1 Introduction

Osteocytes, mature bone cells embedded in mineralized bone matrix, are under dynamic shear fluid in vivo and exhibit significant viscoelastic behavior (Jacobs et al. 2010). Abundant evidence has shown that osteocytes are the key mechanonsensor cells that directly regulate activities of bone-forming osteoblast and bone-removing osteoclast (You et al. 2008). Thus, osteocytes are critical to etiology and new treatments for osteoporosis. One of the common loading mechanisms on cells such as osteocytes is interstitial fluid shear resulting from tissue matrix deformation (Cowin and Telega 2003; Weinbaum et al. 1994). Such deformation introduces mechanical stress and strain inside cell body, such as stretch and compression, and may trigger the activation and transduction of biochemical signals (Guilak 1995). Although the fluid–structure interactions (FSI) at the cellular level appear to be universal and may hold the key to unraveling the mechanisms of cellular mechanotransduction, they are poorly understood (Anderson et al. 2006).

Knowledge of the structural and mechanical property of the cell is necessary to study the spatial distribution and performing process of the stress/strain in the adherent cell under shear flow (Hazel and Pedley 2000). Several traditional in vitro techniques have been developed to determine the viscoelastic properties of various cells, including micropipette aspiration (Baaijens et al. 2005; Evans and Yeung 1989; Hochmuth 2000; Sato et al. 1990; Theret et al. 1988), unconfined cell compression (Wu and Herzog 2006), cyto-indentation (Koay et al. 2003; Shin and Athanasiou 1999), atomic force microscopy (AFM) (Charras and Horton 2002; Darling et al. 2006; Lulevich et al. 2006; Mahaffy et al. 2004; Vargas-Pinto et al. 2015), magnetic bead rheometry (Bausch et al. 1998), and optical traps (Guck 2005). However, cyto-indentation, AFM and magnetic twist methods always apply load locally by probes or beads; thus, the measured mechanical properties only characterize the local nature of the cell. As to micropipette aspiration, unconfined cell compression and optical traps methods, although the entire cell is loaded, there are external objects or energy interposed in the cell during the measurement, which may alter the cell structure and properties.

This study proposes a novel approach combining cell flow experiment and finite element (FE) modeling to noninvasively study the mechanical behavior of a whole cell under fluid flow. The deformation of the osteocytic cell in fluids is recorded by a custom-built dual microscope system named quasi-3D microscopy (Baik et al. 2010), which is able to capture two orthogonal planes of a cell simultaneously at a high temporal resolution. Previous work has validated the accuracy of this observation system (Qiu et al. 2012). An FE model is established to numerically simulate the deformation of an individual cell under shear flow. In the model, the cell configuration is based on the observed data derived from the microscopic images, as opposed to the assumption of a sphere or semi-sphere (Baaijens et al. 2005; Zhao et al. 2009) with statistical dimension or an half-space in the analytical solution (Sato et al. 1990; Theret et al. 1988). Instead of simply assigning the cellular viscoelastic parameters using statistical data from the literatures totally or partially, the parameters of each sample cell are determined individually by an iterative algorithm minimizing the deformation difference between the observed data and the computational results from the FE model, so that the cellular difference induced by biological diversity could be sufficiently considered and reflected.

The contribution of this study is mainly composed of: (1) the establishment of a fluid–structure interactive FE model based on individual cellular data to study the unique mechanical behavior of a particular cell; (2) the determination of the viscoelastic properties of osteocytes using 3D FSI FE model and fluid shear experiments under quasi-3D microscope; (3) the proposal of an index to quantify the cell adhesion on the basis of static friction coefficient; (4) the revelation of an interesting ‘competition phenomenon’ between stress and strain on the cell surface, i.e., the place with high stress has low strain and that with low stress has high strain.

2 Methodology

2.1 Framework

The main difficulty of modeling an adherent cell lies in the estimation of the cellular viscoelasticity, which can be presented by three parameters in standard linear solid (SLS) viscoelastic model, i.e., \(k_{1}, k_{2}\), and \(\eta \). An iterative method based on the experimental observation and finite element model is thus established to obtain the accurate viscoelastic parameters, the frame work of which is illustrated in Fig. 1.

A custom-built quasi-3D observation system composed of two microscopes (Baik et al. 2010; Qiu et al. 2012) as illustrated in Fig. 2 is used to record the deformation of the cell at different moment from both bottom view and side view. The initial cell contour without shear fluid is firstly adopted to construct a 3D numerical cell for FE model. Hence, unlike most of the studies, in which the numerical cell is usually simplified to a hemispheroid or semi-ellipsoid, the numerical cell here is reconstructed on the basis of the real cellular configurations.

