimental studies as well as provide comparability between different studies by appropriate pre-experiment planning. Furthermore, simulations can be used to normalize already published in vivo results by simulating the healing process with the specific set of experimental parameters. Thereby, they help decide if there is an interesting biological factor influencing the healing outcome or if the differences of the healing outcome between studies can be explained solely by mechanical parameters that influence tissue strains (fixation stiffness, bodyweight, fracture geometry).
The aims of this study were (a) to adapt an existing numerical algorithm that reasonably simulates fracture healing in sheep [18], [19] and in humans [20], [21] to the conditions found in rats, which includes the determination of callus tissue properties and the calibration of the underlying mechanoregulatory tissue differentiation rules, (b) to simulate in vivo studies from literature and evaluate their agreement and (c) to predict the time course of the callus stiffness under several combinations of fixation stability, bodyweight, and fracture geometry. For future fracture healing studies using rats, this should promote the replacement of experimental in vivo studies, and allow focus on the influence of biological factors.
Methods
Determining the material properties of the callus tissues
To determine the material properties of the different callus tissues involved (i.e. connective tissue, cartilage, woven and cortical bone), three FE models of different healing stages were created in ANSYS 14.0 (ANSYS Inc., Canonsburg, PA, USA) to calculate the respective callus bending stiffness. All geometries represented transverse osteotomies at the rat femur in accordance with Fig. 1: (1) intact bone cylinder, (2) callus filled with woven bone according to previous histological findings under rigid fixation after 35 days of healing [2], (3) callus filled with woven bone except within the fracture gap, which was filled with cartilage (periosteal callus) and connective tissue (endosteal callus) according to previous histological findings under flexible fixation after 35 days of healing [2]. Conforming to Checa et al. [22], the cortical bones were modeled as hollow cylinders with an inner diameter of 2 mm and an outer diameter of 4 mm. The overall height (l) of the models was 24 mm. All structures were meshed with 10-node tetrahedral elements assuming linear-elastic, isotropic material properties. The models were fixed at the bottom (distal end of the distal cortical bone fragment) in all degrees of freedom. To simulate the bending stiffness of the models in a cantilever bending mode, a shear deflection of 1 mm was applied to the top of the models. Therefore, a tent-like structure made of rigid beam elements [21] was used at the top of the models to allow the application of the shear deflection Us and the calculation of the reaction force Fs at one single node. The bending stiffness, EIFE, was calculated based on the formula for cantilever bending:
Model 1: intact bone cylinder, model 2: fracture callus filled with woven bone and a callus index of 1.5, and model 3: fracture callus with a callus index of 2, where periosteal callus is filled with woven bone up to the fracture line.
(1)These three FE models were used to successively obtain the material properties by comparing the simulated bending stiffness EIFE, resulting from various parameter combinations to the respective ex vivo bending stiffness EIEX measured in rat experiments by Recknagel et al. [2] Thus, model (1) was used to obtain Young’s modulus for cortical bone Ecort, which was then assigned to model (2). This was then used to obtain Young’s modulus for woven bone Ebone, which subsequently was assigned to model (3), to obtain Young’s moduli for endosteal callus Eendost, and gap tissue Egap, as listed in the supporting information (cf. S1 Table).
Numerical simulation of the fracture healing process
To simulate the fracture healing process, the previously obtained material properties were assigned to another callus FE model with an ellipsoid shaped healing region with an outer diameter of 10 mm, which allowed a maximum callus index [23] (ratio between the outer callus diameter and the periosteal diameter of the cortices) of 2.5. The external fixator and the fixator pins were modeled with 2-node beam elements with six degrees of freedom at each node, allowing the simulation of various fixation stabilities by changing the free bending length (offset) of the pins. The model was fixed at the end near the fracture gap of the distal cortical bone fragment in all degrees of freedom. A three dimensional physiological load case (three forces and three moments at that time point during gait of the rats when the axial load component reached its maximum, which was 6 times bodyweight [11]) was applied to the proximal cortical bone fragment using a tent like construct at the end of the proximal cortical bone fragment at the fracture site [21]. The fracture healing process was simulated iteratively over the healing time [18], [19], [21]. At each simulation step, an FE-simulation was performed to calculate the local mechanical stimuli (volumetric strain εv and distortional strain εd) and subsequently predict the change in the local tissue distribution within the elements of the healing region. Last, Young's modulus (Etiss) and Poisson’s ratio (νtiss) of the FE were updated according to the new tissue composition, i.e. the concentration of cartilage (ccart) and bone (cbone), using the following rules of mixtures:(2)
(3)where Econn, νconn, Ecart, νcart, and Ebone, νbone, are Young’s moduli and Poisson’s ratios of connective tissue, cartilage, and woven bone, respectively.
