M. Sato Department of Biomedical Engineering, Institute of Basic Medical Sciences, University of Tsukuba, Tsukuba, Ibaraki 305, Japan
D. P. Theret L. T. Wheeler Department of Mechanical Engineering, University of Houston, Houston, Texas 77004
N. Ohshima Department of Biomedical Engineering, Institute of Basic Medical Sciences, University of Tsukuba, Tsukuba, Ibaraki 305, Japan
R. M. Nerem Biomechanics Laboratory and School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0405, Fellow ASME
Application of the Micropipette Technique to the Measurement of Cultured Porcine Aortic Endothelial Cell Viscoelastic Properties The viscoelastic deformation of porcine aortic endothelial cells grown under static culture conditions was measured using the micropipette technique. Experiments were conducted both for control cells (mechanically or trypsin detached from the substrate) and for cells in which cytoskeletal elements were disrupted by cytochalasin B or colchicine. The time course of the aspirated length into the pipette was measured after applying a stepwise increase in aspiration pressure. To analyze the data, a standard linear viscoelastic half-space model of the endothelial cell was used. The aspirated length was expressed as an exponential function of time. The actin microfilaments were found to be the major cytoskeletal component determining the viscoelastic response of endothelial cells grown in static culture.
Introduction The growth and behavior of the endothelial lining of a blood vessel are strongly influenced by mechanical and hemodynamic forces [1, 2]. In vivo, the cells are found to be oriented with the direction of flow and more elongated in regions of high shear stress [3]. In addition, cell shape may have a strong correlation with the localization of atherogenesis [4]. Recently, the influence of flow on endothelial structure and function has been studied using cultured endothelial cells [4-7]. These studies have demonstrated that in response to a fluid flow-imposed shear stress, cultured vascular endothelial cells elongate and orient their major axis with the direction of flow and reorganize their cytoskeletal structure. Believing that the mechanical properties of a cell may reflect quantitatively the organization of its internal cytoskeletal elements and having observed the cytoskeletal alterations associated with exposure to flow, we have measured the mechanical properties of cultured endothelial cells using the micropipette technique, both for static, no-flow conditions and after exposure to shear stress [8,9]. Our initial studies focused on the elastic properties of endothelial cells, and our results show a higher elastic modulus for cells exposed to shear stress than that for the cells from static, no-flow conditions. We also reported that in using
Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division November 28, 1989; revised manuscript received May 22, 1990.
this technique it was the cytoskeletal elements which provided the major resistance to deformation [8], In applying the micropipette technique to the measurement of the elastic properties of cells, it was necessary to wait for several minutes before the aspirated length of the cell in the micropipette reached its asymptotic value. From this, it appeared that the endothelial cells exhibited a strong viscous behavior. In the present report we describe the viscoelastic behavior of endothelial cells cultured under no-flow conditions. Using cytoskeletal disrupting agents, we investigate the influence of two different cytoskeletal elements, the actin microfilaments and the microtubules, on these viscoelastic properties. Methods Materials. Porcine aortic endothelial cells were cultured in 12-well plastic plates using a modified Dulbecco culture medium with 10 percent fetal bovine serum plus an antibioticantimycotic solution. These cells were used in the second or third passage. For micropipette measurements, endothelial cells were detached from substrate by trypsin or a mechanical method in which a sterile wood stick was used to scratch the substrate. The sheet of cell aggregates were detached further by aspiration through a 1 ml pipette. Prior to measurements, endothelial cell suspensions were stirred for approximately 15 seconds on a test tube stirrer. The cells were divided into four groups: (1) cells detached by trypsin (control(T)); (2)
Journal of Biomechanical Engineering
AUGUST 1990, Vol. 1 1 2 / 2 6 3
Copyright © 1990 by ASME Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 01/29/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Amplifier
1
Hydraulic Manipulator
Invented Microscope
Syringe
Fig. 1 Schematic diagram of the system used for the micropipette experiments
mechanically detached cells (control(M)); (3) cells treated by 2 ^iM cytochalasin B (CB); and (4) cells treated by 2 /tM colchicine. In both of these last two groups the cells were treated for about 14 hours prior to detachment. The concentrations of CB and colchicine and the duration of treatment was decided with reference to the reports by Wong and Gotlieb [10] and Meza et al. [11]. Microfilament assembly of the cytoskeletal elements is disrupted by CB and microtubule coils are depolymerized by colchicine. The cells treated by CB or colchicine were also detached from the substrate by a mechanical method and their viscoelastic properties measured using the micropipette technique. Experimental Apparatus. The experimental system is shown schematically in Fig. 1. The cells detached from the substrate were suspended in the culture medium and observed through an inverted microscope (Model IMT-2, Olympus). The micropipette experiment was performed at room temperature, i.e., 22-24°C. The pressure control system is composed of a large volume air-tight reservoir, a small syringe, a micrometer and a solenoid valve. The reservoir is