Bayesian Calibration of Computational Knee Models to Estimate Subject-Specific Ligament Properties, Tibiofemoral Kinematics, and Anterior Cruciate Ligament Force With Uncertainty Quantification

Abstract

High-grade knee laxity is associated with early anterior cruciate ligament (ACL) graft failure, poor function, and compromised clinical outcome. Yet, the specific ligaments and ligament properties driving knee laxity remain poorly understood. We described a Bayesian calibration methodology for predicting unknown ligament properties in a computational knee model. Then, we applied the method to estimate unknown ligament properties with uncertainty bounds using tibiofemoral kinematics and ACL force measurements from two cadaver knees that spanned a range of laxities; these knees were tested using a robotic manipulator. The unknown ligament properties were from the Bayesian set of plausible ligament properties, as specified by their posterior distribution. Finally, we developed a calibrated predictor of tibiofemoral kinematics and ACL force with their own uncertainty bounds. The calibrated predictor was developed by first collecting the posterior draws of the kinematics and ACL force that are induced by the posterior draws of the ligament properties and model parameters. Bayesian calibration identified unique ligament slack lengths for the two knee models and produced ACL force and kinematic predictions that were closer to the corresponding in vitro measurement than those from a standard optimization technique. This Bayesian framework quantifies uncertainty in both ligament properties and model outputs; an important step towards developing subject-specific computational models to improve treatment for ACL injury.

1 Introduction

Injury to the anterior cruciate ligament (ACL) of the knee is debilitating and has long-term consequences such as early onset of osteoarthritis (OA) [1]. This injury afflicts about 200,000 individuals in the United States annually with increased prevalence in active-duty military personnel and young athletes participating in demanding activities and competitive pivoting sports such as soccer and basketball [2,3]. In sum, ACL injury is a significant public health issue due to the substantial costs of surgical reconstruction as well as the physical and financial burden this injury imposes on the patient over their lifetime [4,5].

Substantial effort has been devoted to identifying factors that increase risk of suffering this lesion. Increased knee laxity, for example, is related to the risk of first-time ACL injury [6,7] and of early ACL graft failure following surgical ligament reconstruction [8,9]. Excessive knee laxity as assessed through complicated multiplanar clinical examinations such as the pivot shift test is particularly worrisome to clinicians because it is not only associated with early ACL graft failure but also function and clinical outcome [8,10,11]. One contributor to the magnitude of the pivot shift are ligament structural properties including their slack lengths (i.e., the length at which the ligament begins to carry force) and stiffness [12–16]. However, the ligaments and ligament properties driving variability in knee kinematics under the multiplanar loads of the pivot shift exam are not well understood [15]. This gap in knowledge impedes the ability to tailor ACL reconstruction and adjunctive techniques to reestablish an individual patient's rotatory laxity to mitigate risk of ACL graft failure [17].

Computational models of the tibiofemoral joint have the potential to identify ligaments and ligament properties driving heterogeneity in kinematics among knees. However, such models often relied on mean ligament stiffnesses with standardized slack lengths [18–21], which does not account for the wide population variability in these properties [14,15]. Moreover, computational modeling integrated with stochastic analysis revealed high sensitivity of knee mechanics to ligament slack length and stiffness [22,23]. Thus, many computational knee models have utilized optimization techniques to identify subject-specific ligament properties by minimizing the differences between model predictions of knee kinematics and those obtained from in vitro experiments [24–30]. While optimization-based calibration (OC) provides an estimate of ligament properties corresponding to a given optimization criterion, OC cannot differentiate among multiple sets of ligament properties that provide equivalent minimum errors for model predictions. Even if there is a unique minimum error set of ligament properties, OC does not provide uncertainty bounds either for the calibrated ligament properties nor for subsequent predictions of joint mechanics made by the computational model.

Bayesian calibration (BC) is an alternative method to OC for estimating unknown ligament properties in a computational knee model [31,32]. A stochastic method, BC has been used extensively in other research fields, but less so in biomechanics applications [33–36]. As does OC, BC integrates training data from runs of a computational model with data from the modeled physical system. In addition, BC incorporates subject matter information about ligament properties as a “prior” distribution, which is updated by the data from the computational model and physical system experiments [32].

In contrast to OC, BC assumes that, due to simplified physics, biology, or choices of numerical methods, the computational model may contain systematic bias compared with the physical system. BC uses a flexible regression [32,37,38] to estimate any systematic bias in the computational model. The estimated bias and the posterior distribution of the calibrated parameters are used to make “calibrated” predictions of the physical system, e.g., tibiofemoral kinematics and ligament forces, with uncertainty quantifications. Thus, BC does not identify single values for the unknown model parameters (i.e., ligament properties) or physical system outputs but instead provides a quantitative measure of uncertainty for each estimated ligament property or predictions of the physical system.

