8 Bartel Chapter 2

8.0.1 Basic concepts

8.0.1.0.1 Viewpoints for analysis of biomechanical systems

Two outcomes are generally sought:

  • Understand the motion (rigid body kinematics)

  • Understand the stress, strain, and deformation

To meet our objective as designers, we need the loads!

  • Loads are necessary for determining load paths, stress, strain, and deformation

  • Two methods can be used to obtain them

    1. Analysis of motion and calculation based on equilibrium

    2. Intelligent assumptions (usually based on #1)

8.0.1.0.2 Typical elements in a rigid body model
  • In the body-as-machine analogy, each body structure has a analog is the engineer’s toolkit
TABLE 2.1: Rigid Body Model Elements (@Bartel2006)
Anatomic.Element Model.Element
Bones or limb segments Rigid links
Joints Standard joints: spherical, revolute, cardan, etc. Rigid contact surfaces (kinematic constraints). Deformable contact surfaces (force constraints)
Muscles + tendons Actuators
Nerves Actuators + elastic + viscous elements
Ligaments + joint capsules Controllers, Elastic or viscoelastic springs

http://www.coaleducation.org/lessons/middle/doe/graphics.htm

8.0.1.0.3 Bones

Bones

  • Bones are deformable, however…

  • bones are relatively rigid and can be modeled as rigid for certain purposes

8.0.1.0.4 Joints

Joints

  • Joints act as kinematic constraints

  • Kinematically can be described as:

    • Articulating (knee, hip, shoulder, etc)

    • Deforming (intervertebral discs in the spine, pubic symphysis)

  • Joints have varying levels of complexity

8.0.1.0.5 Engineering perspective – simple joints

  • Spherical (ball and socket) – hip and shoulder

    @Gray1918

    @Gray1918

  • Revolute (hinge) – humero-ulnar joint

    @Gray1918

    @Gray1918

    @Gray1918

8.0.1.0.6 Engineering perspective – complex joints

  • Most joints are complex

    • Wrist

      @Gray1918

8.0.1.0.7 Kinematics

  • Due to the complexity of joints, defining appropriate kinematics is one of the major challenges of link dynamics problems

    • Imaging studies allow direct direct visualization of joint motion

    • “Exact” motion can be established via kinematic study (precision?)

  • We must use judgment in our kinematic assumptions with respect to the desired outcome of the model

  • Imprecise or inaccurate kinematics affects accuracy

  • Reality recognizes that joints are flexible bodies which deform – modeling as a kinematic constraint may not be appropriate in all cases

8.0.1.0.8 Modeling of muscle/tendon/ligament

  • Tendon and ligament have similar structure and load carrying function

  • They can be modeled in similar ways

  • Alternatively, it is often easier to model muscle and tendon together as one structure

8.0.1.0.9 Muscle/tendon model

@Bartel2006

  • We often choose to model muscle and tendon can be modeled as a unit (circuit analogy)

    • Active control of a contractile element (neural stimulus)

    • Elastic elements required in series and parallel (tendons, active and passive muscle tissue) – thus, energy storage

    • Viscous (damping) element in parallel – thus, energy dissipation

  • The parameters associated with these elements depend on:

    • Muscle length and volume

    • Speed of muscle contraction

    • Etc

  • Activation-contraction dynamics are complex – requires idealization (thus, error introduction!)

  • Sometimes we first compute the link-dynamics problem and back-calculate the required muscular response

    • Could be used as calibration for future models. Use caution.

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8.0.1.0.10 Ligaments and capsules

  • Ligaments carry loads (obvious)

  • Some ligaments are idealized as discrete cables (springs) or groups in series/parallel (collateral ligaments)

    @Gray1918

  • Some arrays of ligaments act more like membranes (interosseous membrane)

    @Gray1918

    @Gray1918

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8.0.1.0.11 Mechanical response of ligaments

@Bartel2006

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Ligaments have non-linear elastic behavior (followed by “plastic” deformation and failure)

Small strains have very low forces (high compliance) – occasionally neglected in normal range of joint motion or modeled as linear spring

Larger ligaments have varying “properties” through the cross section

  • ie the ligamentous structures stiffen at different extensions

  • May be appropriate to model as a set of parallel non-linear springs

8.0.2 Static analysis of the skeletal system

8.0.2.1 Static analysis

8.0.2.1.1 Static analysis

  • We can apply equilibrium at each link (ie at each joint)

    • \(\displaystyle\sum_j F_{ij} = 0\)

    • \(\displaystyle\sum_j r_{ij} \times F_{ij} + \displaystyle\sum_k M_{ik} = 0\)

  • This results in 6 equations per link (6 DOF per link)

