  Implantable wireless bioelectronic devices enable communication and/or power
transfer through RF wireless connections with external nodes. These devices
encounter notable design challenges due to the lossy nature of the host body,
which significantly diminishes the radiation efficiency of the implanted
antenna and tightens the wireless link budget. Prior research has yielded
closed-form approximate expressions for estimating losses occurring within the
lossy host body, known as the in-body path loss. To assess the total path loss
between the implanted transmitter and external receiver, this paper focuses on
the free-space path loss of the implanted antenna, from the body-air interface
to the external node. This is not trivial, as in addition to the inherent
radial spreading of spherical electromagnetic waves common to all antennas,
implanted antennas confront additional losses arising from electromagnetic
scattering at the interface between the host body and air. Employing analytical
modeling, we propose closed-form approximate expressions for estimating this
free-space path loss. The approximation is formulated as a function of the
free-space distance, the curvature radius of the body-air interface, the depth
of the implanted antenna, and the permittivity of the lossy medium. This
proposed method undergoes thorough validation through numerical calculations,
simulations, and measurements for different implanted antenna scenarios. This
study contributes to a comprehensive understanding of the path loss in
implanted antennas and provides a reliable analytical framework for their
efficient design and performance evaluation.
