Abstract. Biomechanical input data of in silico models are subject to uncertainties due to subject variability, experimental protocol and computing technique.
Uncertainty Modeling and Propagation in Musculoskeletal Modeling Tien Tuan Dao and Marie Christine Ho Ba Tho
Abstract. Biomechanical input data of in silico models are subject to uncertainties due to subject variability, experimental protocol and computing technique. Traditional perturbation analysis showed important drawbacks such as unobvious definition of the true range of value for the sensitivity analysis. In this present study, we used a novel framework to model the uncertainties of thigh mass property as well as to quantify their impact on the thigh muscle force estimation. A simplified patient specific musculoskeletal model (3 segments, 2 joints and 8 hip flexor muscles) of a post-polio residual paralysis subject was developed. Knowledge-based fusion pbox was used to model the uncertainties of the thigh mass property. Then, a Monte Carlo simulation was performed to quantify their impact on the thigh muscle force estimation through forward dynamics simulation. The global range of value of the rectus femoris force is from 2327.59 ± 39.32 N to 3353.16 ± 383.8 N at the peak level. The global range of value of the gracilis force is from 143.53 ± 2.35 N to 159.27 ± 8 N at the peak level. Cumulative probability functions of these ranges were presented and discussed. Our study suggested that under input data uncertainties, the musculoskeletal simulation results needs to be determined within a global range of values. Consequently, the clinical use of such global range will make the decision making more reliable. Thus, our study could be used as a guideline for such a purpose. Keywords: Uncertainty Modeling, Uncertainty Propagation, Thigh Mass Property, Muscle Force Estimation, Patient Specific Musculoskeletal Model, OpenSIM, Monte Carlo Simulation. Tien Tuan Dao · Marie Christine Ho Ba Tho UTC CNRS UMR 7338, Biomechanics and Bioengineering (BMBI), University of Technology of Compiègne, BP 20529, 60205 Compiègne Cedex, France
© Springer International Publishing Switzerland 2015 V.-H. Nguyen et al. (eds.), Knowledge and Systems Engineering, Advances in Intelligent Systems and Computing 326, DOI: 10.1007/978-3-319-11680-8_45
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1 Introduction In silico medicine is one of the most challenging research topics in the field of biomechanics for the last decades [1]. Computer-aided numerical models like rigid or deformable musculoskeletal models have been used for better diagnosis of musculoskeletal disorders [2], [3], [4], [5], [6], [7]. However, the development of these models, especially patient specific models deals with engineering challenges such as input data uncertainties due to subject variability, measuring protocol or computing method [8], [9], [10]. In fact, the clinical use of any numerical model should take these input data uncertainties into consideration leading to have more reliable decision makings. Input data of musculoskeletal models deals with random and epistemic uncertainties. The random uncertainty relates to the repeatability, the reproducibility errors and the variability due to the fact that data obtained from different protocols/population (races, origins)/experimental techniques. The epistemic uncertainty concerns the accuracy level of the measuring protocols and computing methods [10]. In the musculoskeletal modeling literature, the sensitivity of input data has been performed using traditional approaches such as variation and perturbation analysis [11], [12], [13], [14], [15], [16], [17] or Monte Carlo analysis [18]. However, there are some drawbacks dealing with these approaches even if they are easy to be implemented and to be used. The first one is the uncertain determination of the true perturbation thresholds which have been selected in a subjective manner according to each study. The second problem is the difficult consideration of data uncertainties derived from multiple acquisition sources (e.g. experimental measurement, literature-based extraction) with their proper accuracy levels. Consequently, there is a lack of a generic mathematical framework to model both random and epistemic uncertainties of input data derived from multiple acquisition sources. Recently, we have initiated a novel framework to model the biomechanical data uncertainties using knowledge-based fusion probability box and their propagation using Monte Carlo simulation [10]. The first example of this framework was performed on the uncertainty modeling of muscle morphological properties (physical cross-sectional area (pCSA) and muscle tension coefficient). Moreover, the impact of these uncertainties on the computing of peak muscle force was quantified. Even if this example is simple, the obtained results showed the potential use of such useful framework for more complex musculoskeletal simulation. The aims of this present study was to apply this novel framework to model the uncertainties of the thigh mass as property well as to quantify their impact on the estimation of thigh muscle forces during swing-like motion of the right leg.
2 Materials and Methods A 3-steps methodology was developed to model the uncertainty of input data as well as to quantify its impact on the musculoskeletal simulation results (Fig. 1). The first step deals with the development of the musculoskeletal model derived from medical
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imaging. The second step includes the data collection, uncertainty representation and computing components. The third step consists of the forward dynamics simulation, Monte Carlo simulation, and the results analysis.
