Computer Methods in Biomechanics and Biomedical Engineering
ISSN: 1025-5842 (Print) 1476-8259 (Online) Journal homepage: http://www.tandfonline.com/loi/gcmb20
A framework for parametric modeling of ankle ligaments to determine the in situ response under gross foot motion Bingbing Nie, Matthew Brian Panzer, Adwait Mane, Alexander Ritz Mait, John-Paul Donlon, Jason Lee Forman & Richard Wesley Kent To cite this article: Bingbing Nie, Matthew Brian Panzer, Adwait Mane, Alexander Ritz Mait, John-Paul Donlon, Jason Lee Forman & Richard Wesley Kent (2015): A framework for parametric modeling of ankle ligaments to determine the in situ response under gross foot motion, Computer Methods in Biomechanics and Biomedical Engineering, DOI: 10.1080/10255842.2015.1125474 To link to this article: http://dx.doi.org/10.1080/10255842.2015.1125474
Published online: 29 Dec 2015.
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Date: 29 December 2015, At: 11:20
Computer Methods in Biomechanics and Biomedical Engineering, 2015 http://dx.doi.org/10.1080/10255842.2015.1125474
A framework for parametric modeling of ankle ligaments to determine the in situ response under gross foot motion Bingbing Nie, Matthew Brian Panzer, Adwait Mane, Alexander Ritz Mait, John-Paul Donlon, Jason Lee Forman and Richard Wesley Kent Center for Applied Biomechanics, University of Virginia, Charlottesville, VA, USA
ARTICLE HISTORY
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ABSTRACT
Ligament sprains account for a majority of injuries to the foot and ankle complex, but ligament properties have not been understood well due to the difficulties in replicating the complex geometry, in situ stress state, and non-uniformity of the strain. For a full investigation of the injury mechanism, it is essential to build up a foot and ankle model validated at the level of bony kinematics and ligament properties. This study developed a framework to parameterize the ligament response for determining the in situ stress state and heterogeneous force–elongation characteristics using a finite element ankle model. Nine major ankle ligaments and the interosseous membrane were modeled as discrete elements corresponding functionally to the ligamentous microstructure of collagen fibers and having parameterized toe region and stiffness at the fiber level. The range of the design variables in the ligament model was determined from existing experimental data. Sensitivity of the bony kinematics to each variable was investigated by design of experiment. The results highlighted the critical role of the length of the toe region of the ligamentous fibers on the bony kinematics with the cumulative influence of more than 95%, while the fiber stiffness was statistically insignificant with an influence of less than 1% under the given variable range and loading conditions. With the flexibility of variable adjustment and high computational efficiency, the presented ankle model was generic in nature so as to maximize its applicability to capture the individual ligament behaviors in future studies.
Introduction The foot and ankle is among the most frequently injured regions of the lower extremity in all levels of sports (Barker et al. 1997; Hinterman 1999), with syndesmosis and lateral ligament sprains accounting for about 85–90% of those injuries (Rubin & Sallis 1996). External rotation of the ankle joint is considered the primary mechanism of the syndesmotic ankle sprain (Boytim et al. 1991; Nussbaum et al. 2001; Waterman et al. 2010) with concomitant eversion and dorsiflexion perhaps associated with particular patterns of ligament trauma (Sarrafian 1983; Boytim et al. 1991; Norkus & Floyd 2001; Wolfe et al. 2001; Waterman et al. 2011; Wei et al. 2011). The precise relationships between gross foot and ankle kinematics and the resultant ligament injury patterns remain unclear. The two main functional joints of the foot and ankle complex are the ankle and subtalar joints (Leardini et al. 2000). The subtalar complex is between the talus and the
CONTACT Bingbing Nie © 2015 Taylor & Francis
[email protected]
Received 11 May 2015 Accepted 23 November 2015 KEYWORDS
Parametric modeling; finite element ankle model; collagen fibers; design of experiment; sensitivity study; computational biomechanics
calcaneus and the ankle joint ascribes the articulations between the talus, tibia, and fibula. Stability of the ankle and subtalar joint is provided by ligaments and tendons as a restraint (Figure 1). The anterior tibiotalar (ATT), posterior tibiotalar (PTT), tibiocalcaneal (CT), and tibionavicular (TiN) ligaments comprise the deltoid, growing from the medial malleolus that stabilizes the ankle joint on the medial side. The lateral ligaments, including the anterior talofibular (ATaF), posterior talofibular (PTaF) and calcaneofibular (CF), stabilize the ankle joint on the lateral side. The high ankle ligaments, including the anterior tibiofibular (ATiF) and posterior tibiofibular (PTiF), are relatively stiff ligaments that comprise the distal syndesmosis and stabilize the ankle mortise. The interosseous membrane (IOM) is broader above than below and extends between the interosseous crests of the tibia and fibula. It runs downwards and laterally for the most part as a pyramid-shaped spatial network and stabilizes the tibia–fibula relationship.
