Optimising muscle parameters in musculoskeletal models using Monte Carlo simulation

The use of musculoskeletal simulation software has become a useful tool for modelling joint and muscle forces during human activity, including in reduced gravity because direct experimentation is difficult. Knowledge of muscle and joint loads can better inform the design of exercise protocols and exercise countermeasure equipment. In this study, the LifeModeler? (San Clemente, CA, USA) biomechanics simulation software was used to model a squat exercise. The initial model using default parameters yielded physiologically reasonable hip-joint forces but no activation was predicted in some large muscles such as rectus femoris, which have been shown to be active in 1-g performance of the activity. Parametric testing was conducted using Monte Carlo methods and combinatorial reduction to find a muscle parameter set that more closely matched physiologically observed activation patterns during the squat exercise. The rectus femoris was predicted to peak at 60.1% activation in the same test case compared to 19.2% activation using default parameters. These results indicate the critical role that muscle parameters play in joint force estimation and the need for exploration of the solution space to achieve physiologically realistic muscle activation.     

Role of biomechanics and muscle activation strategy in the production of endpoint force patterns in the cat hindlimb

We used a musculoskeletal model of the cat hindlimb to compare the patterns of endpoint forces generated by all possible combination of 12 hindlimb muscles under three different muscle activation rules: homogeneous activation of muscles based on uniform activation levels, homogeneous activation of muscles based on uniform (normalized) force production, and activation based on the topography of spinal motoneuron pools. Force patterns were compared with the patterns obtained experimentally by microstimulation of the lumbar spinal cord in spinal intact cats. Magnitude and orientation of the force patterns were compared, as well as the proportion of the types found, and the proportions of patterns exhibiting points of zero force (equilibrium points). The force patterns obtained with the homogenous activation and motoneuron topography models were quite similar to those measured experimentally, with the differences being larger for the patterns from the normalized endpoint forces model. Differences in the proportions of types of force patterns between the three models and the experimental results were significant for each model. Both homogeneous activation and normalized endpoint force models produced similar proportions of equilibrium points as found experimentally. The results suggest that muscle biomechanics play an important role in limiting the number of endpoint force pattern types, and that muscle combinations activated at similar levels reproduced best the experimental results obtained with intraspinal microstimulation.     

Subject-specific tendon-aponeurosis definition in hill-type model predicts higher muscle forces in dynamic tasks

Neuromusculoskeletal models are a common method to estimate muscle forces. Developing accurate neuromusculoskeletal models is a challenging task due to the complexity of the system and large inter-subject variability. The estimation of muscles force is based on the mechanical properties of tendon-aponeurosis complex. Most neuromusculoskeletal models use a generic definition of the tendon-aponeurosis complex based on in vitro test, perhaps limiting their validity. Ultrasonography allows subject-specific estimates of the tendon-aponeurosis complex's mechanical properties. The aim of this study was to investigate the influence of subject-specific mechanical properties of the tendon-aponeurosis complex on a neuromusculoskeletal model of the ankle joint. Seven subjects performed isometric contractions from which the tendon-aponeurosis force-strain relationship was estimated. Hopping and running tasks were performed and muscle forces were estimated using subject-specific tendon-aponeurosis and generic tendon properties. Two ultrasound probes positioned over the muscle-tendon junction and the mid-belly were combined with motion capture to estimate the in vivo tendon and aponeurosis strain of the medial head of gastrocnemius muscle. The tendon-aponeurosis force-strain relationship was scaled for the other ankle muscles based on tendon and aponeurosis length of each muscle measured by ultrasonography. The EMG-driven model was calibrated twice - using the generic tendon definition and a subject-specific tendon-aponeurosis force-strain definition. The use of subject-specific tendon-aponeurosis definition leads to a higher muscle force estimate for the soleus muscle and the plantar-flexor group, and to a better model prediction of the ankle joint moment compared to the model estimate which used a generic definition. Furthermore, the subject-specific tendon-aponeurosis definition leads to a decoupling behaviour between the muscle fibre and muscle-tendon unit in agreement with previous experiments using ultrasonography. These results indicate the use of subject-specific tendon-aponeurosis definitions in a neuromusculoskeletal model produce better agreement with measured external loads and more physiological model behaviour.     

