Tumor Tvol may be described in various ways but in this paper have been described using an Exponential and Logistic model adapted for untreated HCC as demonstrated in Figure 2. Tvol for untreated HCC are described in Figure 1. Small tumors initially

Wnt tumor tend to grow exponentially but eventually with increasing size, blood and nutrients decrease and growth rate slows as represented by the logistic curve in Figure 2. Tumor volume doubling times do not indicate the true ‘birth rate’ of tumor cells which is better described by the potential volume doubling time (Tpot). This is described in more detail in the Discussion section. Radiosensitivity can also be described in many ways. Radiosensitivity is a measure of the fraction of clonogenes (cells capable of infinite reproduction) that survive a given X-ray dose. Here, a common method of using the fraction surviving a 2.0-Gy single dose (SF2) is shown in Figure 3. A more comprehensive measure of radiosensitivity utilizes the Linear-Quadratic (L-Q) equation, survival fraction =  exp[−N · [α · d + β · d2]]. selleck kinase inhibitor N is the number of fractions, d is the dose per fraction, α is a measure of cells killed in the Linear portion of the dose-response curve and β is a measure of cells killed in the Quadratic (dose)2 component of the equation. These two methods

of defining radiosensitivity have been used in Figure 5 to predict the change in tumor control probability (TCP) with tumor size. The dose that normal

tissue can tolerate is very dependent on the volume treated and many models have been developed to quantify this effect. In this paper, the Relative Seriality Model described by Kallman et al.4 is shown in the Appendix (equation 5) and is used to derive Figure 4. The selection criteria for inclusion in this analysis were that each individual case could be identified and that no anticancer treatment was given during the period of observation. There were 11 series with 283 individuals that fulfilled these criteria.5–15 A lognormal distribution, shown in Figure 1, was a significantly better fit than a normal distribution (χ2 = 5.69, P = 0.22). MCE A lognormal rather than a normal distribution is consistent with distributions of doubling times of other human tumors. In this series of 283 cases, the median value was 130 days, geometric mean 129, mode 120, mean 176, minimum 17.5 and maximum 1165 days (standard deviation 153 days). Figure 2 shows a series of exponential growth curves which were generated using Appendix equation 1. Figure 2 also shows a single logistic growth curve and the equations for this are Appendix equations 2, 3 and 4. Exponential growth curves shown in Figure 2 are for a range of Tvol from 0–390 days increasing in increments of 30 days. Of particular interest is the curve corresponding to the 130 days Tvol, which approximates the median Tvol of untreated HCC.