Abstract
In medical treatment it can be necessary to know the position of a motor unit in a muscle. Recent advances in high-density surface electromyography (EMG) measurement have opened the possibility of extracting information about single motor units. We present a mathematical approach to identify these motor units. On the base of an electrostatic forward model, we introduce an adjoint approach to efficiently simulate a surface EMG measurement and an optimal control approach to identify these motor units. We show basic results on existence of solutions and first-order optimality conditions.
Highlights
In the human body, muscles are responsible for movement
Muscles are responsible for movement. These muscles consist of many muscle fibers, which are organized in so-called motor units
Using a quasi-static approach, cf. [26], one can simulate for fixed time t the potential Φ, which is generated from the moving action potential, by solving a Poisson equation of the form (σ(x)∇Φ(x, t)) · ∇v(x) dx + μΦ(s, t)v(s) ds = v(x) dρ(t)
Summary
Muscles are responsible for movement. These muscles consist of many muscle fibers, which are organized in so-called motor units. Using a quasi-static approach, cf [26], one can simulate for fixed time t the potential Φ, which is generated from the moving action potential, by solving a Poisson equation of the form (σ(x)∇Φ(x, t)) · ∇v(x) dx + μΦ(s, t)v(s) ds = v(x) dρ(t) In this setting, the time dependent source density ρ(t), which is given through a moving action potential, is spatially concentrated on the motor unit and modeled as a measure, concentrated on a line. The time dependent source density ρ(t), which is given through a moving action potential, is spatially concentrated on the motor unit and modeled as a measure, concentrated on a line Solutions to this problem in the sense of Stampacchia can be found in W 1,p (Ω) where p > d and 1/p + 1/p = 1, cf e.g., [28] and a discussion concerning uniqueness can be found in [24]. A detailed elaboration of such an approach, will not be subject to the current paper
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