Non-rigid deformation to include subject-specific detail in musculoskeletal models of CP children with proximal femoral deformity and its effect on muscle and contact forces during gait

To account for proximal femoral deformities in children with cerebral palsy (CP), subject-specific musculoskeletal models are needed. Non-rigid deformation (NRD) deforms generic onto personalized bone geometry and thereby transforms the muscle points. The goal of this study was to determine to what extent the models and simulation outcomes in CP patients differ when including subject-specific detail using NRD or Magnetic Resonance Imaging (MRI)-based models. The NRD models slightly overestimated hip contact forces compared to MRI models and differences in muscle point positions and moment arm lengths (MALs) remained, although differences were smaller than for the generic model.     

Subject-specific hip geometry and hip joint centre location affects calculated contact forces at the hip during gait

Hip loading affects the development of hip osteoarthritis, bone remodelling and osseointegration of implants. In this study, we analyzed the effect of subject-specific modelling of hip geometry and hip joint centre (HJC) location on the quantification of hip joint moments, muscle moments and hip contact forces during gait, using musculoskeletal modelling, inverse dynamic analysis and static optimization. For 10 subjects, hip joint moments, muscle moments and hip loading in terms of magnitude and orientation were quantified using three different model types, each including a different amount of subject-specific detail: (1) a generic scaled musculoskeletal model, (2) a generic scaled musculoskeletal model with subject-specific hip geometry (femoral anteversion, neck-length and neck-shaft angle) and (3) a generic scaled musculoskeletal model with subject-specific hip geometry including HJC location. Subject-specific geometry and HJC location were derived from CT. Significant differences were found between the three model types in HJC location, hip flexion–extension moment and inclination angle of the total contact force in the frontal plane. No model agreement was found between the three model types for the calculation of contact forces in terms of magnitude and orientations, and muscle moments. Therefore, we suggest that personalized models with individualized hip joint geometry and HJC location should be used for the quantification of hip loading. For biomechanical analyses aiming to understand modified hip joint loading, and planning hip surgery in patients with osteoarthritis, the amount of subject-specific detail, related to bone geometry and joint centre location in the musculoskeletal models used, needs to be considered.     

Neuromusculoskeletal modeling: estimation of muscle forces and joint moments and movements from measurements of neural command

This paper provides an overview of forward dynamic neuromusculoskeletal modeling. The aim of such models is to estimate or predict muscle forces, joint moments, and/or joint kinematics from neural signals. This is a four-step process. In the first step, muscle activation dynamics govern the transformation from the neural signal to a measure of muscle activation-a time varying parameter between 0 and 1. In the second step, muscle contraction dynamics characterize how muscle activations are transformed into muscle forces. The third step requires a model of the musculoskeletal geometry to transform muscle forces to joint moments. Finally, the equations of motion allow joint moments to be transformed into joint movements. Each step involves complex nonlinear relationships. The focus of this paper is on the details involved in the first two steps, since these are the most challenging to the biomechanician. The global process is then explained through applications to the study of predicting isometric elbow moments and dynamic knee kinetics.     

Biomechanics of the rabbit knee and ankle: muscle, ligament, and joint contact force predictions

Mathematical models of small animals that predict in vivo forces acting on the lower extremities are critical for studies of musculoskeletal biomechanics and diseases. Rabbits are advantageous in this regard because they remodel their cortical bone similar to humans. Here, we enhance a recent mathematical model of the rabbit knee joint to include the loading behavior of individual muscles, ligaments, and joint contact at the knee and ankle during the stance phase of hopping. Geometric data from the hindlimbs of three adult New Zealand white rabbits, combined with previously reported intersegmental forces and moments, were used as inputs to the model. Muscle, ligament, and joint contact forces were computed using optimization techniques assuming that muscle endurance is maximized and ligament strain energy resists tibial shear force along an inclined plateau. Peak forces developed by the quadriceps and gastrocnemius muscle groups and by compressive knee contact were within the range of theoretical and in vivo predictions. Although a minimal force was carried by the anterior cruciate and medial collateral ligaments, force patterns in the posterior cruciate ligament were consistent with in vivo tibial displacement patterns during hopping in rabbits. Overall, our predictions compare favorably with theoretical estimates and in vivo measurements in rabbits, and enhance previous models by providing individual muscle, ligament, and joint contact information to predict in vivo forces acting on the lower extremities in rabbits.     

