-
因为在生物安全性和导入效率等方面的优势, 基于力学方法实现基因导入日益得到学术界的重视. 本文提出了一种基于零质量射流, 对微管道中运动的细胞施加流体作用力, 引起其发生变形, 进而促使细胞膜上力敏通道开启的方法, 并通过数值模拟进行了理论验证. 本文采用浸没有限元法, 对红细胞在微管道内运动过程中受到零质量射流作用的变形情况进行数值模拟, 探讨了如何高效地实现小分子物质导入细胞. 数值模拟的重要参数有微管道内压力梯度Δp, 零质量射流的振幅Am和频率f. 经过对流场特征和红细胞受力情况的分析, 发现当细胞某处表面张力T0大于临界表面张力 τ c时, 细胞表面力敏通道打开, 并得到每一时刻细胞力敏通道打开的百分比Popen. 本文定义了通道开启积分I, 以衡量在不同的流动参数情况下, 细胞膜力敏通道的开启程度, 进而探讨了压力梯度和射流振动频率、振幅对I值的影响, 以寻找优化的工艺参数. 该方法具有微制造工艺简单易行, 在保证高通量导入的同时, 便于对所施加的流体作用力进行精确控制的特点, 这使得蛋白质、基因等物质实现跨膜输运, 进入细胞并发生重编程成为可能.With advantages in biosafety and efficiency, gene delivery based on mechanical approaches has received more and more attention in academic research. In the present paper, a method based on zero-net-mass-flux jet is proposed to apply fluid shear to the moving cells in the microchannel, which causes cell to deform, and then open its mechano-sensitive channel on the cell membrane. This novel method is verified theoretically by numerical simulation in this study. In this paper, an immersed finite element method is utilized to numerically simulate the deformation of red blood cells subjected to zero-net-mass-flux jet during the movement of red blood cells in microchannel, aiming at investigating how to efficiently introduce small molecules into cells. The important parameters of numerical simulation are pressure gradient Δp along the microchannel, the amplitude Am and frequency f of the zero-net-mass-flux jet. Through the analysis of the characteristic of flow field and the stress on the red blood cells, we find that when cell surface tension T0 is greater than critical surface tension τ c, the gating of cell surface mechano-sensitive channel will occur, and the percentage of gating Popen on the cell membrane can be obtained at each moment. Addtionally, the channel opening integral I is defined to measure the gating degree of the membrane mechano-sensitive channel under different flow parameters, and the influences of pressure gradient, jet vibration frequency and amplitude on the I are further discussed in order to find the optimized process parameters, The method we proposed is simpler and easier to implement, and the applied fluid shear stress can be controlled precisely, so that it is possible for proteins, genes and other substances to be transported into the cell across the membrane, and to implement reprogramming.
-
Keywords:
- immersed finite element method /
- zero-net-mass-flux jet /
- mechano-sensitive channel /
- transmembrane transport
[1] Takahashi K, Yamanaka S 2006 Cell 126 663
Google Scholar
[2] Nielsen K M, Bones A M, Smalla K, van Elsas J D 1998 Fems Microbiol. Rev. 22 79
Google Scholar
[3] Chen I, Dubnau D 2004 Nat. Rev. Microbiol. 2 241
Google Scholar
[4] Chen I, Christie P J, Dubnau D 2005 Science 310 1456
Google Scholar
[5] Luo D, Saltzman W M 2000 Nat. Biotechnol. 18 33
Google Scholar
[6] Song Y, Hahn T, Thompson I P, Mason T, Preston G M, Li G, Paniwnyk L, Huang W E 2007 Nucleic Acids Res. 35 129
Google Scholar
[7] Mehierhumbert S, Guy R H 2005 Adv. Drug Deliv. Rev. 57 733
