# Tumor Volume Distributions Based on Weibull Distributions of Maximum Tumor Diameters
## 2 Materials and Methods  
### Maximum Tumor Diameter Distribution  
When individuals are diagnosed with a tumor, usually the maximum tumor diameter at the time of diagnosis is recorded. In the past, it has been suggested that the maximum diameters (in the following, referred to as simply \"diameters\") of tumors of one type are Weibull distributed [22]. The probability density function (pdf) of the diameters is then:  
\[f(d)=\lambda\cdot k\cdot(\lambda\cdot d)^{k-1}\cdot e^{-(\lambda\cdot d)^{k}} \text{ for }d\ >\ 0, \tag{1}\]  
where and \(\lambda\) and \(k\) are the parameters of the Weibull distribution, and d is the maximum diameter of a tumor. However, these studies did not distinguish between the distribution of tumor size in patients (hereafter referred to as the intrinsic distribution) and the distribution of actually diagnosed tumor diameters. This distinction is essential because of the technical limitations of diagnostic methods the occurrence of small tumors is underestimated. Therefore, the diagnosed distributions will change over the years as diagnostic methods tend to improve. The best example of this is the introduction of mammography in the diagnosis of breast cancer. We therefore propose to describe the intrinsic distribution of tumor diameters by a Weibull distribution and to model the limitation of diagnostic methods in the detection of small tumors by a \"diagnostic limit function\" \(1-exp\left[-(d/d_{c})^{k}\right]\) that depends on a critical maximum tumor diameter \(d_{c}\). This \"diagnostic limit function\" represents the probability of detecting a tumor in a population of cancer patients. The simplest possible representation of the diagnostic function was chosen because, to the best of our knowledge, too little information is available to justify a more precise formulation. The introduction of the parameter k will become clear at a later stage, when the volume distributions are derived. The resulting probability density function of maximum tumor diameters in a cohort of tumor patients can then be written as:  
\[f(d)=\left(1-e^{-(\frac{d}{d_{c}})^{k}}\right)\cdot\lambda\cdot k\cdot(\lambda \cdot d)^{k-1}\cdot e^{-(\lambda\cdot d)^{k}}\text{ for }d\ >\ 0, \tag{2}\]  
where \(d_{c}\) is the critical diameter. If the shape of a tumor is represented by an ellipsoid [23], the volume is \(V=\frac{\pi}{6}\cdot d\cdot d_{2}\cdot d_{3}\) with the three diameters \(d\) (maximum diameter), \(d_{2}\) and \(d_{3}\). Obviously, the distribution of the maximum tumor diameters alone is not sufficient to determine the shape of the volumes and, thus, \(d\), \(d_{2}\) and \(d_{3}\). Therefore, it is necessary to introduce a first assumption; it is assumed that the tumor volume is sufficiently defined by the maximum diameter with:  
\[V=\mu\cdot d^{k} \tag{3}\]  
where \(k\) is the parameter from the Weibull distribution of diameters, and \(\mu\) is a scaling factor which is \(\pi/6\) multiplied by the dimension [cm\({}^{3\text{-k}}\)] when the maximum diameter is expressed in cm. In case of ellipsoids, the product of the second and third diameter is determined by \(d_{2}\cdot d_{3}=d^{k-1}\). In a simple case, the ellipsoid is rotationally symmetric about the longest axis \(d\) and forms a spheroid. The relation between maximum and minimum diameter is then fixed: \(d_{2}=d_{3}=d^{(k-1)/2}\). This means that the shape of the spheroid is determined by the choice of \(k\), and the closer \(k\) is to 3, the more spherical the spheroid becomes. If the tumors were perfect spheres, \(k=3\) would sufficiently determine the volume. On the other hand, the tumor volume becomes pencil shaped for \(k=1\), which means that a realistic range of values for \(k\) is between 1 and 3. It should also be mentioned that the shape of the ellipsoid or spheroid depends on the volume. This is a direct consequence of Equation (3) and is exemplarily shown in Figure 1 for \(k=1.5\) and three different tumor diameters (\(d=1.5\), 3 and 6 cm).