Stochastic description of the peak hip contact force during walking free and going upstairs
Introduction
Simulations of the mechanical response of human femurs by finite element (FE) methods have been addressed for more than four decades (Brekelmans et al., 1972, Huiskes and Chao, 1983, Keyak et al., 1990, Yosibash et al., 2007). Despite recent achievements in verifying and validating patient-specific FE models by in-vitro experiments (Schileo et al., 2007, Cristofolini et al., 2010, Trabelsi et al., 2011), these models are rarely used in clinical practice to this day. A missing link with respect to the clinical applicability is the quantification of model uncertainties. Uncertainty quantification is an essential part of model validation (Oberkampf et al., 2004) and will increase the credibility and explanatory power of patient-specific FE models. Reliable FE models require three components: (a) geometry of the femur, (b) material properties and their spatial distribution, and (c) loading conditions (muscle and joint forces). Uncertainties are associated with all three model components and remain a major unexplored topic. In particular when modeling physiological loading conditions, assumptions have to be made on the magnitude and direction of the hip contact force. The applied force is usually representing an average loading condition (Bergmann et al., 2001), which was obtained by repeated in-vivo measurements with telemetric implants in multiple subjects. These in-vivo measurements imply a variability of the hip contact force not only between but also within subjects, which in turn raises the question of how this variability influences the results of a patient-specific FE model. Prior to that, a careful and rigorous stochastic description of the hip contact force is required.
The number of studies proposing a stochastic loading model for the human femur is limited (Nicolella et al., 2001, Nicolella et al., 2006, Grasa et al., 2005, Pérez et al., 2006, Viceconti et al., 2006, Long et al., 2009, Dopico-González et al., 2010), especially after restricting the literature research to three-dimensional models. Details of the models found in the literature are summarized in Table 1. Interestingly, every study investigated a quasi-static loading scenario and referred to peak forces measured in other experiments (with Bergmann et al. (2001) being cited most frequently) for either walking free or going upstairs. Although most of the stochastic loading models are based on the same data, the different number, types and characteristics of random variables demonstrate no consensus on how variability in loading should be modeled.
We therefore statistically analyzed data from the public databases HIP98 (Bergmann, 2001) and OrthoLoad (Bergmann, 2008). Both databases contain in-vivo measurements of hip contact force magnitudes and directions for multiple patients performing various activities, among them walking free and going upstairs. We restrict our attention to these activities as they are considered to be the most common activities in daily life and as such are of importance for testing implants (Bergmann et al., 2010). Another reason is the larger number of public data compared to activities like falling or stumbling. These extreme activities are clinically relevant to femoral fractures, because the involved dynamics cause larger forces. Yet, real stumbling was observed only twice in Bergmann et al. (2004), whereas experimental attempts did not simulate the extreme activity in a realistic way. The scarcity of data for extreme activities limits a statistical analysis and motivates further research. Nevertheless, the same statistical method can be applied to all activities, if sufficient data is available.
A typical approach to characterize maximum values in other fields (e.g. maximum floods, earthquake strengths, insurance claims, etc.) is by fitting generalized extreme value (GEV) distributions (Fasen et al., 2014). In our case, this approach could be used to describe maximum force magnitudes from one patient. An analysis of pooled data from multiple patients, however, is not easily possible, as patients are statistically significantly different from each other. For this reason, we perform regression analysis with a linear mixed-effects model that considers part of the inter-patient variability as random effect.
Based on such a statistical analysis we propose a novel stochastic description of the peak hip contact force (magnitude and direction) during walking free and going upstairs.
Section snippets
Materials and methods
Databases used, retrieved files associated with walking free and going upstairs activities, and extraction procedures for determining the peak loads are presented in Sections 2.1 and 2.2. The statistical analysis is summarized in Section 2.3.
Results
Peak force magnitudes of every patient were found to be statistically significantly different from each other. When separated by database affiliation for walking free and going upstairs as in Fig. 2, some of the 95% confidence intervals of the sample means were non-overlapping. Additionally, some patients had for the same activity significantly different peak forces in each database (e.g. KWR when walking free, or KWR and PFL when going upstairs). This led us to consider the same patient in two
Discussion
A rigorous statistical analysis of the peak hip contact force using pooled experimental data from HIP98 and OrthoLoad (Bergmann, 2001, Bergmann, 2008) was presented. We proposed in (9) a simple stochastic description of the force magnitude considering both walking free and going upstairs activities, depending on BW. Additionally the variability within the force direction was investigated, which resulted in stochastic description of the sagittal plane angle Ax in (10) and frontal plane angle Ay
Conflict of interest statement
None of the authors have any conflict of interest to declare that could prejudice this work.
Acknowledgments
The authors thank the anonymous referees for their valuable and constructive comments leading to improvements in the content and presentation. We acknowledge the generous support of the Institute for Advanced Study of the Technische Universität München, funded by the German Excellence Initiative, and thank Dr. Stephan Haug, Technische Universität München, for helpful discussions on statistical analyses.
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