虚拟手术中人体软组织变形仿真
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摘要
虚拟手术仿真(Virtual Surgery Simulation)是专门用来模拟在手术过程可能遇到的各种现象的虚拟现实(Virtual Reality)应用系统。其研究内容包括对医学数据的交互与可视化、虚拟人体器官在虚拟手术器械作用下的各种变化的模拟及对操作人员的各种感官反馈(如视觉反馈和力反馈等)的模拟等，是虚拟现实技术在医学领域中的重要应用。
虚拟手术仿真研究涉及计算机图形学、计算机视觉、弹性力学、生物力学、机器人学、医学等诸多领域，通过上述领域知识实现组织器官对象的几何模型、物理/计算模型、碰撞检测、对象实时变形(响应、破裂等)、手术过程真实感绘制等功能。其中人体软组织器官的物理/计算建模是实现虚拟手术仿真的关键技术之一。由于跨学科的特点，人体软组织器官的物理/计算建模研究与其它相关关键技术相比相对滞后，已经成为该类系统研究和应用的瓶颈。
本文将人体软组织器官拓扑结构分为厚壁或薄壁的腔体，如血管、膀胱、胃、肠等；块状软组织，如由皮肤、脂肪及肌肉等组成的组织(如人体乳房等)。对应于不同的虚拟手术仿真要求，将人体软组织简化为线性、非线性、粘弹性物理模型，采用组织的弹性模量、阻尼系数、密度等物理特性表征软组织的粘弹性、各向异性、非均匀性等特性。
本文采用弹簧-质点建模方法提出了人体软组织面模型和体模型系统统一的建模方法。将连续的软组织离散为由弹簧-阻尼器及与其连接的质点构成的离散系统，用邻接矩阵表示离散系统的拓扑结构，即质点-弹簧间的邻接关系，依据该信息，可自动计算与每个质点连接的弹簧、阻尼器对接点的粘弹性力的贡献，然后采用4阶Runge-Kutta方法对得到的微分方程组进行数值积分，用OpenGL技术对得到的仿真结果进行图形绘制。
最后，通过对腔体模型(小肠、血管)和块状软组织模型(腿部软组织等)等实例对本文所提出的方法进行了动态仿真验证。
Virtual surgery simulation is a virtual reality system used to simulate all kinds of phenomenon of surgery. For surgical simulation to be a useful training and simulation tool it must he realistic with respect to tissue deformation, tool interactions, visua rendering, force feedback and real-time response.
As an interdiscipline research tcpic, virtual surgery incorporates the researches about Computer Science, Computer Graphics, Sensor Technology, Biomcchanics, Modem Medicine, Image Processing, Computer Vision, Robot Technology, Scientific Computing Visualization, etc. Using all of the above techniques we can implement the geometrical model, physical/compute model, collisior detection, real-time deformation (cutting, tearing etc), and realistic rendering of human soft tissue. The physical/compute modeling of soft tissue is one of the key techniques of surgery simulation. The research of it is dropped behind by other techniques because of the character of interdiscipline. It has become the bottleneck of surgery simulation research development.
This thesis describe human soft tissue using topology of cavity organ for instance blood vessel, bladder, stomach, and intestines etc, and volume soft tissue for instance skin, fattiness, and muscles. Toward different surgery simulation request, we simplify human soft tissue to linear model, nonlinear model, or stiff/elastic model. Using stiffness coefficient, damping coefficient, and mass point density to represent the characters of stiff/elastic, anisotropy, and non homogenous of soft tissue.
An unified soft t.ssue modeling method using mass-spring model was given in this paper. The soft tissues are discreted to a mass-spring-damper system, and the topologies of the system are described using adjacent tables, the dynamical equations can be constructed based on the data stored in the adjacent tables. The 4th order Runge-Kutta method is used to integrate the equations of deformation. Using computer graphical techniques to display the visual feedback of the deformation. Four examples of a surface model and a volumetric model to test the method are given.
