Virtual Power Based Algorithm for Decoupling Large Motions from Infinitesimal Strains: Application to Shoulder Joint Biomechanics

New trends of numerical models of human joints require more and more computation of both large amplitude joint motions and fine bone stress distribution. Together, these problems are difficult to solve and very CPU time consuming. The goal of this study is to develop a new method to diminish the calculation time for this kind of problems which include calculation of large amplitude motions and infinitesimal strains. Based on the Principle of Virtual Power, the present method decouples the problem into two parts. First, rigid body motion is calculated. The bone micro-deformations are then calculated in a second part by using the results of rigid body motions as boundary conditions. A finite element model of the shoulder was used to test this decoupling technique. The model was designed to determine the influence of humeral head shape on stress distribution in the scapula for different physiological motions of the joint. Two versions of the model were developed: a first version completely deformable and a second version based on the developed decoupling method. It was shown that biomechanical variables, as mean pressure and von Mises stress, calculated with the two versions were sensibly the same. On the other hand, CPU time needed for calculating with the new decoupled technique was more than 6 times less than with the completely deformable model.     

Effects of idealized joint geometry on finite element predictions of cartilage contact stresses in the hip

Computational models may have the ability to quantify the relationship between hip morphology, cartilage mechanics and osteoarthritis. Most models have assumed the hip joint to be a perfect ball and socket joint and have neglected deformation at the bone-cartilage interface. The objective of this study was to analyze finite element (FE) models of hip cartilage mechanics with varying degrees of simplified geometry and a model with a rigid bone material assumption to elucidate the effects on predictions of cartilage stress. A previously validated subject-specific FE model of a cadaveric hip joint was used as the basis for the models. Geometry for the bone-cartilage interface was either: (1) subject-specific (i.e. irregular), (2) spherical, or (3) a rotational conchoid. Cartilage was assigned either a varying (irregular) or constant thickness (smoothed). Loading conditions simulated walking, stair-climbing and descending stairs. FE predictions of contact stress for the simplified models were compared with predictions from the subject-specific model. Both spheres and conchoids provided a good approximation of native hip joint geometry (average fitting error ～0.5 mm). However, models with spherical/conchoid bone geometry and smoothed articulating cartilage surfaces grossly underestimated peak and average contact pressures (50% and 25% lower, respectively) and overestimated contact area when compared to the subject-specific FE model. Models incorporating subject-specific bone geometry with smoothed articulating cartilage also underestimated pressures and predicted evenly distributed patterns of contact. The model with rigid bones predicted much higher pressures than the subject-specific model with deformable bones. The results demonstrate that simplifications to the geometry of the bone-cartilage interface, cartilage surface and bone material properties can have a dramatic effect on the predicted magnitude and distribution of cartilage contact pressures in the hip joint.     

Comparison of deformable and elastic foundation finite element simulations for predicting knee replacement mechanics

Rigid body total knee replacement (TKR) models with tibiofemoral contact based on elastic foundation (EF) theory utilize simple contact pressure-surface overclosure relationships to estimate joint mechanics, and require significantly less computational time than corresponding deformable finite element (FE) methods. However, potential differences in predicted kinematics between these representations are currently not well understood, and it is unclear if the estimates of contact area and pressure are acceptable. Therefore, the objectives of the current study were to develop rigid EF and deformable FE models of tibiofemoral contact, and to compare predicted kinematics and contact mechanics from both representations during gait loading conditions with three different implant designs. Linear and nonlinear contact pressure-surface overclosure relationships based on polyethylene material properties were developed using EF theory. All other variables being equal, rigid body FE models accurately estimated kinematics predicted by fully deformable FE models and required only 2% of the analysis time. As expected, the linear EF contact model sufficiently approximated trends for peak contact pressures, but overestimated the deformable results by up to 30%. The nonlinear EF contact model more accurately reproduced trends and magnitudes of the deformable analysis, with maximum differences of approximately 15% at the peak pressures during the gait cycle. All contact area predictions agreed in trend and magnitude. Using rigid models, edge-loading conditions resulted in substantial overestimation of peak pressure. Optimal nonlinear EF contact relationships were developed for specific TKR designs for use in parametric or repetitive analyses where computational time is paramount. The explicit FE analysis method utilized here provides a unique approach in that both rigid and deformable analyses can be run from the same input file, thus enabling simple selection of the most appropriate representation for the analysis of interest.     

