Deep intracranial tumor removal can be achieved if the neurosurgical robot has sufficient flexibility and stability. Towards achieving this goal, we have developed a spring-based continuum robot, namely a Minimally Invasive Neurosurgical Intracranial Robot (MINIR-II) with novel tendon routing and tunable stiffness for use in a magnetic resonance imaging (MRI) environment. The robot consists of a pair of springs in parallel, i.e., an inner inter-connected spring that promotes flexibility with decoupled segment motion and an outer spring that maintains its smooth curved shape during its interaction with the tissue. We propose a shape memory alloy (SMA) spring backbone that provides local stiffness control and a tendon routing configuration that enables independent segment locking. In this work, we also present a detailed local stiffness analysis of the SMA backbone and model the relationship between the resistive force at the robot tip and the tension in the tendon. We also demonstrate through experiments, the validity of our local stiffness model of the SMA backbone and the correlation between the tendon tension and the resistive force. We also performed MRI compatibility studies of the 3-segment MINIR-II robot by attaching it to a robotic platform that consists of SMA spring actuators with integrated water cooling modules.