In contrast to previous simulations of this research group [18], [19], [21], [24], the complexity of the rules for predictions of the tissue differentiation was reduced. Now, bone and cartilage formation were directly linked solely to the mechanical stimuli (εv & εd). Cartilage formation was allowed for volumetric strains in the range of εv,cart,low <εv <εv,cart,high in combination with distortional strains in the range of εd,cart,low <εd <εd,cart,high, whereas bone formation was allowed in the range of the volumetric strains of εv,bone,low <εv <εv,bone,high in combination with distortional strains in the range of εd,bone,low <εd <εd, bone,high. Outside these ranges, resorption of these tissues will occur (cf. Fig. 2).
A) Established for sheep, according to Claes and Heigele [9], B) for rat after calibration of the model in the present study.
To simulate bone and cartilage formation and resorption, the rate of change in both concentrations was set to 7% per day, since this allowed simulation of a bony callus after 35 days of healing with a size according to previous findings [2].
Calibration of the underlying mechanoregulatory rules
To adapt the fracture healing algorithm from sheep to rat, the thresholds within the underlying mechanoregulatory tissue differentiation rules (i.e. εv,cart,low, εv,cart,high, εd,cart,high, and εd,bone,high, cf. Fig. 2) were varied in 600 different combinations (designs) in a design of experiments (DOE) approach using Latin Hypercube Sampling in the software package optiSLang (Dynardo GmbH, Weimar, Germany). The other parameters defining the ranges of the mechanical stimuli were kept constant to the following values: εv,bone,low = εv,cart,high, εv,bone,high = −εv,cart,high, εd,cart,low = 0, εd,bone,low = 1000 µε [25]. The simulations were performed according to the experimental setup of a previously published in vivo monitoring study [26]: the fracture healing process was simulated for each design under rigid (Kax = 102 N/mm) and flexible (Kax = 30 N/mm) fixation and the callus stiffness was calculated at each week until 6 weeks (for rigid fixation), or 12 weeks (for flexible fixation). The mean values of differences between simulated and their respective in vivo measured callus stiffness were calculated and used to find the best design out of the 600.
Evaluation and corroboration of the numerical fracture healing model
To evaluate the validity of the calibrated fracture healing algorithm, it was used to simulate 21 in vivo scenarios from 9 different experimental rat fracture healing studies from literature (Table 1). Simulation outcomes (i.e. simulated callus stiffness) were compared to the ex vivo callus stiffness from the respective experiment. Requirements for the included in vivo studies were transverse osteotomies of rat femurs, stabilization with external fixators with well described experimental parameters (bodyweight of the animals, fixation stiffness and fracture gap size), and well reported ex vivo mechanical testing of the callus stiffness.
Prediction of callus stiffness subject to bodyweight, fixation stiffness and fracture gap size
For numerous combinations of different values for bodyweight (BW), fixation stability (Kax), and fracture gap size (Sfr), fracture healing was simulated with the calibrated and corroborated iterative algorithm described above. For each iteration step, the callus bending stiffness KC was calculated by simulating a cantilever bending test. For these time courses of KC, fit curves with a sigmoidal shape were created using the Curve Fitting Toolbox of Matlab (MathWorks, Ismaning, Germany) with the equation(4)
Using these fit curves, the time to heal (assumed as the time point when the callus stiffness reached 90% of the intact bone stiffness) was calculated for all of these cases and correlated to the initial IFM (defined as the axial internal force in the femoral mid-shaft, which was assumed to be 6 times the bodyweight [11], divided by the axial fixation stiffness) and to the interfragmentary strain (defined as the IFM divided by the fracture gap size). Furthermore, the deviations (minimum Δmin, median Δmedian, and maximum Δmax) of the resulting fit curve in comparison to the original simulated curves were calculated to allow evaluation of the fidelity of the fitting function.
Results
Determining the material properties of the callus tissues
Using a Young’s modulus of 15,750 MPa for the cortical bone of the intact bone model (Fig. 1) led to a bending stiffness of 182,358 Nmm2, which is in good accordance with the ex vivo bending stiffness of intact rat femurs (176,805±51,586 Nmm2), measured by Recknagel et al. [2]. Therefore, the material properties for the cortical shells were incorporated into models 2 and 3 (Fig. 1). Model 2, for the simulation of a callus filled with woven bone with a Young’s modulus of 1,000 MPa, led to a bending stiffness of approx. 100% of the intact bones, typical for a fully bony bridged fracture callus with moderate callus size (callus index = 1.5) [2], [27], [28]. Therefore, the woven bone in model 3 (Fig. 1) was assigned this value for Young’s modulus. Model 3 (Fig. 1) represented a fracture callus with a greater size (callus index = 2) and a fracture gap of 1 mm, filled with connective tissue and cartilage, typical for a healing situation after 35 days under a more flexible fixation [2]. Using a Young’s modulus of 5 MPa for cartilage led to a bending stiffness of 28% to 29% of the intact bones, which is in good agreement with the ex vivo results (25±11%) of Recknagel et al. [2]. The obtained material properties are summarized in Table 2.
