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Themed Issue Papers |
1 Department of Mechanical and Manufacturing Engineering, The University of Melbourne, Victoria 3010, Australia2 Department of Biomedical Engineering, The University of Texas at Austin, 1 University Station, C0800 Austin, TX 78712-0238, USA
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(Received 28 September 2005;
accepted after revision 2 January 2006; first published online 11 January 2006)
Corresponding author J. W. Fernandez: Department of Mechanical and Manufacturing Engineering, The University of Melbourne, Victoria 3010, Australia. Email: justinf{at}unimelb.edu.au
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Introduction |
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One key difficulty in understanding these issues has been the inability to measure bone movements accurately in vivo. Conventional motion capture technologies employ video-based systems to track markers attached to the skin. While these systems are non-invasive, the markers affixed to the skin shift relative to the underlying bone (Reinschmidt et al. 1997). Skin motion artifact can be avoided by using medical imaging techniques such as fluoroscopy. Newer technologies, such as cine magnetic resonance imaging (MRI) and single- and bi-plane fluoroscopy, are capable of accurately recording 3-D joint motion non-invasively. Cine MRI is limited by the restriction it imposes on the movements that the subject can perform, and it cannot be used to investigate weight-bearing activities such as gait (Sheehan et al. 2000). Single-plane X-ray fluoroscopy allows direct visualization of the underlying bones and has been used to track bone movements in normal and reconstructed joints (Banks & Hodge, 1996; Asano et al. 2001; Komistek et al. 2003). However, this method is limited to two-dimensional assessment and is prone to errors due to parallax (Kanisawa et al. 2003). Bi-plane fluoroscopy enables accurate quantitative assessment of 3-D joint motion. In this method, which has been demonstrated on both dogs and humans, radiographic images are captured in two different planes at frame rates as high as 250 Hz (You et al. 2001; Tashman & Anderst, 2003). The main limitation is that radio-opaque markers must be implanted in the bones to enable accurate registration between the two views obtained from the X-ray system.
A second difficulty has been an inability to monitor muscle and bone loading non-invasively in living subjects. Direct measurements have been made of articular contact forces at various joints (Bergmann et al. 1993; Kaufman et al. 1996); however, these data are for patients fitted with prostheses and may not be representative of the loading experienced by patients with osteoarthritis. Mathematical models have been used, in conjunction with data from non-invasive biomechanical experiments, to determine musculoskeletal loading in vivo. Because the muscles, ligaments and joints form a mechanically redundant system (i.e. the number of muscles and ligaments crossing any joint is greater than the number of degrees of freedom prescribing joint motion), the forces developed by the muscles and ligaments cannot be determined uniquely. Early attempts to solve this problem grouped individual muscles and ligaments together, until the number of force-bearing structures equaled the number of degrees of freedom assumed by the model (Morrison, 1970). More recent attempts have applied optimization theory, in which a physiological cost function is minimized subject to the constraints imposed by the musculoskeletal system and the motor task being modelled (Pandy, 2001). Recent advances in computer hardware and software have enabled elaborate simulations of movement to be performed using large-scale anatomical models of the body; models with many segments and a large number of muscles. Even with these advances, however, models used to study whole-body movement still incorporate over-simplified models of the joints; for example, whole-body models developed to simulate walking almost always represent the knee as a one degree-of-freedom hinge joint (Yamaguchi & Zajac, 1990; Anderson & Pandy, 2001; Neptune et al. 2004).
A third difficulty has been the inability to model musculoskeletal anatomy accurately on a subject-specific basis. High-resolution MRI is now being used to develop precise anatomical models of the muscles, bones and joints from magnetic resonance (MR) scans recorded from a single subject (Arnold et al. 2000; DeFrate et al. 2004). Before these models can be used to guide patient-specific treatment decisions, however, methods for building such models and incorporating them in simulations of movement must be established and carefully validated.
