The flow of a Couple stress fluids, representing blood through an artery with mild stenosis is investigated under various approximations. The Blood flow consisting of a core region of suspension of erythrocytes assumed to be a couple stress fluid (Non-Newtonian) & a peripheral layer of plasma (Newtonian fluid) with dusty particles as micro-organisms. Analytical expressions for both blood and microorganisms have been formulated for blood taking the volume fraction of the microorganisms into account. A detailed study will be made as regards the couple stress parameter, pulsatile Reynolds number, volume fraction of microorganisms and stenosis with other parameters in this investigation. The velocity distribution, the relative apparent viscosity and flux for blood and micro-organisms are calculated as function of couple-stress parameter, pulsatile Reynold's number at different phase angles. The velocity is maximum in the core region. The expressions for velocity of blood and microorganisms, flow rate, apparent viscosity, phase-lag between flow rate and pressure gradient will be derived.