8.0.2.1.2 The problem of redundancy
8.0.2.1.2.1 (a mathematical problem)

  • Problem: human body has “redundant” muscle groups which make it difficult or impossible to compute the individual muscle loads (without making assumptions)

    • Agonist/Antagonist muscles can be used to hold a fixed location despite external load

      • ie flex your bicep/triceps

    • This allows the body to have broad function and also to compensate for injury/fatigue/etc

  • “Redundancy” makes for an indeterminate set of equations (more unknowns than equations)

8.0.2.1.3 Example of static analysis

  • Compute the elbow joint loads due to lifting a box

  • Compute the tibio-talar resultant force and moment in the ankle joint when standing

8.0.2.1.4 Indeterminance

@Bartel2006

  • Forces from:

    • Glenoid contact

    • Deltoid

    • Pectoralis major

    • Supraspinatus

    • Infraspinatus

    • Teres minor/major

    • Subscapularis

    • Latissimus dorsi

    • Biceps brachii

    • Triceps brachii

    • Ligaments/capsule

    • …

  • We have 6 equations and \(>\) 6 unknowns… must make assumptions

8.0.2.1.5 Example–rigid link analysis of body segments

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  • Three body segments – \(6 \times 3 = 18\) equations (9 equations for planar model)

  • If our intent is to calculate the inter-segmental resultant forces, we can accomplish this goal without regard to internal muscle forces (known external forces)

  • Calculation of muscle forces requires significant additional work and assumption

  • Stress analysis would require calculation of the muscle forces

8.0.2.1.6

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8.0.2.1.7

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8.0.2.2 The Joint Force Distribution Problem

8.0.2.2.1 The Joint Force Distribution Problem

  • Rigid body mechanics often insufficient for desired purpose

    • – ie a first step that gives input loads to a higher fidelity model

  • Thus, we must distribute the inter-segmental resultant forces to the muscles and other soft tissues

  • Additional information about the neuromusculoskeletal system helps

    • Use information about the load carrying elements to supplement the number of available equations

      • ie Assume relationships between muscle forces

    • Specify performance criterion and assume an optimum is found by the body

      • Minimize energy used

      • Minimize a contact force

    • Both methods can be employed

8.0.2.2.2 Example auxiliary conditions

  • Force reduction

    • Assume several muscles have zero force (one muscle dominates)

    • Assume several muscles act as one at a coincident point of action

    • Assume muscle exertions are proportional to each other

    • Assume muscle exertions are proportional to cross section

      • Muscle scaling – 0.2MPa max achievable stress

  • Use external measurements

    • EMG Measurements (Electromyogram)

      • Electrical impulses of the muscles indicative of contraction effort

    • Use “constitutive” descriptions of tissues. (ie solve the statically indeterminate mechanics problem)

  • These methods add equations but also add questions

    • ie what is the muscle cross sectional area
8.0.2.2.3 Optimization Technique

  • This method attempts to find the “best” solution among an infinite number of possible solutions

  • Assumes that the body chooses the best solution neurologically

    • ie minimize contact forces (through sensation – pain, etc)

  • Must write an objective function

    • \(f(\vec{x}) = f(x_1, x_2, x_3,...)\)

    • \(\vec{x}\) is the vector of unknown variables

  • Minimize wrt a set of constraint equations (\(h(x), g(x)\) – equality and inequality constraints)

    • equilibrium, constitutive, force limits, etc

  • Challenges:

    • Is the objective function “correct”?

    • Is it appropriate to assume that the body finds the optimum?

8.0.3 The musculoskeletal dynamics problem

8.0.3.0.1 The musculoskeletal dynamics problem

image Credit: Chyn Wey Lee/Western Herald 2009

  • Inertial effects cannot always be ignored

    • Automotive accident

    • Collisions between people

    • Falls

  • Equations become:

    • \(\displaystyle\sum_j F_{ij} = m_i \ddot{r}_i\)

    • \(\displaystyle\sum_j r_{ij} \times F_{ij} + \displaystyle\sum_k M_{ik} = \dot{H}_i = I \ddot{\theta}\)

  • The RHS of these equations is the rate of change of inertia

8.0.3.0.2 Methods to solve the dynamics problem

  • Direct solution

    • Internal and external forces are known as a function of time

    • Directly integrate the equations of motion

      • \(\displaystyle\iint \displaystyle\sum_j F_{ij} \, dt \, dt = \displaystyle\iint m_i \ddot{r}_i \, dt \, dt\)

      • \(\displaystyle\iint \displaystyle\sum_j r_{ij} \times F_{ij} \, dt \, dt + \displaystyle\iint \displaystyle\sum_k M_{ik} \, dt \, dt = \displaystyle\iint I \ddot{\theta} \, dt \, dt\)