Subject-specific knee joint geometry improves predictions of medial tibiofemoral contact forces

Estimating tibiofemoral joint contact forces is important for understanding the initiation and progression of knee osteoarthritis. However, tibiofemoral contact force predictions are influenced by many factors including muscle forces and anatomical representations of the knee joint. This study aimed to investigate the influence of subject-specific geometry and knee joint kinematics on the prediction of tibiofemoral contact forces using a calibrated EMG-driven neuromusculoskeletal model of the knee. One participant fitted with an instrumented total knee replacement walked at a self-selected speed while medial and lateral tibiofemoral contact forces, ground reaction forces, whole-body kinematics, and lower-limb muscle activity were simultaneously measured. The combination of generic and subject-specific knee joint geometry and kinematics resulted in four different OpenSim models used to estimate muscle–tendon lengths and moment arms. The subject-specific geometric model was created from CT scans and the subject-specific knee joint kinematics representing the translation of the tibia relative to the femur was obtained from fluoroscopy. The EMG-driven model was calibrated using one walking trial, but with three different cost functions that tracked the knee flexion/extension moments with and without constraint over the estimated joint contact forces. The calibrated models then predicted the medial and lateral tibiofemoral contact forces for five other different walking trials. The use of subject-specific models with minimization of the peak tibiofemoral contact forces improved the accuracy of medial contact forces by 47% and lateral contact forces by 7%, respectively compared with the use of generic musculoskeletal model.     

The effects of electromyography-assisted modelling in estimating musculotendon forces during gait in children with cerebral palsy

Neuro-musculoskeletal modelling can provide insight into the aberrant muscle function during walking in those suffering cerebral palsy (CP). However, such modelling employs optimization to estimate muscle activation that may not account for disturbed motor control and muscle weakness in CP. This study evaluated different forms of neuro-musculoskeletal model personalization and optimization to estimate musculotendon forces during gait of nine children with CP (GMFCS I-II) and nine typically developing (TD) children. Data collection included 3D-kinematics, ground reaction forces, and electromyography (EMG) of eight lower limb muscles. Four different optimization methods estimated muscle activation and musculotendon forces of a scaled-generic musculoskeletal model for each child walking, i.e. (i) static optimization that minimized summed-excitation squared; (ii) static optimization with maximum isometric muscle forces scaled to body mass; (iii) an EMG-assisted approach using optimization to minimize summed-excitation squared while reducing tracking errors of experimental EMG-linear envelopes and joint moments; and (iv) EMG-assisted with musculotendon model parameters first personalized by calibration. Both static optimization approaches showed a relatively low model performance compared to EMG envelopes. EMG-assisted approaches performed much better, especially in CP, with only a minor mismatch in joint moments. Calibration did not affect model performance significantly, however it did affect musculotendon forces, especially in CP. A model more consistent with experimental measures is more likely to yield more physiologically representative results. Therefore, this study highlights the importance of calibrated EMG-assisted modelling when estimating musculotendon forces in TD children and even more so in children with CP.     

Can a reduction approach predict reliable joint contact and musculo-tendon forces?

Musculoskeletal models generally solve the muscular redundancy by numerical optimisation. They have been extensively validated using instrumented implants. Conversely, a reduction approach considers only one flexor or extensor muscle group at the time to equilibrate the inter-segmental joint moment. It is not clear if such models can still predict reliable joint contact and musculo-tendon forces during gait.Tibiofemoral contact force and gastrocnemii, quadriceps, and hamstrings musculo-tendon forces were estimated using a reduction approach for five subjects walking with an instrumented prosthesis. The errors in the proximal-distal tibiofemoral contact force fell in the range (0.3–0.9 body weight) reported in the literature for musculoskeletal models using numerical optimisation. The musculo-tendon forces were in agreement with the EMG envelops and appeared comparable to the ones reported in the literature with generic musculoskeletal models.Although evident simplifications and limitations, it seems that the reduction approach can provided quite reliable results. It can be a useful pedagogical tool in biomechanics, e.g. to illustrate the theoretical differences between inter-segmental and contact forces, and can provide a first estimate of the joint loadings in subjects with limited musculoskeletal deformities and neurological disorders.     