This work had three main objectives: (1) to present an overview of the BC method for bias-correcting simulator model output and selecting unknown ligament parameters using cadaver experimental data; (2) to use BC to estimate unknown ligament properties with uncertainty bounds and compare them to OC estimates; (3) to apply BC to make calibrated predictions of tibiofemoral kinematics and ACL force in knees of differing levels of knee laxity assessed via simulated pivot shift examination.

2 Methods

Our approach had five main steps: (1) select two cadaveric knees exhibiting the greatest difference in tibiofemoral kinematics from a larger set of knees tested on a robotic manipulator; (2) develop multibody dynamic (MBD) computational models of the selected knees; (3) assess the activity of each ligament parameter in the MBD model using a sensitivity analysis; (4) for the most active ligament parameters, estimate the ligament properties and the uncertainty in their estimation; and (5) compare the calibrated predictions of the tibiofemoral kinematics and ACL force with both the corresponding experimental data and OC predictions for the two knees.

2.1 Select Cadaveric Knees Exhibiting Different Levels of Laxity During a Simulated Pivot Shift Exam.

First (Fig. 1), eight fresh-frozen human cadaveric knees (four females; mean age: 36 ± 6 years and four males; mean age: 27 ± 2 years) ranging from 25 to 42 years were loaded using a six-degrees-of-freedom robotic manipulator (ZX165 U; Kawasaki Robotics, Wixom, MI) following previously described methods [39] (see Appendix 1). Mass compensation for the tibial limb segment attached to the end effector accounted for the influence of gravity. Each specimen had no history of ACL trauma, or other knee injury or surgery.

Fig. 1.

Eight cadaveric knees were tested using a robotic manipulator. Multiplanar loads were applied sequentially starting with compression, then a valgus moment, internal rotation moment, and an anterior force (simulated pivot shift exam). The two knees exhibiting the minimum (Knee^Min, leftmost bar) and maximum (Knee^Max, rightmost bar) increase in anterior tibial translation (ATT) and internal tibial rotation (ITR) after ACL sectioning were selected for model development and Bayesian calibration.

The kinematics of the tibia relative to the fixed femur were expressed by adapting the convention of Grood and Suntay [40]. The flexion–extension axis was defined as the transepicondylar axis of the femur. Internal–external (IE) tibial rotation was referenced about the superior–inferior (SI) axis of the tibia [41]. The anteroposterior (AP) axis was defined as the common perpendicular of the flexion axis and the tibial long axis and varus–valgus (VV) was referenced about the AP axis. Bone-fixed axes were identified via anatomic landmarks from axial CT scans of each knee having 0.625 mm slice thickness (Biograph mCT; Siemens, Erlangen, Germany). The anatomic landmarks were registered to the end effector of the robot for testing using a previously published technique [42,43]. Specifically, the bone fixed coordinate systems were registered between the CT scan and robot by digitizing (FARO Gauge; FARO Technologies, Lake Mary, FL) (accuracy, 0.018 mm) the end effector as well as 6.35 mm–diameter radiopaque glass spheres (Winsted Precision Ball) (tolerance, 60.015 mm) fixed to the tibia and to the femur with at least three spheres attached to each bone.

We applied multiplanar loads (Fig. 1) to each knee, which captured key components of the pivot shift test [44], a clinical exam that is highly specific to ACL injury and is associated with clinical outcome and graft tear following ACL reconstruction [8,10,11]. The loads were applied in series and consisted of serially applied compression (100 N), valgus (8 Nm), internal rotation (2 Nm), and an anterior force (30 N) [43,44]. The multiplanar loads were applied at 30° of flexion with the ACL intact and after it was sectioned through a medial parapatellar arthrotomy. The arthrotomy was about 8 cm in length and was made prior to testing and was sutured closed with nonabsorbable sutures (TI.CRON, GS-25, ½, 48 mm; Medtronic, Inc., Dublin) before and after ACL sectioning. The anterior tibial translation (ATT) was tracked using the midpoint of the peaks of the tibial eminences while internal tibial rotation (ITR) was measured about the tibial long axis.

Knees with increased ATT and ITR are associated with high grade pivot shift [8]. Two knees from our cohort (Fig. 1) exhibited both the minimum and maximum increases in ATT and ITR after ACL sectioning. The knee with the minimum increase in laxity had 5.3 mm and 2.1 deg increases in ATT and ITR, respectively (from a 27 year old male and denoted Knee^Min hereafter). The knee with the maximum increase in laxity had 9.0 mm and 13.4 deg increases in ATT and ITR, respectively (from a 33 year old female and denoted Knee^Max hereafter).

Resultant in situ forces carried by the ACL at the peak applied loads were also determined via the principle of superposition and serial sectioning [45]. To this end, tibiofemoral kinematics of the intact knee were repeated immediately before and after the ACL was sectioned. ACL force was then calculated as the vector difference in force measured at the knee just prior to and immediately after sectioning the ACL.

