The use of slack variables in the adjoint method applied to tumor drug dosage

In this article, a novel approach for solving optimal control problems with inequality constraints is presented using a modified gradient method based on the adjoint technique. So far, the incorporation of inequality constraints in the adjoint approach requires the introduction of additional penalty terms in the cost function. However, such an approach may lead to a potential distortion of optimal control due to the need for weighting factors for these terms. The method proposed in this article avoids the use of penalty functions and allows the iterative computation of optimal controls. In order to demonstrate the effectiveness of the method, the algorithm is applied to the optimization problem of anti-angiogenesis of tumors in medicine. Tumor growth is closely related to the formation of new blood vessels. Above a certain tumor size, tumors expand their vascular network, a process known as angiogenesis. Targeting this process with anti-angiogenic therapy holds great promise for cancer treatment. Mathematical modeling and the use of optimal control theory enables the understanding of tumor dynamics and optimize treatment strategies. In this article this mathematical model for computing an optimal treatment therapy is considered. For this purpose, an adjoint gradient approach will be exploited in this article, to solve the optimal control problem iteratively by an optimization algorithm.