Google Scholar
[8] Komatsu Y, Takeuchi D, Tokunaga T, Sakurai H, Makino A, Honda T, Ikeda Y H, Tomonaga K 2019 Mol. Ther. Meth. Clin. D. 14 47
Google Scholar
[9] Zhou H Y, Wu S L, Joo J Y, Zhu S Y, Han D W, Lin T X, Trauger S A, Bien G, Yao S S, Zhu Y, Siuzdak G, Scholer H R, Duan L X, Ding S 2009 Cell Stem Cell 4 381
Google Scholar
[10] Weaver J C, Chizmadzhev Y A 1996 Bioelectrochem. Bioenerg. 41 135
Google Scholar
[11] Michalet X, Pinaud F, Bentolila L A, Tsay J M, Doose S, Li J J, Sundaresan G, Wu A M, Gambhir S S, Weiss S 2005 Science 307 538
Google Scholar
[12] de Vry J, Martinezmartinez P, Losen M, Temel Y, Steckler T, Steinbusch H W M, de Baets M H, Prickaerts J 2010 Prog. Neurobiol. 92 227
Google Scholar
[13] Fetterman H R, Larsen D M, Stillman G E, Tannenwald P E, Waldman J 1971 Phys. Rev. Lett. 26 1290
Google Scholar
[14] Sharei A, Zoldan J, Adamo A, Sim W Y, Cho N, Jackson E L, Mao S, Schneider S, Han M, Lyttonjean A K R, Basto P, Jhunjhunwala S, Lee J, Heller D A, Kang J W, Hartoularos G C, Kim K, Anderson D, Langer R, Jensen K F 2013 P. Natl. Acad. Sci. USA. 110 2082
Google Scholar
[15] Stewart M P, Sharei A, Ding X Y, Sahay G, Langer R, Jensen K F 2016 Nature 538 183
Google Scholar
[16] Azagury A, Khoury L, Enden G, Kost J 2014 Adv. Drug Deliv. Rev. 72 127
Google Scholar
[17] Lentacker I, de Cock I, Deckers R, de Smedt S C, Moonen C T W 2014 Adv. Drug Deliv. Rev. 72 49
Google Scholar
[18] Delalande A, Kotopoulis S, Postema M, Midoux P, Pichon C 2013 Gene 525 191
Google Scholar
[19] Miller D L, Pislaru S V, Greenleaf J F 2002 Somat. Cell Molec. Gen. 27 115
Google Scholar
[20] Newman C, Bettinger T 2007 Gene Ther. 14 465
Google Scholar
[21] Hamill O P, Martinac B 2001 Physiol. Rev. 81 685
Google Scholar
[22] Sackin H 1995 Annu. Rev. Physiol. 57 333
Google Scholar
[23] Martinac B, Kloda B (edited by Egelman E H) 2012 Comprehensive Biophysics (Vol. 6) (Amsterdam: Elsevier) pp108–141
[24] Sankin G, Yuan F, Zhong P 2010 Phys. Rev. Lett. 105 078101
Google Scholar
[25] Yuan F, Yang C, Zhong P 2015 Proc. Natl. Acad. Sci. USA. 112 7039
Google Scholar
[26] Yuan F, Sankin G, Zhong P 2011 J. Acoust. Soc. Am. 130 3339
Google Scholar
[27] Modaresi S, Pacelli S, Subham S, Dathathreya K, Paul A 2020 Adv. Ther. 3 1900130
Google Scholar
[28] Kizer M E, Deng Y X, Kang G Y, Mikael P E, Wang X, Chung A J 2019 Lab on a Chip 19 1747
Google Scholar
[29] Ingard U, Labate S 1950 J. Acoust. Soc. Am. 22 211
Google Scholar
[30] Travnicek Z, Vit T, Tesař V 2006 Phys. Fluids 18 48
Google Scholar
[31] Smith B L, Glezer A 1998 Phys. Fluids 10 2281
Google Scholar
[32] Wiltse J M, Glezer A 1993 J. Fluid Mech. 249 261
Google Scholar
[33] Luo Z, Xia Z, Liu B 2006 AIAA J. 44 2418
Google Scholar
[34] Luo Z, Xia Z 2005 Sensor. Actuat. A-phys. 122 131
Google Scholar
[35] Zhang P, Wang J 2007 AIAA J. 45 1058
Google Scholar
[36] Wang J, Feng L, Chaojun X U 2007 Sci. China-Technol. Sci. 50 550
Google Scholar
[37] Lockerby D A, Carpenter P W, Davies C 2002 AIAA J. 40 67