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Figure 1. The foot/ankle complex and the major ligaments.
The ligament and tendons are soft connective tissues consisting of densely packed collagen fibers and serve as the main part to transfer load among bones (Simon 1994). The relationship between the ankle/subtalar joint rotation and ligament loading has been investigated in several previous experimental studies (Siegler et al. 1988; Rudd et al. 2004). However, pre-stress, slack, and the complexity of ligament deformation patterns make it difficult to determine the in situ stress–strain behavior of a ligament experimentally. Computational models have proven to be a useful tool to investigate the joint mechanics and ligament properties. While the multi-body rigid ankle models can provide rapid solutions for motion-based mechanics (Kwak et al. 2000; Wei et al. 2011, 2015), finite element (FE) models have the ability to obtain the stress and strain information of the ligaments (Cheung et al. 2006; Reggiani et al. 2006). Several FE foot and ankle models have been developed by comparing the global model response with the experimental results (Tannous et al. 1996). The existing modeling implementations have been limited due to the absence of considerations of complex loading path and ligamentous structure, and were validated under only a few simple loading conditions. Shin et al. developed a FE model of the foot and leg and the model was validated for kinematic and kinetic behavior against available PMHS experimental data in several loading conditions, which included forefoot impact, axial rotation, dorsiflexion, and combined loadings (Shin et al. 2012). Structural-level validation of the model indicated improved biofidelity relative to previous FE models, but no attempt was made to validate the model at the ligament level. A parametric modeling approach on the ligament response, such as the one developed here, can provide a link between gross structural behaviors and the underlying bone and ligament mechanics that generate them, including consideration of the in situ ligament states and non-uniform strain built in the microstructure of the ligaments. The objective of this study was to develop a parametric framework for ankle ligament modeling that can be
used to define in situ ligament mechanics during gross motions of the ankle and subtalar joints. The ankle model was based on the FE model of Shin et al. (2012). The nine major ankle ligaments and the IOM were included with consideration of the microstructure as a bundle of collagen fibers. The fibers’ mechanical behaviors were modeled with design variables (DV) describing the toe region, the linear stiffness and the distribution of the IOM. A sensitivity study was conducted based on design of experiment (DOE) to investigate the contributions of the DV to the bony kinematics during external loading of the ankle and subtalar joints.
Methodology The detailed model of the ankle and subtalar joints was built up that aimed to include the ligamentous microstructure and the precise bone geometry. The model was developed to be compatible with the LS-DYNA non-linear FE solver (LSTC, Livermore, CA, Version 971 R7). The FE model of the foot and leg by Shin et al. (2012) served as the baseline model for this study (subsequently referred as the ‘baseline model’) (Shin et al. 2012) and was modified focusing on the foot and ankle structures. Parametric modeling of the ankle ligaments Ligament structure The nine ligaments and the IOM in the ankle model were numbered from 1 to 10 (i = 1, … 10). The geometry of the nine ligaments was refined based on dissection studies and human anatomy information (Leardini et al. 2000; Norkus & Floyd 2001; Wei et al. 2010; Agur & Dalley 2012; Clanton et al. 2014), including the insertion points and width on the connecting bones (Table 1). The ligament or the IOM was represented as a group of fibers (Weiss & Gardiner 2001; Lucas 2007), modeled by one-dimensional (1-D) discrete elements, evenly distributed along the pre-determined insertion widths. The number of the fibers included in each of them was recorded as ni. Within each ligament, the average length of the bundle of fiber elements was used to represent the initial length, l0,i (Table 2). Skewness of the bundle of IOM fibers was set to be a distribution with geometric progression with a common ratio, α, to reflect the fact that the collagen fibers are concentrated distally. The distributing length of all the IOM fiber elements, recorded as L, was measured from the baseline model (L = 306 mm). Mechanical behavior of the ankle ligaments Ankle ligaments were clinically reported to exhibit a structural ‘functional laxity,’ indicating the length difference when measured in vivo or on the isolated
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Table 1. Insertion geometry of the ankle ligaments in the model. Insertion geometry Ligament/IOM list Deltoid ATaT PTaT CT TiN Lateral ATaF
High ankle
No. (i) 1 2 3 4 5
Bone Talus Talus Calcaneus Tibia Talus
W1 (mm) 6.4 6.4 6.0 12.6 3.5
Bone Tibia Tibia Tibia Navicular Fibula
W2 (mm) 7.0 7.0 6.0 9.3 4.7
PTaF
6
Talus
4.8
Fibula
4.3
CF ATiF
7 8
Calcaneus Tibia
10.0 15.0
Fibula Fibula
10.0 15.0
PTiF
9
Tibia
15.0
Fibula
10.0
Reference Dissection study Agur and Dalley (2012) Dissection study Agur and Dalley (2012) Agur and Dalley (2012), Leardini, O’Connor et al. (2000), Norkus and Floyd (2001) Agur and Dalley (2012), Leardini et al. (2000), Norkus and Floyd (2001) Dissection study Dissection study; Bloemers and Bakker (2006), Lin et al. (2006) Dissection study; Bloemers and Bakker (2006), Lin et al. (2006)
Note: W1 and W2 indicated the ligament insertion width on the bones.
