Basic concepts

Viewpoints for analysis of biomechanical systems

Two outcomes are generally sought:

• Understand the motion (rigid body kinematics)
  • This is typically musculoskeletal multi-body static/dynamic simulation
• Understand the stress, strain, and deformation
  • This is typically finite element modeling
• The above are general categories. There are many variations!

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
  3. Typically both are required!

Typical elements in a rigid body model

• In the body-as-machine analogy, each body structure has a analog in the engineer’s toolkit

Bones

• Bones are deformable, however…
• bones are relatively rigid and can be modeled as rigid for certain purposes
• Example: musculoskelatal multibody dynamic simulation (AKA Link Dynamics Models)

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

Engineering perspective – simple joints

• Spherical (ball and socket) – hip and shoulder

• Revolute (hinge) – humero-ulnar joint

Engineering perspective – complex joints

• Many joints are complex
  • Wrist

Kinematics

• Due to the complexity of joints, defining appropriate kinematics is one of the major challenges of link dynamics problems
  • Imaging studies allow direct (but limited) 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

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

Muscle/tendon model

• We often choose to model muscle and tendon 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.

Ligaments and capsules

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

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

Mechanical response of ligaments

• 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

Application of Link Dynamics Models

• A common task is to determine the forces/moments transmitted at joints
  • Typically in this context, bony deformation is not a concern (as so bone assumed to be rigid)
• Two types of external boundary conditions are possible:
  • Kinematic constraints (constrained motion)
  • Surface tractions (external forces/pressures)

Example:

• Typical Outputs
  • Muscle/tendon/ligament forces
  • Energy consumption, power output, work
  • Joint reactions, loads, accelerations
  • Floor and other external loads
  • Range of motion, functional deficit due to injury
  • Insights into overall functional mechanics

• Choice of how to include a boundary condition may depend on the analysis
  • Foot may not penetrate floor
  • Floor imparts a force/moment on the foot
• In many cases, we can apply a statically equivalent load in place of a motion constraint (or vice-versa)
• Our assumptions must be appropriate for the goal of the model

Three types of solution are available for any problem

• Static analysis
  • Fixed configuration – no motion allowed
• Quasi-static analysis
  • A range of motion is considered (as an allowed configuration change – ie joint rotation)
  • However, dynamics are ignored or inertia forces assumed to be constant
• Dynamic analysis
  • Inertia in included

Static analysis of the skeletal system

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)

Computation of reaction forces

Equilibrium revisited

Force diagrams, statically equivalent forces, and “free body diagrams”

• Draw structural bodies in equilibrium, ie properly constrained or with balanced forces

• To compute reaction forces \(R\) and \(B\), use statically equivalent forces (ie place the weight of the forearm at its center of gravity)

Force diagrams, statically equivalent forces, and “free body diagrams”

• Complete calculations using the equilibrium equations

• Equillibrium: \[\begin{split} \sum F_y =& \; 0 \\ \sum M =& \; 0 \\ \end{split}\]

Force diagrams, statically equivalent forces, and “free body diagrams”

• Equillibrium: \[\begin{array} \sum F_y =& \; +B - G - W -R = 0 \\ \sum M_R =& \; +(0 \, {\mathrm cm}) (R) + (3 \, {\mathrm cm}) (B) - (15 \, {\mathrm cm}) (G) - (30 \, {\mathrm cm}) (W) = 0 \\ \end{array}\]

Force diagrams, statically equivalent forces, and “free body diagrams”

• Use the moment equation first: \[\begin{split} (0\, {\mathrm cm})(R) + (3\, {\mathrm cm})(B) - (15\, {\mathrm cm})(G) - (30\, {\mathrm cm})(W) =& \; 0 \\ \end{split}\] \[\begin{split} (3\, {\mathrm cm})(B) =& \; (15\, {\mathrm cm})(G) + (30\, {\mathrm cm})(W) \\ \end{split}\]

• Use known values \[\begin{split} G =&\; 15 \, \mathrm{N}\\ W =&\; 20 \, \mathrm{N}\\ \end{split}\]

• Therefore: \[\begin{split} B = \frac{(15 \, {\mathrm cm})(15 \mathrm{N}) + (30\, {\mathrm cm})(20 \mathrm{N})}{3\, {\mathrm cm}} = 275 \, \mathrm{N} \end{split}\]

Force diagrams, statically equivalent forces, and “free body diagrams”