Google Scholar
[38] Peskin C S 1977 J. Comput. Phys. 25 220
Google Scholar
[39] Mittal R, Iaccarino G 2005 Annu. Rev. Fluid Mech. 37 239
Google Scholar
[40] Goldstein D, Handler R A, Sirovich L 1993 J. Comput. Phys. 105 354
Google Scholar
[41] Peskin C S 2002 Acta Numer. 11 479
Google Scholar
[42] Wang X, Liu W K 2004 Comput. Method Appl. Mech. Eng. 193 1305
Google Scholar
[43] Zhang L, Gerstenberger A, Wang X, Liu W K 2004 Comput. Method Appl. Mech. Eng. 193 2051
Google Scholar
[44] Lee T, Choi M, Kopacz A M, Yun S H, Liu W K, Decuzzi P 2013 Sci. Rep. 3 2079
Google Scholar
[45] Liu W K, Liu Y, Farrell D, Zhang L, Wang X S, Fukui Y, Patankar N A, Zhang Y, Bajaj C L, Lee J, Hong J, Chen X, Hsu H 2006 Comput. Method. Appl. M. 195 1722
Google Scholar
[46] Lü H, Tang S L, Zhou W P 2011 Chin. Phys. Lett. 28 84708
Google Scholar
[47] Roy S, Heltai L, Costanzo F 2015 Comput. Math. Appl. 69 1167
Google Scholar
[48] Tezduyar T E 1991 Adv. Appl. Mech. 28 1
Google Scholar
[49] Tezduyar T E (edited by Stein E) 2004 Encyclopedia of Computational Mechanics (Vol. 3) (Hoboken: John Wiley & Sons, Ltd.) p545
[50] Hughes T J R, Franca L P, Balestra M 1986 Appl. Mech. Engin. 59 85
Google Scholar
[51] Liu W K, Jun S, Zhang Y F 1995 Int. J. Numer. Meth. Fluids 20 1081
Google Scholar
[52] Zhang P, Wang J, Feng L 2008 Sci. China-Technol. Sci. 51 1315
Google Scholar
[53] Glezer A, Amitay M 2002 Ann. Rev. Fluid Mech. 34 503
Google Scholar
[54] Mallinson S G, Johnson G, Gaston M, Hong G 2004 Proc. SPIE 5276 341
Google Scholar
[55] Maas S A, Ellis B J, Ateshian G A, Weiss J A 2012 ASME J. Biomech. Eng. 134 011005
Google Scholar
[56] Liu F, Wu D, Chen K 2013 J. Mech. Behav. Biomed. 24 1
Google Scholar
[57] Wang Q, Manmi K, Liu K 2015 Interface Focus 5 20150018
Google Scholar
[58] He B, Zhang C H, Ding A 2017 Chin. Phys. B 26 126102
Google Scholar
[59] 邵纯, 曹燕飞, 邹龙, 陈伟芳 2015 工程力学 32 206
Google Scholar
Shao C, Cao Y F, Zou L, Chen W F 2015 Engineering Mechanics 32 206
Google Scholar
[60] Sabass B, Stone H A 2016 Phys. Rev. Lett. 116 258101
Google Scholar
[61] Wiggins P A, Phillips R 2004 P. Natl. Acad. Sci. USA 101 4071
Google Scholar
[62] Pak O S, Young Y, Marple G R, Veerapaneni S, Stone H A 2015 Proceedings of the National Academy of Sciences of the United States of America 112 9822
Google Scholar
[63] Chasan B 2008 Physical Biology of the Cell (New York: Taylor and Francis Group)
[64] Zhang L P, Pak O S, Feng Z, Liu A P, Young Y N 2016 Acta Mech. Sinica 32 1012
Google Scholar
[65] Skalak R, Tozeren A, Zarda R P, Chien S 1973 Biophysical J. 13 245
Google Scholar
-
图 1 零质量射流示意图. Dc表示空腔直径, H表示空腔高度, D0表示孔口直径, h表示孔口高度. 典型的合成射流流场可划分为三个明显流动区域, 即近场区域、过渡区域和远场区域
Fig. 1. Schematic diagram of ZNMF jet flow. Dc stands for the cavity diameter, H for the cavity height, D0 for the orifice diameter, and h for the orifice height. The typical flow field of synthetic jet can be divided into three flow zones: Near field zone, transition zone, and far field zone.