Motion of groups of atoms in DNA studied by molecular dynamics simulation

The analysis of Molecular Dynamics simulations of two double stranded oligonucleotides is presented in terms of motions of quasi rigid subunits. First, a strategy is presented for grouping atoms submitted to concerted internal motions. The method is based on the analysis of the interatomic distance RMS matrix. It is found that each nucleotide can reasonably be decomposed into 3 or 4 rigid groups of atoms depending on the tolerance of the definition of a rigid body. In the second part, the different kinds of motions of the subunits (deformation, translation and rotation) are studied in terms of correlation using the canonical correlation analysis of data. It is shown that the residual deformation of any subunit does not influence the translational and rotational motions of the others, except perhaps for long time dynamics. Received: 3 October 1997 / Revised version: 22 December 1997 / Accepted: 23 December 1997     

Evaluation of a subject-specific finite-element model of the equine metacarpophalangeal joint under physiological load

The equine metacarpophalangeal (MCP) joint is frequently injured, especially by racehorses in training. Most injuries result from repetitive loading of the subchondral bone and articular cartilage rather than from acute events. The likelihood of injury is multi-factorial but the magnitude of mechanical loading and the number of loading cycles are believed to play an important role. Therefore, an important step in understanding injury is to determine the distribution of load across the articular surface during normal locomotion. A subject-specific finite-element model of the MCP joint was developed (including deformable cartilage, elastic ligaments, muscle forces and rigid representations of bone), evaluated against measurements obtained from cadaver experiments, and then loaded using data from gait experiments. The sensitivity of the model to force inputs, cartilage stiffness, and cartilage geometry was studied. The FE model predicted MCP joint torque and sesamoid bone flexion angles within 5% of experimental measurements. Muscle–tendon forces, joint loads and cartilage stresses all increased as locomotion speed increased from walking to trotting and finally cantering. Perturbations to muscle–tendon forces resulted in small changes in articular cartilage stresses, whereas variations in joint torque, cartilage geometry and stiffness produced much larger effects. Non-subject-specific cartilage geometry changed the magnitude and distribution of pressure and the von Mises stress markedly. The mean and peak cartilage stresses generally increased with an increase in cartilage stiffness. Areas of peak stress correlated qualitatively with sites of common injury, suggesting that further modelling work may elucidate the types of loading that precede joint injury and may assist in the development of techniques for injury mitigation.     

Computationally efficient finite element evaluation of natural patellofemoral mechanics

Finite element methods have been applied to evaluate in vivo joint behavior, new devices, and surgical techniques but have typically been applied to a small or single subject cohort. Anatomic variability necessitates the use of many subject-specific models or probabilistic methods in order to adequately evaluate a device or procedure for a population. However, a fully deformable finite element model can be computationally expensive, prohibiting large multisubject or probabilistic analyses. The aim of this study was to develop a group of subject-specific models of the patellofemoral joint and evaluate trade-offs in analysis time and accuracy with fully deformable and rigid body articular cartilage representations. Finite element models of eight subjects were used to tune a pressure-overclosure relationship during a simulated deep flexion cycle. Patellofemoral kinematics and contact mechanics were evaluated and compared between a fully deformable and a rigid body analysis. Additional eight subjects were used to determine the validity of the rigid body pressure-overclosure relationship as a subject-independent parameter. There was good agreement in predicted kinematics and contact mechanics between deformable and rigid analyses for both the tuned and test groups. Root mean square differences in kinematics were less than 0.5 deg and 0.2 mm for both groups throughout flexion. Differences in contact area and peak and average contact pressures averaged 5.4%, 9.6%, and 3.8%, respectively, for the tuned group and 6.9%, 13.1%, and 6.4%, respectively, for the test group, with no significant differences between the two groups. There was a 95% reduction in computational time with the rigid body analysis as compared with the deformable analysis. The tuned pressure-overclosure relationship derived from the patellofemoral analysis was also applied to tibiofemoral (TF) articular cartilage in a group of eight subjects. Differences in contact area and peak and average contact pressures averaged 8.3%, 11.2%, and 5.7% between rigid and deformable analyses in the tibiofemoral joint. As statistical, probabilistic, and optimization techniques can require hundreds to thousands of analyses, a viable platform is crucial to component evaluation or clinical applications. The computationally efficient rigid body platform described in this study may be integrated with statistical and probabilistic methods and has potential clinical application in understanding in vivo joint mechanics on a subject-specific or population basis.     