2.2 Develop Multibody Dynamic-Knee Models for the Selected Knees.

In the second step (Fig. 2), we integrated the geometries of the two selected knees into a previously-published [46] MBD computational knee modeling pipeline (MSC software; ADAMS 2019.2, Hexagon AB, Stockholm, Sweden). The model previously predicted ligament forces and kinematics of the intact tibiofemoral joint that agreed with corresponding cadaveric measurements during passive flexion [46].

Fig. 2.

Knee geometries were extracted from MRI and CT scans of Knee ^Min and Knee^Max. These geometries were integrated into our MBD computational model of the tibiofemoral joint. Multiplanar forces and moments (simulated pivot shift exam) were applied to the MBD model and the resulting tibiofemoral kinematics and ACL force were predicted.

In the first of three main steps in building the computational models, we scanned each specimen via computed tomography (CT) (Biograph mCT; Siemens, Erlangen, Germany) and 3 Tesla magnetic resonance imaging (MRI) (General Electric, Inc.) and extracted the tibiofemoral bony and soft tissue geometries (Fig. 2) [47]. These CT and MRI images were imported into image processing software (Mimics; Materialize, Inc, Leuven, Belgium) and segmented using brightness thresholding, region growing, and manual editing methods to obtain three-dimensional renderings of the distal femur, proximal tibia, tibiofemoral articular cartilage, and menisci. We modeled the soft tissue envelope including the meniscal root and coronary attachments and cruciate, collateral, and capsular structures using 42 tension-only springs as described previously [46]. In particular, the ACL consisted of the anteromedial, intermediate, and posterolateral bundles using a total of six elements [48–50]. Second, we used average structural stiffness of the ligaments as obtained from the literature [49,51–53]. Ligament toe-regions were modeled with a second-order power function also using values from the literature. A previously described optimization calibration method, denoted OC, was used for both knees to tune the slack lengths of the cruciate and collateral ligaments, and capsule [46]. The algorithm adjusted ligament slack length to achieve experimentally measured in situ ligament forces of the intact knee at full extension. The medial and lateral menisci were modeled as wedge-shaped elements attached by three-dimensional linear springs providing circumferential, radial and superior-inferior stiffness properties [20]. Intersections of opposing cartilage surfaces and between the menisci and cartilage were identified using a contact detection algorithm [54]. Reaction forces at the contacting surfaces were described by a power function, y = Ax^b, where “y” is the reaction force and “x” is the penetration depth. The coefficients for cartilage–cartilage contact and menisci–cartilage contact was taken to be A = 327 N/mm and 19 N/mm, respectively, and b = 2.07 and 3.37, respectively [20,46].

Multiplanar loads simulating the pivot shift clinical exam were applied at 30 deg of flexion as in the cadaveric experiment (Fig. 2). The kinematics of the tibia relative to the fixed femur were also defined in the same manner as the cadaveric tests [40]. MBD model predictions of tibiofemoral kinematics and ACL force at the peak applied multiplanar pivoting loads for the two selected knees were compared to the resultant in vitro kinematics and in situ ACL force from the cadaveric experiment.

2.3 Screening to Determine the Set of Most Active Input Ligament Parameters.

The MBD model contains 26 ligament parameters (i.e., model inputs) that are potential calibration inputs (Sec. 2.4). These ligament parameters are the slack length, stiffness, and toe region of the major ligaments and the slack length and stiffness of the meniscal root attachments (Table 1). Each ligament parameter was scaled above and below the value determined by OC using physiological ranges reflecting population variability [49,51–53,55–57] (Table 1). The upper limit of ACL stiffness was selected to be greater than the upper limit of the other ligaments in Table 1 due to the wide variability in this parameter that is reported in the literature [49,56].

Table 1.

Scale factors used for varying ligament parameters (model inputs) consisting of ligament slack length, stiffness, and toe region

	Slack length		Stiffness		Toe region
Ligament	Lower limit	Upper limit	Lower limit	Upper limit	Lower limit	Upper limit
Anterior cruciate ligament (ACL)	0.85	1.15	0.7	4.25	0.2	1.8
Posterior cruciate ligament (PCL)	0.9	1.1	0.4	1.6	0.2	1.8
Posterior oblique ligament (POL)	0.85	1.15	0.1	1.9	0.2	1.8
Anterolateral ligament (ALL)	0.85	1.15	0.1	1.9	0.2	1.8
Lateral collateral ligament (LCL)	0.9	1.1	0.4	1.6	0.2	1.8
Medial collateral ligament (MCL)	0.85	1.15	0.4	1.6	0.2	1.8
Lateral meniscus ant. root (LMA)	0.9	1.1	0.4	1.6	NA	NA
Lateral meniscus post. root (LMP)	0.9	1.1	0.4	1.6	NA	NA
Medial meniscus ant. root (MMA)	0.9	1.1	0.4	1.6	NA	NA
Medial meniscus post. root (MMP)	0.9	1.1	0.4	1.6	NA	NA

NA, not applicable. The meniscal roots were represented with a slack length and stiffness; no toe region was included. Ant, anterior; Post, posterior.