虚拟手术仿真研究涉及计算机图形学、计算机视觉、弹性力学、生物力学、机器人学、医学等诸多领域，通过上述领域知识实现组织器官对象的几何模型、物理/计算模型、碰撞检测、对象实时变形(响应、破裂等)、手术过程真实感绘制等功能。其中人体软组织器官的物理/计算建模是实现虚拟手术仿真的关键技术之一。由于跨学科的特点，人体软组织器官的物理/计算建模研究与其它相关关键技术相比相对滞后，已经成为该类系统研究和应用的瓶颈。
本文将人体软组织器官拓扑结构分为厚壁或薄壁的腔体，如血管、膀胱、胃、肠等；块状软组织，如由皮肤、脂肪及肌肉等组成的组织(如人体乳房等)。对应于不同的虚拟手术仿真要求，将人体软组织简化为线性、非线性、粘弹性物理模型，采用组织的弹性模量、阻尼系数、密度等物理特性表征软组织的粘弹性、各向异性、非均匀性等特性。
本文采用弹簧-质点建模方法提出了人体软组织面模型和体模型系统统一的建模方法。将连续的软组织离散为由弹簧-阻尼器及与其连接的质点构成的离散系统，用邻接矩阵表示离散系统的拓扑结构，即质点-弹簧间的邻接关系，依据该信息，可自动计算与每个质点连接的弹簧、阻尼器对接点的粘弹性力的贡献，然后采用4阶Runge-Kutta方法对得到的微分方程组进行数值积分，用OpenGL技术对得到的仿真结果进行图形绘制。
最后，通过对腔体模型(小肠、血管)和块状软组织模型(腿部软组织等)等实例对本文所提出的方法进行了动态仿真验证。
Virtual surgery simulation is a virtual reality system used to simulate all kinds of phenomenon of surgery. For surgical simulation to be a useful training and simulation tool it must he realistic with respect to tissue deformation, tool interactions, visua rendering, force feedback and real-time response.
As an interdiscipline research tcpic, virtual surgery incorporates the researches about Computer Science, Computer Graphics, Sensor Technology, Biomcchanics, Modem Medicine, Image Processing, Computer Vision, Robot Technology, Scientific Computing Visualization, etc. Using all of the above techniques we can implement the geometrical model, physical/compute model, collisior detection, real-time deformation (cutting, tearing etc), and realistic rendering of human soft tissue. The physical/compute modeling of soft tissue is one of the key techniques of surgery simulation. The research of it is dropped behind by other techniques because of the character of interdiscipline. It has become the bottleneck of surgery simulation research development.
This thesis describe human soft tissue using topology of cavity organ for instance blood vessel, bladder, stomach, and intestines etc, and volume soft tissue for instance skin, fattiness, and muscles. Toward different surgery simulation request, we simplify human soft tissue to linear model, nonlinear model, or stiff/elastic model. Using stiffness coefficient, damping coefficient, and mass point density to represent the characters of stiff/elastic, anisotropy, and non homogenous of soft tissue.
An unified soft t.ssue modeling method using mass-spring model was given in this paper. The soft tissues are discreted to a mass-spring-damper system, and the topologies of the system are described using adjacent tables, the dynamical equations can be constructed based on the data stored in the adjacent tables. The 4th order Runge-Kutta method is used to integrate the equations of deformation. Using computer graphical techniques to display the visual feedback of the deformation. Four examples of a surface model and a volumetric model to test the method are given.
引文
[1] Richard M. Satava, Medical Virtual Reality: The Current Status of the Future, in Proc. 4th Conf. Medicine Meets Virtual Reality, SanDiego, CA, 1996, pp. 100-106
[2] Http://www. nlm. nil. gov/research/visible/visible human, html
[3] 王强，基于医学图象的曲面重构基础算法研究，浙江大学计算机科学系，国家CAD&CG重点实验室博士学位论文，2001．9
[4] 魏迎梅，虚拟环境中碰撞检测问题的研究，国防科技大学，计算机学院博士学位论文，2000，10
[5] Morten Bro-Nielsen, Stephane Cotin. Real-time Volumetric Deformable Models for Surgery Simulation using Finite Element and Condensation. In EUROGRAPHICS' 96, p57-p66, 1996.