图 2 数值模拟物理模型示意图, 其中L1表示射流小孔左端距离入口距离, L2表示射流小孔右端距离入口距离, L2–L1 = 2 μm, H表示管道的高度, v示意红细胞运动方向, 速度单位为 mm/s. 红细胞(red blood cell, rbc)在压力梯度作用下通过射流上方
Fig. 2. Physical model of numerical simulation, in which L1 stands for the distance between the left end of the jet hole and the inlet, while L2 for the distance between the right end of the jet hole and the inlet, L2–L1 = 2 μm, H is the height of the channel, v is the movement direction of the red blood cells, the units of velocity is mm/s, which passes above the synthetic jet driven by pressure gradient.
图 3 压力梯度作用下零质量射流流场分布(Am = 78.125π mm, f = 625 Hz, Δp = 60 Pa), 其中速度的量纲为mm/s. 截取的截面Y轴坐标范围170 μm ≤ Y ≤ 230 μm, 小孔位于截面正中心Y = 200 μm. 图中箭头表示速度方向, 颜色表示速度大小, 颜色越深, 则速度值越大. 四幅图描述了流场运行稳定后, 一个周期(T = 1.6 × 10–3 s)内沿着流动方向截面上的流场分布情况 (a) t = T/4; (b) t = 2T/4; (c) t = 3T/4; (d) t = T
Fig. 3. Flow field distribution under the action of ZNMF jet and pressure gradient (Am = 78.125π mm, f = 625 Hz, Δp = 60 Pa), in which the unit for velocity magnitude is mm/s. The cross-section depicted ranges 170 μm ≤ Y ≤ 230 μm in Y coordinate, and the hole is in the center of the cross-section Y = 200 μm. The arrow in the figure represents the direction of velocity vectors, and the color represents the magnitude of velocity. The darker the color, the greater the velocity magnitude. The four figures describe the distribution of flow field in a period: (a) t = T/4; (b) t = 2T/4; (c) t = 3T/4; (d) t = T.
图 4 红细胞经过射流作用区域时, 质心的Y轴和Z轴坐标随时间t变化规律(Am = 78.125π mm, f = 625 Hz, Δp = 30 Pa). 图中绿线表示质心Z轴坐标变化, 蓝线表示Y轴坐标变化, 三条黑线分别表示红细胞被拉伸到最长时的时刻, 对应为ta = 125 × 10–5 s, tb = 285 × 10–5 s, tc = 445 × 10–5 s, 任意两条黑线之间时间间隔为一个振动周期, T = 1.6 × 10–3 s. 红蓝两条垂直线表示红细胞进入孔口正上方的时刻, 红细胞经过射流孔口的时间Δt = 58 × 10–5 s
Fig. 4. The variation of Y-axis and Z-axis coordinates of cell centroid with time (Am = 78.125π mm, f = 625 Hz, Δp = 30 Pa). The green line represents for the change of Z-axis coordinate of the centroid, while the blue line represents for the change of its Y-axis coordinate. The three black lines represent the moment when red blood cells are stretched to the maximum, corresponding to ta = 125 × 10–5 s, tb = 285 × 10–5 s, tc = 445 × 10–5 s, and any two black lines represent a jet period, T = 1.6 × 10–3 s. The red and blue vertical lines indicate the moment when the red blood cells enter the orifice directly above, the interal of the red blood cells pass through the orifice Δt = 58 × 10–5 s.