A musculoskeletal foot model for clinical gait analysis

Several full body musculoskeletal models have been developed for research applications and these models may potentially be developed into useful clinical tools to assess gait pathologies. Existing full-body musculoskeletal models treat the foot as a single segment and ignore the motions of the intrinsic joints of the foot. This assumption limits the use of such models in clinical cases with significant foot deformities. Therefore, a three-segment musculoskeletal model of the foot was developed to match the segmentation of a recently developed multi-segment kinematic foot model. All the muscles and ligaments of the foot spanning the modeled joints were included. Muscle pathways were adjusted with an optimization routine to minimize the difference between the muscle flexion–extension moment arms from the model and moment arms reported in literature. The model was driven by walking data from five normal pediatric subjects (aged 10.6±1.57 years) and muscle forces and activation levels required to produce joint motions were calculated using an inverse dynamic analysis approach. Due to the close proximity of markers on the foot, small marker placement error during motion data collection may lead to significant differences in musculoskeletal model outcomes. Therefore, an optimization routine was developed to enforce joint constraints, optimally scale each segment length and adjust marker positions. To evaluate the model outcomes, the muscle activation patterns during walking were compared with electromyography (EMG) activation patterns reported in the literature. Model-generated muscle activation patterns were observed to be similar to the EMG activation patterns.     

Static optimization underestimates antagonist muscle activity at the glenohumeral joint: A musculoskeletal modeling study

Static optimization is commonly employed in musculoskeletal modeling to estimate muscle and joint loading; however, the ability of this approach to predict antagonist muscle activity at the shoulder is poorly understood. Antagonist muscles, which contribute negatively to a net joint moment, are known to be important for maintaining glenohumeral joint stability. This study aimed to compare muscle and joint force predictions from a subject-specific neuromusculoskeletal model of the shoulder driven entirely by measured muscle electromyography (EMG) data with those from a musculoskeletal model employing static optimization. Four healthy adults performed six sub-maximal upper-limb contractions including shoulder abduction, adduction, flexion, extension, internal rotation and external rotation. EMG data were simultaneously measured from 16 shoulder muscles using surface and intramuscular electrodes, and joint motion evaluated using video motion analysis. Muscle and joint forces were calculated using both a calibrated EMG-driven neuromusculoskeletal modeling framework, and musculoskeletal model simulations that employed static optimization. The EMG-driven model predicted antagonistic muscle function for pectoralis major, latissimus dorsi and teres major during abduction and flexion; supraspinatus during adduction; middle deltoid during extension; and subscapularis, pectoralis major and latissimus dorsi during external rotation. In contrast, static optimization neural solutions showed little or no recruitment of these muscles, and preferentially activated agonistic prime movers with large moment arms. As a consequence, glenohumeral joint force calculations varied substantially between models. The findings suggest that static optimization may under-estimate the activity of muscle antagonists, and therefore, their contribution to glenohumeral joint stability.     

To what extent is joint and muscle mechanics predicted by musculoskeletal models sensitive to soft tissue artefacts?

Musculoskeletal models are widely used to estimate joint kinematics, intersegmental loads, and muscle and joint contact forces during movement. These estimates can be heavily affected by the soft tissue artefact (STA) when input positional data are obtained using stereophotogrammetry, but this aspect has not yet been fully characterised for muscle and joint forces. This study aims to assess the sensitivity to the STA of three open-source musculoskeletal models, implemented in OpenSim.A baseline dataset of marker trajectories was created for each model from experimental data of one healthy volunteer. Five hundred STA realizations were then statistically generated using a marker-dependent model of the pelvis and lower limb artefact and added to the baseline data. The STA׳s impact on the musculoskeletal model estimates was finally quantified using a Monte Carlo analysis.The modelled STA distributions were in line with the literature. Observed output variations were comparable across the three models, and sensitivity to the STA was evident for most investigated quantities. Shape, magnitude and timing of the joint angle and moment time histories were not significantly affected throughout the entire gait cycle, whereas magnitude variations were observed for muscle and joint forces. Ranges of contact force variations differed between joints, with hip variations up to 1.8 times body weight observed. Variations of more than 30% were observed for some of the muscle forces.In conclusion, musculoskeletal simulations using stereophotogrammetry may be safely run when only interested in overall output patterns. Caution should be paid when more accurate estimated values are needed.     