A screening process (Fig. 3) was used to identify the most active ligament parameters (i.e., model inputs, Table 1) for each MBD model output (i.e., AP translation, IE rotation, VV rotation, and ACL force). A sensitivity analysis was conducted to rank order the 26 ligament parameters according to their effect on the range of each model output [32]. The sensitivity analysis was based on computed model outputs from a 400 run space-filling maximin Latin Hypercube (LH) design of the 26-dimensional input space [58].

Fig. 3.

Workflow for variable screening of the 26 ligament parameters (i.e.,, model inputs). The workflow used runs from a 400-by-26 Latin-hypercube (LH) design over the model input ranges to conduct 400 evaluations of the multibody dynamics (MBD) model. For each vector of ligament parameters, the kinematics and ACL forces (i.e., model outputs) were computed using the MBD model. These 400 model simulations and their associated model outputs were used to estimate the total effect sensitivity indices of each ligament and associated ligament parameter on each model output.

Using a previously published technique [32], total effect sensitivity indices were estimated for each of the model inputs summarized in Table 1 and each of the four model outputs (AP translation, IE rotation, VV rotation, and ACL force) yielding a total of 104 sensitivity indices. In brief, the total effect sensitivity indices measure the range of the average value of the model output as either a single model input varies over its domain or an interaction of that model input with one or more additional model inputs. Total effect sensitivity indices use the variance of model outputs with respect to a model input of interest as a surrogate for the range of the model output values. The model input that has the largest total effect sensitivity index for a given model output is interpreted to be the most active input for that output.

For both Knee^Min and Knee^Max, Fig. 4 describes the total effect sensitivity indices for the four model outputs: AP translation, IE rotation, VV rotation, and ACL force versus all model inputs that exceed a threshold value of 0.015. The threshold of 0.015 was chosen because model inputs with total effect sensitivity indices less than or equal to this threshold yielded minimal changes in model outputs. Model inputs with total effect sensitivity indices greater than the threshold were declared calibration inputs. For both Knee^Min and Knee^Max, model outputs of tibiofemoral translations and rotations and ACL force were ≤0.08 mm, ≤0.3 deg, and ≤1.4 N, respectively, for model inputs with sensitivity indices ≤0.015. For Knee^Min, eight ligament parameters were retained as calibration inputs: the slacks and stiffnesses of the ACL, ALL, and POL, plus the slacks of the MCL and POL. For Knee^Max, ten ligament parameters were retained as calibration inputs: the slacks, stiffnesses, and toe regions of both the ACL and the POL, plus the slacks of the ALL, LCL, MCL, and PCL. In the general description of the BC model (next section) the vector of calibration parameters as determined above for Knee^Min and Knee^Max are denoted by x^c.

Fig. 4.

Total effect sensitivity indices for selected ligament parameter model inputs on the kinematic and ACL force outputs of Knee^Min and Knee^Max. The eight and seven parameters plotted for Knee^Min and Knee^Max, respectively, were the only ligament parameters of the 26 studied that had total effect sensitivity indices exceeding 0.015.

2.4 Bayesian Calibration to Estimate Subject-Specific Ligament Parameters.

To achieve our 2nd objective, we used BC to estimate subject-specific ligament properties. To explain BC, consider a generic setting in which one or more experimental runs are made on a physical system. Let ${{\mathrm{y}}^p(\mathbf{x}}^p)$ denote the measured physical system output (here the measured kinematics and ACL force from the robotic cadaver experiment) and ${\mathbf{x}}^p$ denotes the inputs on which the physical system depends (here the peak applied loads of the simulated pivot shift). In general, the BC method assumes that ${{\mathrm{y}}^p(\mathbf{x}}^p)$ can be described by a mean model, ${\mu (\mathbf{x}}^p),$ perturbed by measurement error.

The outputs from the MBD simulator are assumed to depend on the same ${\mathbf{x}}^p$ as the mean of the physical system plus additional computational model inputs, ${\mathbf{x}}^c$ that are used to bring the model outputs nearer to those of the physical system. In our application these additional model inputs are the parameters that determine the ligament properties (Table 1). Let ${{\mathrm{y}}^s(\mathbf{x}}^p,{\mathbf{x}}^c)$ denote the simulator output from the MBD model.