  • Inverse solution

    • External forces are known, internal unknown

    • Motion has been measured

      • \(\displaystyle\sum_j \left(F_{ij}\right)_\mathrm{internal} = m_i \ddot{r}_i -\displaystyle\sum_j \left(F_{ij}\right)_\mathrm{external}\)

      • \(\displaystyle\sum_k \left(M_{ik}\right)_\mathrm{internal} = I \ddot{\theta}-\displaystyle\sum_k \left(M_{ik}\right)_\mathrm{external}\)

  • The inverse method is far more common in biomechanics – internal forces are rarely known

  • Unfortunately, redundancy remains and we must make assumptions

8.0.3.0.3 Body segment mass and geometric properties

  • Accuracy requires:

    • Accurate measurement of anatomical segments and mass distributions

    • Lines of action of muscles, tendons, ligaments

      • Can be determined approximately by origins and insertions of tendon and ligament

  • Empirical data is often used

8.0.3.0.4 Anthropometric Data

  • Approximate properties can be generalized from living subjects and cadavers

  • These properties are averages from a small number of samples

  • Significant variability should be expected from the published data

  • You must recognize and acknowledge the limitations

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8.0.3.0.5

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8.0.3.0.6

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8.0.3.0.7

@Bartel2006

8.0.3.0.8

@Bartel2006

8.0.3.0.9 Average segment lengths

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8.0.3.0.10 Mathematical models for mass properties

  • If we assume body segments can be approximated by common shapes, we can empirically define mass properties

    • Ellipsoidal cones (truncated)

    • Ellipsoids

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8.0.3.0.11 Muscle and ligament forces

  • A critical aspect of modeling muscles and ligaments is to determine appropriate lines of action

  • Cadaveric measurements are available

  • In-vivo reconstructions are possible with current medical imaging, however, appropriate ethical practices are a must

8.0.3.0.12 Lines of action

imagePublic Domain work of the US federal government

  • Simple method: muscles can be modeled as strings connecting two points by a straight line

  • More complex: muscle and tendon wrap around bones (ie patella)

  • In some cases, parts of muscles can be independently activated

8.0.3.0.13 Muscles cross joints

@Bartel2006

  • Muscles cross joints, thus, we must account for the joint motion on the line of action

    • Example: cam-like action of knee
8.0.3.0.14 Efficient use of muscles

  • Muscle redundancy suggests that more than one muscle can perform the same task

  • It is theorized that efficiency might drive “muscle selection”

    • Size of muscle (force limit) and moment arm both effect which muscle is recruited for a task

  • Thus, an accurate estimate of muscle generating area is required

  • Length of muscle changes during contraction, however, volume is nearly conserved

  • Thus, we can calculate an effective area (and force) on the basis of measured length and volume

8.0.4 Joint stability

8.0.4.0.1 Joint stability

  • Thus far, we have discussed the resultant forces at joints without considering how those forces are passed

  • Joint stability is a critical concept whereby the joint must be structurally sound to the applied loads

  • Joint stability requires the joint to maintain functional position throughout its range of motion

    • with “normal” loads

    • with “normal” contact forces

  • The relationship between these loads and “normal” motion must be maintained or negative consequences can occur (locally and globally within the body)

8.0.4.0.2 Idealized stability in synovial joints

Capsule of right knee-joint (distended). Lateral aspect. @Gray1918

Capsule of right knee-joint (distended). Posterior aspect. @Gray1918

  • Small changes in the magnitude or direction of the functional load do not lead to large changes in the position of the joint (or its contact points)

  • Joint contact occurs between surfaces covered with articular cartilage

  • Peripheral loading doesn’t occur

  • Their exists a unique equilibrium position for each set of loads

8.0.4.0.3 Mechanisms for maintaining joint stability

  • Contact at the articular surfaces (passive)

    • ie hip socket provides significant contact surfaces for stability

  • Muscle Action (active – voluntary)

    • ie muscle contraction increase the contact forces adding stability

  • Stretching of the ligaments and capsules (passive)

  • Healthy joints have near frictionless contact

  • Curvature of the surfaces generally enhances the stability of the joint

    • Opportunity for lateral force components

  • Bicondylar joints provide two effective contact points, thus, they transmit moments

  • The position of the joint contact can be another unknown in a problem (under determined system)

    • Muscle force and contact position are often interrelated… ie change the force and the contact position must change

  • Muscles have a finite reaction time… thus they may not react quickly enough to counteract an unexpected set of forces

  • Ligaments limit the range of motion/contact within a joint

    • May apply limited forces for the normal range of motion

    • A health balance exists between stability and laxity (limited range of motion vs dislocation)

    • Trauma or disease can upset the balance

  • Surgeons attempt to maintain a healthy balance of stability and laxity during surgical repair, must consider all three stabilizing mechanisms

8.0.4.0.4 Range of healthy joint contact forces

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8.0.4.0.5

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