A 3D fluid–structure interaction FE model is then established using the numerical cell under the flow conditions as in experiments. A group of estimated cellular viscoelastic parameters is input to the FE model, and the calculated load-deformation path is compared to the records from the quasi-3D observation system. The group of viscoelastic parameters will be updated if there is a large gap between the simulated deformation process and the experimental observations. The iteration stops until the load-deformation path resulted from the FE model approaches the observations, when the viscoelastic parameters of the cell are determined.

Fig. 1
figure 1

Framework to determine the force and deformation of an adherent cell under shear flow

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Fig. 2
figure 2

Schematic of a the quasi-3D microscope observation system and b cell flow chamber experiment

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With the viscoelastic parameters of the adherent cell, the force as well as the deformation of the cell under shear fluid can be worked out by the 3D fluid–structure interaction FE model.

2.2 In vitro experiments

2.2.1 Cell culture and cell flow chamber experiments

Osteocytic MLO-Y4 cells (Bonewald and Kato 2002) growing in culture media consisting of \(\upalpha \)-modified Eagle’s medium \((\upalpha \hbox {-MEM})\), 5 % fetal bovine serum, and 5 % calf serum, at \(37^{\,\circ }\hbox {C}\) in a humidified atmosphere of 95 % air and 5 % \(\hbox {CO}_{2}\) are selected in this study.

Syringe pump (New Era Pump Systems, Wantagh, NY) and gas-tight syringe (Hamilton, Reno, NV) are used to provide steady laminar flow with a wall shear stress (WSS) of \(10\,\hbox {dynes/cm}^{2}\) in a flow chamber with width of \(700\,{{\upmu }\hbox {m}}\) and height of \(550\,{{\upmu }\hbox {m}}\), comprising a square glass micro-tube with a glass micro-slide inside. Cells are seeded on fibronectin coated glass micro-slides for 20 min to ensure partially adherent status as in vivo and at low density to reduce the interaction between cells. Cells are stained before flow starts with Alexa Fluor 594 wheat germ agglutinin plasma membrane dye.

2.2.2 Image acquisition

The images of individual osteocytes are captured simultaneously by two microscopes in the quasi-3D microscope system introduced by Baik et al. (2010), as shown in Fig. 2. This system consists of an upright microscope and an inverted one, and a mirror is aligned at \(45^{\circ }\) in the light path of the upright microscope to obtain side-view image, while the inverted microscope gets regular bottom-view image of the same cell. Fluorescent images with a resolution of \(0.215\,{{\upmu }\hbox {m}}/\hbox {pixel}\) can be captured at a frequency of \(12\,\hbox {Hz}\), and then processed by digital image correlation techniques to determine the displacement of the cell contour.

2.3 3D fluid–structure interaction FE model

The assumptions below are set in the fluid–structure coupling finite element model: (1) osteocyte is a homogeneous, isotropic, and incompressible viscoelastic continuum; (2) only the passive deformation of osteocyte is analyzed without consideration of any active force; (3) the hydrodynamic properties of culture medium are equivalent to those of the water at room temperature; and (4) the environmental temperature is constant since the experiment is finished within a few minutes.

2.3.1 Cell reconstruction

Cell images in flow chamber are obtained in two perpendicular directions by the quasi-3D microscopic system. After image enhancement, the bottom and side outlines of the cell on every frame can be identified, and the 3D configurations of the cell can be reconstructed by a lofting algorithm based on cellular convexity assumptions. Microscopic observations show that the overall configuration of bone cells is quite close to ellipsoid, falling into the category of convex structures. Thus, the transition area from side to bottom outline also accords with the convexity characteristics. The reconstruction of cellular 3D configuration by bottom-view and side-view images is indicated in Fig. 3.

Fig. 3
figure 3

Reconstruction of 3D cell from 2D microscopic images. a, b Two simultaneously obtained views of the plasma membrane of an osteocyte cell, where the white solid lines represent the contour of cell. c A typical cell profile. d The reconstructed 3D numerical cell. Scale bar is \(5~\upmu \hbox {m}\)

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The white lines in Fig. 3a, b represent cell contours detected in the deformation process by digital image correlation algorithm. Subsequently, cell contours are smoothed and initialized by MATLAB (MathWorks, Natick, MA), and fitted by spline function in Slidworks (Dassault Systèmes, Concord, MA), as shown in Fig. 3c. The cell contour is finally lofted into 3D structure in Solidworks, as in Fig. 3d.