[6] S. Roth, M. Gross, S. Turello, F. Carls, A Bernstein-Bezier Based Approach to Soft Tissue Simulation, EUROGRAPHICS'98, p285-p294
[7] D. Terzopoulos, K. Fleischer, Modeling inelastic deformation: Viscoelasticity, plasticity, fracture."SIGGRAPH'88, 1988. p269-278.
[8] Y. C. Fung. Biomechanics: mechanical properties of living tissues. Springcr-Vcrlag, New York, NY, USA, 1981
[9] D. d Aulignac and C. Laugier(INRIA), M. C. Cavusoglu. Towards a realistic ecbograpbic simulator with force feedback. In the Proc. Of the IEEE/RSJ International Conference on Intelligent Robots and Systems(IROS' 99), Kyongju, Korea, Oct. 1999, pp 727-732
[10] Morten Bro-Nielsen, Fast finite elements for surgery simulation. In J.D. Westwood et al.,editors, Medicaine Meets Virtual Reality:5, Amsterdam, 1997, IOS Press.
[11] S. Cotin, H. Delingette, and N. Ayache. Real-time elastic deformations of soft tissues for surgery simulation. IEEE Transactions on Visualization and Computer Graphics, 5(1):62-73, January-March 1999.
[12] D. Terzopoulos and K. Waters. Physically-Based Facial Modeling, Analysis, and Animation. J. Visualization and Computer Animation, 1, 1990, pp. 73-80．
[13] S. A. Cover, N. F. Ezquerra, J. F. O' Brien, R. Rowe, T. Gadacz, and E.Palm. Interactively Deformable Models for Surgical Simulation. IEEE Computer Graphics Applications, 13, 1993, pp.68-75．
[14] http://iregtl. iai.fzk.de/KISMET/UGK. home, html
[15] R. M. Koch, Mothods for Physics Based Facial Surgery Predication, Copmuter Graphics Lab. , Institute of Scientific Computing, ETH Zurich, PhD Dissertation, 2001
[16] Joel Brown, Real-Time Soft Tissue and Suture Simulation, PhD Dissertation, Department of Computer Science of Stanford University, May, 2003
[17] W. Larrabee. A Finite Element Model of Skin Deformation. Biomechanics of Skin and Tissue: A Review. Laryngoscope, 96, 1986, pp. 399-405．
[18] S. Pieper, J. Rosen, and D. Zeltser. Interactive Graphics for Plastic Surgery: A Task-Level Analysis and Implementation. Computer Graphics, 25(2) , 1992, pp. 127-134．
[19] M. Bro-nielsen and S.Cotin. Real-Time Volumetric Deformable Models for Surgery Simulation Using Finite Elements and Condensation. Proc.Eurographics' 96, Vol.15, 1996, pp.57-66．
[20] Stephane Cotin, Modeles Anatomiques Deformables en Temps-Reel, INRA Sophia antipolis, Universite de Nice Sophia Antipolis, Phd Dissertation, Nov. 1997
[21] Han-Wen Nienhuys, Cutting in Deformable Objects PhD Dissertation, Faculteit Wiskunde en Informatica Universiteit Utrecht, June 2003
[22] Christina Simone, Modeling of Needle Insertion Forces for Percutaneous Therapies, The Department of Engineering of John Hopkins University, Master Thesis, May, 2002,2
[23] Andrew B. Mor, Progrissive Cutting with Monimal New Element Creation of Soft Tissue Models for Interactive Surgical Simulation, The
Robotics Institute, Carnegie Mellon University, PhD Dissertation, Oct. 2001
[24] J. berkeley, S. Weghorst, H.Gladstone, G. Raugi, D. Berg, and M. Ganter. Fast Finite Element Modeling for Surgical Simulation. Proc.Meets Virtual Reality(MMVR' 99) , ISO Press, 999, pp. 55-61．
[25] S. Cotin, H. Delingette, J.M.Clement, L. Soler, N. Ayache, and J. Marescaux. Geometrical and Physical Representations for a Simulator of Hepatic Surgery .Proc. Medicine Meets Virtual Reality (MMVR' 96) , 1996．
[26] S. Cotin, H. Delingette, and N. Ayache. Real-time Non-Linear Elastic Deformations of Soft Tissues for Surgery Simulation. IEEE Tr.on Visualization and Computer Graphics, to appear.