A multibody knee model with discrete cartilage prediction of tibio-femoral contact mechanics

Combining musculoskeletal simulations with anatomical joint models capable of predicting cartilage contact mechanics would provide a valuable tool for studying the relationships between muscle force and cartilage loading. As a step towards producing multibody musculoskeletal models that include representation of cartilage tissue mechanics, this research developed a subject-specific multibody knee model that represented the tibia plateau cartilage as discrete rigid bodies that interacted with the femur through deformable contacts. Parameters for the compliant contact law were derived using three methods: (1) simplified Hertzian contact theory, (2) simplified elastic foundation contact theory and (3) parameter optimisation from a finite element (FE) solution. The contact parameters and contact friction were evaluated during a simulated walk in a virtual dynamic knee simulator, and the resulting kinematics were compared with measured in vitro kinematics. The effects on predicted contact pressures and cartilage–bone interface shear forces during the simulated walk were also evaluated. The compliant contact stiffness parameters had a statistically significant effect on predicted contact pressures as well as all tibio-femoral motions except flexion–extension. The contact friction was not statistically significant to contact pressures, but was statistically significant to medial–lateral translation and all rotations except flexion–extension. The magnitude of kinematic differences between model formulations was relatively small, but contact pressure predictions were sensitive to model formulation. The developed multibody knee model was computationally efficient and had a computation time 283 times faster than a FE simulation using the same geometries and boundary conditions.     

Effect of image registration on 3D absorbed dose calculations in 177Lu-DOTATOC peptide receptor radionuclide therapy

Peptide receptor radionuclide therapy (PRRT) is an effective MRT (molecular radiotherapy) treatment, which consists of multiple administrations of a radiopharmaceutical labelled with ¹⁷⁷Lu or ⁹⁰Y. Through sequential functional imaging a patient specific 3D dosimetry can be derived. Multiple scans should be previously co-registered to allow accurate absorbed dose calculations. The purpose of this study is to evaluate the impact of image registration algorithms on 3D absorbed dose calculation.A cohort of patients was extracted from the database of a clinical trial in PRRT. They were administered with a single administration of ¹⁷⁷Lu-DOTATOC. All patients underwent 5 SPECT/CT sequential scans at 1 h, 4 h, 24 h, 40 h, 70 h post-injection that were subsequently registered using rigid and deformable algorithms. A similarity index was calculated to compare rigid and deformable registration algorithms. 3D absorbed dose calculation was carried out with the Raydose Monte Carlo code.The similarity analysis demonstrated the superiority of the deformable registrations (p < .001).Average absorbed dose to the kidneys calculated using rigid image registration was consistently lower than the average absorbed dose calculated using the deformable algorithm (90% of cases), with percentage differences in the range [−19; +4]%. Absorbed dose to lesions were also consistently lower (90% of cases) when calculated with rigid image registration with absorbed dose differences in the range [−67.2; 100.7]%. Deformable image registration had a significant role in calculating 3D absorbed dose to organs or lesions with volumes smaller than 100 mL.Image based 3D dosimetry for ¹⁷⁷Lu-DOTATOC PRRT is significantly affected by the type of algorithm used to register sequential SPECT/CT scans.     