Table 2. DV definition and range selection of the parametric ankle model.
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DV definition (xp , p = 1, …, 20) Basis value Ligament/IOM list Deltoid ATaT PTaT CT TiN Lateral ATaF PTaF CF High ankle ATiF PTiF Fibers IOM Skewness
No. (i) 1 2 3 4 5 6 7 8 9 10
l0,i (mm) 20.8 11.9 22.7 33.2 16.5 25.0 15.2 16.5 14.8 – –
Ki (kN/mm) 0.0838 0.8263 0.0239 0.0391 0.0692 0.5711 0.4524 1.5161 0.0406 – –
Toe region (ci) (mm) R2 0.9061 0.7968 0.9901 1 0.9373 0.9267 0.8175 0.9213 0.9895 – –
Lower bound 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 – –
Upper bound 10.38 5.95 11.35 16.60 8.25 12.52 7.59 8.25 7.40 – –
Fiber number (ni) Lower bound 2 21 1 1 2 15 12 38 1 10 1
Upper bound 8 82 2 4 6 58 46 152 4 100 1.20
Note: R2 indicated the R-square value of the linear regression when defining the fiber material (Figure 3).
bone-ligament–bone complex (Wirth et al. 1978). The stress–strain curve from tensile tests of isolated ligaments is upwardly concave, as collagen fibers within ligament are crimped to varying degrees while unloaded and the crimping will progressively straighten when load is applied (Mertz 1984; Bloemers & Bakker 2006). For each fiber element in our model, this overall mechanical behavior of the ligament was characterized with a bilinear force–elongation curve having two zones: a zero-force toe region representing the functional laxity and the low force regime associated with the sequential uncrimping of the collagen fascicles, and a region of constant stiffness corresponding to the stretching of a collagen fiber to build internal force rapidly (Irwin & Mertz 1997; Weiss & Gardiner 2001; Lucas 2007) (Figure 2(a)). The in situ stress (or slack) state of individual ligaments within the ankle joint has not been documented in detail, largely due to the difficulty of replicating it in excised samples used, for example, in experimental studies of ligament tensile behavior (Kastelic et al. 1980; Weiss & Gardiner 2001). In this study, a computational approach was taken to determine these states. The initial toe region of the nine ligaments was defined as a design variable (DV),
that is, ci, (i = 1, …, 9). For the IOM, the toe region was set to be zero, i.e. no initial slack was considered. This decision was made after dissection studies of the IOM in situ, which revealed no slack and the rapid generation of force when manual attempts were made to open the interosseous space. The same fiber stiffness, recorded as k, was adopted across all the ligaments, and the value was set to be k = 0.02 kN/mm (Lucas 2007). The number of the fibers, ni, included in each ligament or IOM was set to a DV. This resulted in the total stiffness (Ki) determined by:
Ki = ni ∗ k (i = 1 … 10)
(1)
The common ratio of the distribution of the IOM fibers, α, was defined as another DV (Figure 2(b)). In total, 20 DVs, recorded as xp ( p = 1, …, s, s = 20), were included as a parametric model characterizing the response built in the microstructure of the ligaments. The DV range was determined in an effort to explore the possible physical solutions and to capture a global trend of the ligament response. To define the range of ni for the nine ligaments, the rapid increase in force generation along the force–elongation behavior in the baseline model was estimated via linear approximation. The
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Figure 2. (a) Parametric modeling of the fiber toe region and the linear stiffness. Each of the nine ligament or IOM, i, was modeled as a bundle of ni fibers, each having a toe region ci. The fiber stiffness, k, was a constant. (b) Distribution of the IOM fibers as a geometric progression with a common ratio, α. δj indicated the distance between neighboring elements and the initial value, δ1, was determined by L, n10, and α.