[27] H. Delingette. Towards Realistic Soft Tissue Modeling in Medical Simulation. Rapport de Recherche NO. 3506, INRLA, Sophia-Antipolis, September 1998
[28] S.Gibson and B. Mirtich. A Surgery of Deformable Modeling in Computer Graphics, Tech. Report No.TR-97-19, Mitsubishi Electric Research Lab. , Combrige, MA, November 1997．
[29] B. Girod, E. Keere, and S. Girod. Craniofacial Surgery Simulation. Proc. 4th Into Conf. Visualization in Biomedical Computing (VBC' 96) , Hamberg, Germany, 1996, pp. 541-546．
[30] R.M.Koch, M. H. Gross, F.R.Carls, E. F. von Biiren, G. Fankhouser, and Y. Parish. Simulating Facial Surgery Using Finite Element Methods. Proc.SIGGRAPH' 96, ACM Press, August, 1996, pp.421-428．
[31] C. Kuhn, U. Kuhnapfel, H. G. Krumm, and B. Neisius. A 'Virtual Reality' Based Training System for Minimally Invasive Surgery. Proc. Computer Assisted Radiology (CAR'96) , June 1996, pp. 764-769．
[32] T. McInerney and D. Terzopoulos. Deformable Models in Medical Image Analysis: A Survey. Medical Image Analysis, 1(2) , Oxford University Press, 1996, pp. 91-108．
[33] D. Terzopoulos, J. Platt, A. Barr, and K. Fleischer. Elastically Deformable Models. Computer Graphics, 21(4) , December 1994．
[34] K. Waters. A Physical Model of Facial Tissue and Muscle Articulation Derived from Computer Tomography Data. Proc. Visualization in Biomedical Computing (VBC), Chapel Hill, NC, 1992．
[35] Keeve, E. , Girod, S. , Pfeifle, P. and Girod, B. , "Anatomy-Based Facail Tissue Modeling Using the Finite Element Method", Proc. Visualization'96, IEEE Computer Society Press, pp. 21-28, 1996．
[36] Koch, R. M., Gross, M. H., Carls, F. R. , Von Buren, S. F., Fankhauser, G. and Parish, Y.I.H., "Simulating Facial Surgery Using Finite Element Models", Proc. SIGGRAPH'96, Addison-Wesley Publishing Co., pp. 421-428, 1996．
[37] James D. L , Pai D. K., Accurate Real-Time Deformable Objects.,Proc. SIGGRAPH '99 (Los Angeles, USA, August 8-13, 1999) , Computer Graphics Proceedings, 65-72, 1999,
[38] Stephane Gottschalk, Collision Queries Using Oriented Bounding Boxes, PhD Dissertation, Department of Computer Science, The University of North Carolina, 2000
[39] Luciana Porcher Nedel, Anatomic Modeling of Human Bodies Using Physically Based Muscle Simulation, Computer Science, Federal University of Rio Grande do Sul, EPFL, PhD Dissertation, 1998
[40] Yan Zhuang, Real-Time Simulation of Physically Based Global Deformation, Computer Science, University of California at Berkeley, PhD Dissertation, Fall, 2000
[41] 阎丽霞，王健宁，石教英．基于粘弹性模型的生物体软组织变形仿真． ChinaGraph’2000
[42] 阎丽霞，虚拟手术关键技术研究，浙江大学计算机科学系博士学位论文， 2001． 2
[43] H. Abe, K. Hayashi, M. Sato(Eds.), Data Book on Mechanical Properties of Living Cells, Tissues, and Organs, Springer-Verlag, 1996