Bayesian calibration assumes the MBD output can be described as a draw from a flexible Gaussian Process (a random function). Lastly, BC assumes that expert prior knowledge, summarized as a “prior” probability distribution, is available that describes the parameters governing the random function generator as well as the values of the unknown ${\mathbf{x}}^c$ model inputs. In this application the prior was selected to be the default prior used by the Los Alamos Laboratories software gmpsa (see Eq. (8.4.51) in Ref. [30]); this prior has proven to be robust in many applications [32].

Additionally, BC takes the statistical model for the measured output for the cadaveric specimen ${{\mathrm{y}}^p(\mathbf{x}}^p)$ to be the regression-like expression

$${{\mathrm{y}}^p(\mathbf{x}}^p)={\mu (\mathbf{x}}^p)+{ϵ(\mathbf{x}}^p)$$ (1)

where the measurement errors for different ${\mathbf{x}}^p$, ${ϵ(\mathbf{x}}^p),$ are assumed to be independent and Gaussian distributed with zero mean and unknown variance ${\sigma }_{ϵ}^2$. BC takes the simulator ${{\mathrm{output}\hspace{0.17em}\mathrm{y}}^s(\mathbf{x}}^p,{\mathbf{x}}^c)$ for a given output (a tibiofemoral kinematic value or the ACL force) to be a draw from a different Gaussian Process. BC allows the possibility that a bias, ${\delta (\mathbf{x}}^p)$, may be needed to correct the simulator code with the mean of the physical system; the bias is defined by

$${\delta (\mathbf{x}}^p)={\mu (\mathbf{x}}^p)-{{\mathrm{y}}^s(\mathbf{x}}^p,{\mathbf{x}}_{\mathbf{B}}^c)$$ (2)

where ${\mathbf{x}}_{\mathbf{B}}^c$ is “best” choice of ligament parameters. In words, the bias ${\delta (\mathbf{x}}^p)$ measures the discrepancy between the true mean of the experimental cadaveric output when run with inputs, ${\mathbf{x}}^p$, and the MBD-model run at the same ${\mathbf{x}}^p$ (assuming the unknown ligament parameter is chosen to be ${\mathbf{x}}_{\mathbf{B}}^c$). For example, the bias ${\delta (\mathbf{x}}^p$) is zero if the computational simulator perfectly emulates the mean of the physical experiment when the simulator is run at ${\mathbf{x}}_{\mathbf{B}}^c$.

Equivalently, Eq. (2) states

$${\mu (\mathbf{x}}^p)={{\mathrm{y}}^s(\mathbf{x}}^p,{\mathbf{x}}_{\mathbf{B}}^c)+{\delta (\mathbf{x}}^p)$$ (3)

which relates the mean of the selected cadaveric output to the MBD simulator output.

The BC predictions of the ligament parameter including uncertainty limits are based on draws from the posterior of these parameters conditioned on both the computed tibiofemoral kinematics and ACL force. Specifically, uncertainty limits for the ligament parameters are determined from 10,000 Monte Carlo draws from the joint posterior distribution of the calibration parameters plus the ${{\mathrm{y}}^s(\mathbf{x}}^p,{\mathbf{x}}_{\mathbf{B}}^c)$ and ${\delta (\mathbf{x}}^p)$ model parameters. We conditioned on both kinematic and ACL force data because this choice parallels that used by OC, which also utilized positional and ligament force information [46]. BC takes the draws from the posterior to be the likely values for these unknown parameters. BC predicts the value of a given ligament parameter to be the “center” of the draws from the posterior distribution of that ligament parameter, taken here to be the median of these draws [32] and measures the uncertainty of each prediction to be the central 90% of the posterior draws for that parameter.

To compare BC with OC, the knee-specific ligament parameters estimated by BC were divided by the OC predictions and calculated as a percentage; thus, values less than or greater than 100% indicated a decrease or an increase, respectively, relative to those determined by OC. In Figs. 5–8 (see Results section), these normalized values were compared to those of OC, whose standardized values were unity. The results for four ligaments that are key stabilizers against pivoting loads were examined; namely, the ACL, ALL, MCL, and POL [59–61]. We focused on the slack lengths of these ligaments because this property plays an important role in dictating tibiofemoral kinematics [14,22].

Fig. 5.

Posterior distribution of scaled anterior cruciate ligament (ACL) slack for Knee^Min and Knee^Max conditional on the MBD simulator data and the measured cadaver kinematic (AP, IE, VV) and ACL force data. The left side of each graph is a (vertical) histogram of the posterior draws of ACL slack and to the right side of the vertical line is a smoothed histogram of these draws. The dashed blue line is the normalized optimized calibration (OC) prediction of ACL slack. The extremes of the box in the center of each histogram are the 5% and 95% quantiles of the ACL draws; the height of the horizontal line in the interior of the box is the median; the 5% and 95% quantiles form two-sided 90% uncertainty quantification limits for ACL slack.