• Use the force equation next: \[\begin{split} +B - G - W -R = 0 \\ R = +B - G - W \\ \end{split}\]

• Use known values \[\begin{split} G =&\; 15 \, \mathrm{N}\\ W =&\; 20 \, \mathrm{N}\\ B =&\; 275 \, \mathrm{N}\\ \end{split}\]

• Therefore: \[\begin{split} R = 275 \, \mathrm{N} - 20 \, \mathrm{N} - 15 \, \mathrm{N}= 240 \, \mathrm{N}\\ \end{split}\]

The problem of redundancy (a mathematical problem)

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

• For example:

  • Agonist/Antagonist muscles can be used to hold a fixed location despite external load
    • ie flex your bicep/triceps
    • Hold position with minimal or maximal force!
  • This allows the body to have broad function and also to compensate for injury/fatigue/etc

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

Additional examples 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

Indeterminance

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

Example–rigid link analysis of body segments

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

Auxiliary conditions

• We can assume auxiliary conditions to enable the calculation
  • 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

• We can 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)
    • Almost always measured on separate tissue, assumed accurate
• These methods add equations but also add questions
  • ie what is the muscle cross sectional area

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?

The musculoskeletal dynamics problem

• 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

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
    • \(\displaystyle\iint \displaystyle\sum_j F_{ij} \, dt \, dt = \displaystyle\iint m_i \ddot{r}_i \, dt \, dt\)
    • \(\displaystyle\iint \displaystyle\sum_j r_{ij} \times F_{ij} \, dt \, dt + \displaystyle\iint \displaystyle\sum_k M_{ik} \, dt \, dt = \displaystyle\iint I \ddot{\theta} \, dt \, dt\)
  • Note: the mathematics of the discrete calculation are interesting… covered in my advanced finite element class

• 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
• Further, typically substantial error associated with calculation of position!

Body segment mass and geometric properties

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

Anthropometry

• noun
  • – the scientific study of the measurements and proportions of the human body.

Proportionality Constants¹

• Proportionality Constants
  • Proportionality constants describe the average ratio of a given body segment length to stature.
  • Can be problematic, however, since segment ratios of any individual are not likely to be “average”.
    • The ratios for a given segment vary widely across individuals within a given population.
  • Some measures well correlated (“shoulder height” \(R^2\) = 0.93), other measures do not (“shoulder breadth” \(R^2\) = 0.15) do not.
  • As a result, this is a helpful tool for exploring relationships, but limits must be acknowledged in real design/analysis.

Sources of Anthropometric Data

• ANSUR II
  • The Anthropometric Survey of US Army Personnel
  • data were published internally in 2012, publicly in 2017.
  • Likely the most comprehensive publicly available data set on body size and shape.
  • Includes 93 measures for over 6,000 adult US military personnel (4,082 men and 1,986 women).
  • It is still not an approximation of the US Civilian population. * Consequently, while there is useful information here, designs and standards based on these data will not accommodate most user populations in the intended manner.

• NHANES
  • The National Health and Nutrition Examination Survey (NHANES) is conducted continuously in the United States.
  • The studies started in the mid-20th century and originally targeted specific groups (adults, children, Hispanics, etc.).
  • Beginning in 1999 the data were gathered continuously and released in two year cycles.
  • Objective is to obtain an overall health picture of the US civilian population, the large data set only contain a few measures of body size and shape.
    • Stature, mass, BMI, and waist circumference.

• University Data
  • OpenLab @ PSU
  • OpenLab Explorer
  • University of Michigan

Deficiencies And Shortcomings Of Anthropometric Data

• Differences in the data exist across different segments of society
  • For example the statistics on height differ for Caucasian and people of Asian descent
• These statistics also change over time with nutritional and other changes
  • Go to the Air Zoo and examine the military uniforms
  • (especially from World War I).

Specific Examples Of Deficiencies

• It is taken on a subject population, such as 4000 Male United States Air Force personnel (ANSUR)
  • It is a small sample size relative to the entirety of the population
  • Military personnel are mostly young, healthy, adults
  • It may not include people who are very small or very large
  • That’s it may not be generalizable to the population at Large

• Gender bias
  • Much of the existing data is from the military
  • Under represents women (due to their historical under-representation in the military)
• Racial bias
  • Differences in Height Around the World
  • As designed previously there are differences among the races in their anthropometric data
  • If you are designing a products for a specific market, then you may consider using only using data from that region
  • However, there are many things to consider as you do that.