If subject-specific musculoskeletal computer models and non-invasive experiments could be used to diagnose joint disease accurately, then this approach could have a major impact on patient management and presurgical healthcare. The overall system must be practical, cost effective and user friendly if it is to find widespread use among clinicians. Furthermore, the models must be thoroughly validated against experiments if clinicians are to have confidence in the model predictions. We approach these challenges by proposing a hierarchical rigid-body and finite-element (FE) modelling framework using anatomically based subject-specific models. This paper outlines our own previous and on-going work in this area, as well as some of the latest developments reported by others.
Generic computer models and simulations of movement
The redundant problem in movement biomechanics is often addressed using optimization methods. There are two approaches: static optimization and dynamic optimization or optimal control theory. Most studies have used these methodologies independently (Zajac & Gordon, 1989; Pandy, 2001), but they may also be concatenated, as described by Shelburne et al. (2004a) and Pandy (2005). In the static optimization approach, inverse dynamics is used first to calculate the net muscular torques exerted about each of the joints. Parameter optimization is then applied to resolve the muscle force redundancy at each time step along the movement trajectory. In human movement studies, the criterion most often used as the cost function is minimizing the sum of the muscle stresses squared (Brand & Crowninshield, 1981). While static optimization is computationally fast, it is prone to measurement errors in joint motion that can lead to significant errors in the estimated values of the joint torques. Seth & Pandy (2005a,b) showed recently that muscle force estimates obtained from static optimization cannot produce a stable forward simulation of the model in a task such as vertical jumping. In addition, because a different problem is solved at each time step, the effects of muscle excitation–contraction coupling cannot be taken into account in the calculations of muscle force. Finally, the static optimization method cannot handle performance criteria that depend explicitly on time, such as metabolic energy consumption (Bhargava et al. 2004).
Dynamic optimization, which is based on the forward dynamics method in mechanics, is a more powerful approach for estimating muscle forces during movement. This approach solves the optimization problem over the entire time interval of the task, and so can handle time-dependent performance criteria such as jump height (Pandy et al. 1990), minimum time (Hatze, 1976; Pandy et al. 1995) and metabolic energy consumption (Anderson & Pandy, 2001). Muscle excitation–contraction coupling is easily incorporated into the formulation of the problem, because the model equations are integrated forwards in time. The most serious limitation of dynamic optimization is its high computational cost; the method is computationally expensive because the governing equations of motion must be integrated a large number of times for each iteration of the computational algorithm (Pandy et al. 1992; Anderson et al. 1995).
To illustrate the forward dynamics approach, we review the 3-D gait simulation reported by Anderson & Pandy (2001). In this study, the body was represented as a 10-segment, 23 degree-of-freedom (dof) mechanical linkage, actuated by 54 muscle–tendon units. The pelvis was modelled as a rigid segment that could translate and rotate in three dimensions relative to the ground. Each hip was modelled as a ball-and-socket joint, each knee as a hinge joint, each ankle–subtalar joint as a universal joint and each metatarsal joint as a hinge joint. The head, arms and torso were represented as a single segment that articulated with the pelvis via a ball-and-socket joint located at approximately the third lumbar vertebra (Fig. 1, left panel). Contact between each foot and the ground was characterized by five stiff springs distributed under the sole of the foot. Each of the 54 muscle–tendon units was represented as a three-element, Hill-type muscle in series with tendon. Muscle excitation–contraction coupling was modelled as a first-order process (Zajac, 1989). The model was used to simulate vertical jumping (Anderson & Pandy, 1999), as well as walking at normal speeds (Anderson & Pandy, 2001). For walking, the problem was to minimize the amount of metabolic energy consumed per unit of distance travelled, subject to the dynamical equations of motion and a set of initial and terminal boundary conditions.
