When treating a vasospasm brought on by a malfunctioning coronary artery system (CAS), the goal is to enhance the physiology of the CAS by synchronizing its dynamic behavior with that of a healthy CAS, according to the perspective of cardiac medicine. This treatment is supported by a sophisticated automated detection and visualization system for the identification of a diseased CAS. For the biomathematical model of a CAS, we provide, in this study, a novel fractional-powered-integral-type finite-time-convergent sliding mode (FFSM), which is composed of the fractional-powered and integral terms of state variables. To achieve almost finite-time synchronization between two CASs, an adaptive sliding mode (SM) control approach in terms of the stated FFSM is introduced. Upon the implementation of the devised control, a special form of stability is demonstrated. That is, because of a feature of the model for a CAS, the two error states are stabilized sequentially on the sliding surface, resulting in an almost finite-time stability. The nonlinear dynamical consequences of the synchronized error system can be countered by the adaptive FFSM control scheme, which consists of three online updated gains with the principles of adaptation. To demonstrate the validity of the current scheme, numerical experiments are carried out.