图 5 一个射流周期内(T = 1.6 × 10–3 s)红细胞变形所受应力和流体压力场(Am = 78.125π mm, f = 625 Hz, Δp = 30 Pa) (a)−(d)分别表示不同时刻红细胞质心所在XZ截面的流体压力分布以及红细胞受到的压力 (a) t = T/4 (Y = 197.7 μm); (b) t = T/2 (Y = 199.5 μm); (c) t = 3T/4 (Y = 200.8 μm); (d) t = T (Y = 202.4 μm)
Fig. 5. Diagram of stress of red blood cell and fluid pressure field in one peroid for Am = 78.125π mm, f = 625 Hz, Δp = 30 Pa. (a)−(d) represents the fluid pressure distribution in the XZ section of the RBC center of mass at different times and the stress on the RBC membrane: (a) t = T/4 (Y = 197.7 μm); (b) t = T/2 (Y = 199.5 μm); (c) t = 3T/4 (Y = 200.8 μm); (d) t = T (Y = 202.4 μm).
图 6 不同振幅Am 作用下红细胞变形时力敏通道开启百分比Popen随运动时间t的变化(f = 625 Hz, Δp = 30 Pa), 图中竖直绿线表示红细胞刚好经过孔口正中心位置的时刻
Fig. 6. Variation of Popen for mechano-sensitive channel gating in cell deformation with time t under different amplitude Am (f = 625 Hz, Δp = 30 Pa). The green line in the figure stands for the moment when the cell passes above the orifice.
图 7 通道开启程度I随着压力梯度Δp和射流振动频率f的变化. 颜色代表I的大小, 颜色越深, I值越大. 图中竖直黑线表示Δp = 30 Pa, 两条水平横线分别代表f = 2500 Hz和f = 625 Hz
Fig. 7. The variation of channel gating integral I with the pressure gradient Δp and the jet vibration frequency f. The color represents the magnitude of I, the darker the color, the greater of I. The black line stands for Δp = 30 Pa, while the two horizontal lines for f = 2500 Hz and f = 625 Hz, respectively.
-
[1] Takahashi K, Yamanaka S 2006 Cell 126 663
Google Scholar
[2] Nielsen K M, Bones A M, Smalla K, van Elsas J D 1998 Fems Microbiol. Rev. 22 79
Google Scholar
[3] Chen I, Dubnau D 2004 Nat. Rev. Microbiol. 2 241
Google Scholar
[4] Chen I, Christie P J, Dubnau D 2005 Science 310 1456
Google Scholar
[5] Luo D, Saltzman W M 2000 Nat. Biotechnol. 18 33
Google Scholar
[6] Song Y, Hahn T, Thompson I P, Mason T, Preston G M, Li G, Paniwnyk L, Huang W E 2007 Nucleic Acids Res. 35 129
Google Scholar
[7] Mehierhumbert S, Guy R H 2005 Adv. Drug Deliv. Rev. 57 733
Google Scholar
[8] Komatsu Y, Takeuchi D, Tokunaga T, Sakurai H, Makino A, Honda T, Ikeda Y H, Tomonaga K 2019 Mol. Ther. Meth. Clin. D. 14 47
Google Scholar
[9] Zhou H Y, Wu S L, Joo J Y, Zhu S Y, Han D W, Lin T X, Trauger S A, Bien G, Yao S S, Zhu Y, Siuzdak G, Scholer H R, Duan L X, Ding S 2009 Cell Stem Cell 4 381
Google Scholar
[10] Weaver J C, Chizmadzhev Y A 1996 Bioelectrochem. Bioenerg. 41 135
Google Scholar
[11] Michalet X, Pinaud F, Bentolila L A, Tsay J M, Doose S, Li J J, Sundaresan G, Wu A M, Gambhir S S, Weiss S 2005 Science 307 538
Google Scholar
