Absement: Quantitative Assessment of Metabolic Cost during Quasi-Isometric Muscle Loading

Abstract: Accurate quantitative assessment of metabolic cost during static posture holding is a strategically important problem in biomechanics and physiology. Traditional metrics such as ``time under tension'' are fundamentally insufficient, because they are scalar quantities that ignore the temporal history of deviations, that is, the microdynamics of posture, which has nontrivial energetic consequences. In this work, we propose a theoretically grounded methodology to address this problem by introducing the concept of the \textbf{deviation absement} ($Δ\mathcal{A}\ell$), defined as the time integral of the deviation of the muscle--tendon unit length from a reference value. We rigorously prove that, for a broad class of quasi-static models, absement appears as the leading first-order state variable. For small deviations in a neighbourhood of a reference posture, the total metabolic cost $\mathcal{E}{\mathrm{met}}(\ell)$ admits a universal asymptotic expansion of the form \begin{equation*} \mathcal{E}{\mathrm{met}}(\ell) = P_0 T + C_1 Δ\mathcal{A}\ell + C_2 \int_0^T(\ell(t)-\ell_0)^2\,dt + O(|\ell-\ell_0|{L^\infty}^3), \end{equation*} where $T$ is the duration of loading, and $P_0, C_1, C_2$ are constants determined by local properties of the system. Thus, the deviation absement ($Δ\mathcal{A}\ell$) is the \textbf{unique first-order sufficient statistic} that allows one to quantify and separate the energetic contribution of systematic drift of the mean posture from the contribution of micro-oscillations (tremor), which is described by the quadratic term. This result has direct consequences for parameter identification: the proposed formalism makes it possible to recover physically meaningful coefficients $(P_0, C_1, C_2)$ by means of linear regression of experimental data obtained from standard kinematic measurements and indirect calorimetry.