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Viscoelastic Finite Element Modeling of Deformation Transients of Single Cells Viscoelastic finite element modeling of deformation transients of single cells Soo Kng Teo A thesis submitted to Imperial College London for the degree of Doctor of Philosophy Imperial College London Department of Bioengineering 2008 Viscoelastic finite element modeling of deformation transients of single cells Abstract The objective of this thesis is to use computational modeling to study the deformation of single cells subjected to mechanical stresses. Our motivation stems from experimental observations that cells are subjected to mechanical stresses arising from their environment throughout their lifetime, and that such stresses can regulate many important biological processes. While the exact mechanotransduction mechanisms involved are not well understood, quantitative models for cell deformation can yield important insights. In this thesis, we developed an axisymmetric finite element model to study the deformation of suspended fibroblasts in the optical stretcher and neutrophils in tapered micropipettes. The key feature of our model is the use of a viscoelastic constitutive equation whose parameters can be varied both spatially and temporally so as to mimic the experimentally-observed spatio-temporal heterogeneity of cellular material properties. Our model suggested that cellular remodeling, in the form of an increased cellular viscosity, occurred during optical stretching of fibroblasts. The increase would have to be approximately 20-fold to explain the experimental data for different loading time-scales. We also showed that cell size is a more important factor in determining the strain response of the optically-stretched fibroblasts compared to the thickness of the actin cortical region. This result can explain the higher optical deformability observed experimentally for malignant fibroblasts. In addition, our simulations showed that maximal stress propagates into the nuclear region for malignant fibroblasts whereas for normal fibroblasts, the maximal stress does not. Finally, results from modeling the tapered micropipette experiments also suggested that cellular remodeling, in the form of a decreased cellular elasticity and viscosity, occurred during the process of neutrophil aspiration. Taken together, our simulation results on optically-stretched fibroblasts and aspirated neutrophils suggested that cells in general are able to sense mechanical stresses and respond by varying their material properties during deformation. 2 Viscoelastic finite element modeling of deformation transients of single cells Acknowledgement I have been very fortunate to be given this chance to purse a doctorate degree, one of my many dreams and a route less well traveled among my friends. There are a lot of people whom I want to thank, for their help, support and encouragement during the past 4 years of my life as I struggle along this long and demanding path. First and foremost, I would like to thank my supervisor, Professor Kim Parker, for his patience, encouragement and guidance during the course of this research work. I am grateful for all the knowledge that you had shared with me and I have definitely learnt a lot from you. I would also like to thank my co-supervisor, Dr. Chiam Keng Hwee, who taught me how to think independently. Special thanks also go to my ex-supervisor, Dr. Andrew Goryachev. I would have never started on this research project without him. I would also like to express my gratitude to my ex-colleagues in the Bioinformatics Institute and my current colleagues in the biophysics team in the Institute of High Performance Computing. All of you helped to create an environment conducive for research, where I do not have to worry about any administrative or logistics issues. I gratefully acknowledge the financial support from the Agency of Science, Technology and Research. Many thanks to my scholarship officers, Serene, Swee Bee, Joanne, Cindy and others for their support with all the administrative matters as well as my fellow friends on the same scholarship program: Kwang Hwee, Sing Yang, Zi En and Ho Huat, for all the suffering and enjoyment we all shared along this journey. Friends have always been a very important part of my life especially during times where the going gets tough. For that, I have to thank Dave, James, Seath, Han Hui and Phey Leng. You kept me sane during those times when my research was going nowhere and the problems that I faced seem insurmountable. 3 Viscoelastic finite element modeling of deformation transients of single cells Lastly, I want to thank my mum for being there for me throughout all my life. Her patience and support has been my pillar of strength. Without her, I would never have the courage to dream. 4 Viscoelastic finite element modeling of deformation transients of single cells Table of Content Abstract.............................................................................................................................. 2 Acknowledgement............................................................................................................. 3 Table of Content................................................................................................................ 5 List of Figures.................................................................................................................... 9 List of Tables ................................................................................................................... 16 Chapter 1. Introduction............................................................................................. 18 Chapter 2. Literature review .................................................................................... 22 2.1 Structure of an eukaryotic cell ............................................................................ 23 2.1.1 Cytoskeleton..................................................................................................... 25 2.2 Experimental Techniques..................................................................................... 27 2.2.1 Localized perturbations ................................................................................... 28 2.2.1.1 Atomic force microscopy........................................................................... 28 2.2.1.2 Magnetic twisting cytometry..................................................................... 29 2.2.1.3 Cytoindentation......................................................................................... 30 2.2.2 Entire cell perturbations .................................................................................. 31 2.2.2.1 Laser/optical tweezers .............................................................................. 31 2.2.2.2 Microplate stretcher.................................................................................. 32 2.2.2.3 Microfabricated post array detector......................................................... 32 2.2.2.4 Micropipette aspiration ............................................................................ 33 2.2.2.5 Shear flow ................................................................................................. 34 2.2.2.6 Substrate stretcher .................................................................................... 34 2.2.2.7 Optical stretcher ....................................................................................... 35 2.3 Overview of Mechanical Models ......................................................................... 36 2.3.1 Micro/Nanostructural Approach ..................................................................... 37 2.3.1.1 Open-cell foam model............................................................................... 38 2.3.1.2 Stress-supported structures....................................................................... 39 2.3.1.3 Tensegrity architecture ............................................................................. 40 2.3.2 Continuum approach........................................................................................ 41 2.3.2.1 Cortical shell-liquid core models.............................................................. 41 5 Viscoelastic finite element modeling of deformation transients of single cells 2.3.2.1.1 Newtonian liquid drop model ............................................................ 42 2.3.2.1.2 Compound Newtonian liquid drop model.......................................... 42 2.3.2.1.3 Shear thinning liquid drop................................................................. 43 2.3.2.1.4 Maxwell liquid drop model................................................................ 44 2.3.2.2 Solid models.............................................................................................. 44 2.3.2.2.1 Linear elastic solid model.................................................................. 44 2.3.2.2.2 Viscoelastic solid model..................................................................... 45 2.3.2.2.3 Power-law dependent model.............................................................. 46 Chapter 3. Numerical method................................................................................... 47 3.1 Linear finite element scheme ............................................................................... 48 3.1.1 Preliminary ...................................................................................................... 48 3.1.2 Linear elastic constitutive equations ............................................................... 49
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