2.3.2 Structure model

Previous studies demonstrate that adherent cells exhibit viscoelastic properties (Bausch et al. 1998; Evans et al. 1984; Ward and Pinnock 1966; Yamada et al. 2000). The cell is thus assumed to be deformable with isotropic and homogeneous viscoelastic properties, which is characterized by an incompressible finite strain model (Haider and Guilak 2000; Jones et al. 1999). The relaxation shear modulus G(t) of the model is represented by a one-term Prony series:

$$\begin{aligned} G(t)=G_\infty +G_1 e^{-\beta _1 t} \end{aligned}$$
(1)

The parameters of Prony series can be derived from three parameters in standard linear solid (SLS) viscoelastic model, i.e., \(k_{1}, k_{2}\), and \(\eta \), as illustrated in Fig. 4a.

Fig. 4
figure 4

Schematic of a standard linear solid (SLS) model and b the boundary condition for fluid model

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2.3.3 Fluid model

The properties of the fluid medium are assumed to be equivalent to water at room temperature, i.e., an incompressible Newtonian fluid having a dynamic viscosity of \(0.89\times 10^{-3}\,\mathrm{Pa}\cdot \mathrm{s}\) and density of \(997\,\hbox {kg/m}^{3}\).

Reynolds numbers \((R_{e})\) are calculated for the channel region using the formula

$$\begin{aligned} Re=\frac{\rho vL}{\eta } \end{aligned}$$
(2)

where \(\nu \) is the mean velocity and L is the hydraulic diameter of the chamber, which is assumed to be as high as the chamber, i.e., \(550\,{\upmu }\hbox {m}\). The mean velocity is 57 mm/s when WSS is \(10~\hbox {dyn/cm}^{2}\), and the calculated \(R_{e}\) is around 35, indicating laminar flow and helping to determine the calculation domain.

2.3.4 Boundary conditions

The boundary conditions agree with those in the experiment. The local CFD domain is \(700\,\upmu \hbox {m}\times 550\,\upmu \hbox {m}\times 200\,\upmu \hbox {m}\), covering the cell deformable domain, which is adherent to the bottom of the chamber, as shown in Fig. 4b. The flow is unidirectionally in X-direction with parabolic velocity at the entrance and free of pressure at the exit.

2.3.5 Fluid–structure interaction (FSI) algorithm

In the FSI model, it is the cell membrane that acts as the fluid–solid interface, allowing the exchange of data (i.e.,  forces and displacements) between the fluid and solid domains. The coupled simulation adopts the staggered iteration approach (Bathe 2009). Fluid equations are firstly solved to get the fluid stress tensor, which then acts on the fluid–solid interface, resulting in cellular deformation. Such deformation relays back to the fluid domain, and the fluid equations are recalculated for next iteration.

3 Results

3.1 Cell creep behavior

The adherent cell presents typical viscoelastic deformation under the fluid shear stress in the experiment. Compared with other area, the deformation on the top of the cell is the most significant, where the load-deformation path at the flow direction is shown in Fig. 5a. The steady shear flow is triggered on at the instantaneous moment \(t_{0}\) and turned off at the equilibrium reaching time \(t_{1}\). The flow lasts for 120 s, covering cellular instantaneous and equilibrium response. Three subgraphs (i, ii, iii) beside the curve in Fig. 5a show the specific side-view configuration of the cell at the corresponding moments. A sharp deformation can be observed at \(t_{0}\), named initial jump. The deformation gradually accumulates under fluid flow and reaches equilibrium state at \(t_{1}\). After the flow is off at \(t_{1}\), the cellular deformation gradually recovers. It has to be illustrated that throughout the entire deformation process, the cell adheres to the base steadily without any debonding.

Fig. 5
figure 5

a Viscoelastic response of cell deformation and b the loading profile of the flow

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3.2 3D cell configuration

Figure 6 shows the topological information of the reconstructed 3D cell, described by the relationship between the surface area and the volume of the cell. The black, red, and blue dash lines in Fig. 6 represent disk plate (the diameter is set as ten times of the height, i.e., \(h=0.1\)D), semi-sphere, and sphere, respectively. The eight sample cells fall in between spherical and hemispherical, and has big disparity with disk plate, which represents completely adhesive state. It is a validation that the sample cells in experiments are in the state of partial adhesion. The volume of MLO-Y4 osteocyte cell is \(4023\pm 1487\,\upmu \hbox {m}^{3}\) with the surface area of \(1377\pm 366\,\upmu \hbox {m}^{2}\, (\hbox {Mean}\pm \hbox {STD}, {n}=8)\).

