12.1.2 Iterative model for calculating trunk-muscle forces

During load lifting, especially asymmetric lifting, trunk muscles are activated to help sustain the posture. This muscular redundancy results in more unknown forces than force- and moment-equilibrium equations, making the problem statically indeterminate. The three-dimensional model used by Ergoniza Intelligent to calculate compression forces in the L4/L5 intervertebral disc (Section 12.1) contains 13 unknown forces—those corresponding to muscle contractions and intervertebral-disc reactions—but only 6 equations.

To solve this system, constraints are introduced regarding how the muscles act76F¹. These constraints establish that muscle-contraction intensities—force per unit cross-sectional area—and intervertebral-disc compression force are minimized during exertion. This is equivalent to assuming that, during load lifting, the central nervous system attempts to protect the muscles and intervertebral discs by minimizing the stresses imposed on them77F².

Because both compressive forces and muscle intensities must be minimized, the problem can be solved using a two-stage linear-programming optimization procedure78F³. First, maximum muscle intensity is minimized while satisfying moment-equilibrium conditions. The muscle forces that minimize the muscular contribution to vertebral compression are then determined while respecting the intensity limits.

The first stage minimizes the maximum contraction intensity among all muscles. For each muscle i, Fᵢ represents the magnitude of its contraction force. From the position and line of action of the muscle, the model determines the moment generated about L4/L5 by a unit force from that muscle. This vector coefficient is represented by cᵢ and depends on the geometry of the analyzed posture. The moment generated by the muscle is therefore cᵢFᵢ, expressed in N·m, and varies linearly with its contraction force. At equilibrium:

\[\sum_{i = 1}^{10}{c_{i}F_{i} = - M_{ext,corr}}\]

where M_ext,corr is the external moment that the muscles must balance, expressed in the L4/L5 local coordinate system after incorporating the effect of intra-abdominal pressure and, when applicable, the backrest simplification. The preceding equation actually represents three independent equations, one for each axis of the L4/L5 local coordinate system.

The linear-programming problem is formulated as:

min λ

subject to:

\[\sum_{i = 1}^{10}{c_{i}F_{i} = - M_{ext,corr}}\]

\[F_{i}\ \geq 0\ \ \ \ \ i = 1,\ldots,10\]

\[\frac{F_{i}}{A_{i}}\ \leq \lambda\ \ \ \ \ i = 1,\ldots,10\]

where λ is the greatest muscle intensity, expressed in N/cm², and Aᵢ is the physiological cross-sectional area of muscle i, in cm².

In the second stage, once λ has been determined, a second linear-programming problem is solved to minimize the total compressive force on the intervertebral disc. The second problem is formulated as:

\[\min{\sum_{i}^{}F_{i,}}\]

subject to the same equilibrium equations, Fᵢ ≥ 0, and:

\[\frac{F_{i}}{A_{i}}\ \leq \lambda\]

Once the muscle forces have been determined, they are projected onto the L4/L5 local coordinate system. Their components perpendicular to the disc plane contribute to axial compression, while the components within the plane contribute to shear.

After obtaining the muscle forces, the disc reaction is calculated by imposing force equilibrium. Projecting this reaction onto the axis normal to the L4/L5 plane gives axial compression, while its projections onto the anteroposterior and lateral axes give the two shear components.

Ergoniza Intelligent displays the calculation results in a window like the one shown in Figure 64. In this example, estimated axial compression is 2609.5 N, below the operational threshold of 3400 N applicable to the selected model. The shear components are 289.4 N in the anteroposterior direction and 0.0 N in the lateral direction; their resultant is also below the 500 N threshold. The displayed diagnosis is therefore Acceptable.

The Muscle Activity section presents, for each of the ten muscles considered, the calculated total force and its components in the L4/L5 local coordinate system: X, lateral; Y, anteroposterior; and Z, axial or normal to the disc plane. The window also shows the three components of the moment at L4/L5, the moment arm of intra-abdominal pressure, and its associated equivalent force.

Ergoniza Intelligent verifies that both optimization problems are feasible, that muscle forces are nonnegative, and that moment-equilibrium residuals remain within the established tolerance. If no usable solution is obtained, the window displays Unable to Calculate.