partly filled with water and connected to the micropipette through the solenoid valve. The pressure was controlled by adjusting the height of the reservoir by the use of the micrometer and aspirating the air from the upper part of reservoir by the use of the syringe. A part of the line between the solenoid valve and the reservoir is connected to a pressure transducer (Model LPU-0.1, Toyo Baldwin), and the pressure level of the micropipette system is recorded on a multi-channel recorder (Nihon-Kohden, Inc.). The transducer is calibrated with a water manometer before each experiment. The tip of the micropipette was made to approach the surface of an endothelial cell by controlling the micropipette and the sample holder with two separate hydraulic micro-manipulators. When a negative pressure is applied to the tip of the micropipette, the cell is drawn toward and into the pipette. In order to determine the zero pressure level, the air pressure in the reservoir is changed by a slight adjustment of the small syringe so as to just maintain the cell in its initial position. In the viscoelastic experiments, the desired negative pressure was produced by aspirating the air from the reservoir in the same way and then applied stepwise to the tip of the micropipette by opening the solenoid valve. It was confirmed from a movement of a small particle recorded on video tape that the negative pressure extended to the tip of the micropipette within 1/30 second after opening the solenoid valve. The deformation process of the cell in the micropipette was observed through a TV camera, and the image and elapsed time were recorded on video tape for approximately 10 minutes. Theoretical Model.
From experimental observations, it
2 6 4 / V o l . 112, AUGUST 1990
appears that the major resistance to the deformation of a detached endothelial cell is provided by a cortical layer of cytoskeletal elements, lying just beneath the membrane [8]. This suggests that a possible model for use in the analysis and interpretation of the experiments reported here is that of a shell of a thickness corresponding to that of the cortical layer. However, if the micropipette radius is small compared to the cell radius, then this can be approximated as a plate of thickness equal to that of the cortical layer. Furthermore, if the stresses within this layer decay rapidly with distance below the surface, then as a'first approximation the geometry may be represented by a half space. This point will be considered further in the discussion section; however, such a model is convenient since the analysis for linear elastic deformation of a halfspace geometry was carried out by Theret et al. [12, 13] and the extension of these results to the viscoelastic case is straightforward. The details of a theoretical analysis corresponding to a purely elastic response as measured by the micropipette technique have been reported previously [12, 13], where the endothelial cell is modeled as a homogeneous, incompressible elastic halfspace. Using a "punch" model [12, 13], the loading is idealized as a combination consisting of a tensile stress applied over a circular region, representing the reduced pressure within the micropipette, equilibrated by the stress distribution in an annular zone, which is the contact region between the cell and the micropipette. In this zone, the normal component of displacement is assumed to vanish, which means that the punch is assumed to be rigid and square ended. The entire surface of the half-space is assumed to be free of tangential traction, and as a consequence the contact between the cell and the micropipette is assumed to be perfectly lubricated. The pipette inner radius and wall thickness are the characteristic lengths, and the applicability of the present approach is contingent upon, among other things, these dimensions being small compared to the radius of curvature of the cell. In the present paper, we employ the standard linear model (see Fig. 2) to account for the viscoelastic response of a cell [13]. The governing equations are divs = 0,e = —— [gradu + (gradu) T ), divu = 0, 2E s + T 6 S = — (e'+T„e'), s = s ' - p l ,
(1)
where the prime denotes the deviatoric part and the dot indicates differentiation with respect to time. In equation (1) s denotes the stress tensor and u the displacement vector, whereas e stands for the infinitesimal strain tensor. The first of equations (1) is the equation of equilibrium, the second expresses the strain in terms of the symmetric part of the displacement gradient, and the third is the incompressibility condition. The fourth is a form of the constitutive relation appropriate to the standard linear solid. The quantity TC is the relaxation time of the load at constant deformation, T„ is the relaxation time of the deformation at constant load, and E denotes the Young's modulus associated with long term equilibrium. These constants are related to the quantities of Fig. 2 through
•
T
-vT°~(1+^r),E=^/:i-
(2)
Finally, it is important to note that the quantity p in the last of equations (1) is indeterminate in the sense that it is not determined by e, but rather enters the theory as an unknown. The model under consideration possesses an axis of symmetry and as a result it is convenient to employ cylindrical coordinates (r, z) as indicated in Fig. 3. With respect to this coordinate system, the displacement components of interest Transactions of the ASME
Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 01/29/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
k2
C\lltund PorCine
EndOfhe:',o~
Cell Oetacned by TrYPSI" Ap
Fig. 2
=
a
5zz =
20 rT'(l'IH2'O
Fig. 4 A sequence of photomicrographs of progressive detormatlon of a trypsln·detached endothelial cell into a micropipette. Aspiration pressure of 20 mmH 2 0 was stepwise applied to the tip of micropipette at time 1=0.