Validation of lumbar spine loading from a musculoskeletal model including the lower limbs and lumbar spine

Low back mechanics are important to quantify to study injury, pain and disability. As in vivo forces are difficult to measure directly, modeling approaches are commonly used to estimate these forces. Validation of model estimates is critical to gain confidence in modeling results across populations of interest, such as people with lower-limb amputation. Motion capture, ground reaction force and electromyographic data were collected from ten participants without an amputation (five male/five female) and five participants with a unilateral transtibial amputation (four male/one female) during trunk-pelvis range of motion trials in flexion/extension, lateral bending and axial rotation. A musculoskeletal model with a detailed lumbar spine and the legs including 294 muscles was used to predict L4-L5 loading and muscle activations using static optimization. Model estimates of L4-L5 intervertebral joint loading were compared to measured intradiscal pressures from the literature and muscle activations were compared to electromyographic signals. Model loading estimates were only significantly different from experimental measurements during trunk extension for males without an amputation and for people with an amputation, which may suggest a greater portion of L4-L5 axial load transfer through the facet joints, as facet loads are not captured by intradiscal pressure transducers. Pressure estimates between the model and previous work were not significantly different for flexion, lateral bending or axial rotation. Timing of model-estimated muscle activations compared well with electromyographic activity of the lumbar paraspinals and upper erector spinae. Validated estimates of low back loading can increase the applicability of musculoskeletal models to clinical diagnosis and treatment.     

A computationally efficient strategy to estimate muscle forces in a finite element musculoskeletal model of the lower limb

Concurrent multiscale simulation strategies are required in computational biomechanics to study the interdependence between body scales. However, detailed finite element models rarely include muscle recruitment due to the computational burden of both the finite element method and the optimization strategies widely used to estimate muscle forces. The aim of this study was twofold: first, to develop a computationally efficient muscle force prediction strategy based on proportional-integral-derivative (PID) controllers to track gait and chair rise experimental joint motion with a finite element musculoskeletal model of the lower limb, including a deformable knee representation with 12 degrees of freedom; and, second, to demonstrate that the inclusion of joint-level deformability affects muscle force estimation by using two different knee models and comparing muscle forces between the two solutions. The PID control strategy tracked experimental hip, knee, and ankle flexion/extension with root mean square errors below 1°, and estimated muscle, contact and ligament forces in good agreement with previous results and electromyography signals. Differences up to 11% and 20% in the vasti and biceps femoris forces, respectively, were observed between the two knee models, which might be attributed to a combination of differing joint contact geometry, ligament behavior, joint kinematics, and muscle moment arms. The tracking strategy developed in this study addressed the inevitable tradeoff between computational cost and model detail in musculoskeletal simulations and can be used with finite element musculoskeletal models to efficiently estimate the interdependence between muscle forces and tissue deformation.     

An EMG-based, muscle driven forward simulation of single support phase of gait

This paper describes the process used to generate lower limb kinematics during single limb stance phase of gait, using musculoskeletal modelling, muscle driven forward simulation and gradient based optimisation techniques (including design of experiment techniques).Initial inputs to the forward simulation process were the normalised quantified muscle activation patterns of 22 muscles, and the initial segmental configuration (both angles and angular velocity) derived from Winter (The biomechanics and motor control of human gait, 1987, University of Waterloo Press, pp. 1-72). Two distinct musculoskeletal models (one including 6 DOF, the other 7 DOF) were defined and a muscle driven forward simulation was implemented.A series of optimisation sequences then were executed to modify the muscle activation patterns and initial segmental configuration, until the system output of the forward simulation approximated the angle data reported by. The accuracy and effectiveness of the analysis sequence proposed and the model response obtained using two distinct musculoskeletal models were verified and analysed with respect to the kinesiology of normal walking.     

Accuracy of generic musculoskeletal models in predicting the functional roles of muscles in human gait