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Inverse dynamics was used to determine joint-contact loading and the relative positions of the bones at the knee at each instant during the gait cycle. The joint angles, ground reaction forces and muscle forces obtained from the walking simulation were applied to the lower-limb model (Fig. 1, middle panel), and a static equilibrium problem was solved to find the anterior–posterior displacement, medial–lateral displacement, varus–valgus orientation, joint loading and ligament forces at the knee. Knee-joint pressure was also approximated by solving a Hertz contact problem. The lower limb model shown in Fig. 1 has also been used to calculate shear forces and ligament loading in the anterior cruciate ligament (ACL)-deficient knee during gait (Shelburne et al. 2004b).
The aforementioned rigid-body models assume a generic geometry obtained from an average healthy adult male. Generic models are useful for studying general trends in large populations. However, if modelling results are to be used to plan surgical procedures, guide treatment decisions and trial implant designs, they must take into account a subject's individual musculoskeletal anatomy. Neuromuscular diseases, such as cerebral palsy, are often responsible for producing differences in bony geometry and muscle attachment sites relative to the healthy population. To understand the effects of joint disease, a subject's unique articular joint shape and cartilage thickness must be known, both of which contribute to the contact pressure distribution. The need to customize musculoskeletal anatomy is an essential step in the modelling process, if the predictions of computer models are to be useful to clinicians.
Subject-specific musculoskeletal computer models
Computer-generated reconstructions from computed tomography (CT) and MRI provide an accurate, non-invasive method to quantify musculoskeletal anatomy in living subjects. MRI has been used to develop subject-specific biomechanical models of normal subjects and of patients with movement disorders, such as cerebral palsy gait. Several studies have investigated the accuracy with which soft tissue excursions, such as muscle moment arms and muscle–tendon lengths, can be estimated in individual subjects using models built from MRI.
Murray et al. (2002) developed a scaling method to determine the moment arms of muscles in individual subjects. This method scaled the shapes of the bones and the origin and insertion sites of the muscles based upon the height and weight of the subject. Arnold et al. (2001) found that a ‘deformable’ musculoskeletal model built from MRI, in combination with a few subject-specific parameters and simple scaling techniques, gave accurate estimates of the lengths and moment arms of the hamstrings and psoas muscles in persons with cerebral palsy who walked with a crouch gait. Attempts have also been made to use more detailed morphing methods, such as free-form deformation (FFD), which is used to customize specific features of bone identified from CT and MRI (Fernandez et al. 2004). The FFD technique can also introduce curvature into bones, which is an important consideration when modelling the bones of persons with neuromuscular disorders, such as cerebral palsy.
To illustrate application of the FFD technique, Fig. 4 (left panel) shows a femur embedded in a host mesh, which is then morphed to minimize the distance between control points. The control points are selected from markers that are easily identified on the MR or CT images. A generic bone (lightly shaded) is morphed inside a host to a specific femur (darkly shaded), which has all of the target points matched. The details of the fitting process used to derive the musculoskeletal models from the Visible Human Male dataset, as well as the details involved in customizing the models to subject-specific data using host-mesh fitting, are presented by Fernandez et al. (2004).
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In this approach, commercial software (e.g. 3D Doctor, Able Software, Lexington, MA, USA) may be used to segment tissue geometry data automatically and generate an unstructured surface STL-mesh (an Industry standard computer-aided design format). Considerable computing time is spent using an FE-preprocessor to tranform the unstructured surface mesh into a structured FE volume mesh, which is required for a robust well-behaved FE analysis. The geometry obtained is more accurate, since the method is able to capture unique features of an individual's anatomy, but it is also more time consuming compared to general scaling, and therefore can only be applied to a small number of subjects. Alternatively, FFD begins with a predefined structured generic FE volume mesh, which is then morphed to data obtained from automatic segmentation of MRI or CT. This technique is better than general scaling, but it is not as detailed as complete mesh generation in the region of contact.
