INTRODUCTION Most cancer modeling techniques developed up to now adopt the straightforward “bottom-up” approach, focusing on the better understanding and quantication of rather microscopic tumor dynamics mechanisms and the investigation of crucial biological entity interdependences including tumor response to treatment in the generic investigational context. To

CONTENTS Introduction 407 e Oncosimulator 411 Model-A Symbolic Mathematical Formulation of DEBCaST 414

Hypermatrix of the Anatomical Region of Interest 414 Practical Considerations Regarding the Construction of the Discretization Mesh 415 Hypermatrix and Operator Formulation of DEBCaST 415

Results 420 Model A: Tumor Response to Chemotherapeutic Schemes 420 Model B: Tumor Response to Radiotherapeutic Schemes 422

Discussion and Future Perspectives 426 Acknowledgments 431 References 431

this end several combinations of mathematical concepts, entities, and techniques have been developed and/or recruited and appropriately adapted. ey include population dynamics models (Guiot et al. 2006), diusion-related continuous and nite mathematics treatments (Murray 2003; Swanson et al. 2002; Breward et al. 2003; Cristini et al. 2005; Frieboes et al. 2006; Enderling et al. 2007; Ramis-Conde et al. 2008), cellular automata and hybrid techniques (Duechting and Vogelsaenger 1981; Duechting et al. 1992; Ginsberg et al. 1993; Kansal et al. 2000; Stamatakos et al. 2001a, 2001b; Zacharaki et al. 2004), agent-based techniques (Mansury and Deisboeck 2003), etc. Additionally, a number of bulky clinical tumor models focusing mainly on invasion and tumor growth morphology rather than on tumor response to concrete therapeutic schemes as administered in the clinical setting have appeared. Finite dierence and nite element-based solutions of the diusion and classical mechanics equations constitute the core working tools of the corresponding techniques (Murray 2003; Swanson et al. 2002; Clatz et al. 2005).