Standard linear model for a viscoelastic solid Uz
II:
where cI>(TJ) denotes the wall function. This function is a result of the solution of the boundary problem analyzed in [12] and [13]. Since the micropipettes used in the present experiments have a wall parameter close to TJ = 0.5, it is clear from the reported results [12, 13J that we may take cI>(TJ) = 2. Thus, the expression for L(t) reduces to
a
3aAp L(t)= -uz(O, 0, t)=-- x 1TE z Fig. 3 Half·space model. Boundary conditions around a tip of micropipette.
are u r and u z' and the stress components are s", szz' and srz' The region of interest is the half-space 0= {(r,Z) Ir?O,Z?Oj.
[l+(k);k
uz(r, 0, t) = 0 for a ~ r ~ band t? 0, where h(t) denotes the unit step function. As for initial conditions, we assume u=u=OonOfort=O.
(4)
Fortunately, the solution of the present problem can be found in a simple manner with the aid of the Laplace transform and an application of the Correspondence Principle [14] from the solution of the elastic problem dealt with previously [12]. For this reason, we here deal with those features of the solution which are needed in the present research. Further details are found in [12] and [13]. Introducing the wall parameter, TJ = (b - a)/a, then the aspirated length, L(t), is given by 3aAp L(t) = - uz(O, 0, t) = - - cI>(TJ) x 21TE (5)
Journal of Biomechanical Engineering
for
7 a'
[1- k] k+k 2
(7)
exp(--t-)Jh(t). 2
(6)
It is convenient to in-
2aAp k] L s = - - ' Lo=---L s ' 1Tk) k) +k 2 L(t)=L s
=0 for r? band t? 0,
7
L s = lim L(t), L o = lim L(t) /-00 /-0+ From equation (6),
srz(r, 0, t) = 0 for r? 0 and t? 0, (3)
-1) eXP(-+)Jh(t),
where for simplicity, we write troduce the quantities
With regard to boundary conditions, we assume that at points of the boundary II of 0
szz(r, 0, t) = Aph(t) for 0 ~ r ~ a and t? 0,
2
(8)
(9)
7
Statistical Analysis. We used an unpaired t-test to statistically compare two experimental groups. Data are presented as the mean ± SD, and differences are significant when p ~ 0.05.
Results A sequence of photographs of the progressive deformation of a trypsin-detached endothelial cell under a constant negative pressure, Ap = 20 mmH 2 0, is shown in Fig. 4. The diameter of the cell is approximately 16 ~m and the internal diameter of the micropipette is approximately 4 ~m. The lefthand side photograph in the top row shows the cell has made contact with the micropipette tip at time t = O. The small arrow head in each photograph indicates the aspirated portion of the cell which is increasing with elapsed time. The magnitude of the deformation appears to approach its maximum at around 4 min. The time course of the normalized aspirated length, L/a, under constant aspiration pressure is shown in Fig. 5(a) for cells treated by the four different methods. The cells already show some deformation at time t = 0+ . After this initial deformation, which is considered to be an elastic response, the cell surface shows a creep displacement that is nonlinear with time. For example, in the case of the control (M) cell, the AUGUST 1990, Vol. 112/265
Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 01/29/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
CB (Ap=5 mmH20) Colchicine (Ap=IO mmHsO)
Control (T) 200 400 Time t (sec)
Control (M)
A CB
Fig. 7 Elastic constant k2 obtained for four different groups of endothelial cells. Mean±SD.