An optimal approach to developing accurate subject-specific computer models of the musculoskeletal system might involve a combination of the aforementioned methods. In developing a subject-specific model of the knee, for example, where the aim is to quantify the stress distributions at the tibiofemoral and patellofemoral joints, an accurate mesh may be generated in the region of the contact for each subject. This may be accomplished by using automatic segmentation of MRI images followed by complete mesh generation using an FE preprocessor. For the remainder of the femur and tibia, the geometry of each bone could be scaled from a generic FE mesh to match fiducial bone features via the FFD method. This approach can also reduce the cost of model development, since MRI is a relatively expensive imaging modality to employ on a subject-specific basis. For example, 1 mm MR slices could be recorded in the neighbourhood of the joint, while 5–10 mm slices could be obtained along the shafts of the femur and tibia.
Accurate knowledge of joint contact stresses is fundamental for improving the diagnosis and treatment of persons with arthritis. Quantifying stress concentrations along the articulating regions of a joint can help to identify likely sites of cartilage degeneration and pain. This information, in turn, can be used to plan more effective surgical procedures, or to design appropriate non-surgical rehabilitation therapies. Finite-element modelling is a useful tool for quantifying joint contact stresses and soft tissue strains that result from muscle loading.
As part of the International Union of Physiological Sciences (IUPS) Physiome Project (Hunter & Borg, 2003), organ models, FE modelling tools and mark-up languages have been developed to describe the human body from the organ systems level down to the single cell. A number of anatomically based geometries (see Fig. 5) have been developed from the Visible Human Male Project (Ackerman, 1998), including detailed 3-D reconstructions of bones, muscles, tendons, ligaments and the menisci. The soft tissues were manually digitized from 2-D images to create 3-D data sets for geometric fitting to high-order FEs. The high-order elements were useful for describing smooth anatomical bodies and their interactions with other tissues. The bones were digitized using a hand-held Polhemus Fastscan laser scanner (Applied Research Associates, NZ Ltd, Christchurch, NZ). An anatomically accurate physical model (SOMSO, http://www.somso.de), typically found in medical school teaching laboratories, was used to obtain the muscle origin and insertion sites. Muscle attachment sites were measured using a digitizing device known as a FARO arm (FARO Technologies, Inc., Lake Mary, FL, USA).
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New computational methods for simulating movement
Computer simulations provide a powerful means for understanding muscle and joint function. Model simulations provide information that cannot be obtained directly from a movement experiment; specifically, computer simulations can yield estimates of the muscle forces responsible for the observed movement pattern.
The 3-D model shown in Fig. 1 is one of the most elaborate models of walking to appear in the literature (Anderson & Pandy, 2001). This model was used, in conjunction with dynamic optimization theory, to calculate leg muscle forces for walking at normal speeds. The dynamic optimization problem was solved using a parameter optimization approach that required the model equations to be integrated hundreds of times for each iteration of the computational algorithm (Pandy et al. 1992). Consequently, months of computer time were needed to solve the optimization problem, making this approach intractable.
Recent work has demonstrated that accurate estimates of muscle forces can be obtained with smaller investments of computer time by solving a so-called ‘tracking problem’ (Thelen et al. 2003; Neptune et al. 2004; Seth & Pandy, 2005a,b). In this formulation, the model solution is constrained to follow a prescribed path. To simulate walking, for example, the joint angles and ground reaction forces can be treated as constraints that the solution must satisfy within a prescribed tolerance. The problem is to calculate the muscle excitation histories (and muscle forces) that correspond to the measured patterns of body motions and ground reaction forces. The main limitation of the tracking method is that it compromises the predictive power of the dynamic optimization approach; that is, the tracking method cannot be used to predict how changes in body structure affect tissue function and motor task performance. Nonetheless, the tracking method can be used to perform pre- and postsurgery comparisons on patients with movement disorders. Also, because the joint angles and ground reaction forces are prescribed, the solution space is reduced, and computation time is much less.
