# Tumor Volume Distributions Based on Weibull Distributions of Maximum Tumor Diameters
## 1 Introduction  
Tumor volume is an important parameter in various aspects of cancer research, including tumor growth modeling, predictions of therapeutic outcomes in cancer treatment [1, 2, 3] and modeling of tumor control probability (TCP) in radiotherapy.  
In particular, in TCP modeling, the knowledge of the tumor volume is of high relevance. The classical TCP model for an individual patient is of sigmoidal shape and depends on patient individual characteristics, such as tumor volume, cell radiation sensitivity, clonogenic cell density and others [4, 5, 6]. Characteristically, the TCP shows a steep gradient with dose. If the tumor control in a cohort of patients with the same tumor is epidemiologically observed, the gradient of the TCP-dose relation decreased due to variations of the patient-specific characteristics in the cohort [7, 8]. Standard practice is therefore to fit empirical TCP models as, e.g., logistic models to those data [9, 10]. Since there is no direct relation between the empirical cohort TCP models and the mechanistically based patient individual TCP models, the obtained fitting parameters cannot be used to predict patient individual TCP. However, this is necessary for the use of TCP calculation in radiotherapy treatment planning.  
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. However, since the parameters of thenew logistic representation of cohort TCP are mechanistically based, it allows determination of the patient individual model parameters. This is the first step towards the clinical use of TCP models.  
Hence, it would be helpful to know the distribution of tumor volumes for the individual tumor types. Unfortunately, tumor volume is never documented in tumor registries today. If anything at all the maximum tumor diameter is documented. In the database of the National Cancer Institute, in its Surveillance, Epidemiology, and End Results (SEER) program database, for example, the maximum tumor diameters of about half a million patients per year are documented. It would be useful if tumor volumes could somehow be derived from reported maximum tumor diameters. Although there have been several attempts to estimate tumor volumes from tumor diameters [13; 14; 15; 16; 17; 18; 19; 20; 21], there is no known method to calculate tumor volume distributions from tumor diameter distributions.  
This paper presents a first attempt at deriving tumor volume distributions from tumor diameter distributions. The paper is mainly divided into three parts. In the first part, the method for deriving tumor volume distributions from tumor diameter distributions is presented. It is shown that, under certain assumptions, an exponential distribution of tumor volumes follows from a Weibull distribution of maximum tumor diameters. In the second part, a three-dimensional dataset of segmented tumor volumes is used to verify the method. In the third and final part, tumor volume distributions are calculated from the maximum diameters of the SEER database for breast, NSCLC and liver as examples.