Appendix

Muscle Formulation

Muscle Formulation

There are two types of muscle formulations available in LifeMOD/BodySIM: a trainable or "effective force" formulation and a Hill-type muscle formulation.

The trainable formulation for the muscles consist of "training" elements for inverse-dynamics simulations and active (contractile) elements for forward-dynamics simulations. Muscles record shortening/lengthening patterns while the model is being driven by the motion capture data in an inverse-dynamics simulation. They then repeat those patterns and serve as actuators for the forward-dynamics simulations. The muscle actuators are trained not to exceed the physiological capability of the individual muscle.

The Hill-type muscle formulation is the traditional combination of contractile elements (CE) and a parallel passive, elastic element (PE) that approximated standard muscle tissue in the human body. The contractile element contains an muscle activation state that controls the active muscle force capability for the muscle. Data from an EMG test may be used as activations for the contractile element.

Various muscle parameters may be adjusted. See Parameters to tune the model for simulation. See also Choosing Model Parameters for data sources and information on how to select the parameters mentioned in this section.

Muscle Formulation - Effective Muscle Force (Trainable)

LifeMOD/BodySIM™ uses an effective force approach to muscle modeling. Muscles produce the necessary forces to replicate the desired motion of the body while staying within each muscle's physiological limit. If enough muscles are included, the calculated muscle forces will be very close to the physical muscle force values for the same activity. The problem of redundancy, or several muscles contributing to the motion of the joint, is handled by allowing the user to filter the output of any muscle from 0 to 200%.

If EMG data is available the user can modify the filtering function by comparing the muscle response to the data.

A two step process is used for LifeMOD™ musculoskeletal simulation. In the first step, the muscles are created on the body in the form of training elements or passive non-contributing elements. The body is then manipulated using motion agents which move via motion capture data, or user-entered curve data. In this inverse-dynamics step, muscle shortening/lengthening (l_desired)patterns are recorded. The recorded patterns then serve as actuators to drive the motion. The formulation of the active muscle is displayed in the equation below.

where,
f _max = pCSA * M_stress

Actuators are used to produce an effective force, F_eff, that minimizes the error between the desired instantaneous muscle length and the actual length. If this force is beyond the physiological limit for a particular muscle, the force becomes the limit value. The force is further refined, F_muscle, by a user-specified filter function.

Physiological properties for each muscle include:

physiological cross sectional area (pCSA)
maximum tissue stress (M_stress)
resting load (F_resting)
force output filter % (f_filter)
overall muscle tone (M_tone)

This data establishes an upper limit on the muscle force (f_max) for each muscle in the model. The values are calculated as follows:

1. The original pCSA and M_stress are extracted from the LifeMOD™ database and scaled to the model based on height, weight, age, and gender.

2. The pCSA is further scaled by the user-entered muscle tone (M_tone%)

3. f _max = pCSA * M_stress

The algorithm in Figure 1 calculates the force in each muscle using the following process:

1. The instantaneous muscle length and velocity is recorded and retrieved.

2. It is compared to the desired instantaneous muscle length/velocity calculated from the inverse-dynamics simulation.

3. The difference (error) is corrected by the P_gain. The derivative of the error is multiplied by the D_gain. This results in the muscle force, F_eff, necessary to minimize the error.

4. If the resulting muscle force F_eff is greater or equal to the physiological maximum f _max, f _eff = f _max.

5. F_eff is then filtered with the user-specified filter function F_filter to become F_muscle.

See Choosing Model Parameters for data sources and information on how to select the parameters mentioned in this section.

Muscle Formulation - Hill-Based Muscle Force

The Hill-type muscle model is developed from the material behavior of the muscle model adapted from the original work by [Hill, '38] which results in a generally-accepted state equation applicable to skeletal muscle that has been stimulated to show tetanus. Reviews of this model and extensions can be found in [Winters, '88] and [Zajac, '89]. The Hill-type muscle model (Figure 2), consists of a contractile element (CE) which is in series with a series elastic element (SEE) both of which are in parallel with a passive element (PE). The SEE, shown in grey in Figure 2 is often neglected when a series tendon is added. The main assumptions of the Hill model are that the contractile element is entirely stress free and freely distensible in the resting state, as described by Hill's equation. When the muscle is activated, the series and parallel elements are elastic, and the whole muscle is a simple combination of identical sarcomeres in series and parallel.

