Current leads used to supply electrical energy from a room-temperature power supply to a superconducting application represent a major thermal load and therefore largely determine the operating cost. This paper presents a concept to minimize the thermal load at the cold end by integrating the recuperative heat exchanger of a cryogenic mixed-refrigerant cycle (CMRC). A mixture of non-flammable components is used to absorb the distributed thermal load continuously over the entire temperature range. The paper describes the numerical model that consists of an electric model coupled with a thermal modelling framework, allowing the optimization of the mixture composition and the temperature profiles. Simulations yield a reduction of the thermal load at the cold end by 45 % compared to conventional conduction-cooled current leads.