Neptune et al. (2004) used the tracking method to perform a dynamic simulation of gait, in which the body was constrained to move in the sagittal plane. Thelen et al. (2003) applied the method of feedback linearization to calculate the net joint torques needed to produce a set of desired joint angle trajectories observed for a pedalling task. In this study, static optimization theory was used to decompose the net joint torques into individual muscle forces. One limitation of the static optimization approach is that time-dependent performance criteria, such as metabolic energy consumption, are difficult to include in the formulation of the problem. Also, the original formulation of the computed muscle control (CMC) method did not take into account muscle excitation–contraction coupling. Thelen & Anderson (2005) recently improved the CMC method by taking into account the delay in muscle force production. In this study, a model of the body was used to track experimental gait data (joint angles and ground reaction forces), and the muscle forces responsible for the observed movement pattern were found. The calculations were verified by comparing the time histories of the muscle forces with measurements of the muscle activation patterns obtained from EMG.
Recently, Seth & Pandy (2005a) reported on a similar method to the one described by Thelen et al. (2003). Seth and Pandy's neuromusculoskeletal tracking (NMT) method concatenates skeletal motion tracking and optimal neuromuscular tracking to enable forward simulations of movement to be performed more quickly and accurately. The skeletal motion tracker calculates the net joint torques needed to actuate a skeletal model and track prescribed joint angles and external forces in a forward simulation consistent with the observed task. The optimal neuromuscular tracker calculates a set of muscle excitations that tracks the net joint torques directly and resolves the muscle redundancy problem dynamically. Because the optimal neuromuscular tracker uses dynamic optimization theory to calculate muscle forces, muscle excitation–contraction coupling (i.e. the delay between muscle excitation and the development of muscle force) is easily incorporated into the formulation of the problem. By tracking the external forces applied by the environment (e.g. ground reaction forces during gait), the NMT method yields more accurate estimates of the joint angular accelerations generated during a task. Measurements of muscle activations may also be included in the formulation of the tracking problem as a means of further improving estimates of muscle force (Lloyd & Besier, 2003). The NMT method has been shown to produce muscle force estimates three orders-of-magnitude (1000 times) faster than the conventional parameter optimization approach (Seth & Pandy, 2005a).
Solving the contact problem
Some studies have calculated joint contact stresses using either a Hertz contact model (Pandy et al. 1998) or an elastic foundation model (Fregly et al. 2003). The Hertz contact model makes explicit assumptions about the nature of the contact; specifically, that the contact area is elliptical in shape; that it is much greater in size than the depth of the cartilage layer being modelled; and that a contact force may be found by integrating the displacements of a set of springs distributed over the contact area. Perhaps the major limitations of this approach are that: (1) the material is assumed to be homogeneous throughout; and (2) it does not account for any complex biphasic (solid and fluid) behaviour.
The most accurate method for determining joint contact stresses is FE modelling. Detailed constitutive laws, such as the biphasic behaviour of cartilage, can be taken into account in the development of an FE model. FE modelling can also be applied to estimate stress distributions within a tissue, such as cartilage or ligament, and complex contact models with anatomical articular profiles can be built using this approach. Most studies have used static and quasi-static FEs to calculate joint contact stresses during activities such as walking. In this approach, the equations of static equilibrium are enforced at each time step during a simulation of the model, where a large set of non-linear equations must be solved simultaneously.
Detailed 3-D representations of joint geometry are commonly obtained by digitizing the shapes of articulating surfaces in cadaver specimens. Donahue et al. (2003) developed an FE model to calculate joint pressures associated with meniscal replacements. They validated their model calculations against measurements of pressure distributions obtained from cadaver specimens using pressure-sensitive film. Some studies have also reproduced 3-D joint geometry using MRI. Song et al. (2004) used tissue insertion sites with wrapping surfaces to take into account the normal forces transmitted by the muscles and ligaments to the bones. Model validation was performed through extensive cadaver testing. Recent studies have treated soft tissues such as cartilage, ligaments and the menisci as deformable continuums in the development of FE joint models (Pena et al. 2005). The advantage of this approach is that detailed stress and strain distributions can be studied near ligament insertion sites, by treating objects as continuums rather than as simple springs.
