Experimental evidence and model explanation for cell population characteristics modification when applying sequential photodynamic therapy

We present experimental evidence of the existence of cell variability in terms of threshold light dose for Hep G2 (liver cancer cells) cultured. Using a theoretical model to describe the effects caused by successive photodynamic therapy (PDT) sessions, and based on the consequences of a partial response we introduce the threshold dose distribution concept within a tumor. The experimental model consists in a stack of flasks, and simulates subsequent layers of a tissue exposed to PDT application. The result indicates that cells from the same culture could respond in different ways to similar PDT induced-damages. Moreover, the consequence is a partial killing of the cells submitted to PDT, and the death fraction decreased at each in vitro PDT session. To demonstrate the occurrence of cell population modification as a response to PDT, we constructed a simple theoretical model and assumed that the threshold dose distribution for a cell population of a tumor is represented by a modified Gaussian distribution.

Exact penalties for variational inequalities with applications to nonlinear complementarity problems

In this paper, we present a new reformulation of the KKT system associated to a variational inequality as a semismooth equation. The reformulation is derived from the concept of differentiable exact penalties for nonlinear programming. The best theoretical results are presented for nonlinear complementarity problems, where simple, verifiable, conditions ensure that the penalty is exact. We close the paper with some preliminary computational tests on the use of a semismooth Newton method to solve the equation derived from the new reformulation. We also compare its performance with the Newton method applied to classical reformulations based on the Fischer-Burmeister function and on the minimum. The new reformulation combines the best features of the classical ones, being as easy to solve as the reformulation that uses the Fischer-Burmeister function while requiring as few Newton steps as the one that is based on the minimum.

On Approximate KKT Condition and its Extension to Continuous Variational Inequalities

In this work, we introduce a necessary sequential Approximate-Karush-Kuhn-Tucker (AKKT) condition for a point to be a solution of a continuous variational inequality, and we prove its relation with the Approximate Gradient Projection condition (AGP) of Garciga-Otero and Svaiter. We also prove that a slight variation of the AKKT condition is sufficient for a convex problem, either for variational inequalities or optimization. Sequential necessary conditions are more suitable to iterative methods than usual punctual conditions relying on constraint qualifications. The AKKT property holds at a solution independently of the fulfillment of a constraint qualification, but when a weak one holds, we can guarantee the validity of the KKT conditions.