Figure 2: Discretized model for muscle contraction dynamics based on a Hill-type muscle.. The total force F_MUSCLE is the sum of a passive force F_PE and an active force F_CE. The passive element (PE) is a function of the instantaneous muscle length, l_curr. The contractile element (CE) is a function of the instantaneous muscle length, l_curr, instantaneous muscle shortening velocity v_curr and the time dependent activation state A(t).

When ignoring the SEE, the muscle force F_MUSCLE is the sum of both forces thus,

F_MUSCLE = F_CE + F_PE

Passive Element F_PE

The muscle input parameters for the passive element are displayed in Figure 3.

Figure 3: Section of the Hill muscle dialog box with descriptions of the input parameters for the passive element.

The passive element force F_PE is modeled with a passive muscle stress (σ) value multiplied by the physilogical cross sectional area, pCSA, of the particular muscle.

F_PE = σ · pCSA , where

σ = passive muscle stress
pCSA = physiological cross sectional area

Passive muscle stress, σ, modeled by the nonlinear stress-strain relationship [Deng, '87]:

σ = (k · ε)/(1-ε/asym), where

ε = strain defined as the elongation relative to the resting length of the muscle,
k = passive muscle stiffness
asym = strain asymptote

The strain is defined by

ε = (l_curr - l_free)/ l_free , where

l_curr = current (instantaneious) length of the muscle
l_free = free length of the muscle at rest when it is removed from the body

The l_free results in a smaller free length than the muscle length in its initial position in the model. The initial position in the body is defined by l_{rest .}Assuming a linear relationship between the sarcomere length s [Magid, '85, Meyers, '98, Rack, '69], and the muscle length, the free length of the muscle can be calculated as:

l_free = l_rest· (S_free )/ (S_rest ),

where the muscle reference length l_ref is based on

l_ref = l_rest· (S_ref )/ (S_rest ), where

S_free = average sarcomere length of "free" muscle
S_rest = length of sarcomere at rest
S_ref = length of sarcomere at muscle optimal length [Brelin-Fornari, '98]

Active Element F_CE

The muscle input parameters for the contractile element are displayed in Figure 4.

Figure 4: Section of the Hill muscle dialog box with descriptions of the input parameters for the contractile element.

Active muscle behavior (contractile element) is modeled with a normalized activation state and a maximum muscle force at activation.

F_CE = A(t) · F_max · f_H (v_r) · f_L(l_r) , where

A(t) = activation state (normalized between 0 -- resting -- to 1 -- maximum activation)
F_max= muscle force at maximum activation isometric conditions
f_H= the normalized active force-velocity relation (Hill-curve)
f_L= the normalized active force-length relation
v_r= dimensionless lengthening velocity
l_r= dimensionless muscle length

The muscle force at maximum activation is calculated by:

F_max = σ_max · pCSA, where

σ_max = is the maximum isometric muscle stress
pCSA = physiological cross sectional area

The function f_H is the normalized active force-velocity relation (Hill-curve). Separate functions are defined for lengthening and shortening of the CE-element.

, where

v_r = v_curr/V_max
v_curr = the instantaneous lengthening velocity V_max = the maximum shortening velocity of the muscle
CE_sh= shape force-velocity curve (shortening)
CE_shl= shape force-velocity curve (lengthening)
CE_ml= maximum relative force (lengthening)

The shape is determined by the parameters CE_sh and CE_shl, where CE_ml defines the maximum force the muscle can generate during lengthning relative to the maximum isometric force F_max.

Figure 5: Normalized muscle force as a function of normalized muscle length (l_r) for both active, f_l(l_r), (blue curve) and passive, F_PE, (other) element curves.

Figure 6: Standard force-velocity curve f_h for the conditions CE_sh= .5, CE_shl= .075, C_ml= 1.5

The function f_lis the normalized active fore-length relation.

f_l(l_r) = e -((l_r-1)/S_k)**2 , where

l_r = l_curr/l_ref
l_curr = the instantaneous muscle length
l_ref = optimum reference length at which the active force is generate most efficiently
S_k = determines the shape of the curve (figure 2)

Muscle activation, A(t), is described by a data spline. The data spline uses time as the independent variable and the normalized activation A as the dependent variable. A library of EMG data is avaliable via the Xchange function in the main LifeMOD panel.