Fig. 6
figure 6

Comparison of topological information among the reconstructed 3D cell, dis plate, semi-sphere, and sphere

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Fig. 7
figure 7

Displacement–time curves of three sample cells (a–c) from experiment and FE simulation in the iteration process to determine cellular viscoelastic parameters

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3.3 Determination of viscoelastic parameters

The creep behavior presented in Fig. 5 has demonstrated that the cell can be regarded as viscoelastic material. Three sample cells are then randomly selected to reproduce the in vitro experiments within much shorter time, 20 s, to determine the viscoelastic parameters, using the iterative method introduced in Sect. 2.1 based on the experiment in Sect. 2.2 and FE model in Sect. 2.3. The flow only lasts for 10 s for it is neither too long to induce unnecessary interference nor too short to cover up cellular viscoelasticity. The largest displacement of the three sample cells during their deformation are 1.5, 2.5 and \(1.3\,\upmu \hbox {m}\), respectively, as shown in Fig. 7. The calculated displacement curves with the initial trials have a big deviation to the experimental results, as presented in the first column in Fig. 7. Continuous iterations finally fix a set of viscoelastic parameters, which makes the displacement curves consistent with the experimental ones, as shown in the last column in Fig. 7. The viscoelastic parameters of the osteocytic cell are: the equilibrium elasticity modulus \(k_{1}=0.15\pm 0.038\,\hbox {kPa}\), instantaneous elasticity modulus \((k_{1}+k_{2})=0.77\pm 0.23\,\hbox {kPa}, k_{2}=0.62\pm 0.14\,\hbox {kPa}\), creep characteristic time \({\tau }_{\sigma }=11.30\pm 2.34\,\hbox {s}\), viscosity coefficient \(\eta =1.38\pm 0.33\,\hbox {kPa}\,\hbox {s}\).

Table 1 Calculated fluid mechanical stimulation on the surface of adherent cell by FE model
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3.4 Mechanical stimulation and response

The 3D numerical cell is constructed with the microscopic topology information and the viscoelastic parameters. The fluid–structure interaction FE model is used to analyze the interaction between cell and flow under different WSS. The WSS of the venules of human beings is about \(10\,\hbox {dyn/cm}^{2}\) (Koutsiaris 2015; Koutsiaris et al. 2007); thus, the cellular mechanical characteristics under different WSS conditions are studied, as shown in the third column in Table 1.

3.4.1 Mechanical stimuli on the cell flow interface

Normal pressure and shear force are two perspectives to investigate the force on the cell. The force on an adherent cell with moderate size (length in X direction\(\,=\,19.90\,\upmu \hbox {m}\), height in Z direction\(\,=\,14.84\,\upmu \hbox {m}\)) under different flow velocity is studied, as shown in Table 1 and Fig. 8.

Table 1 presents the calculated value of the maximum and minimum pressure, and maximum shear stress, as well as the corresponding positions under different WSS. It can be seen that the pressure and shear stress on the cell are basically within 4–7 times of the WSS from Table 1. Figure 8 shows the pressure and shear stress distribution on the surface of the cell. Both the pressure and shear stress in Fig. 8 have been normalized (divided by WSS). The pressure and shear stress along the central section line of the cell in the flow direction are extracted and shown as black curves in Fig. 8b, d, respectively. The green curves represent the projection of the black curves on XZ plane, essentially the central section line of the cell. The red ones in the bottom plane represent the normalized pressure in Fig. 8b and normalized shear stress in Fig. 8d.

The extremum pressure on the cell approximately distributes in the center of the upstream side and downstream side, respectively, as shown in Fig. 8a. The position with extremum pressure almost remains the same under different flow intensity, as given in Table 1. The pressure along the central section line has a distribution similar to sinusoidal variation, as the red curve shown in Fig. 8b. The maximum shear stress appears at the top of the cell for all the WSS studied, as given in Table 1. Low shear stress (<0.45) appears at the bottom of the cell, forming a surround band, as shown in Fig. 8c. The distribution curve of the shear stress along the center section line of the cell surface is analogous to a downward parabola, as the red one in Fig. 8d.