Impact responses of the flexed human knee using a deformable impact interface

Blunt impact trauma to the patellofemoral joint during car accidents, sporting activities, and falls can produce a range of injuries to the knee joint, including gross bone fracture, soft tissue injury, and/or microinjuries to bone and soft tissue. Currently, the only well-established knee injury criterion applies to knee impacts suffered during car accidents. This criterion is based solely on the peak impact load delivered to seated cadavers having a single knee flexion angle. More recent studies, however, suggest that the injury potential, its location, and the characteristics of the damage are also a function of knee flexion angle and the stiffness of the impacting structure. For example, at low flexion angles, fractures of the distal patella are common with a rigid impact interface, while at high flexion angles splitting of the femoral condyles is more evident. Low stiffness impact surfaces have been previously shown to distribute impact loads over the anterior surface of the patella to help mitigate gross and microscopic injuries in the 90 deg flexed knee. The objective of the current study was to determine if a deformable impact interface would just as effectively mitigate gross and microscopic injuries to the knee at various flexion angles. Paired experiments were conducted on contralateral knees of 18 human cadavers at three flexion angles (60, 90, 120 deg). One knee was subjected to a fracture level impact experiment with a rigid impactor, and the opposite knee was impacted with a deformable interface (3.3 MPa crush strength honeycomb material) to the same load. This (deformable) impact interface was effective at mitigating gross bone fractures at approximately 5 kN at all flexion angles, but the frequency of split fracture of the femoral condyles may not have been significantly reduced at 120 deg flexion. On the other hand, this deformable interface was not effective in mitigating microscopic injuries observed for all knee flexion angles. These new data, in concert with the existing literature, suggest the chosen impact interface was not optimal for knee injury protection in that fracture and other minor injuries were still produced. For example, in 18 cadavers a total of 20 gross fractures and 20 subfracture injuries were produced with a rigid interface and 5 gross fractures and 21 subfracture injuries with the deformable interface selected for the current study. Additional studies will be needed to optimize the knee impact interface for protection against gross and microscopic injuries to the knee.     

A theoretical model of the knee and ACL: theory and experimental verification.

A three-dimensional mathematical model of the human knee joint was developed to examine the role of single ligaments, such as an anterior cruciate ligament (ACL) graft in ACL reconstruction, on joint motion and tissue forces. The model is linear and valid for small motions about an equilibrium position. The knee joint is modeled as two rigid bodies (the femur and the tibia) interconnected by deformable structures, including the ACL or ACL graft, the cartilage layer, and the remainder of the knee tissues (modeled as a single element). The model was demonstrated for the equilibrium condition of the knee in extension with an anterior tibial force, causing anterior drawer and hyperextension. The knee stiffness matrix for this condition was measured for a human right knee in vitro. Predicted model response was compared with experimental observations. Qualitative agreement was found between model and experiment, validating the model and its assumptions. The model was then used to predict the change in graft and cartilage forces and joint motion of the knee due to an increment of load in the normal joint both after ACL removal and with various altered states simulating ACL reconstructions. Results illustrate the interdependence between loads in the ACL graft, other knee structures, and contact force. Stiffer grafts and smaller maximum unloaded length of the ligament lead to higher graft and contact forces. Changes in cartilage stiffness alter load sharing between ACL graft and other joint tissues.     

'Equivalent geometry' of the knee and the prediction of tensions along the cruciates: an experimental study

In this paper we introduce the concept of the functional (or equivalent) geometry of the knee, which is an attempt to reduce the natural knee with its complex geometry, frictional resistance and deformable cartilage into a two-dimensional joint comprising rigid femur and tibia in frictionless contact. An apparatus and method are described to measure the slope of the tangent to the surfaces of the 'equivalent' bones at their 'point' of contact. An antero posterior force of +/-300-500 N and axial compressive load of twice body weight were applied on cadaveric knee joints. The corresponding displacement of the tibia in the saggital plane was measured firstly with both cruciates intact and then when each was severed in turn. From the data obtained both the slope of the tangent mentioned above and the tensions developed along the cruciates under the influence of the forces applied were calculated. The results showed that the functional geometry of the knee in the saggital plane can be represented by a convex femur and a concave tibia. The tensions along the cruciates calculated on the basis of the experimental measurements were nearly always lower than the antero posterior force applied, and although this corroborated the trend demonstrated in a previous theoretical analysis, they were lower still. The reason for this may be the deformation of the cartilage under load, thus modifying the geometry of contact resulting in a more concave tibia of the 'equivalent' knee joint, than that of the rigid model used in the theoretical analysis.     

Estimating joint kinematics from skin motion observation: modelling and validation

Modelling of soft tissue motion is required in many areas, such as computer animation, surgical simulation, 3D motion analysis and gait analysis. In this paper, we will focus on the use of modelling of skin deformation during 3D motion analysis. The most frequently used method in 3D human motion analysis involves placing markers on the skin of the analysed segment which is composed of the rigid bone and the surrounding soft tissues. Skin and soft tissue deformations introduce a significant artefact which strongly influences the resulting bone position, orientation and joint kinematics. For this study, we used a statistical solid dynamics approach which is a combination of several previously reported tools: the point cluster technique (PCT) and a Kalman filter which was added to the PCT. The methods were tested and evaluated on controlled human-arm motions, using an optical motion capture system (Vicon^TM).