[44] Cornelius Leondes(Edited), Computer Techniques and Computational Methods in Biomechanics, New York,2001
[45] Guibaa L., Stolfi J. : Primitives for the Manipulation of General
Subdivision and Computation of Voronoi Diagrams. ACM Trans, on Graphics, Vol. 4, 1985．
[46] DeHaemer Jr. M., Zyda M. J. : Simplification of Objects Rendered by Polygonal Approximations, Computers & Graphics, 15(2) , 175-184,1991．
[47] Schroeder W. J., Zarge J. A. et al: Decimation of Triangle Meshes, Computer Graphics, 26(2) , 65-70, 1992．
[48] Turk G. : Re-Tilling Polygonal Surface, Computer Graphics, 28(2) , 55-64, 1992．
[49] Rossignac J., Borrel P. : Multi-Resolution 3D Approximation for Rendering Complex Scences, In Falcidieno B., Kunii T. editors, Geometric Modeling in Computer Graphics, Springer Verlag, 455-465, 1993．
[50] Hoppe H., DeRose T. et al. : Mesh optimization, Computer Graphics, (27) 19-26, 1993．
[51] Isler Veysi, Lau R. W. H., Green Mark: Real-Time Multi-Resolution Modeling for Complex Virtul environment, Proceedings of VRST' s 96, 11-19, Hong Kong, 1996．
[52] Fabrice Neyret Marie-Paule Cani: Pattern_Based Texturing Revisited. SIGGRAPH 99 Conference Proceedings.
[53] D. R. Peackey: Solid Texturing of Complex Surfaces. Computer Graphics, Volume 19, No. 3, July 1985．
[54] K. Perlin: An Image Synthesizer. Computer Graphics, Volume 19, No. 3, 1985．
[2] Http://www. nlm. nil. gov/research/visible/visible human, html
[3] 王强，基于医学图象的曲面重构基础算法研究，浙江大学计算机科学系，国家CAD&CG重点实验室博士学位论文，2001．9
[4] 魏迎梅，虚拟环境中碰撞检测问题的研究，国防科技大学，计算机学院博士学位论文，2000，10
[5] Morten Bro-Nielsen, Stephane Cotin. Real-time Volumetric Deformable Models for Surgery Simulation using Finite Element and Condensation. In EUROGRAPHICS' 96, p57-p66, 1996.
[6] S. Roth, M. Gross, S. Turello, F. Carls, A Bernstein-Bezier Based Approach to Soft Tissue Simulation, EUROGRAPHICS'98, p285-p294
[7] D. Terzopoulos, K. Fleischer, Modeling inelastic deformation: Viscoelasticity, plasticity, fracture."SIGGRAPH'88, 1988. p269-278.
[8] Y. C. Fung. Biomechanics: mechanical properties of living tissues. Springcr-Vcrlag, New York, NY, USA, 1981
[9] D. d Aulignac and C. Laugier(INRIA), M. C. Cavusoglu. Towards a realistic ecbograpbic simulator with force feedback. In the Proc. Of the IEEE/RSJ International Conference on Intelligent Robots and Systems(IROS' 99), Kyongju, Korea, Oct. 1999, pp 727-732
[10] Morten Bro-Nielsen, Fast finite elements for surgery simulation. In J.D. Westwood et al.,editors, Medicaine Meets Virtual Reality:5, Amsterdam, 1997, IOS Press.
[11] S. Cotin, H. Delingette, and N. Ayache. Real-time elastic deformations of soft tissues for surgery simulation. IEEE Transactions on Visualization and Computer Graphics, 5(1):62-73, January-March 1999.