Fig. 8
figure 8

Distribution of a normalized pressure and c normalized shear stress on the surface of cell from FE analysis and distribution of b normalized pressure and d normalized shear stress along the central section line of the cell

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The traction force at the mesh node in FE model is the integral value of pressure and shear stress on the cell flow interface with the magnitude of pN \((10^{-12}\,\hbox {N})\). Although such nodal traction force depends on the uniform size of the meshes on the interface, it can indicate the force distribution. Figure 9a–c shows the magnitude and distribution of the nodal traction force at different directions. The nodal traction force along the flow direction is dominant compared to that in other two directions.

The sum of the nodal traction force on the entire cell flow interface is the flow resultant force, the magnitude of which is determined by the interaction between flow and cell without dependence on the mesh size. The normalized flow force is defined as the ratio between the flow resultant force and the cell weight. Figure 9d presents the relationships between the normalized flow force and the WSS, indicating that normalized flow force has a positive linear correlation to WSS. With the WSS of \(10\,\hbox {dyn/cm}^{2}\), equivalent to that in capillaries of human beings, the flow dragging force is almost 85 times of the cell weight in the flow direction and 3.98 times in the gravitational direction. The resultant force in flow direction is significant dominant compared to that in other two directions.

The relationship between static friction force F and normal pressure N can be expressed as \(F=\mu {\cdot }N\) where \(\mu \) is static friction coefficient. Analogy to mechanical friction, the \(\mu \) between the cell and the fibronectin coated base under five WSS conditions are thus worked out to be 27.3, 25.9, 21.4, 20.1, and 18.0, respectively.

Fig. 9
figure 9

Distribution of the nodal traction force on the interface between cell and flow from FE analysis in a flow direction, b gravity direction, and c transverse direction; d the relationship between the nominal flow force and WSS, the sample cell weight is 29 pN

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3.4.2 Intracellular strain

Figure 10 shows the distribution of intracellular strain from FE analysis, including three normal strain components in Fig. 10a–c and three shear strain components in Fig. 10d–f. The distance between two cutting sections is \(4\,\upmu \hbox {m}\). The shear strain component \(E_{xz}\) is largest among the six strain components, about one order higher than others. The maximum \(E_{xz}\) is distributed on the upstream and downstream sides of the adherent area, where the flow shear stress is almost zero as illustrated in Fig. 8c of Sect. 3.4.1. The \(E_{xz}\) in the center of upstream and downstream surface as well as on the top of the cell is much smaller. It can be learned from Sect. 3.4.1 that the normal pressure in the center of the upstream and downstream surface is highest, and the shear stress on the top of the cell is also higher than other areas. These contradictory phenomena indicate that the stress and the strain of the adherent cell have interesting competitive relations.

Fig. 10
figure 10

Distribution of intracellular strain

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4 Discussion

In this paper, a method coupling in vitro experiment and fluid–structure interaction FE model is proposed to determine the viscoelastic properties of an adherent cell under shear flow and characterize the corresponding mechanical stimuli and the intracellular deformation. The cellular deformation in vitro experiment is recorded by a quasi-3D microscopic system. The experiment proves that the cell can be represented by viscoelastic materials. A 3D numerical cell is reconstructed on the basis of the observed cellular configuration instead of an ideal assumption of semi-sphere or sphere in the literature. Considering the fluid characteristics and boundary conditions, an FSI FE model is established. A group of viscoelastic parameters are input into this FE model to calculate the cellular deformation, which is compared to the experimental results. These parameters will be updated if the calculated deformation is greatly deviated from the observed results. Such iteration continues until the deviation is satisfying, when the viscoelastic parameters are determined. The established FSI FE model with the these parameters is then used to analyze the mechanical stimulation and response of the cell under different flow intensity.

Compared with traditional techniques measuring cellular mechanical properties, such as AFM, magnetic twist, micropipettes, and optical tweezers, the method presented in this study is nonlocal, noninvasive, and also close to the physiological conditions in vivo; cellular responses under shear flow are the mechanical behavior of the entire cell instead of a particular local part; no external objects such as probes, beads, or pipettes are used to apply loads onto the sample cells; last but not least, the force is applied naturally to the cellular surface by fluid flow, the process of which is similar to the physiological interaction between osteocytes and their microenvironment.

