Tumour control probability: a formulation applicable to any temporal protocol of dose delivery

Abstract

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).

Derivation of the Tumour Control Probability (TCP) from a Cell Cycle Model

“…In section 1.3, we review existing TCP models, in particular the work of Zaider and Minerbo [25]. In section 2, we give a detailed, physically based derivation of the radiation cell cycle model (1).…”

A TCP‐NTCP estimation module using DVHs and known radiobiological models and parameter sets

“…Our NTCP and TCP calculation module incorporates a total of four radiobiological models. Included are two NTCP models: a sigmoidal dose response (SDR) model introduced by Lyman (9) and individual‐based and population‐based variants of the CV model; and two TCP models: a two‐parameter Poisson‐based model and a model employing linear‐quadratic cell kill and the formalism developed by Zaider and Minerbo (8) to account for repopulation. The simple SDR and Poisson models have been most frequently applied in the analysis of normal tissue complication and tumor response data, respectively.…”

“…(15) is mainly phenomenological, we believed it would be useful to include a second TCP model that is parametrized in terms of fundamental cellular radiation response characteristics. Recently, Zaider and Minerbo (8) derived a conceptually robust expression for TCP that incorporates the effect of tumor repopulation. The original Zaider‐Minerbo expression, valid for any temporal protocol of dose delivery, has been adapted for the case of a fractionated delivery with varying time intervals between fractions by Stavreva et al (15) This adapted expression predicts that the TCP after the delivery of n fractions is $\text{TCP}={\left[1-\frac{p_s(T_n)e^{\lambda T_n}}{\left(1-p_s(T_n)e^{\lambda T_n}\underset{k=1}{\overset{n-1}{false\sum }}\frac{1}{p_{s}false(T_{k}false)}[e^{-\lambda T_{k+1}}-e^{-\lambda T_k}]\right)}\right]}^N,$ where λ is the rate of cellular repopulation, $T_k$ is the time between the kth fraction and the first fraction, and $p_{\mathrm{normals}}(T_k)$ is the cell survival after the kth fraction.…”

“…Although the two packages share some similarities, a number of differences also exist. For example, our module includes the Critical Volume (CV) NTCP model ( 5 – 7 ) and the recent Zaider‐Minerbo/LQ TCP model, (8) which are not available in BIOPLAN. TCP_NTCP_CALC also includes several different parameter databases and provides a convenient method of archiving (and using) newly published parameter databases.…”

The use of TCP based EUD to rank and compare lung radiotherapy plans: in-silico study to evaluate the correlation between TCP with physical quality indices

“…These models are multiple and based on different mathematical and statistical concepts. Some of them are directly available in the treatment planning systems (TPS), which are used to calculate the dose distribution (1)(2)(3)(4)(5)(6)(7). However, several precautions should be observed for a safe use of these TCP models to predict radiotherapy outcomes.…”