Calibration of the underlying mechanoregulatory rules
The best agreement between in vivo data and simulations were obtained by shifting the threshold values of the underlying mechanoregulatory rules towards higher levels (εv,cart,low = −5.3%, εv,cart,high = −1.2%, εd,cart,high = 23%, and εd,bone,high = 29%, cf. Fig. 2B) in comparison to the thresholds that were used in previous simulations (εv,cart,low = −5%, εv,cart,high = −0.85%, εd,cart,high = 16%, and εd,bone,high = 7%, cf. Fig. 2A) for sheep and humans [18], [19], [21]. Applying these calibrated thresholds, the time course of the callus stiffness was predicted in good accordance with the in vivo data from the previous study [29] (Fig. 3). Regarding the tissue distribution after 35 days of healing, the simulations yield a bridged, bony callus of moderate size under rigid fixation (callus index approximately 1.5), and a larger bony callus (callus index approximately 2.0) under flexible fixation, with a remaining fracture gap that contains connective tissue and cartilage (Fig. 4). These predictions correspond well to respective, previous findings [2], [29].
Bars indicate in vivo data (statistical means and 95% confidence intervals) from the rat experiment [29], that was used to calibrate the numerical model, solid line represents the outcome of the numerical simulation after calibration.
Evaluation and corroboration of the numerical fracture healing model
The simulations of in vivo rat studies from literature showed good agreement for several combinations of fixation stiffness and fracture gap size qualitatively (successful healing or not) and quantitatively (direct comparison of the simulated and the ex vivo measured callus stiffness at a specific time point – Fig. 5). However, some large differences exist (Table 1). First, at large fracture gap sizes (3 mm & 5 mm), a successful healing process with high callus stiffness was predicted, whereas there was no measureable callus stiffness observed in vivo [30], [31]. Second, a successful healing was simulated for old female rats after 42 days with a predicted callus stiffness of 107%, whereas the ex vivo measured callus stiffness was still low at 25±9% [3]. Furthermore, the simulated stiffness of fracture calluses, which are completely filled with woven bone, were in some cases lower [3], [5], [30] (differences up to 76% of intact bone stiffness) or higher [1] (differences up to 47% of intact bone stiffness) than the measured in vivo results (Table 1).
The points and error bars indicate mean and standard deviations or median and interquartile ranges. Numbers of the single in vivo data points refer to Table 1. The solid line represents perfect agreement and the dashed lines indicate uncertainty ranges of the simulations due to variations in gap size, free bending length of the pins, and bodyweight of the rats (±14% of the intact bone stiffness, determined in a preceded sensitivity analysis as described in the discussion section). In some cases, numerical predictions clearly underestimated the callus stiffness after bony bridging (group A) and in some cases, successful healing with high callus stiffness was predicted although no healing was observed in vivo and low callus stiffness was measured ex vivo (group B).
Prediction of callus stiffness subject to bodyweight, fixation stiffness and fracture gap size
Using fit curves with a sigmoidal shape, the simulated course of callus stiffness over the healing time could be replicated well with a maximal error of <5% of the intact bone stiffness (cf. S2 Table, which provides fit curve parameters for numerous different combinations of bodyweight, fixation stiffness and fracture gap size). The time to heal (assumed to be the time point when the callus stiffness reached 90% of the intact bone stiffness) was prolonged for larger gap sizes, lower fixation stiffness and higher bodyweights of the rats (Fig. 6).
S2 Table).
Discussion
In this study, a simplified version of an already existing fracture healing algorithm was successfully adapted to the healing process in rats by calibrating the thresholds of the underlying mechanoregulatory tissue differentiation rules. To simulate the time course of the callus stiffness and the patterns of bone and cartilage formation according to in vivo findings in rats [2], [29], the threshold values of the mechanical stimuli (volumetric and distortional strains) had to be shifted towards higher load levels, which is in contrast to the simulations of Checa and colleagues [22]. In their study, the mechanical conditions to allow bone formation had to be shifted towards higher load levels for the sheep. However, the authors used a different numerical model for simulating the healing processes and used a different in vivo dataset for evaluating the simulations. Due to the differences of both numerical models (Checa et al. used a poroelastic description of the material properties, and a linear combination of the fluid flow and the distortional strain as mechanical stimulus and simulated several cell activities such as migration and proliferation), it is hard to directly compare their simulation results with the findings of this study. Therefore, it remains unclear if the differences in the necessary adaptation of the load levels from the sheep to the rat are due to differences between either the numerical models or the in vivo data used for evaluation. Further simulations with both models using the same in vivo data are required to clarify this issue.