Fig. 5(a)
p
D. P. Theret L. T. Wheeler Department of Mechanical Engineering, University of Houston, Houston, Texas 77004
N. Ohshima Department of Biomedical Engineering, Institute of Basic Medical Sciences, University of Tsukuba, Tsukuba, Ibaraki 305, Japan
R. M. Nerem Biomechanics Laboratory and School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0405, Fellow ASME
Application of the Micropipette Technique to the Measurement of Cultured Porcine Aortic Endothelial Cell Viscoelastic Properties The viscoelastic deformation of porcine aortic endothelial cells grown under static culture conditions was measured using the micropipette technique. Experiments were conducted both for control cells (mechanically or trypsin detached from the substrate) and for cells in which cytoskeletal elements were disrupted by cytochalasin B or colchicine. The time course of the aspirated length into the pipette was measured after applying a stepwise increase in aspiration pressure. To analyze the data, a standard linear viscoelastic half-space model of the endothelial cell was used. The aspirated length was expressed as an exponential function of time. The actin microfilaments were found to be the major cytoskeletal component determining the viscoelastic response of endothelial cells grown in static culture.
Introduction The growth and behavior of the endothelial lining of a blood vessel are strongly influenced by mechanical and hemodynamic forces [1, 2]. In vivo, the cells are found to be oriented with the direction of flow and more elongated in regions of high shear stress [3]. In addition, cell shape may have a strong correlation with the localization of atherogenesis [4]. Recently, the influence of flow on endothelial structure and function has been studied using cultured endothelial cells [4-7]. These studies have demonstrated that in response to a fluid flow-imposed shear stress, cultured vascular endothelial cells elongate and orient their major axis with the direction of flow and reorganize their cytoskeletal structure. Believing that the mechanical properties of a cell may reflect quantitatively the organization of its internal cytoskeletal elements and having observed the cytoskeletal alterations associated with exposure to flow, we have measured the mechanical properties of cultured endothelial cells using the micropipette technique, both for static, no-flow conditions and after exposure to shear stress [8,9]. Our initial studies focused on the elastic properties of endothelial cells, and our results show a higher elastic modulus for cells exposed to shear stress than that for the cells from static, no-flow conditions. We also reported that in using
Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division November 28, 1989; revised manuscript received May 22, 1990.
this technique it was the cytoskeletal elements which provided the major resistance to deformation [8], In applying the micropipette technique to the measurement of the elastic properties of cells, it was necessary to wait for several minutes before the aspirated length of the cell in the micropipette reached its asymptotic value. From this, it appeared that the endothelial cells exhibited a strong viscous behavior. In the present report we describe the viscoelastic behavior of endothelial cells cultured under no-flow conditions. Using cytoskeletal disrupting agents, we investigate the influence of two different cytoskeletal elements, the actin microfilaments and the microtubules, on these viscoelastic properties. Methods Materials. Porcine aortic endothelial cells were cultured in 12-well plastic plates using a modified Dulbecco culture medium with 10 percent fetal bovine serum plus an antibioticantimycotic solution. These cells were used in the second or third passage. For micropipette measurements, endothelial cells were detached from substrate by trypsin or a mechanical method in which a sterile wood stick was used to scratch the substrate. The sheet of cell aggregates were detached further by aspiration through a 1 ml pipette. Prior to measurements, endothelial cell suspensions were stirred for approximately 15 seconds on a test tube stirrer. The cells were divided into four groups: (1) cells detached by trypsin (control(T)); (2)
Journal of Biomechanical Engineering
AUGUST 1990, Vol. 1 1 2 / 2 6 3
Copyright © 1990 by ASME Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 01/29/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Amplifier
1
Hydraulic Manipulator
Invented Microscope
Syringe
Fig. 1 Schematic diagram of the system used for the micropipette experiments
mechanically detached cells (control(M)); (3) cells treated by 2 ^iM cytochalasin B (CB); and (4) cells treated by 2 /tM colchicine. In both of these last two groups the cells were treated for about 14 hours prior to detachment. The concentrations of CB and colchicine and the duration of treatment was decided with reference to the reports by Wong and Gotlieb [10] and Meza et al. [11]. Microfilament assembly of the cytoskeletal elements is disrupted by CB and microtubule coils are depolymerized by colchicine. The cells treated by CB or colchicine were also detached from the substrate by a mechanical method and their viscoelastic properties measured using the micropipette technique. Experimental Apparatus. The experimental system is shown schematically in Fig. 1. The cells detached from the substrate were suspended in the culture medium and observed through an inverted microscope (Model IMT-2, Olympus). The micropipette experiment was performed at room temperature, i.e., 22-24°C. The pressure control system is composed of a large volume air-tight reservoir, a small syringe, a micrometer and a solenoid valve. The reservoir is partly filled with water and connected to the micropipette