Biomechanical assessments of muscle function are often performed using a generic musculoskeletal model created from anatomical measurements obtained from cadavers. Understanding the validity of using generic models to study movement biomechanics is critical, especially when such models are applied to analyze the walking patterns of persons with impaired mobility. The aim of this study was to evaluate the accuracy of scaled-generic models in determining the moment arms and functional roles of the lower-limb muscles during gait. The functional role of a muscle was described by its potential to contribute to the acceleration of a joint or the acceleration of the whole-body center of mass. A muscle's potential acceleration was defined as the acceleration induced by a unit of muscle force. Dynamic simulations of walking were generated for four children with cerebral palsy and five age-matched controls. Each subject was represented by a scaled-generic model and a model developed from magnetic resonance (MR) imaging. Calculations obtained from the scaled-generic model of each subject were evaluated against those derived from the corresponding MR-based model. Substantial differences were found in the muscle moment arms computed using the two models. These differences propagated to calculations of muscle potential accelerations, but predictions of muscle function (i.e., the direction in which a muscle accelerated a joint or the center of mass and the magnitude of the muscle's potential acceleration relative to that of other muscles) were consistent between the two modeling techniques. Our findings suggest that scaled-generic models and image-based models yield similar assessments of muscle function in both normal and pathological gait.     

Are Subject-Specific Musculoskeletal Models Robust to the Uncertainties in Parameter Identification?

Subject-specific musculoskeletal modeling can be applied to study musculoskeletal disorders, allowing inclusion of personalized anatomy and properties. Independent of the tools used for model creation, there are unavoidable uncertainties associated with parameter identification, whose effect on model predictions is still not fully understood. The aim of the present study was to analyze the sensitivity of subject-specific model predictions (i.e., joint angles, joint moments, muscle and joint contact forces) during walking to the uncertainties in the identification of body landmark positions, maximum muscle tension and musculotendon geometry. To this aim, we created an MRI-based musculoskeletal model of the lower limbs, defined as a 7-segment, 10-degree-of-freedom articulated linkage, actuated by 84 musculotendon units. We then performed a Monte-Carlo probabilistic analysis perturbing model parameters according to their uncertainty, and solving a typical inverse dynamics and static optimization problem using 500 models that included the different sets of perturbed variable values. Model creation and gait simulations were performed by using freely available software that we developed to standardize the process of model creation, integrate with OpenSim and create probabilistic simulations of movement. The uncertainties in input variables had a moderate effect on model predictions, as muscle and joint contact forces showed maximum standard deviation of 0.3 times body-weight and maximum range of 2.1 times body-weight. In addition, the output variables significantly correlated with few input variables (up to 7 out of 312) across the gait cycle, including the geometry definition of larger muscles and the maximum muscle tension in limited gait portions. Although we found subject-specific models not markedly sensitive to parameter identification, researchers should be aware of the model precision in relation to the intended application. In fact, force predictions could be affected by an uncertainty in the same order of magnitude of its value, although this condition has low probability to occur.     

Musculoskeletal model choice influences hip joint load estimations during gait

The prevalence of musculoskeletal modeling studies investigating hip contact forces and the number of models used to conduct such investigations has increased in recent years. However, the consistency between models remain unknown and differences in model predicted hip contact forces between studies are difficult to distinguish from natural inter-individual differences. The purpose of this study was therefore to evaluate differences in hip joint contact forces during gait between four OpenSim models. These models included the generic models gait2392 and the Arnold Lower Limb Model, as well as the hip specific models hip2372 and London Lower Limb Model. Data from four individuals who have had a total hip replacement with instrumented hip implants performing slow, normal, and fast walking trials were taken from the HIP98 database to evaluate the various models effectiveness at estimating hip loads. Muscle forces were estimated using static optimization and hip contact forces were calculated using the JointReaction analysis in OpenSim. Results indicated that, for gait, the hip specific London Lower Limb Model consistently predicted peak push-off hip joint contact forces with lower magnitude and timing errors compared to the other models. Likewise, root mean square error values were lowest and correlation coefficients were highest for the London Lower Limb Model. These results suggest that the London Lower Limb Model is the most appropriate model for investigations focused on hip joint loading.     

Biomechanical model calculation of muscle contraction forces: a double linear programming method

This paper presents a novel scheme for the use of linear programming to calculate muscle contraction forces in models describing musculoskeletal system biomechanics. Models of this kind are frequently found in the biomechanics literature. In most cases they involve muscle contraction force calculations that are statically indeterminate, and hence use optimization techniques to make those calculations. We present a linear programming optimization technique that solves a two-objective problem with two sequential linear programs. We use the technique here to minimize muscle intensity and joint compression force, since those are commonly used objectives. The two linear program model has the advantages of low computation cost, ready implementation on a micro-computer, and stable solutions. We show how to solve the model analytically in simple cases. We also discuss the use of the dual problem of linear programming to gain understanding of the solution it provides.