[12] D. Terzopoulos and K. Waters. Physically-Based Facial Modeling, Analysis, and Animation. J. Visualization and Computer Animation, 1, 1990, pp. 73-80．
[13] S. A. Cover, N. F. Ezquerra, J. F. O' Brien, R. Rowe, T. Gadacz, and E.Palm. Interactively Deformable Models for Surgical Simulation. IEEE Computer Graphics Applications, 13, 1993, pp.68-75．
[14] http://iregtl. iai.fzk.de/KISMET/UGK. home, html
[15] R. M. Koch, Mothods for Physics Based Facial Surgery Predication, Copmuter Graphics Lab. , Institute of Scientific Computing, ETH Zurich, PhD Dissertation, 2001
[16] Joel Brown, Real-Time Soft Tissue and Suture Simulation, PhD Dissertation, Department of Computer Science of Stanford University, May, 2003
[17] W. Larrabee. A Finite Element Model of Skin Deformation. Biomechanics of Skin and Tissue: A Review. Laryngoscope, 96, 1986, pp. 399-405．
[18] S. Pieper, J. Rosen, and D. Zeltser. Interactive Graphics for Plastic Surgery: A Task-Level Analysis and Implementation. Computer Graphics, 25(2) , 1992, pp. 127-134．
[19] M. Bro-nielsen and S.Cotin. Real-Time Volumetric Deformable Models for Surgery Simulation Using Finite Elements and Condensation. Proc.Eurographics' 96, Vol.15, 1996, pp.57-66．
[20] Stephane Cotin, Modeles Anatomiques Deformables en Temps-Reel, INRA Sophia antipolis, Universite de Nice Sophia Antipolis, Phd Dissertation, Nov. 1997
[21] Han-Wen Nienhuys, Cutting in Deformable Objects PhD Dissertation, Faculteit Wiskunde en Informatica Universiteit Utrecht, June 2003
[22] Christina Simone, Modeling of Needle Insertion Forces for Percutaneous Therapies, The Department of Engineering of John Hopkins University, Master Thesis, May, 2002,2
[23] Andrew B. Mor, Progrissive Cutting with Monimal New Element Creation of Soft Tissue Models for Interactive Surgical Simulation, The
Robotics Institute, Carnegie Mellon University, PhD Dissertation, Oct. 2001
[24] J. berkeley, S. Weghorst, H.Gladstone, G. Raugi, D. Berg, and M. Ganter. Fast Finite Element Modeling for Surgical Simulation. Proc.Meets Virtual Reality(MMVR' 99) , ISO Press, 999, pp. 55-61．
[25] S. Cotin, H. Delingette, J.M.Clement, L. Soler, N. Ayache, and J. Marescaux. Geometrical and Physical Representations for a Simulator of Hepatic Surgery .Proc. Medicine Meets Virtual Reality (MMVR' 96) , 1996．
[26] S. Cotin, H. Delingette, and N. Ayache. Real-time Non-Linear Elastic Deformations of Soft Tissues for Surgery Simulation. IEEE Tr.on Visualization and Computer Graphics, to appear.