through the solenoid valve. The pressure was controlled by adjusting the height of the reservoir by the use of the micrometer and aspirating the air from the upper part of reservoir by the use of the syringe. A part of the line between the solenoid valve and the reservoir is connected to a pressure transducer (Model LPU-0.1, Toyo Baldwin), and the pressure level of the micropipette system is recorded on a multi-channel recorder (Nihon-Kohden, Inc.). The transducer is calibrated with a water manometer before each experiment. The tip of the micropipette was made to approach the surface of an endothelial cell by controlling the micropipette and the sample holder with two separate hydraulic micro-manipulators. When a negative pressure is applied to the tip of the micropipette, the cell is drawn toward and into the pipette. In order to determine the zero pressure level, the air pressure in the reservoir is changed by a slight adjustment of the small syringe so as to just maintain the cell in its initial position. In the viscoelastic experiments, the desired negative pressure was produced by aspirating the air from the reservoir in the same way and then applied stepwise to the tip of the micropipette by opening the solenoid valve. It was confirmed from a movement of a small particle recorded on video tape that the negative pressure extended to the tip of the micropipette within 1/30 second after opening the solenoid valve. The deformation process of the cell in the micropipette was observed through a TV camera, and the image and elapsed time were recorded on video tape for approximately 10 minutes. Theoretical Model.
From experimental observations, it
2 6 4 / V o l . 112, AUGUST 1990
appears that the major resistance to the deformation of a detached endothelial cell is provided by a cortical layer of cytoskeletal elements, lying just beneath the membrane [8]. This suggests that a possible model for use in the analysis and interpretation of the experiments reported here is that of a shell of a thickness corresponding to that of the cortical layer. However, if the micropipette radius is small compared to the cell radius, then this can be approximated as a plate of thickness equal to that of the cortical layer. Furthermore, if the stresses within this layer decay rapidly with distance below the surface, then as a'first approximation the geometry may be represented by a half space. This point will be considered further in the discussion section; however, such a model is convenient since the analysis for linear elastic deformation of a halfspace geometry was carried out by Theret et al. [12, 13] and the extension of these results to the viscoelastic case is straightforward. The details of a theoretical analysis corresponding to a purely elastic response as measured by the micropipette technique have been reported previously [12, 13], where the endothelial cell is modeled as a homogeneous, incompressible elastic halfspace. Using a "punch" model [12, 13], the loading is idealized as a combination consisting of a tensile stress applied over a circular region, representing the reduced pressure within the micropipette, equilibrated by the stress distribution in an annular zone, which is the contact region between the cell and the micropipette. In this zone, the normal component of displacement is assumed to vanish, which means that the punch is assumed to be rigid and square ended. The entire surface of the half-space is assumed to be free of tangential traction, and as a consequence the contact between the cell and the micropipette is assumed to be perfectly lubricated. The pipette inner radius and wall thickness are the characteristic lengths, and the applicability of the present approach is contingent upon, among other things, these dimensions being small compared to the radius of curvature of the cell. In the present paper, we employ the standard linear model (see Fig. 2) to account for the viscoelastic response of a cell [13]. The governing equations are divs = 0,e = —— [gradu + (gradu) T ), divu = 0, 2E s + T 6 S = — (e'+T„e'), s = s ' - p l ,
(1)
where the prime denotes the deviatoric part and the dot indicates differentiation with respect to time. In equation (1) s denotes the stress tensor and u the displacement vector, whereas e stands for the infinitesimal strain tensor. The first of equations (1) is the equation of equilibrium, the second expresses the strain in terms of the symmetric part of the displacement gradient, and the third is the incompressibility condition. The fourth is a form of the constitutive relation appropriate to the standard linear solid. The quantity TC is the relaxation time of the load at constant deformation, T„ is the relaxation time of the deformation at constant load, and E denotes the Young's modulus associated with long term equilibrium. These constants are related to the quantities of Fig. 2 through
•
T
-vT°~(1+^r),E=^/:i-
(2)
Finally, it is important to note that the quantity p in the last of equations (1) is indeterminate in the sense that it is not determined by e, but rather enters the theory as an unknown. The model under consideration possesses an axis of symmetry and as a result it is convenient to employ cylindrical coordinates (r, z) as indicated in Fig. 3. With respect to this coordinate system, the displacement components of interest Transactions of the ASME
Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 01/29/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
k2
C\lltund PorCine
EndOfhe:',o~
Cell Oetacned by TrYPSI" Ap
Fig. 2
=
a
5zz =
20 rT'(l'IH2'O
Fig. 4 A sequence of photomicrographs of progressive detormatlon of a trypsln·detached endothelial cell into a micropipette. Aspiration pressure of 20 mmH 2 0 was stepwise applied to the tip of micropipette at time 1=0.
