Bayesian Analysis of a One Compartment Kinetic Model Used in Medical Imaging.

Kinetic models are used extensively in science, engineering, and medicine. Mathematically, they are a set of coupled differential equations including a source function, otherwise known as an input function. We investigate whether parametric modeling of a noisy input function offers any benefit over the non-parametric input function in estimating kinetic parameters. Our analysis includes four formulations of Bayesian posteriors of model parameters where noise is taken into account in the likelihood functions. Posteriors are determined numerically with a Markov chain Monte Carlo simulation. We compare point estimates derived from the posteriors to a weighted non–linear least squares estimate. Results imply that parametric modeling of the input function does not improve the accuracy of model parameters, even with perfect knowledge of the functional form. Posteriors are validated using an unconventional utilization of the chi square test. We demonstrate that if the noise in the input function is not taken into account, the resulting posteriors are incorrect.

动力学模型在科学、工程和医学中广泛应用。从数学角度来说，它们是一组包含源函数（也称为输入函数）的耦合微分方程。我们研究对有噪声的输入函数进行参数化建模在估计动力学参数方面是否比非参数化输入函数更有优势。我们的分析包括模型参数的贝叶斯后验的四种公式，其中在似然函数中考虑了噪声。后验通过马尔可夫链蒙特卡罗模拟以数值方式确定。我们将从后验得出的点估计与加权非线性最小二乘估计进行比较。结果表明，即使完全了解函数形式，对输入函数进行参数化建模也不会提高模型参数的准确性。使用卡方检验的一种非常规应用对后验进行了验证。我们证明，如果不考虑输入函数中的噪声，所得的后验是不正确的。