technical note How well are clinical gross tumor volume DVHs approximated by an analytical function? Pavel Stavrev2, Colleen Schinkel1,2, Nadia Stavreva2 and B. Gino Fallone1-3 department of Physics, University of Alberta, 2Department of Medical Physics, Cross Cancer Institute, 3Department of Oncology, University of Alberta, Edmonton, Alberta, Canada The dose heterogeneity in the tumor is often described as being normally distributed. Besides the normal distribution we propose the Fermi function describing Fermi statistics as a possible dose heterogeneity descriptor. In order to demonstrate the adequacy of the proposed functions as dose distribution descriptors 30 clinical gross tumor volume (GTV) dose-volume histograms (DVHs) are gathered and fit with the examined functions. Key words: dose-volume histograms; gross tumor volume; Gaussian and Fermi statistics In order to theoretically investigate a given tumor control probability (TCP) model for the case of heterogeneous irradiation, it is often necessary to simulate tumor dosevolume histograms (DVHs) that closely resemble clinical ones. Some authors1-3 have assumed that tumor dose inhomogeneities are normally distributed around the target dose. In this case the integral DVH, iDVH, is represented by the erfc function: iDVH : v(d\ fi,0) = O.5erfc Ojlx (d-n) [1] where v is the relative tumor volume irradiated to a maximum dose d, |i is a parameter Received 14 January 2009 Accepted 15 February 2009 Correspondence to: Pavel Stavrev, Department of Medical Physics, Cross Cancer Institute, 11560 University Ave, Edmonton, Alberta, Canada. Phone: + 1 780 989 4334; Fax: +1 780 432-8615; E-mail: pavel.stavrev@gmail.com corresponding to the mean (target) dose delivered to the tumor, and d is a parameter related to the slope of the erfc function. However, no investigations of how well this function describes the clinical tumor DVHs are reported in the literature. We propose the parallel use of the Fermi statistics function for the description of clinical tumor DVHs: iDVH: v(d\ju,0) = - 1 1 + exp d-fi 8 [2] This function describes the filling up of free energy levels in a Fermi system. The parameters d, /J, and 0 have the same meaning as in eq. [1]. To investigate this problem, we gathered 30 clinical gross tumor volume (GTV) DVHs for different treatment sites - lung, head & neck, prostate, etc., that were either obtained in the treatment planning process at the Cross Cancer Institute (CCI) or reported in the literature.4-14 They were fit with the erfc [Eq. 1] and Fermi [Eq. 2] functions. The fit was performed using the %2 criterion for goodness of fit, presuming a log-log-normal distribution for the integral DVHs. Correspondingly, the function to be minimized is: x2=X The log-log form of the %2 criterion was used to account for the fact that an integral DVH is defined in the interval [0,1], while the standard %2 criterion deals with normally distributed random variables defined in (—oo,+oo).15 The experimental error is ^experimental which unfortunately is not reported in the literature. Therefore, we substituted ^experimental with a percentage band - In (- In (Vtheoretical )) + ln(-ln (v;xperimental)) log experimental -CT, experimental log experimental experimental ^ (^experimental) Figure 1. Fits to four clinical (head & neck) DVHs from CCI with the erfc function - a) and with the Fermi function - b) for the case of a 2% error band (^^^m = 2%). On each subplot the p-value of the fit are shown, along with the statistics (number of data points, N^) and the corresponding best fit values of the model parameters (ß mid 0). 10 - 8 - m