Standard linear model for a viscoelastic solid Uz
II:
where cI>(TJ) denotes the wall function. This function is a result of the solution of the boundary problem analyzed in [12] and [13]. Since the micropipettes used in the present experiments have a wall parameter close to TJ = 0.5, it is clear from the reported results [12, 13J that we may take cI>(TJ) = 2. Thus, the expression for L(t) reduces to
a
3aAp L(t)= -uz(O, 0, t)=-- x 1TE z Fig. 3 Half·space model. Boundary conditions around a tip of micropipette.
are u r and u z' and the stress components are s", szz' and srz' The region of interest is the half-space 0= {(r,Z) Ir?O,Z?Oj.
[l+(k);k
uz(r, 0, t) = 0 for a ~ r ~ band t? 0, where h(t) denotes the unit step function. As for initial conditions, we assume u=u=OonOfort=O.
(4)
Fortunately, the solution of the present problem can be found in a simple manner with the aid of the Laplace transform and an application of the Correspondence Principle [14] from the solution of the elastic problem dealt with previously [12]. For this reason, we here deal with those features of the solution which are needed in the present research. Further details are found in [12] and [13]. Introducing the wall parameter, TJ = (b - a)/a, then the aspirated length, L(t), is given by 3aAp L(t) = - uz(O, 0, t) = - - cI>(TJ) x 21TE (5)
Journal of Biomechanical Engineering
for
7 a'
[1- k] k+k 2
(7)
exp(--t-)Jh(t). 2
(6)
It is convenient to in-
2aAp k] L s = - - ' Lo=---L s ' 1Tk) k) +k 2 L(t)=L s
=0 for r? band t? 0,
7
L s = lim L(t), L o = lim L(t) /-00 /-0+ From equation (6),
srz(r, 0, t) = 0 for r? 0 and t? 0, (3)
-1) eXP(-+)Jh(t),
where for simplicity, we write troduce the quantities
With regard to boundary conditions, we assume that at points of the boundary II of 0
szz(r, 0, t) = Aph(t) for 0 ~ r ~ a and t? 0,
2
(8)
(9)
7
Statistical Analysis. We used an unpaired t-test to statistically compare two experimental groups. Data are presented as the mean ± SD, and differences are significant when p ~ 0.05.
Results A sequence of photographs of the progressive deformation of a trypsin-detached endothelial cell under a constant negative pressure, Ap = 20 mmH 2 0, is shown in Fig. 4. The diameter of the cell is approximately 16 ~m and the internal diameter of the micropipette is approximately 4 ~m. The lefthand side photograph in the top row shows the cell has made contact with the micropipette tip at time t = O. The small arrow head in each photograph indicates the aspirated portion of the cell which is increasing with elapsed time. The magnitude of the deformation appears to approach its maximum at around 4 min. The time course of the normalized aspirated length, L/a, under constant aspiration pressure is shown in Fig. 5(a) for cells treated by the four different methods. The cells already show some deformation at time t = 0+ . After this initial deformation, which is considered to be an elastic response, the cell surface shows a creep displacement that is nonlinear with time. For example, in the case of the control (M) cell, the AUGUST 1990, Vol. 112/265
Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 01/29/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
CB (Ap=5 mmH20) Colchicine (Ap=IO mmHsO)
Control (T) 200 400 Time t (sec)
Control (M)
A CB
Fig. 7 Elastic constant k2 obtained for four different groups of endothelial cells. Mean±SD.
Fig. 5(a)
p
