  In this paper we develop theoretical analysis and numerical reconstruction
techniques for the solution of an inverse boundary value problem dealing with
the nonlinear, time-dependent monodomain equation, which models the evolution
of the electric potential in the myocardial tissue. The goal is the detection
of a small inhomogeneity $\omega_\varepsilon$ (where the coefficients of the
equation are altered) located inside a domain $\Omega$ starting from
observations of the potential on the boundary $\partial \Omega$. Such a problem
is related to the detection of myocardial ischemic regions, characterized by
severely reduced blood perfusion and consequent lack of electric conductivity.
In the first part of the paper we provide an asymptotic formula for electric
potential perturbations caused by internal conductivity inhomogeneities of low
volume fraction, extending the results published in [7] to the case of
three-dimensional, parabolic problems. In the second part we implement a
reconstruction procedure based on the topological gradient of a suitable cost
functional. Numerical results obtained on an idealized three-dimensional left
ventricle geometry for different measurement settings assess the feasibility
and robustness of the algorithm.