Fig. 1 Workflow used for the model development, uncertainty modeling and propagation
2.1 Development of a Musculoskeletal Model Derived from Medical Imaging A musculoskeletal model was developed for a post-polio residual paralysis (PPRP) subject (male, 26 years old, 1m70 height, and 66kg body mass). Computed tomography (CT) scanner images were acquired using a spiral-imaging scanner (GE Light Speed VCT 64) at the Polyclinique St Cme of Compiegne (France). The subject signed an informed consent agreement before participating into this present study. The CT scan protocol included 3mm thickness and a matrix of 512512 pixels leading to have 384 joint slices (Fig. 2A). Semi-automatic segmentation was applied using ScanIP software (Simpleware, UK) to extract the bone and soft tissues of the lower limbs (Fig. 2B).
Fig. 2 Draw CT images (A) and (B) segmented image
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A simplified 3-segment (pelvis, right femur and thigh envelop, right tibia and leg envelop) musculoskeletal model with 6 degrees of freedom was developed using the OpenSIM 3.1 API (Application Programming Interface) [19] (Fig. 3A). The model includes 2 joints (i.e. right hip and knee) modelled as ball-and-socket and hinge joints respectively. The model includes 8 right hip flexor muscles: rectus femoris, sartorius, psoas, tensor fasciae latae, gluteus minimus, gluteus medius, pectineus, gracilis. Each muscle was modelled as a Thelen2003Muscle model [20].
Fig. 3 Simplified musculoskeletal model: initial position (A) and final position of swing-like motion (B)
2.2 Uncertainty Modeling of the Thigh Mass Property The thigh mass property of the developed musculoskeletal model was collected from 3 acquisition sources. The first one relates to the in vivo geometrical computing using segmented images, mass density of the soft tissue and accumulated pixels principle. This procedure was published in our previous study on the same patient [5]. The second data source is the use of a regression table [21] to compute the thigh mass in proportion to the body mass. The third data source is the use of a regression equation and geometrical characteristics [22] to compute the thigh mass. Based on the collected data, the knowledge-based fusion p-box structure [10] was used to create an uncertainty representation of the thigh mass property. The knowledge-based fusion p-box is a fused structure from multiple parametric p-boxes enveloping of four normal distributions as follows: (D(µli , σli ), D(µli , σui ), D(µui , σli ), D(µui , σui ))
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where mean [µli , µui ] and standard deviation [σli , σui ] intervals (l means lower bound and u means upper bound) were computed from the used three data sources. The computing of the knowledge-based fusion p-box was performed using Matlab R2010.b (Mathworks, USA).
2.3 Uncertainty Propagation of the Thigh Mass Property through the Forward Dynamics To estimate the distribution of the muscle force estimation results, an interval-based Monte Carlo simulation with 20 samples was performed using the lower and upper probability non-decreasing functions of the knowledge-based fusion p-box of the thigh mass property [10]. At each iteration with a thigh mass value selected randomly, a forward simulation was performed using computed muscle excitation to estimate the different thigh muscle forces during a swing-like (85-degrees hip flexion and 55-degrees knee flexion) motion [20], [23] (Fig. 3B). Note that the forward dynamics and Monte Carlo simulations were realized using OpenSIM 3.1 API [19] and Visual C++ (Microsoft, USA) on a Dell computer (Precision T3500, 2.8Ghz, 3GB RAM). The results were analyzed using Matlab R2010.b (Mathworks, USA).
3 Computational Results Computed thigh mass from 3 different acquisition sources is from 4.462 to 7.419 kg. Note that the computing methods are very different from one to other source. The regression table was established from the measurements on 7 cadavers (69 ± 17 yo, 61.1 ± 10.9 kg) with standard measuring protocol and devices (dissection, balance). The thigh ratio is about 10.008 ± 1.197 % of the body mass. The regression equation (mthigh = 0.074 ∗ M + 0.138P25 − 4.641) (M is the whole body mass and P25 is the circumference of upper thigh (= 52 cm for this present patient)) was created from the measurements on 13 cadavers. The patient specific procedure used in vivo medical images, image processing and geometrical computing. The thigh ratio is about 7.145 ± 0.386 % of the body mass. Using the collected data, the knowledge-based fusion p-box of the right thigh mass property was established. Each range of value from each acquisition source allows a knowledge-based p-box to be established (Fig. 4). Then a conservative rule was applied to create a fused probability box (Fig. 5). The estimated forces of rectus femoris and gracilis muscles at the peak level corresponding to 85-degrees hip flexion and 55-degrees knee flexion considering the uncertainties of thigh mass into account are shown in Fig. 6 and Fig. 7. The muscle forces were expressed by cumulative probability functions established from the results of Monte Carlo simulations. The range of value of the rectus femoris force is from 2327.59 ± 39.32 N to 3353.16 ± 383.8 N at the peak level. The range of value of the gracilis force is from 143.53 ± 2.35 N to 159.27 ± 8 N at the peak level. The total run time of all Monte Carlo simulations (20 samples) is around 200 minutes.