Figure 3. Determination of the DV range for the ligament (e.g. ATiF, i = 8). A linear regression was conducted with a largest possible R-Squared value to account for the rapid increase in force generation. The resultant function was y = 1.51x − 1.37, therefore, the overall ligament stiffness was determined as K8 = 1.51 kN/ mm.
low-force region was excluded. E.g. for the ligament ATiF, which was numbered as i = 8, the total stiffness was calculated as K8 = 1.51 kN/mm (Figure 3). Following this, the basis value of the fiber number within ATiF, n0,8, was calculated as:
n0,8 =
K8 = 76 k
(2)
The range of ni (i = 1, … 9) were determined by scaling the basis value from 0.5 to 2. With limited experimental data on IOM, a wide range of the fiber numbers, n10, was set to be from 10 to 100.
A wide range of the DV representing the toe region was selected to account for the high variability of the ligament properties as reported in the existing experimental studies. The functional laxity of the ankle ligaments was among 20–60% of the ultimate strain at the occurrence of failure (Nigg et al. 1990), with the reported ultimate strain ranging from 10 to 84% (Noyes & Grood 1976; Attarian et al. 1985; Siegler et al. 1988; Parenteau et al. 1996). The initial low force region for each ligament, which was excluded in estimating the linear stiffness of the force–elongation curve, was 10% on average (Figure 3). Therefore, the upper and lower bounds of the toe region (ci) was determined by scaling the initial ligament length, l0,i, from 0 to 0.5 as a reasonable approximation. For the skewness, α, a range of 1–1.2 was adopted. Bone geometry The bone geometry was reconstructed from CT and Magnetic Resonance Imaging scans of the lower limb of a male volunteer with anthropometric characteristics (175.3 cm height and 77.1 kg weight) close to the 50th percentile male (Shin et al. 2012). Refined meshing was performed using HyperMesh 12.0 (Altair Inc., PA, USA). Hexahedral FE meshes were introduced to allow for the modeling of precise geometry of the ankle bones and ligament insertion. The bones were treated as rigid bodies. Influence of the flesh on the bony kinematics was found to be insignificant, as the load among bones was mainly transmitted by the ligaments serving as the connective tissues. Therefore, flesh was excluded to improve the computational efficiency. The surface-to-surface contact
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that the motion was dictated by the ligaments. The resultant yaw–pitch-roll angle of the fibula, talus and navicular, i.e. the Euler angle description (z–y–x), relative to the tibia, was calculated from the direction cosine matrix as a quantitative measure to describe the bony kinematics. Consequently, a total of nine angles (3 angles × 3 bones) were available as the model response.
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Figure 4. LCS for each bone (left foot).