One example of an FE quasi-static model developed using FFD morphing (see section on Subject-specific musculoskeletal computer models above) is the patellofemoral joint model reported by Fernandez & Hunter (2005). This model predicted patellar kinematics, muscle forces, surface contact pressures and stresses internal to cartilage and bone using the Physiome Project database of anatomical geometries and muscle attachment points (see Fig. 5). Subject-specific anatomical features were derived from hospital MRI data using host-mesh fitting. The mesh data were used to fit the outer cortical bone, inner cancellous bone, cartilage layer and articulating femur profile. The model included the separate portions of vasti as well as rectus femoris. These muscles were modelled with pennate fibre directions to provide an anatomically based resistance for the patella during flexion. The tibia was driven by prescribed kinematic profiles obtained from video-based motion capture. Estimates of the surface contact pressure distributions were obtained with the knee flexed from 10 to 90 deg (see Fig. 6).
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A number of recent studies have used explicit FEs, in which the kinematics is explicitly advanced forward in time from one step to the next. The equations of dynamic equilibrium are solved at each time step during a simulation, and this approach can handle the high non-linearity of contact mechanics with increased numerical stability. A further benefit is that the model does not have to be constrained in order to prevent rigid body motions, as required in quasi-static simulations. This method also allows joint kinematics and stress distributions to be predicted simultaneously, similar to a rigid-body model simulation. Giddings et al. (2001) and Godest et al. (2002) used the explicit approach to estimate the distribution of contact stresses on a total knee replacement for gait. The model simulation results were validated against experimental data obtained from the Stanmore knee simulator. More recently, Halloran et al. (2005) developed an explicit FE knee model that included both the patellofemoral and tibiofemoral joints. They found little difference in the predicted joint kinematics between a deformable-body model and one that modelled the bones as rigid.
In general, subject-specific joint geometries are best obtained from MRI. Not only is this imaging modality safe for use on humans, but it also allows contours of the bones, muscles and cartilage to be recorded simultaneously. Thresholding geometries is a difficult process, and manual segmentation still causes a variation of about 10% in thickness, which leads to differences in cartilage stress (Li & Lopez, 1999). The challenge for future studies is to develop practical, semi-automated segmentation processes that are accurate, especially since the use of cartilage in tibiofemoral joint models has been shown to yield more accurate results than bone-only models (DeFrate et al. 2004).
Experimental validation of subject-specific model simulations
High-resolution MRI is now being used to develop precise anatomical models of the muscles, bones and joints from data recorded on a single subject (Arnold et al. 2000; DeFrate et al. 2004). Before these models can be used to guide subject-specific treatment decisions, however, methods for building such models and incorporating them in simulations of movement must be thoroughly tested and validated. Both in vitro and in vivo experiments can be performed to evaluate the accuracy of methods used to build subject-specific models from MRI. Muscle moment arms calculated in a customized model of a cadaver specimen built from MR data can be compared against moment arm measurements obtained directly from the same specimen (Arnold et al. 2000; Murray et al. 2002). The cadaver experiments can also be used to develop cost-effective methods of building subject-specific computer models from image data, for example, by devising suitable anthropometric scaling laws and/or by optimizing image slice widths. Cost-effective methods are necessary if subject-specific modelling is to find wide application in clinical medicine.
