Purves R D
Department of Pharmacology, University of Otago Medical School, Dunedin, New Zealand.
J Pharm Sci. 1995 Jan;84(1):71-4. doi: 10.1002/jps.2600840117.
Numerical inversion of the Laplace transform is a useful technique for pharmacokinetic modeling and parameter estimation when the model equations can be solved in the Laplace domain but the solutions cannot be inverted back to the time domain. The accuracy of numerical inversion of the Laplace transform using an infinite series approximation due to Hosono was systematically studied by reference to 17 widely differing functions having known inverse transforms. The error of inversion was found to be very sensitive to the details of the computer implementation of the method; for example, double-precision artihmetic is essential. The method used to sum the series in the least-squares program Multi(Filt) was often unable to achieve a relative error of less than 10(-4), and a Monte Carlo simulation showed that this method is insufficiently accurate for reliable least-squares parameter estimation. Improvements to the algorithm are described whereby a better method of applying Euler's transformation is used and the number of terms summed is determined automatically by the rate of convergence of the series. The improved algorithm is more efficient in inverting easy functions and more reliable in inverting difficult functions, especially those involving a time lag. With its use, pharmacokinetic parameter estimation can be performed with essentially the same accuracy as when the function is defined in the time domain.
当模型方程能在拉普拉斯域求解但解无法逆变换回时域时，拉普拉斯变换的数值反演是药代动力学建模和参数估计的一种有用技术。参照17个具有已知逆变换的差异很大的函数，系统地研究了使用因细野提出的无穷级数近似进行拉普拉斯变换数值反演的准确性。发现反演误差对该方法计算机实现的细节非常敏感；例如，双精度算术是必不可少的。用于在最小二乘程序Multi(Filt)中求和级数的方法通常无法实现小于10(-4)的相对误差，蒙特卡罗模拟表明该方法对于可靠的最小二乘参数估计不够准确。描述了对算法的改进，其中使用了应用欧拉变换的更好方法，并且求和的项数由级数的收敛速度自动确定。改进后的算法在反演简单函数时更高效，在反演困难函数（尤其是那些涉及时间滞后的函数）时更可靠。使用该算法进行药代动力学参数估计时，其准确性与函数在时域定义时基本相同。
