  We present a direct numerical method for the solution of an optimal control
problem controlling the growth of LDL, HDL and plaque. The optimal control
problem is constrained with a system of coupled nonlinear free and mixed
boundary partial differential equations consisting of three parabolics one
elliptic and one ordinary differential equations. In the first step, the
original problem is transformed from a free boundary problem into a fixed one
and from the mixed boundary condition to a Neumann one. Then, employing a fixed
point-collocation method, we solve the optimal control problem. In each step of
the fixed point iteration, the problem is changed to a linear one and then, the
equations are solved using the collocation method bringing about an NLP which
is solved using sequential quadratic programming. Then, the obtained solution
is verified using indirect methods originating from the first-order optimality
conditions. Numerical results are considered to illustrate the efficiency of
methods.