The addition of a Kalman filter to the PCT for rigid body motion estimation results in a smoother signal that better represents the joint motion. Calculations indicate less signal distortion than when using a digital low-pass filter. Furthermore, adding a Kalman filter to the PCT substantially reduces the dispersion of the maximal and minimal instantaneous frequencies. For controlled human movements, the result indicated that adding a Kalman filter to the PCT produced a more accurate signal. However, it could not be concluded that the proposed Kalman filter is better than a low-pass filter for estimation of the motion. We suggest that implementation of a Kalman filter with a better biomechanical motion model will be more likely to improve the results.     

Estimating joint kinematics from skin motion observation: modelling and validation

Modelling of soft tissue motion is required in many areas, such as computer animation, surgical simulation, 3D motion analysis and gait analysis. In this paper, we will focus on the use of modelling of skin deformation during 3D motion analysis. The most frequently used method in 3D human motion analysis involves placing markers on the skin of the analysed segment which is composed of the rigid bone and the surrounding soft tissues. Skin and soft tissue deformations introduce a significant artefact which strongly influences the resulting bone position, orientation and joint kinematics. For this study, we used a statistical solid dynamics approach which is a combination of several previously reported tools: the point cluster technique (PCT) and a Kalman filter which was added to the PCT. The methods were tested and evaluated on controlled human-arm motions, using an optical motion capture system (Vicon(TM)). The addition of a Kalman filter to the PCT for rigid body motion estimation results in a smoother signal that better represents the joint motion. Calculations indicate less signal distortion than when using a digital low-pass filter. Furthermore, adding a Kalman filter to the PCT substantially reduces the dispersion of the maximal and minimal instantaneous frequencies. For controlled human movements, the result indicated that adding a Kalman filter to the PCT produced a more accurate signal. However, it could not be concluded that the proposed Kalman filter is better than a low-pass filter for estimation of the motion. We suggest that implementation of a Kalman filter with a better biomechanical motion model will be more likely to improve the results.     

Calculation of the instantaneous centre of rotation for a rigid body

The mathematical procedure presented is used for the calculation of the instantaneous centre of rotation for a rigid body based on kinematic measurement data. For every rotation angle interval observed an average centre of rotation is calculated by an appropriate optimizing calculation similar to data approximation by moving average. Practical applicability of this procedure has been proved by means of two planar and one three-dimensional test measurements of known movements (pure rotation, rolling motion and motion of a ball-and-socket joint.     

Rotational stiffness of football shoes influences talus motion during external rotation of the foot

Shoe-surface interface characteristics have been implicated in the high incidence of ankle injuries suffered by athletes. Yet, the differences in rotational stiffness among shoes may also influence injury risk. It was hypothesized that shoes with different rotational stiffness will generate different patterns of ankle ligament strain. Four football shoe designs were tested and compared in terms of rotational stiffness. Twelve (six pairs) male cadaveric lower extremity limbs were externally rotated 30 deg using two selected football shoe designs, i.e., a flexible shoe and a rigid shoe. Motion capture was performed to track the movement of the talus with a reflective marker array screwed into the bone. A computational ankle model was utilized to input talus motions for the estimation of ankle ligament strains. At 30 deg of rotation, the rigid shoe generated higher ankle joint torque at 46.2?±?9.3 Nm than the flexible shoe at 35.4?±?5.7 Nm. While talus rotation was greater in the rigid shoe (15.9?±?1.6 deg versus 12.1?±?1.0 deg), the flexible shoe generated more talus eversion (5.6?±?1.5 deg versus 1.2± 0.8 deg). While these talus motions resulted in the same level of anterior deltoid ligament strain (approxiamtely 5%) between shoes, there was a significant increase of anterior tibiofibular ligament strain (4.5± 0.4% versus 2.3?±?0.3%) for the flexible versus more rigid shoe design. The flexible shoe may provide less restraint to the subtalar and transverse tarsal joints, resulting in more eversion but less axial rotation of the talus during foot∕shoe rotation. The increase of strain in the anterior tibiofibular ligament may have been largely due to the increased level of talus eversion documented for the flexible shoe. There may be a direct correlation of ankle joint torque with axial talus rotation, and an inverse relationship between torque and talus eversion. The study may provide some insight into relationships between shoe design and ankle ligament strain patterns. In future studies, these data may be useful in characterizing shoe design parameters and balancing potential ankle injury risks with player performance.     

