  Cardiac pulsations in the human brain have received recent interest due to
their possible role in the pathogenesis of neurodegenerative diseases. Further
interest stems from their possible application as an endogenous signal source
that can be utilized for brain imaging in general. The (pulse-)wave describing
the blood flow velocity along an intracranial artery consists of a forward
(anterograde) and a backward (retrograde, reflected) part, but measurements of
this wave usually consist of a superposition of these components. In this
paper, we provide a mathematical framework for the inverse problem of
estimating the pulse wave velocity, as well as the forward and backward
component of the pulse wave separately from MRI measurements on the middle
cerebral artery. After a mathematical analysis of this problem, we consider
possible reconstruction approaches, and derive an alternate direction approach
for its solution. The resulting methods provide estimates for
anterograde/retrograde wave forms and the pulse wave velocity under specified
assumptions on a cerebrovascular model system. Numerical experiments on
simulated and experimental data demonstrate the applicability of the proposed
methods.
