Path prediction of flexible needles based on Fokker-Planck equation and disjunctive Kriging model

基于福克普朗克方程与析取克里金模型的柔性针路径预测

Path prediction of flexible needles based on the Fokker-Planck equation and disjunctive Kriging model is proposed to improve accuracy and consider the nonlinearity and anisotropy of soft tissues.The stochastic differential equation is developed into the Fokker-Planck equation with Gaussian noise,and the position and orientation probability density function of flexible needles are then optimized by the stochastic differential equation.The probability density function obtains the mean and covariance of flexible needle movement and helps plan puncture paths by combining with the probabilistic path algorithm.The weight coefficients of the ordinary Kriging are extended to nonlinear functions to optimize the planned puncture path,and the Hermite expansion is used to calculate nonlinear parameter values of the disjunctive Kriging optimization model.Finally,simulation experiments are performed.Detailed comparison results under different path planning maps show that the kinematics model can plan optimal puncture paths under clinical requirements with an error far less than 2 mm.It can effectively optimize the path prediction model and help improve the target rate of soft tissue puncture with flexible needles through data analysis and processing of the mean value and covariance parameters derived by the probability density and disjunctive Kriging algorithms.

为提高柔性针穿刺软组织的精度并充分考虑软组织的非线性和各向异性,提出了一种基于福克-普朗克方程和析取克里金模型的柔性针路径预测方法.首先,将随机微分方程演化为加入高斯随机噪声的福克-普朗克方程,优化含有位置和方向的柔性针概率密度函数.其次,由概率密度函数求得柔性针运动的均值和协方差,结合概率路径算法规划穿刺路径.然后,将普通克里金方法中的权重系数推广为非线性函数,由埃尔米特展开式计算析取克里金优化模型的非线性参数值.最后,对不同路径规划模拟图进行仿真试验分析.结果表明,在非线性的软组织环境下,该运动学模型能够在柔性针穿刺误差远小于2 mm的临床要求下规划出最优穿刺路径.通过对概率密度算法和析取克里金算法推导的均值、协方差参数的数据分析处理可有效地优化路径预测模型,有助于提高柔性针穿刺软组织的命中靶向率.