Fig. 8.

Posterior distribution of posterior oblique ligament (POL) slack for Knee^Min and Knee^Max conditional on the computed simulator data and on the measured cadaver kinematic (AP, IE, VV) and ACL force data. The left side of each graph is a (vertical) histogram of the posterior draws of POL slack and the right side is a smoothed histogram of these draws. The dashed blue line is the normalized optimized calibration (OC) prediction of POL slack. The extremes of the box in the center of each histogram are the 5% and 95% quantiles of the POL draws; the height of the horizontal line in the interior of the box is the median; the 5% and 95% quantiles form two-sided 90% uncertainty quantification limits for POL slack.

2.5 Bayesian Calibrated Predictor.

To achieve our 3rd objective, we use the BC predictor of the mean of the physical system ${\mu (\mathbf{x}}^p)$, which is the sum of the individual estimators of ${{\mathrm{y}}^s(\mathbf{x}}^p,{\mathbf{x}}_{\mathbf{B}}^c)$ and ${\delta (\mathbf{x}}^p)$ from Eq. (3)

$${\widehat{\mu }(\mathbf{x}}^p)={\widehat{{\mathrm{y}}^s}(\mathbf{x}}^p,{\mathbf{x}}_{\mathbf{B}}^c)+{\widehat{\delta }(\mathbf{x}}^p)\hspace{0.17em}$$ (4)

Specifically, ${\widehat{{\mathrm{y}}^s}(\mathbf{x}}^p,{{\mathbf{x}}_{\mathbf{B}}}^c)$ is the conditional mean of the process generating ${{\mathrm{y}}^s(\mathbf{x}}^p,{\mathbf{x}}^c)$ with respect to the joint posterior distribution of ${\mathbf{x}}^c\hspace{0.17em}\mathrm{and}\hspace{0.17em}\mathrm{the}\hspace{0.17em}\mathrm{unknown}\hspace{0.17em}\mathrm{model}\hspace{0.17em}\mathrm{parameters}$ (given the two data sources). Similarly, ${\widehat{\delta }(\mathbf{x}}^p)$ is the mean of the posterior distribution of ${\delta (\mathbf{x}}^p)$ based on the process model parameters.

The BC predictor was determined for both Knee^Min and Knee^Max. We subsequently calculated the absolute relative error (ARE) to compare the predictions of ACL force and tibiofemoral kinematics (ATT, ITR, and valgus angulation) for the BC and OC predictions and to the corresponding in vitro measurements. ARE was calculated as the difference between each prediction (from BC and from OC) and the corresponding in vitro measurement divided by the in vitro measurement. The magnitudes of the differences between each prediction and the corresponding in vitro measurement were also calculated. All comparisons were made at the peak applied pivoting loads.

3 Results

Regarding our second objective, Fig. 5 shows the BC predicted ACL slack for Knee^Min and Knee^Max differed by 1.3%. For Knee^Min, the BC prediction for ACL slack was 1.8% greater than that for OC prediction; for Knee^Max, the BC prediction of ACL slack was 3.1% larger than that provided by OC prediction. Further, the two-sided 90% uncertainty limits for Knee^Min (8.2%) are 2.4% less than Knee^Max (10.6%).

Figure 6 shows an 8.4% difference in the BC predicted ALL slack between Knee^Min and Knee^Max. The BC predicted ALL slack length was 4.2% less than that of OC predicted for Knee^Min and 4.2% more for Knee^Max. The two-sided 90% uncertainty limits for Knee^Min (9.3%) are 2.8% less than for Knee^Max (12.1%).

Fig. 6.

Posterior distribution of anterolateral ligament (ALL) slack for Knee^Min and Knee^Max conditional on the MBD simulator data and the measured cadaver kinematic (AP, IE, VV) and ALL force data. The left side of each graph is a (vertical) histogram of the posterior draws of ALL slack and the right side is a smoothed histogram of these draws. The dashed blue line is the normalized optimized calibration (OC) prediction of ALL slack. The extremes of the box in the center of each histogram are the 5% and 95% quantiles of the ALL draws; the height of the horizontal line in the interior of the box is the median; the 5% and 95% quantiles form two-sided 90% uncertainty quantification limits for ALL slack.

Figure 7 shows a 2.0% difference in the BC predicted MCL slack for Knee^Min and Knee^Max. Both knees exhibited smaller slack lengths than OC calibration with a 6.5% smaller predicted value for Knee^Min and 4.5% smaller predicted value for Knee^Max. The two-sided 90% uncertainty limits for Knee^Min (6.3%) are 3.8% less than for Knee^Max (10.1%).

Fig. 7.

