  One of the ways natural and synthetic systems regulate temperature is via
circulating fluids through vasculatures embedded within their bodies. Because
of the flexibility and availability of proven fabrication techniques,
vascular-based thermal regulation is attractive for thin microvascular systems.
Although preliminary designs and experiments demonstrate the feasibility of
thermal modulation by pushing fluid through embedded micro-vasculatures, one
has yet to optimize the performance before translating the concept into
real-world applications. It will be beneficial to know how two vital design
variables -- host material's thermal conductivity and fluid's heat capacity
rate -- affect a thermal regulation system's performance, quantified in terms
of the mean surface temperature. This paper fills the remarked inadequacy by
performing adjoint-based sensitivity analysis and unravels a surprising
non-monotonic trend. Increasing thermal conductivity can either increase or
decrease the mean surface temperature; the increase happens if countercurrent
heat exchange -- transfer of heat from one segment of the vasculature to
another -- is significant. In contrast, increasing the heat capacity rate will
invariably lower the mean surface temperature, for which we provide
mathematical proof. The reported results (a) dispose of some misunderstandings
in the literature, especially on the effect of the host material's thermal
conductivity, (b) reveal the role of countercurrent heat exchange in altering
the effects of design variables, and (c) guide designers to realize efficient
microvascular active-cooling systems. The analysis and findings will advance
the field of thermal regulation both on theoretical and practical fronts.