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Fig. 4 Knowledge-based fusion p-box of the right thigh mass property: 3 separate sources
Fig. 5 Knowledge-based fusion p-box of the right thigh mass property: fused result
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Fig. 6 Cumulative probability of the estimated rectus femoris forces at the peak level corresponding to the 85-degrees hip flexion and 55-degrees knee flexion
Fig. 7 Cumulative probability of the estimated gracilis forces at the peak level corresponding to the 85-degrees hip flexion and 55-degrees knee flexion
4 Discussion The reliability of the input and output data of the musculoskeletal simulations plays an essential role in the promotion of musculoskeletal model in clinical routine practices [5]. Traditional approaches commonly used perturbation analysis to quantify the sensitivity of input data on the simulation results. In addition to the unobvious choice of the true range of value for the variation process, these approaches couldn’t provide enough output information to establish a probabilistic distribution of the output simulation results [24]. In this present study, the use of a more generic mathematical structure as knowledge-based fusion probability box [10] allows a global
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range of value to be taken into consideration in the uncertainty modeling of a parameter of interest (e.g. thigh mass in this present study). Then, its impact on the musculoskeletal simulation results could be quantified using a Monte Carlo method leading to establish a probabilistic distribution of the simulation results (e.g. thigh estimated muscle forces in this present study). Such useful information is of great interest to provide more reliable result interpretations for clinical decision making. In fact, the use of one-value input data could lead to unreliable musculoskeletal simulation results and their interpretation needs to be performed with caution, especially in the case of clinical applications [25]. There are many engineering approaches for accounting data uncertainty into numerical models such as perturbation-based sensitivity, interval analysis, statistical inference method as bootstrapping, worst-case analysis, fuzzy arithmetic, possibility theory, belief theory, and probability boxes. Among these approaches, probability boxes (p-boxes) are flexible mathematical structures for the modeling of both random and epistemic uncertainties. The advantage of this approach deals with the use of an objective integration of multiple data uncertainties from different data sources. Data set (mean ± standard deviation) derived from each data acquisition source could be combined into a global set expressed in the probabilistic space. In particular, for a specific case with critical constraint of computing time, our approach could be combined with a traditional sensitivity approach in which a perturbation study could allow to identify which parameter is the most important, and then to model its uncertainty and its impact on the output response. This allows unnecessary computing resources to be avoided for a specific case [10]. For the study of thigh mass uncertainties, our results are in agreement with those reported in the literature on the significant impact of thigh segmental mass on the muscle force estimation [8], [26], [24]. The findings showed the increasing behavior of thigh muscle force when increasing the thigh mass. A preserved profile of the muscle force pattern was also observed [8], [26]. Moreover, the same muscle activation pattern was found between our study and Barrett et al. (2007) work [27]. In fact, activation pattern of the rectus femoris decreases over time during the swinglike motion. Moreover, estimated forces of gluteus medius and posterior gluteus minimus muscle decrease from the initial position to the final position (85-degrees hip flexion and 55-degrees knee flexion) of a swing-like motion. This profile is concordant with the finding reported by Redl et al. (2007) [28]. However, our approach estimated thigh muscle forces within a global range through a probabilistic distribution function.
5 Conclusions In conclusions, our present study used a novel mathematical framework to model the uncertainties of musculoskeletal modeling input data as well as to quantify their impact on the musculoskeletal simulation results. Our study suggested that under input data uncertainties, the musculoskeletal simulation results needs to be determined within a global range of values expressed by a more powerful mathematical
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structure (e.g. probability box in our present study). Consequently, the clinical use of such global range will make the decision making more reliable. Thus, our study could be used as a guideline for such a purpose. Acknowledgments. This work was carried out in the framework of the Labex MS2T, which was funded by the French Government, through the program Investments for the future managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02).
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