# Tumor Volume Distributions Based on Weibull Distributions of Maximum Tumor Diameters
## 4 Discussion  
In this work, it was shown that it is possible to derive tumor volume distributions from the distributions of maximum tumor diameters in a cohort of patients. It can be seen that spherical tumors (\(k\) = 3) are always exponentially distributed. If the assumption of spherical shape is omitted, then the transformation holds under the assumption the tumor is an ellipsoidal with maximum diameter d and the smaller diameters \(d_{2}\cdot d_{3}=d^{(k-1)}\). As a consequence, the parameter \(k\) from the Weibull distribution of diameters determines the shape of the tumor ellipsoids. 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]. This presented study has several limitations that warrant consideration. The most important thing to mention is that the shape of the tumor volumes must follow a certain regularity. This seems to be the case for the cohort of NSCLC tumor patients studied here, for which both the maximum diameters of the tumors and the tumor volumes are available. However, the number of patients in this dataset is small, and, therefore, the statistical power is limited. Therefore, additional tests of the model with larger patient cohorts and various tumor types for which the volumes and maximum tumor diameters are available are needed in the future. Another disadvantage of the volume data taken from the Cancer Imaging Archive, in addition to its statistical limitations, is that it only includes patients selected for radiotherapy, and, therefore, there is a selection bias (e.g., surgery for small tumors). This selection bias has a direct impact on tumor volume distributions, as selection criteria for radiotherapy may also depend indirectly on tumor sizes (stage). Indeed, the fitted parameter \(k\) differs from that obtained from the SEER data. However, the selection bias does not affect the tested relation between volume and maximum diameter, as shown in Figure 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Fit of Modified Weibull to Maximum Diameter** & \multicolumn{2}{c}{**Results for Exponential Distribution of**} \\  & **(Equation (1))** & & **Volumes** & \\ \hline  & \(d_{c}\) **(cm)** & \(\lambda\) & \(k\) & \(V_{c}\) **(cm3)** & \(\overline{V}\) **(cm3)** \\ \hline Breast cancer & & 0.57 \(\pm\) 0.04 & 1.35 \(\pm\) 0.05 & & 1.12 \(\pm\) 0.02 \\
1983–1989 & 2.17 \(\pm\) 0.24 & & 1.49 \(\pm\) 0.23 & \\
1990–1999 & 0.90 \(\pm\) 0.09 & & 0.45 \(\pm\) 0.06 & \\
2000–2009 & 0.70 \(\pm\) 0.09 & & 0.32 \(\pm\) 0.05 & \\
2010–2019 & 0.60 \(\pm\) 0.10 & & 0.26 \(\pm\) 0.10 & \\ \hline NSCLC & & 0.25 \(\pm\) 0.07 & 1.49 \(\pm\) 0.03 & & 4.26 \(\pm\) 0.004 \\
1983–1989 & 2.04 \(\pm\) 0.13 & & 1.52 \(\pm\) 0.15 & & \\
1990–1999 & 1.48 \(\pm\) 0.10 & & 0.94 \(\pm\) 0.09 & \\
2000–2009 & 0.98 \(\pm\) 0.13 & & 0.51 \(\pm\) 0.10 & \\
2010–2019 & 0.70 \(\pm\) 0.21 & & 0.31 \(\pm\) 0.14 & \\ \hline Liver & & 0.22 \(\pm\) 0.02 & 1.45 \(\pm\) 0.09 & & 4.70 \(\pm\) 0.01 \\
1983–1999 & 18.6 \(\pm\) 16 & & 36.3 \(\pm\) 47 & \\
2000–2009 & 2.00 \(\pm\) 0.33 & & 1.43 \(\pm\) 0.35 & \\
2010–2019 & 1.08 \(\pm\) 0.50 & & 0.59 \(\pm\) 0.38 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Overview of the fitted parameters. It should be noted that Vc and \(\overline{V}\) of the exponential volume distribution are not separately fitted, but calculated from the diameter fits. An open question is how to correctly specify a cohort of individuals with tumors for which the presented intrinsic distributions for tumor diameters and tumor volumes apply. Does this include only patients with macroscopic tumors or also those with microscopic tumors? If microscopic tumors are included, the questions arise: what is the minimum size of such tumors? Are only those microscopic tumors counted which later necessarily transform into a macroscopic tumor?