[12] de Vry J, Martinezmartinez P, Losen M, Temel Y, Steckler T, Steinbusch H W M, de Baets M H, Prickaerts J 2010 Prog. Neurobiol. 92 227
Google Scholar
[13] Fetterman H R, Larsen D M, Stillman G E, Tannenwald P E, Waldman J 1971 Phys. Rev. Lett. 26 1290
Google Scholar
[14] Sharei A, Zoldan J, Adamo A, Sim W Y, Cho N, Jackson E L, Mao S, Schneider S, Han M, Lyttonjean A K R, Basto P, Jhunjhunwala S, Lee J, Heller D A, Kang J W, Hartoularos G C, Kim K, Anderson D, Langer R, Jensen K F 2013 P. Natl. Acad. Sci. USA. 110 2082
Google Scholar
[15] Stewart M P, Sharei A, Ding X Y, Sahay G, Langer R, Jensen K F 2016 Nature 538 183
Google Scholar
[16] Azagury A, Khoury L, Enden G, Kost J 2014 Adv. Drug Deliv. Rev. 72 127
Google Scholar
[17] Lentacker I, de Cock I, Deckers R, de Smedt S C, Moonen C T W 2014 Adv. Drug Deliv. Rev. 72 49
Google Scholar
[18] Delalande A, Kotopoulis S, Postema M, Midoux P, Pichon C 2013 Gene 525 191
Google Scholar
[19] Miller D L, Pislaru S V, Greenleaf J F 2002 Somat. Cell Molec. Gen. 27 115
Google Scholar
[20] Newman C, Bettinger T 2007 Gene Ther. 14 465
Google Scholar
[21] Hamill O P, Martinac B 2001 Physiol. Rev. 81 685
Google Scholar
[22] Sackin H 1995 Annu. Rev. Physiol. 57 333
Google Scholar
[23] Martinac B, Kloda B (edited by Egelman E H) 2012 Comprehensive Biophysics (Vol. 6) (Amsterdam: Elsevier) pp108–141
[24] Sankin G, Yuan F, Zhong P 2010 Phys. Rev. Lett. 105 078101
Google Scholar
[25] Yuan F, Yang C, Zhong P 2015 Proc. Natl. Acad. Sci. USA. 112 7039
Google Scholar
[26] Yuan F, Sankin G, Zhong P 2011 J. Acoust. Soc. Am. 130 3339
Google Scholar
[27] Modaresi S, Pacelli S, Subham S, Dathathreya K, Paul A 2020 Adv. Ther. 3 1900130
Google Scholar
[28] Kizer M E, Deng Y X, Kang G Y, Mikael P E, Wang X, Chung A J 2019 Lab on a Chip 19 1747
Google Scholar
[29] Ingard U, Labate S 1950 J. Acoust. Soc. Am. 22 211
Google Scholar
[30] Travnicek Z, Vit T, Tesař V 2006 Phys. Fluids 18 48
Google Scholar
[31] Smith B L, Glezer A 1998 Phys. Fluids 10 2281
Google Scholar
[32] Wiltse J M, Glezer A 1993 J. Fluid Mech. 249 261
Google Scholar
[33] Luo Z, Xia Z, Liu B 2006 AIAA J. 44 2418
Google Scholar
[34] Luo Z, Xia Z 2005 Sensor. Actuat. A-phys. 122 131
Google Scholar
[35] Zhang P, Wang J 2007 AIAA J. 45 1058
Google Scholar
[36] Wang J, Feng L, Chaojun X U 2007 Sci. China-Technol. Sci. 50 550
Google Scholar
[37] Lockerby D A, Carpenter P W, Davies C 2002 AIAA J. 40 67
Google Scholar
[38] Peskin C S 1977 J. Comput. Phys. 25 220
Google Scholar