Posterior distribution of medial collateral ligament (MCL) slack for Knee^Min and Knee^Max conditional on the MBD simulator data and the measured cadaver kinematic (AP, IE, VV) and ACL force data. The left side of each graph is a (vertical) histogram of the posterior draws of MCL slack and the right side is a smoothed histogram of these draws. The dashed blue line is the normalized optimized calibration (OC) prediction of MCL slack. The extremes of the box in the center of each histogram are the 5% and 95% quantiles of the MCL draws; the height of the horizontal line in the interior of the box is the median; the 5% and 95% quantiles form two-sided 90% uncertainty quantification limits for MCL slack.

Figure 8 indicates a 3.7% difference in the BC predictions of POL slack for Knee^Min and Knee^Max. Both knees exhibited greater median slack lengths than OC calibration with a 5.5% greater predicted value for Knee^Min and a 9.2% greater predicted value for Knee^Max. The two-sided 90% uncertainty limits for Knee^Min (19.0%) are 4.8% greater than for Knee^Max (14.2%).

Concerning objective three, Bayesian calibration for both Knee^Min and Knee^Max produced ACL force and kinematic predictions of ITR and valgus rotation that were closer to the corresponding in vitro measurement than did OC predictions (Fig. 9). For Knee^Min the ARE for the IR, VV, and ACL force measurements are 0.094, 0.69, and 0.98, respectively, for OC calibrated model and 0.03, 0.00, and 0.06, respectively, for the BC calibration. Comparing ARE shows the BC calibrated MBD provides at least a 2.7 times improvement in ARE for Knee^Min. For Knee^Max the ARE between the OC calibrated model and the BC calibrated model for the IR, VV, and ACL force measurements are 0.19, 0.8, and 0.20, respectively, and 0.04, 0.17, and 0.03, respectively. Thus, the BC calibrated MBD provides at least a five times improvement in ARE over the OC calibrated MBD for Knee^Max.

Fig. 9.

Comparison of the predictions given by optimization calibration and Bayesian calibration of the MBD computational simulator and the in vitro measurements of three motions and ACL force under pivot shift loading (red dotted line) for the Knee^Min (left column) cadaver and the Knee^Max (right column) cadaver. Whiskers represent the 5% and 95% quantiles, which form two-sided 90% uncertainty quantification limits.

4 Discussion

We described a Bayesian calibration methodology for bias-correcting simulator model output and selecting unknown ligament parameters using cadaver experimental data and computational data (objective one). Our specific use case incorporated measurements of tibiofemoral kinematics and ACL force in two cadaver knees that spanned a range of laxities under multiplanar pivoting loads (Fig. 1). The experimental measurements of tibiofemoral kinematics and ACL force were compared to compute values of the same outputs from a physics-based MBD knee model. We predicted unknown knee ligament properties with uncertainty bounds (objective two). Predictions and uncertainty bounds were obtained from the Bayesian set of plausible predicted ligament properties, as determined by their (posterior) uncertainty bounds (Figs. 5–8). Finally, we developed a calibrated predictor of tibiofemoral kinematics and ACL force with its own uncertainty bounds (objective three). The calibrated predictor was developed by collecting the posterior draws of the kinematics and ACL force that are induced by the posterior draws of the ligament properties and model parameters (Fig. 9) and taking a “center” (i.e., median) value of the draws as the prediction and the 90% quantiles of the draws as a measure of the uncertainty.

Our most important finding regarding objective two was that subject-specific calibration of ligament properties is needed to account for variations in knee kinematics under pivot shift loads. Bayesian calibration identified unique ligament properties for the MBD models of two knees exhibiting the minimum (Knee^Min) and maximum (Knee^Max) increase in ATT and ITR after ACL sectioning (Fig. 1). Key differences in ligament properties between the two MBD models were the predicted median slack lengths of the ALL and the POL (Figs. 6 and 8), which are known restraints to internal tibial rotation [59,60,62,63]. There were −6.5% to 9.2% differences in the estimated median ligament slack parameters for the ACL, ALL, MCL, and POL from the Bayesian assessment of these values compared with the optimization calibration (OC) of these parameters (Figs. 5–8). For objective three, considering absolute relative error (ARE) for ITR and VV rotations and ACL force, the BC prediction had over 2.7 times less ARE compared with OC prediction for the knee with lowest rotatory laxity and over five-fold less ARE compared with BC for the knee with greatest rotatory laxity. Our finding that BC improved predictions of tibiofemoral kinematics and ACL force is expected since OC did not use subject-specific information about the kinematics and ACL force of each cadaver knee to tune ligament properties. Other models utilizing OC have also found unique ligament properties based on subject specific experimental data [28,30].

