An analytic expression for the tumour control probability (TCP), valid for any temporal distribution of dose, is discussed. The TCP model, derived using the theory of birth-and-death stochastic processes, generalizes several results previously obtained. The TCP equation is [equation: see text] where S(t) is the survival probability at time t of the n clonogenic tumour cells initially present (at t = 0), and b and d are, respectively, the birth and death rates of these cells. Equivalently, b = 0.693/Tpot and d/b is the cell loss factor of the tumour. In this expression t refers to any time during or after the treatment; typically, one would take for t the end of the treatment period or the expected remaining life span of the patient. This model, which provides a comprehensive framework for predicting TCP, can be used predictively, or--when clinical data are available for one particular treatment modality (e.g. fractionated radiotherapy)--to obtain TCP-equivalent regimens for other modalities (e.g. low dose-rate treatments).
Weak forms are derived for Maxwell's equations which are suitable for implementation on conventional C O elements with scalar bases. The governing equations are %xpressed in terms of general vector and scalar potentials for E. Gauge theory is invoked to close the system and dictates the continuity requirements for the potentials at material interfaces as well as the blend of boundary conditions at exterior boundaries. Two specific gauges are presented, both of which lead to Helmholtz weak forms which are parasite-free and enjoy simple, physically meaningful boundary conditions. The extended weak form introduced by Lynch and Paulsen along with associated boundary conditions, is recovered in greater generality from the first gauge considered, where the vector potential is discontinuous at material interfaces and when the scalar potential vanishes. The second and preferred gauge allows the use of continuous vector and scalar potentials at the expense of introducing coupling among the two potentials. A general and numerically efficient procedure for enforcing the jump discontinuities on the normal components of vector fields at dielectric interfaces and boundary conditions on curved surfaces is given in the Appendix.
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