In the present study, several published in vivo studies using rats were simulated to test the validity of the applied algorithm for other combinations of the experimental parameters bodyweight, fixation stiffness and fracture gap size (Table 1). With the exception of healing scenarios in which the healing process was seemingly more influenced by biological factors (large fracture gaps [30], [31] and age [3]), the achieved numerical predictions agreed particularly well with the respective in vivo findings (Table 1). Taking the variations of the ex vivo measured callus stiffness into account, which showed standard deviations of up to 47% [30] and interquartile ranges of 139% [32] of the intact bone stiffness, there were only 6 of 21 cases showing a considerable mismatch between simulations and in vivo data.
A preceding sensitivity study using the calibrated numerical model showed that varying the gap size by 0.1 mm, the offset of the fixator body by 0.5 mm, and the bodyweight by 25 g (all standard deviations) led to standard deviations of up to 14% for the rigid group and up to 20% for the flexible group and a maximum range of the callus stiffness of up to 57% (rigid) and up to 77% (flexible) of the intact bone stiffness (S1 Fig.). Aside from biological differences between the rats, which also have an influence on the healing process, these simulation results explain variations of the ex vivo measured callus stiffness due to variations of experimental parameters. For example, the range of the bodyweight of the rats used in the study of Mark et al. [5] was 350 to 450 g. Simulating the healing process with a bodyweight of 400 g led to a difference of 45% of the intact bone stiffness at the 6 weeks healing time point (50±18% [5] vs. 95%) whereas this difference was minimal (50±18% [5] vs. 52%), when the healing process was simulated with 450 g bodyweight (Table 1) indicating a good agreement of the achieved numerical predictions considering variations in the experimental parameters.
However, there were several necessary assumptions and simplifications that were made. Firstly, material behavior was simplified to linear elastic, isotropic behavior. In particular, this limits the model outcomes in the first phases of healing when tissues are still very flexible and show a distinct fluid phase. Secondly, the adjusted rules for tissue differentiation are simplified to a minimum, compared to previous numerical simulation studies of this group [18], [19], [33], [34] and completely focus on mechanoregulation without considering any biological influences. Thirdly, as previously mentioned, the predictions for healing processes accompanied by large fracture gaps (3 & 5 mm) did not fit well to the in vivo results (Table 1). Enlarging the fracture gap in the applied numerical model leads to a delayed healing since it takes longer to bridge the fracture gap with the same rate of bone formation (S2 Table). However, as long as this bone formation rate will not be set to zero and the IFM will be high enough to prevent a mechanical understimulation of the callus tissue, the model will predict successful healing for all gap sizes. This suggested that the causality of a critical size defect cannot be explained by mechanical stimuli solely and that biological factors must be considered. To include the simulation of non-unions in this model as well, the bone formation rate could be described as a function of the healing time. Thereby a decay of the biological factors promoting tissue development over the healing time could be incorporated allowing the simulation of non-unions as well. Since the exact, underlying biological mechanisms are, so far, not known to predict the tissue development in those cases, the present simulations were focused on the influence of the biomechanical factors on the healing process.
In conclusion, the present numerical model was considered valid for normal healing processes of transverse osteotomies in rats with fracture gap sizes in the range of 0.5 to 2 mm and bodyweights in the range of 250 to 550 g, stabilized by external fixators with an axial fixation stiffness in the range of 10 to 119 N/mm. The data presented in this study might help to improve new investigations of fracture healing in rats by predicting the time course of the healing process prior to the animal studies and allowing the transfer of new in vivo findings to previous studies with different bodyweight, fixation stiffness and fracture gap sizes.
Supporting Information
S1 Fig.
Course of callus stiffness (KC) over the healing time (tH) under rigid and flexible fixation.
https://doi.org/10.1371/journal.pone.0115695.s001
(PDF)
S1 Table.
Bending stiffness calculated with the three finite element models in relation to the Young’s moduli of the different tissue types.
https://doi.org/10.1371/journal.pone.0115695.s002
(PDF)
S2 Table.
Parameters of the sigmoid function (a, b, t0 - cf. equation 4), used to predict the time course of the callus stiffness (Kc) in percent (%) of the intact bones under different combinations of BW, Kax, and Sfr.
https://doi.org/10.1371/journal.pone.0115695.s003
(PDF)
Author Contributions
Conceived and designed the experiments: TW MS AI LC. Performed the experiments: TW MS. Analyzed the data: TW MS. Contributed reagents/materials/analysis tools: TW MS AI LC. Wrote the paper: TW MS LC.
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