Arnold et al. (2000) developed detailed MR-based models of the lower limb to estimate the lengths and moment arms of several leg muscles. Wrapping surfaces were used to model the paths of certain muscles more accurately. The calculated values of muscle moment arms were compared against measurements obtained from cadaver specimens using the tendon excursion method. Krevolin & Pandy (2003) highlighted the need to develop models on a subject-specific basis. These researchers found that generic models of the knee did not match results obtained from cadaver experiments, and also that there was significant variation in the data recorded from different cadaver specimens. They hypothesized that differences between models and experiments are explained mainly by inaccuracies in reproducing bone geometry and soft tissue properties in the models. Bone geometry was reproduced in the model using CT, while ligament origin and insertion sites were identified from MRI. Soft tissue properties such as ligament stiffnesses and reference strains were calculated by simulating anterior–posterior drawer experiments performed on the cadaver specimens. Lloyd & Besier (2003) developed a subject-specific model of the human knee that was actuated by measured activation signals obtained from EMG. The model was used to predict muscle forces over a range of tasks, including knee extension exercise. These researchers also concluded that it is important to customize a model to a subject's muscle strength with respect to a specific task.
Integrating computer modelling, imaging and in vivo experiments to assess musculoskeletal function
Magnetic resonance imaging, bi-plane X-ray fluoroscopy and biomechanical modelling are enabling technologies for the non-invasive evaluation of muscle, ligament and joint function during dynamic activity. As discussed above, subject-specific computer models of the muscles, ligaments, cartilage and bones can be developed from MRI. Bi-plane X-ray fluoroscopy holds the promise of being able to measure the relative movements of the bones at a joint in three dimensions with submillimetre accuracy (Tashman & Anderst, 2003; Li et al. 2005). Highly complex 3-D dynamic simulations of movement can be performed using computational methods, such as computed muscle control (Thelen & Anderson, 2005) and neuromusculoskeletal tracking (Seth & Pandy, 2005a,b). Finally, muscle forces derived from these simulations can be applied to high-fidelity, subject-specific FE models of the joint to calculate the pressure patterns induced during a task. These various elements can be integrated into a framework such as the one illustrated in Fig. 7.
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Next, joint kinematics, ground reaction forces (GRF) and muscle EMG data would be recorded simultaneously during a standard gait analysis experiment performed on the volunteer subject. Three-dimensional bone movements at the knee would also be recorded simultaneously using bi-plane X-ray fluoroscopy. Bi-plane images of the subject's distal femur, proximal tibia/fibula and patella would be obtained for one gait cycle. The 3-D movements of the femur, tibia and patella would be found by projecting the solid bone models obtained from MR onto the fluoroscopic images and matching the positions of the solid models with the projections of the bones in the fluoroscopic images (Banks & Hodge, 1996). The pose-estimation step would begin with an estimate of the pose obtained from the motion capture data. The distance between the 3-D model and 2-D X-ray features would then be minimized to obtain an accurate estimate of in vivo joint kinematics, which we shall term ‘X-ray kinematics’.
The 3-D X-ray joint kinematics would then be input into the musculoskeletal model of the subject's lower limbs, and a tracking problem solved to calculate muscle loading throughout the gait cycle (Thelen & Anderson, 2005; Seth & Pandy, 2005a,b). The tracking problem would be solved using the NMT method, which uses non-linear control theory to track joint kinematics, ground forces and muscle activation signals subject to a performance criterion. The performance criterion could be to minimize metabolic energy expenditure (Bhargava et al. 2004), but other physiological constraints such as joint stability could also be taken into account in the formulation of the dynamic optimization problem. Finally, the muscle force histories and GRF obtained from a solution of the tracking problem would be applied to an FE model of the knee to calculate articular contact stresses and the compressive displacements of cartilage and the menisci throughout the gait cycle. The FE model would begin in the rigid-body X-ray pose determined from X-ray fluoroscopy, and the positions of the model bones would then be adjusted slightly to account for soft tissue interactions (cartilage and menisci). The result would be contact loads and FE deformable body kinematics based on the applied muscle forces and ligament constraints. Most of the steps outlined in Fig. 7 have been previously developed by the authors. Methods related to X-ray imaging and the pose-estimation problem have been reported by others (Banks & Hodge, 1996; Fregly et al. 2004; Li et al. 2005). Our immediate goal is to integrate the various steps described above to produce a practical and economical diagnostic tool for non-invasively assessing musculoskeletal function on a subject-specific basis.
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