algorithms were assigned to simulate interactions among bones. Sensitivity analysis on the bony kinematics Simulation setup For the bony kinematics analysis of the ankle model, Local Coordinate Systems (LCSs) were defined for each of the tibia, fibula, talus, calcaneus and navicular to quantify the relative motions among the ankle bones (Figure 4). A similar procedure as proposed by Camacho et al. (2002) was used. The tibia coordinate system served as the global reference frame and had its origin at the intersection of the tibia’s longitudinal centroidal axis (the tibia’s Z axis) with the plafond. The tibia’s YZ-plane contained the Z-axis and the apex of the medial malleolus and the X axis was defined by the cross product of the unit vectors along the tibia’s Y and Z axes. The fibula coordinate system had its origin at the centroid of the lateral malleolus, and its axes were parallel to those of the tibia when the foot was in the neutral posture. The origins of the calcaneus, talus and navicular coordinate systems were located at the centroid of the bones, with the axes parallel to those of the tibia when the foot was in the neutral posture. External rotation was applied at an angular velocity of 10°/s to investigate the bony sensitivity to the DVs (Figure 4). This was accomplished by fixing all six degrees of freedom (DOF) of the proximal tibia and applying Z-axis external rotation to the calcaneus up to 30°, which was considered to be a non-injury range of ankle motion (Siegler et al. 1988). Other DOF of the calcaneus were either fixed (Z-axis translation, X- and Y-axis rotation) or free (X- and Y-axis translation). The other three bones, the fibula, talus and navicular, were free in all six DOF so
Algorithm of the sensitivity study The sensitivity was computed based on a DOE study using the pre-defined 20 DVs. The Latin Hypercube (LH) was chosen as the sampling scheme for the DOE. The LH algorithm was first introduced by McKay in 1979 as a statistical method to generate a stratified sample of parameter values from a multi-dimensional distribution (McKay et al. 1979). As a structured quasi-random sampling strategy, the LH can generate random sample points ensuring that all portions of the vector space are sampled, and potentially nonlinear input and output relationships can be represented in resulting response surfaces (Neal 2004; Neal et al. 2008). Assume that N sample points were to be determined for the s-vector variable X = (x1 …, xs), which was composed of all the DVs. Then the range of each xp was divided into N equally probable intervals and sampled once from each interval. Let this sample be xpt (t = 1, …, N). For the xp ( p = 1, …, s) component in Xt (t = 1, …, N ), the components of the various xp’s were matched at random. One of most recent LH techniques, the maximin design, sought to maximize the minimum statistical distance between model inputs to increase the multidimensional uniformity (Johnson et al. 1990). The maximin LH design implementation by Carnell was used in this study (Carnell 2012) to generate a DOE plan of 1000 sampling points (N = 1000). Simulations with the resultant models were conducted and all the runs were normally terminated. The computation time of each simulation was approximately 7 min on a Xeon E5-1620 processor with 3.60 GHz frequency using the LS-DYNA solver version R7.1.1. An analysis of variance (ANOVA) was used as a statistical tool to rank the DVs with regards to the model response for screening purposes (Cox & Reid 2000). Each model response was taken as a multi-variant linear regression model, yr, with r indicating the number of the model response, i.e. r = 9. For each DV, the regression coefficient, bp ( p = 1, …, s), were calculated as: yr = b0 +
s bp ∑ p=1
Δxp
xp = b0 +
b1 b b x1 + 2 x2 + … + s xs Δx1 Δx2 Δxs
(3)
The regression coefficients showed the influence of each DV on the individual model response. A positive
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correlation meant a higher DV value would result in a larger value of the response and vice versa. Furthermore, the Sobol global sensitivity analysis (GSA) method, which was based on ANOVA, was introduced to conduct nonlinear analysis of DV significance for higher order models (Stander et al. 2014). It can provide a normalized sensitivity among different DVs with consideration of their contribution on more than one model response (Sobol 1993; Saltelli et al. 2008). A response surface-based function was generated as a representive of the model response, and this function was decomposed in the sub-functions of different variables, in which the mean of each sub-function was zero and each variable combination appears once, as below: s s s ( ) ∑ ( ) ∑ ∑ ( ) fp xp + fpq xp , xq f x1 , … xs = f0 + Downloaded by [Adwait Mane] at 11:20 29 December 2015
p=1
The interested bones were treated separately. Each DV was evaluated by summing up its average influence on the three rotation angles, with the cumulative Sobol index of all DVs for each bone being 1 (100%). The accuracy of the response surface-based functions was evaluated based on the difference between the actual model response, y, and the predicted value, ŷ . For each model response, two error indicators, the root mean square (RMS) error, 𝜀RMS, and the coefficient of multiple determination, mR2, were calculated using the Equations (8) and (9). The indicator, mR2, which varies between 0 and 1, represents the ability of the response surface to identify the variability of the design response and the value of 1 indicates a perfect fit.
𝜀RMS
p=1 q=p+1
( ) + … + f1,2, …, s x1 , … xs
(4)
Then, the variance of each sub-function represented the variance of the function with respect to that variable combination. 1
Vp, …, q =
1
∑N
mR = ∑t=1 N 2
t=1
(̂yt − ȳ t )2 (yt − ȳ t )2
(8)
(9)
where ŷ is the mean of the response.
( ) … fp2 , … q x1 , … xs dxp … dxq ∫ ∫ 0
√ √ N √1 ∑ )2 ( yt − ŷ t = √ N t=1
Results
0
Development of the ankle model
V=
∑
Vp +
∑
Vpq + … + V1,2, …, s
p