Cytosine ribose flexibility in DNA: a combined NMR 13C spin relaxation and molecular dynamics simulation study

Using (13)C spin relaxation NMR in combination with molecular dynamic (MD) simulations, we characterized internal motions within double-stranded DNA on the pico- to nano-second time scale. We found that the C-H vectors in all cytosine ribose moieties within the Dickerson-Drew dodecamer (5'-CGCGAATTCGCG-3') are subject to high amplitude motions, while the other nucleotides are essentially rigid. MD simulations showed that repuckering is a likely motional model for the cytosine ribose moiety. Repuckering occurs with a time constant of around 100 ps. Knowledge of DNA dynamics will contribute to our understanding of the recognition specificity of DNA-binding proteins such as cytosine methyltransferase.     

Precision measurements of the RSA method using a phantom model of hip prosthesis

Radiostereometric analysis (RSA) has become one of the recommended techniques for pre-market evaluation of new joint implant designs. In this study we evaluated the effect of repositioning of X-ray tubes and phantom model on the precision of the RSA method. In precision measurements, we utilized mean error of rigid body fitting (ME) values as an internal control for examinations. ME value characterizes relative motion among the markers within each rigid body and is conventionally used to detect loosening of a bone marker. Three experiments, each consisting of 10 double examinations, were performed. In the first experiment, the X-ray tubes and the phantom model were not repositioned between one double examination. In experiments two and three, the X-ray tubes were repositioned between one double examination. In addition, the position of the phantom model was changed in experiment three. Results showed that significant differences could be found in 2 of 12 comparisons when evaluating the translation and rotation of the prosthetic components. Repositioning procedures increased ME values mimicking deformation of rigid body segments. Thus, ME value seemed to be a more sensitive parameter than migration values in this study design. These results confirmed the importance of standardized radiographic technique and accurate patient positioning for RSA measurements. Standardization and calibration procedures should be performed with phantom models in order to avoid unnecessary radiation dose of the patients. The present model gives the means to establish and to follow the intra-laboratory precision of the RSA method. The model is easily applicable in any research unit and allows the comparison of the precision values in different laboratories of multi-center trials.     

Time-correlation analysis of simulated water motion in flexible and rigid gramicidin channels.

Molecular dynamics simulations have been done on a system consisting of the polypeptide membrane channel former gramicidin, plus water molecules in the channel and caps of waters at the two ends of the channel. In the absence of explicit simulation of the surrounding membrane, the helical form of the channel was maintained by artificial restraints on the peptide motion. The characteristic time constant of the artificial restraint was varied to assess the effect of the restraints on the channel structure and water motions. Time-correlation analysis was done on the motions of individual channel waters and on the motions of the center of mass of the channel waters. It is found that individual water molecules confined in the channel execute higher frequency motions than bulk water, for all degrees of channel peptide restraint. The center-of-mass motion of the chain of channel waters (which is the motion that is critical for transmembrane transport, due to the mandatory single filing of water in the channel) does not exhibit these higher frequency motions. The mobility of the water chain is dramatically reduced by holding the channel rigid. Thus permeation through the channel is not like flow through a rigid pipe; rather permeation is facilitated by peptide motion. For the looser restraints we used, the mobility of the water chain was not very much affected by the degree of restraint. Depending on which set of experiments is considered, the computed mobility of our water chain in the flexible channel is four to twenty times too high to account for the experimentally measured resistance of the gramicidin channel to water flow. From this result it appears likely that the peptide motions of an actual gramicidin channel embedded in a lipid membrane may be more restrained than in our flexible channel model, and that these restraints may be a significant modulator of channel permeability. For the completely rigid channel model the "trapping" of the water molecules in preferred positions throughout the molecular dynamics run precludes a reasonable assessment of mobility, but it seems to be quite low.