[27] H. Delingette. Towards Realistic Soft Tissue Modeling in Medical Simulation. Rapport de Recherche NO. 3506, INRLA, Sophia-Antipolis, September 1998
[28] S.Gibson and B. Mirtich. A Surgery of Deformable Modeling in Computer Graphics, Tech. Report No.TR-97-19, Mitsubishi Electric Research Lab. , Combrige, MA, November 1997．
[29] B. Girod, E. Keere, and S. Girod. Craniofacial Surgery Simulation. Proc. 4th Into Conf. Visualization in Biomedical Computing (VBC' 96) , Hamberg, Germany, 1996, pp. 541-546．
[30] R.M.Koch, M. H. Gross, F.R.Carls, E. F. von Biiren, G. Fankhouser, and Y. Parish. Simulating Facial Surgery Using Finite Element Methods. Proc.SIGGRAPH' 96, ACM Press, August, 1996, pp.421-428．
[31] C. Kuhn, U. Kuhnapfel, H. G. Krumm, and B. Neisius. A 'Virtual Reality' Based Training System for Minimally Invasive Surgery. Proc. Computer Assisted Radiology (CAR'96) , June 1996, pp. 764-769．
[32] T. McInerney and D. Terzopoulos. Deformable Models in Medical Image Analysis: A Survey. Medical Image Analysis, 1(2) , Oxford University Press, 1996, pp. 91-108．
[33] D. Terzopoulos, J. Platt, A. Barr, and K. Fleischer. Elastically Deformable Models. Computer Graphics, 21(4) , December 1994．
[34] K. Waters. A Physical Model of Facial Tissue and Muscle Articulation Derived from Computer Tomography Data. Proc. Visualization in Biomedical Computing (VBC), Chapel Hill, NC, 1992．
[35] Keeve, E. , Girod, S. , Pfeifle, P. and Girod, B. , "Anatomy-Based Facail Tissue Modeling Using the Finite Element Method", Proc. Visualization'96, IEEE Computer Society Press, pp. 21-28, 1996．
[36] Koch, R. M., Gross, M. H., Carls, F. R. , Von Buren, S. F., Fankhauser, G. and Parish, Y.I.H., "Simulating Facial Surgery Using Finite Element Models", Proc. SIGGRAPH'96, Addison-Wesley Publishing Co., pp. 421-428, 1996．
[37] James D. L , Pai D. K., Accurate Real-Time Deformable Objects.,Proc. SIGGRAPH '99 (Los Angeles, USA, August 8-13, 1999) , Computer Graphics Proceedings, 65-72, 1999,
[38] Stephane Gottschalk, Collision Queries Using Oriented Bounding Boxes, PhD Dissertation, Department of Computer Science, The University of North Carolina, 2000
[39] Luciana Porcher Nedel, Anatomic Modeling of Human Bodies Using Physically Based Muscle Simulation, Computer Science, Federal University of Rio Grande do Sul, EPFL, PhD Dissertation, 1998
[40] Yan Zhuang, Real-Time Simulation of Physically Based Global Deformation, Computer Science, University of California at Berkeley, PhD Dissertation, Fall, 2000
[41] 阎丽霞，王健宁，石教英．基于粘弹性模型的生物体软组织变形仿真． ChinaGraph’2000
[42] 阎丽霞，虚拟手术关键技术研究，浙江大学计算机科学系博士学位论文， 2001． 2
[43] H. Abe, K. Hayashi, M. Sato(Eds.), Data Book on Mechanical Properties of Living Cells, Tissues, and Organs, Springer-Verlag, 1996
[44] Cornelius Leondes(Edited), Computer Techniques and Computational Methods in Biomechanics, New York,2001
[45] Guibaa L., Stolfi J. : Primitives for the Manipulation of General
Subdivision and Computation of Voronoi Diagrams. ACM Trans, on Graphics, Vol. 4, 1985．
[46] DeHaemer Jr. M., Zyda M. J. : Simplification of Objects Rendered by Polygonal Approximations, Computers & Graphics, 15(2) , 175-184,1991．
[47] Schroeder W. J., Zarge J. A. et al: Decimation of Triangle Meshes, Computer Graphics, 26(2) , 65-70, 1992．
[48] Turk G. : Re-Tilling Polygonal Surface, Computer Graphics, 28(2) , 55-64, 1992．
[49] Rossignac J., Borrel P. : Multi-Resolution 3D Approximation for Rendering Complex Scences, In Falcidieno B., Kunii T. editors, Geometric Modeling in Computer Graphics, Springer Verlag, 455-465, 1993．
[50] Hoppe H., DeRose T. et al. : Mesh optimization, Computer Graphics, (27) 19-26, 1993．
[51] Isler Veysi, Lau R. W. H., Green Mark: Real-Time Multi-Resolution Modeling for Complex Virtul environment, Proceedings of VRST' s 96, 11-19, Hong Kong, 1996．
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