Researches on the mechanical properties of osteocytes are relatively rare. Previous studies reported the elastic constant of MLO-Y4 round but partially adherent osteocytes is below 1 kPa by AFM (Bacabac et al. 2008). In our previous work, the modulus of the same kind of cell is \(0.49\pm 0.11\,\hbox {kPa}\) based on an computational optimization method (Qiu et al. 2014), the objective function of which is conducted by cell deformation configuration. In this study, the instantaneous modulus of partially adherent MLO-Y4 osteocytes is \(0.77\pm 0.23\,\hbox {kPa}\), close to our reported data but with much lower computational cost. In addition to the instantaneous modulus, we also studied the elastic modulus in equilibrium state \((K_1=0.15\pm 0.038\,\hbox {kPa})\) and the creep characteristic time \((\tau _\sigma =11.30\pm 2.34\,\hbox {s})\), which reflects cellular viscosity coefficient \((\eta =1.38\pm 0.33\,\hbox {kPa}\,\hbox {s})\).

Through this study, osteocytes (<1 kPa) are found to be softer than osteoblasts (>1 kPa) (Darling et al. 2008) with smaller elastic modulus. Previous studies have demonstrated that osteocytes are evolved from osteoblasts by burying itself in lacunae within the bone matrix (Guilak and Mow 2000). It implies that some changes might occur to the cytoskeleton of the osteoblasts after being embedded in bone mineral to improve its sensitivity to extracellular stimulations. Furthermore, another interesting thing is that the mechanical properties of osteocytes and chondrocytes (<1 kPa) (Trickey et al. 2000) are ranged in the same level although they belong to different organizations. Osteocytes and chondrocytes have similar external mechanical environment, i.e., they are embedded in bone lacuna and cartilage lacuna, respectively. Thus, the cellular mechanical properties might have strong correspondence with the external microenvironment of cells.

According to the calculated results from FSI FE model, the flow dragging force on the cell is about 85 times of the cell weight under the WSS of \(10\,\hbox {dyn/cm}^{2}\), and the corresponding static friction coefficient is over 20. It is one order higher than the friction coefficient between pure surfaces of inorganic materials under high-vacuum state. The biological binding, focal adhesion, between the cell and the substrate must be the reason for such high friction coefficients. The approach proposed in this study also provides an accurate method to quantify such bonding strength using experiments and FE models.

Identification of whether the cellular response is a strain-activated or stress-activated mechanism under fluid flow appears difficult since it is hard to acquire stress and strain simultaneously. De et al. (2008) discussed whether the cellular orientation responds to strain or stress of the matrix, and reported that the cellular orientation is a strong function of the Poisson’s ratio \(\upnu \) of the matrix when cell activity is governed by the matrix strain, while if cell activity is governed by the matrix stress, the orientation depends only weakly on \(\upnu \). McGarry et al. (2005) tested the hypothesis that bone cells respond differently to 0.6 Pa fluid shear stress and \(1000\,{\upmu }{\upvarepsilon }\) substrate strain stimulation because of the qualitative and quantitative differences of the cellular deformation. Our study highlights the distribution properties of the stress and strain of MLO-Y4 osteocytes under fluid flow in vitro. A significant phenomenon is that the distribution of stress on the cell is inconsistent with that of the intracellular strain. The highest shear stress is found at the top of the cell as shown in Fig. 8c, while the highest intracellular strain distributes at the upstream and downstream interface between the cell and the substrate as in Fig. 10d. Based on the assumption that cell could sense the extracellular mechanical signals more easily from the membrane domain where the signals are the strongest, there might exist two mechanical signaling channels. One is sensitive to stress from apical region to the nucleus area, and the other is sensitive to strain from the bottom of the cell to the nucleus area. This provides a unique perspective to understand the mechanisms of mechanotransduction particularly for the partially adherent cells under fluid flow.

The current FSI FE model combined with experiments provides a quantitative approach to determine cell mechanical properties and illustrate general features of the stress and strain distributions of adherent cells under fluid flow. Potential limitations of this study are mainly concerned with the numerical model, in particular the representation of a single cell as homogeneous and isotropic continuum. Future studies could enrich the model with certain crucial details, such as cell nucleus, cytoplasm, cytomembrane, and primary cilia (Jacobs et al. 2010). Furthermore, a certain degree of viscoplastic behavior has also been observed in our experiments rather than completely viscoelastic property, which can also be taken into account in the future. It is believed that this work not only quantifies the cellular mechanical characteristics with higher accuracy but also helps to elucidate the mechanisms of mechanotransduction for various adherent cells.