# Tumor Volume Distributions Based on Weibull Distributions of Maximum Tumor Diameters
## 3 Results  
### Verification Using Real Lung Tumor Volumes  
The method derived in this work to determine tumor volume distributions from tumor diameter distributions is based on the assumption that tumors can be represented as ellipsoids and that their volume scales with the diameter to the power of \(k\). The NSCLC dataset allows verification of this scaling. On the one hand, k can be calculated from the fit of the modified Weibull distribution (Equation (1)) to the distribution of maximum diameters; on the other hand, \(k\) can be determined directly from a fit of the volumes to \(d^{k}\). A match would support the methodology.  
Figure 2 plots the volume data for the 176 tumors as a function of maximum diameter. The solid line corresponds to a fit to the relationship \(\frac{\gamma}{6}\cdot d^{k}\). The parameter \(k\) was determined to be 2.21 \(\pm\) 0.25. Figure 3a shows the fit of the Weibull distribution from Equation (1) to the histogrammed maximum diameters. The resulting parameters are \(\lambda\) = 0.13 \(\pm\) 0.01, \(k\) = 2.14 \(\pm\) 0.36 and \(d_{c}\) = 1.5 \(\pm\) 0.9 cm.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Diameter Interval (cm)** & \multicolumn{4}{c}{**Patient Number in Time Period**} \\ \hline
**Breast Cancer** & **1983–1989** & **1990–1999** & **2000–2009** & **2010–2019** \\ \hline
0.0–0.5 & 2172 & 6165 & 9267 & 12,710 \\
0.5–1.0 & 5481 & 18,122 & 24,458 & 31,366 \\
1.0–1.5 & 10,668 & 25,907 & 30,770 & 34,998 \\
1.5–2.0 & 11,471 & 23,032 & 26,788 & 30,885 \\
2.0–2.5 & 10,726 & 17,013 & 18,191 & 21,124 \\
2.5–3.5 & 12,802 & 18,069 & 20,012 & 25,335 \\
3.5–4.5 & 5742 & 7636 & 8897 & 12,414 \\
4.5–5.5 & 2979 & 4102 & 5342 & 7128 \\ \hline
**NSCLC** & **1983–1989** & **1990–1999** & **2000–2009** & **2010–2019** \\ \hline
0.0–1.0 & 2134 & 2060 & 1705 & 2199 \\
1.0–2.0 & 10,312 & 11,962 & 8825 & 11,586 \\
2.0–3.0 & 18,678 & 17,578 & 10,293 & 11,132 \\
3.0–4.0 & 18,809 & 15,894 & 8449 & 8663 \\
4.0–5.0 & 15,007 & 12,410 & 6629 & 6670 \\
5.0–6.0 & 12,021 & 9547 & 5066 & 4935 \\
6.0–7.0 & 8206 & 6836 & 3845 & 3905 \\
7.0–8.0 & 5468 & 4802 & 2770 & 2934 \\
8.0–9.0 & 4011 & 3318 & 1964 & 2026 \\
9.0–10.0 & 1794 & 1142 & 473 & 412 \\ \hline
**Liver** & **1983–1999** & **2000–2009** & **2010–2019** & \\ \hline
0.0–1.0 & 47 & 98 & 159 & \\
1.0–2.0 & 147 & 693 & 2179 & \\
2.0–3.0 & 241 & 1304 & 3566 & \\
3.0–4.0 & 363 & 1234 & 2516 & \\
4.0–5.0 & 392 & 1076 & 1893 & \\
5.0–6.0 & 455 & 867 & 1418 & \\
6.0–7.0 & 335 & 705 & 1146 & \\
7.0–8.0 & 355 & 630 & 872 & \\
8.0–9.0 & 347 & 515 & 798 & \\
9.0–10.0 & 162 & 185 & 176 & \\ \hline \end{tabular}
\end{table}
Table 1: Overview of analyzed SEER data.  
Another indication that the method of estimating volume distributions from diameter distributions is applicable is shown in Figure 3b. Here, the volume distribution is plotted, which was generated directly from the fitted parameters \(\lambda\), \(k\) and \(d_{c}\) of the diameter distribution. For the mean tumor volume, \(\overline{V}=\frac{\pi}{6\cdot\lambda^{k}}=41.2\) cm\({}^{3}\), and for the critical volume, \(V_{c}=\frac{\pi}{6}\cdot d_{c}{}^{k}=1.3\) cm\({}^{3}\) was used.