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Plot of the TCP model of equation 8 for individual patients as the dotted line for three different average tumour volumes 10, 100 and 1000 cm 3 , respectively. The solid lines shows the TCP model for a volume averaged patient cohort according to equation 9.
Source publication
Background
It is shown that tumour volume distributions, can yield information on two aspects of cancer research: tumour induction and tumour control.
Materials and methods
From the hypothesis that the intrinsic distribution of breast cancer volumes follows an exponential distribution, firstly the probability density function of tumour growth time...
Context in source publication
Context 1
... Fig. 4 the TCP model of equation 8 is plotted as the dotted line for three different average tumour volumes 10, 100 and 1000 cm 3 , respectively. The solid lines represent the TCP calculations including an exponential distribution of tumour volumes from equation 9. Clearly, a broadening of the TCP curve is ...
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Citations
... Through this procedure, model parameters were obtained which enabled the calculation of TCP. In [39,42], the population TCP model was derived by analytically incorporating variations of tumor volume sizes. As in [42], it was assumed that the underlying frequency distribution of the present tumor volume sizes in the population is exponentially distributed. ...
... In [39,42], the population TCP model was derived by analytically incorporating variations of tumor volume sizes. As in [42], it was assumed that the underlying frequency distribution of the present tumor volume sizes in the population is exponentially distributed. However, it was conjectured that there is a minimal tumor volume below which tumors are unlikely to be clinically observed [39,42]. ...
... As in [42], it was assumed that the underlying frequency distribution of the present tumor volume sizes in the population is exponentially distributed. However, it was conjectured that there is a minimal tumor volume below which tumors are unlikely to be clinically observed [39,42]. Thus the frequency distribution is given by ...
... Recently, it has been shown that the tumor control for an individual patient can be integrated over the volume by assuming that the tumor volumes in the patient cohort are exponentially distributed [11,12]. This results in an analytic representation of the cohort TCP which, interestingly, results in a logistic curve. ...
... A second important result, particularly with respect to modeling tumor control in radiotherapy, is that tumor volumes are exponentially distributed. As shown in other work, this allows relation of the tumor control of an individual patient to the tumor control in a cohort of radiotherapy patients [11,12]. ...
(1) Background: The distribution of tumor volumes is important for various aspects of cancer research. Unfortunately, tumor volume is rarely documented in tumor registries; usually only maximum tumor diameter is. This paper presents a method to derive tumor volume distributions from tumor diameter distributions. (2) Methods: The hypothesis is made that tumor maximum diameters d are Weibull distributed, and tumor volume is proportional to dk, where k is a parameter from the Weibull distribution of d. The assumption is tested by using a test dataset of 176 segmented tumor volumes and comparing the k obtained by fitting the Weibull distribution of d and from a direct fit of the volumes. Finally, tumor volume distributions are calculated from the maximum diameters of the SEER database for breast, NSCLC and liver. (3) Results: For the test dataset, the k values obtained from the two separate methods were found to be k = 2.14 ± 0.36 (from Weibull distribution of d) and 2.21 ± 0.25 (from tumor volume). The tumor diameter data from the SEER database were fitted to a Weibull distribution, and the resulting parameters were used to calculate the corresponding exponential tumor volume distributions with an average volume obtained from the diameter fit. (4) Conclusions: The agreement of the fitted k using independent data supports the presented methodology to obtain tumor volume distributions. The method can be used to obtain tumor volume distributions when only maximum tumor diameters are available.
... Often empirical models such as the logit are used, they provide a good fit to clinical data while only using a small number of parameters [12]. In Schneider and Besserer [13], Radonic et al. [14] a closed form population TCP model was obtained which characteristically matches the logistic model used by Okunieff et al. [12]. This was done by assuming that the tumor volumes in the patient population are exponentially distributed. ...
In radiation therapy tumor size, and thus also volume, has a significant impact on the local control of tumors. Moreover, tumor volume is a significant prognostic factor for modelling and predicting therapeutic outcomes in cancer treatment. In research, the distribution of tumor volumes in patient populations has so far remained widely unexplored. In this work, the frequency distributions of maximum diameter of tumors for various types of cancer was studied based on SEER data and it was explored if they can be modelled using Weibull distributions. Further, actual volume data were obtained directly from computer tomography (CT) datasets and the frequency distributions of tumor volumes and maximum diameters were explored and a link between them was found. In cancer research, tumors are often modelled as ellipsoids. In order to verify the appropriateness of using ellipsoids as a model, to the obtained three-dimensional data, ellipsoids were fitted and the resulting volumes and diameters analysed. Finally, NSCLC tumor diameters were Monte Carlo simulated using tumor growth and incidence models. A comparison of the simulated tumor diameter distributions with observed SEER data yielded the determination of tumor growth rates.
... The population TCP model leverages the gross tumor volume (GTV) information to incorporate the tumor volume size variation within a cohort into the Poisson statistics based individual TCP model. 10 In the context of this study. it was extended to also incorporate the cell density variation. ...
... In Okunieffs logit model the TCP model is essentially characterized by 2 parameters. One is the 50% tumour Table 1 Clinical patient survival data, In [10] and [11] various fractionation schemes within the denoted ranges were applied. In [12] two distinct fractionation schemes (20 x 2.5 Gy and 10 x 4 Gy) were used exclusively, there is no follow-up data after 36 months; In [13] control dose (TCD 50 ), which is the dose where half of the tumors are controlled. ...
... The average GTV size was fixed to the mean value of the observed clinical data V avg ¼ 3:37 cm 3 . The value of V C ¼ 1:6cm 3 was determined by fitting Equation 10 to the clinical volume data. ...
Purpose: Tumor control probability (TCP) models based on Poisson statistics characterize the distribution of surviving clonogens. Thus enabling the calculation of TCP for individuals. To mathematically describe clinically observed survival data of patient cohorts it is necessary to extend the Poisson TCP model. This is typically done by either incorporating variations of model parameters or by using an empirical logistic model. The purpose of this work is the development of an analytical population TCP model by mechanistic extension of the Possion model. Methods and materials: The frequency distribution of gross tumor volumes was used to incorporate tumor volume variations into the TCP model. Additionally the tumor cell density variation was incorporated. Both versions of the population TCP model were fitted to clinical data and compared to existing literature. Results: It was shown that clinically observed brain tumor volumes of dogs undergoing radiotherapy are distributed according to an exponential distribution. The average gross tumor volume size was 3.37 cm3. Fitting the population TCP model including the volume variation using linear-quadratic and track-event model yieldedα=0.36Gy--1a, β=0.045Gy--2, a=0.9yr--1, TD=5.0d,and p=.36Gy--1, q=0.48Gy--1, a=0.80yr--1, TD=3.0d, respectively. Fitting the population TCP model including both the volume and cell density variation yielded α=0.43Gy--1, β=0.0537Gy--2, a=2.0yr--1, TD=3.0d, σ=2.5,and p=.43Gy--1, q=0.55Gy--1, a=2.0yr--1, TD=2.0d, σ=3.0,respectively. Conclusions: Two sets of radiobiological parameters were obtained which can be used for quantifying the TCP for radiation therapy of brain tumors in dogs. We established a mechanistic link between the poisson statistics based individual TCP model and the logistic TCP model. This link can be used to determine the radiobiological parameters of patient specific TCP models from published fits of logistic models to cohorts of patients.
