The application of the sinusoidal model to lung cancer patient respiratory motion

Accurate modeling of the respiratory cycle is important to account for the effect of organ motion on dose calculation for lung cancer patients. The aim of this study is to evaluate the accuracy of a respiratory model for lung cancer patients. Lujan et al. [Med. Phys. 26(5), 715-720 (1999)] proposed a model, which became widely used, to describe organ motion due to respiration. This model assumes that the parameters do not vary between and within breathing cycles. In this study, first, the correlation of respiratory motion traces with the model f(t) as a function of the parameter n (n = 1, 2, 3) was undertaken for each breathing cycle from 331 four-minute respiratory traces acquired from 24 lung cancer patients using three breathing types: free breathing, audio instruction, and audio-visual biofeedback. Because cos2 and cos4 had similar correlation coefficients, and cos2 and cos1 have a trigonometric relationship, for simplicity, the cos1 value was consequently used for further analysis in which the variations in mean position (z0), amplitude of motion (b) and period (tau) with and without biofeedback or instructions were investigated. For all breathing types, the parameter values, mean position (z0), amplitude of motion (b), and period (tau) exhibited significant cycle-to-cycle variations. Audio-visual biofeedback showed the least variations for all three parameters (z0, b, and tau). It was found that mean position (z0) could be approximated with a normal distribution, and the amplitude of motion (b) and period (tau) could be approximated with log normal distributions. The overall probability density function (pdf) of f(t) for each of the three breathing types was fitted with three models: normal, bimodal, and the pdf of a simple harmonic oscillator. It was found that the normal and the bimodal models represented the overall respiratory motion pdfs with correlation values from 0.95 to 0.99, whereas the range of the simple harmonic oscillator pdf correlation values was 0.71 to 0.81. This study demonstrates that the pdfs of mean position (z0), amplitude of motion (b), and period (tau) can be used for sampling to obtain more realistic respiratory traces. The overall standard deviations of respiratory motion were 0.48, 0.57, and 0.55 cm for free breathing, audio instruction, and audio-visual biofeedback, respectively.