[39] Mittal R, Iaccarino G 2005 Annu. Rev. Fluid Mech. 37 239
Google Scholar
[40] Goldstein D, Handler R A, Sirovich L 1993 J. Comput. Phys. 105 354
Google Scholar
[41] Peskin C S 2002 Acta Numer. 11 479
Google Scholar
[42] Wang X, Liu W K 2004 Comput. Method Appl. Mech. Eng. 193 1305
Google Scholar
[43] Zhang L, Gerstenberger A, Wang X, Liu W K 2004 Comput. Method Appl. Mech. Eng. 193 2051
Google Scholar
[44] Lee T, Choi M, Kopacz A M, Yun S H, Liu W K, Decuzzi P 2013 Sci. Rep. 3 2079
Google Scholar
[45] Liu W K, Liu Y, Farrell D, Zhang L, Wang X S, Fukui Y, Patankar N A, Zhang Y, Bajaj C L, Lee J, Hong J, Chen X, Hsu H 2006 Comput. Method. Appl. M. 195 1722
Google Scholar
[46] Lü H, Tang S L, Zhou W P 2011 Chin. Phys. Lett. 28 84708
Google Scholar
[47] Roy S, Heltai L, Costanzo F 2015 Comput. Math. Appl. 69 1167
Google Scholar
[48] Tezduyar T E 1991 Adv. Appl. Mech. 28 1
Google Scholar
[49] Tezduyar T E (edited by Stein E) 2004 Encyclopedia of Computational Mechanics (Vol. 3) (Hoboken: John Wiley & Sons, Ltd.) p545
[50] Hughes T J R, Franca L P, Balestra M 1986 Appl. Mech. Engin. 59 85
Google Scholar
[51] Liu W K, Jun S, Zhang Y F 1995 Int. J. Numer. Meth. Fluids 20 1081
Google Scholar
[52] Zhang P, Wang J, Feng L 2008 Sci. China-Technol. Sci. 51 1315
Google Scholar
[53] Glezer A, Amitay M 2002 Ann. Rev. Fluid Mech. 34 503
Google Scholar
[54] Mallinson S G, Johnson G, Gaston M, Hong G 2004 Proc. SPIE 5276 341
Google Scholar
[55] Maas S A, Ellis B J, Ateshian G A, Weiss J A 2012 ASME J. Biomech. Eng. 134 011005
Google Scholar
[56] Liu F, Wu D, Chen K 2013 J. Mech. Behav. Biomed. 24 1
Google Scholar
[57] Wang Q, Manmi K, Liu K 2015 Interface Focus 5 20150018
Google Scholar
[58] He B, Zhang C H, Ding A 2017 Chin. Phys. B 26 126102
Google Scholar
[59] 邵纯, 曹燕飞, 邹龙, 陈伟芳 2015 工程力学 32 206
Google Scholar
Shao C, Cao Y F, Zou L, Chen W F 2015 Engineering Mechanics 32 206
Google Scholar
[60] Sabass B, Stone H A 2016 Phys. Rev. Lett. 116 258101
Google Scholar
[61] Wiggins P A, Phillips R 2004 P. Natl. Acad. Sci. USA 101 4071
Google Scholar
[62] Pak O S, Young Y, Marple G R, Veerapaneni S, Stone H A 2015 Proceedings of the National Academy of Sciences of the United States of America 112 9822
Google Scholar
[63] Chasan B 2008 Physical Biology of the Cell (New York: Taylor and Francis Group)
[64] Zhang L P, Pak O S, Feng Z, Liu A P, Young Y N 2016 Acta Mech. Sinica 32 1012
Google Scholar
[65] Skalak R, Tozeren A, Zarda R P, Chien S 1973 Biophysical J. 13 245
Google Scholar
下载:
计量
- 文章访问数: 5960
- PDF下载量: 96
- 被引次数: 0