Our finding that BC calibration modified the slack lengths of the POL and the ALL to achieve the disparate magnitudes of rotational laxity measured in our two knee specimens is consistent with previous anatomical and biomechanical data. Anatomically, the posterior-distal orientation of the POL and the anterior-distal orientation of the ALL on the medial and lateral sides of the knee, respectively, enable them to resist anterior-posterior shear forces during ITR [59,60,62,63]. Moreover, their locations on the periphery of the joint maximize their lever arms against ITR. The BC-predicted reduction in ALL slack in the least rotationally lax knee (Knee^Min) compared to the most rotationally lax knee (Knee^Max) agrees with biomechanical findings that surgical augmentation of the lateral soft tissues decreases ITR under simulated pivot shift loads while sectioning the lateral soft tissues increases ITR [64]. Moreover, our finding that the POL modulates rotational laxity with lesser ITR requiring decreased POL slack (Fig. 8) corroborates previous biomechanical studies documenting that the POL carries force during a simulated pivot shift [59].

Regarding the ACL and MCL, our sensitivity analyses (Fig. 4) reveal that slack in these tissues regulates ATT and valgus angulation, respectively, under pivot shift loading. These findings corroborate previous cadaveric data where slack in an ACL graft was positively associated with ATT under applied pivoting loads [12] (Fig. 4). Similarly, in situ slack of the MCL was positively correlated with valgus angulation under applied valgus moments [14].

An important practical advantage of BC over OC is the ability of BC to provide uncertainty estimates of both ligament properties (objective two) and model predictions (objective three). Uncertainty quantification is an important step toward developing more credible modeling practices because it stimulates further investigation into the source of this uncertainty. Regarding objective two, uncertainty limits for the ligament parameters are determined from 10,000 Monte Carlo draws from the joint posterior distribution of the calibration parameters given the ${{\mathrm{y}}^s(\mathbf{x}}^p,{{\mathbf{x}}_{\mathbf{B}}}^c)$ and ${\delta (\mathbf{x}}^p)$ model parameters (Figs. 5–8). The uncertainty limits of each calibration parameter are determined to capture a given proportion of the posterior probability. For example, the 90% uncertainty interval for POL slack for Knee^Min is from 94.9% to 113.9% of the POL slack obtained by optimized calibration (Fig. 8).

Regarding objective three, uncertainty limits for a mean model ${\mu (\mathbf{x}}^p)$ prediction are obtained by recognizing that each posterior draw of the model and calibration parameters produces a specific posterior distribution for ${\mu (\mathbf{x}}^p)$ from which a draw is made. As for the calibration parameters, the predicted model mean is the median of the posterior draws and the uncertainty limits are selected to capture 95% of the ${\mu (\mathbf{x}}^p)$ draws (see Fig. 9 for the predicted means for three motions and the ACL force for the Knee^Min and Knee^Max cadavers) based on 10,000 draws from the joint posterior distribution of the calibration and model parameters. While our Bayesian model requires the user to specify priors for calibration and model parameters, the predictions and uncertainty limits are often robust to the specific choice of the prior [32].

This study had limitations. First, we used a multibody dynamics model of the knee, which relied on simplified formulations for articular contact and meniscus properties, compared to more complicated finite element formulations. Nevertheless, our model contained all major anatomical stabilizers including the cruciates, collaterals, and capsular structures and the menisci. In contrast, many previous models exclude the menisci or rely on a rigid body formulation of this structure [28,65,66]. Moreover, simulation using the MBD framework allowed us to efficiently complete the simulations required to perform Bayesian calibration due to the reduced degrees-of-freedom of the model system compared to a finite element approach utilizing deformable soft tissue formulations. Second, the BC predictor focused on tibiofemoral kinematics and ACL force at one specific loading condition; namely, the peak applied pivot loads. However, the BC prediction is flexible enough to incorporate a larger array of loading conditions. Third, we utilized in vitro load–displacement data for model calibration, but our calibration workflow could also incorporate in vivo data, such as those obtained from a knee arthrometer, to develop image-based, patient-specific knee models [67–69]. Finally, we chose two knees at the extremes of our dataset in terms of change in ITR and change in ATT with ACL sectioning to test the ability of BC calibration to differentiate ligament properties. The framework presented here can easily consider additional loading scenarios and knee conditions to better calibrate the properties of all the soft tissues in the model.

In conclusion, we have presented a Bayesian Calibration technique to estimate unknown knee ligament properties and to provide calibrated predictions of tibiofemoral kinematics and ACL force, all with uncertainty bounds. Bayesian calibration identified unique ligament properties for the two MBD models including slack lengths of known rotational restraints located on the medial and lateral sides of the knee, respectively; namely, the ALL and the POL, which corroborates their anatomy and their biomechanical function. Bayesian calibration also produced ACL force and kinematic predictions that were closer to the corresponding in vitro measurement than the standard optimization technique. This framework explicitly acknowledges uncertainty in both suggested model parameters and mechanical predictions, which is an important step toward developing more credible modeling practices and more personalized treatments for knee ligament injury to reduce rates of early ACL graft failure.