A Computational Approach to Understand Phenotypic Structure and Constitutive Mechanics Relationships of Single Cells

Abstract

The goal of this study is to construct a representative 3D finite element model (FEM) of individual cells based on their sub-cellular structures that predicts cell mechanical behavior. The FEM simulations replicate atomic force microscopy (AFM) nanoindentation experiments on live vascular smooth muscle cells. Individual cells are characterized mechanically with AFM and then imaged in 3D using a spinning disc confocal microscope. Using these images, geometries for the FEM are automatically generated via image segmentation and linear programming algorithms. The geometries consist of independent structures representing the nucleus, actin stress fiber network, and cytoplasm. These are imported into commercial software for mesh refinement and analysis. The FEM presented here is capable of predicting AFM results well for 500 nm indentations. The FEM results are relatively insensitive to both the exact number and diameter of fibers used. Despite the localized nature of AFM nanoindentation, the model predicts that stresses are distributed in an anisotropic manner throughout the cell body via the actin stress fibers. This pattern of stress distribution is likely a result of the geometric arrangement of the actin network.

INTRODUCTION

The ability to model the mechanical responses of cells to physical stimuli presents many opportunities to the world of medical research. Chief among these is the ability to further our understanding of many diseases.^13,35 There are a wide variety of diseases whose clinical presentation is either known or suspected to be related to abnormal cellular mechanics, alteration of cellular mechanotransduction processes, or changes in tissue structure.^{2,13,14,18,20,23,26,38,39} Because physical distortion can affect cell growth, differentiation, contractility, motility, and apoptotic tendency,^5,12 the ability to predict the mechanical behavior of cells in response to pathological conditions and medical treatments may be critical to prevention and treatment of many of these diseases.^{2,13,14,18,20,23,26,38,39} The goal of this study is to construct a representative 3D finite element model (FEM) of individual biological cells based on the sub-cellular structures that provide the cells with their mechanical properties.

The use of image-based finite element modeling, a powerful engineering analytical tool, is considered a state-of-the-art methodology for researching the mechanics of organs and tissues, however its use for single cell mechanics research has been limited until recently.³⁵ This is largely due to the highly complex geometries and material nonlinearity exhibited by cells, which leads to models that are more computationally expensive than was feasible on the average personal computer (PC) of a few years ago.²⁸ There have been attempts to create finite element models of single cells since PC computing power began increasing in the 1990 s^28,31,35,36; however, despite the ever-growing availability of sufficient computing power for more complex models, only a few 3D confocal image-based structural finite element models of single cells have been published^7,35 but those have not been validated by comparing their predictions to experimental data of the cells they model. Furthermore, of those models that have been published, only one employs large strain mechanics and uses a compound structure.³⁵ To date, no models have been able to automatically generate realistic geometries and meshes from 3D cell images. Current models instead rely on human tracing of cell boundaries which is time consuming and invites a certain degree of inefficiency and subjectivity that automatically generated geometries may be less prone to.

The model presented in this study provides a small step toward allowing finite element analysis to become more of a predictive tool for the determination of cellular mechanical properties based solely on images of subcellular structures. The methods described in this study could also eventually be incorporated into multiscale models, providing a key link between the prediction of mechanical responses of cells to pathological conditions and the development of new medical treatments from the tissue level down to the molecular level.

MATERIALS AND METHODS

System Considered

Vascular smooth muscle cells (VSMCs) were chosen as the model system for this study. These cells are the most predominant cell type in the thickest layer (the tunica media) of blood vessels and are constantly under dynamic load due to arterial pressure in normal healthy conditions. They can undergo significant cytoskeletal remodeling in response to injurious mechanical loading^3,4 that may have pathological implications for the blood vessel.^3,4 In addition, VSMC phenotypic shifts lead to significant changes in cellular mechanical properties in vitro.¹⁰

Actin microfilaments (f-actin) are 7–9 nm diameter polarized polymers comprised of globular actin (g-actin) monomers. These microfilaments exhibit a highly dynamic behavior regulated by the proteins profilin and cofilin. The microfilaments are relatively stiff and have a persistence length of 15 µm and an elastic modulus of 1.3–2.5 GPa in dilute solution.^8,15,27 In adherent cells, actin microfilaments are found bound together by actin binding proteins to form closely packed arrays known as actin stress fibers. The quantity and orientation of these stress fibers play a significant role in establishing the mechanical properties of VSMCs. They orient themselves largely along the direction of the stress field applied to the cell by its surroundings. A key feature of actin stress fibers is the prestress that is actively exhibited upon them by the actomyosin complexes which are formed by the association of myosin motor proteins with the actin filaments.^17,32

Aortic VSMCs were isolated from female Sprague–Dawley rats by means approved by the Clemson University Institutional Review Board following previously reported methods.¹¹ Cells used in experiments were passage 6. Briefly, the cells were cultured in Dulbecco’s Modified Eagle’s Medium (HyClone Laboratories, Logan, UT USA) supplemented with 10% fetal bovine serum (Sigma, St. Louis, MO) and 1% antibiotic/antimycotic (HyClone). The cells were seeded at subconfluent densities on glass coverslips coated with 50 µL of 1 mg/mL type I collagen (BD Biosciences, Bedford, MA, USA). The cells were cultured at 37 °C in a 5% CO₂ environment for 5 days prior to mechanical testing. Three cells were analyzed in detail in this study with the images in Fig. 1 showing the actin stress fiber network and nucleus within each cell. The specific cells used in this study were chosen based on their phenotypic morphologies to have elastic moduli values that were anticipated to be at the high end, low end, and roughly average values for VSMCs based on observations from previous studies.^10,11

FIGURE 1.

Orthogonal views of 3D reconstruction of the actin stress fiber network (green) and nucleus (blue) of (a) Cell A, (b) Cell B, and (c) Cell C; images generated in Olympus MetaMorph^®, all scale bars are 25 µm. All subsequent similar images are oriented identically.

Mechanical Characterization of Live Cells

Atomic force microscopy (AFM) nanoindentation was utilized in this study to characterize the mechanics of the VSMCs.^10,11 20 cells were indented to a depth of ~1 µm at 0.5 µm/s with a 5 µm diameter borosilicate spherical tip on a 0.18 nN/nm cantilever (sQube, Bickenbach, Germany) using an Asylum Molecular Force Probe 3D (Asylum Research, Goleta, CA) set to a 2 kHz sampling rate. Cells were maintained in media at 37 °C for the duration of the experiment using the MFP Bioheater (Asylum Research). For indentation, the AFM probe was placed above the estimated center of the cell (Fig. 2) to avoid the thin cell edges where substrate effects dominate the observed stiffness of the cells during AFM nanoindentation. In a prior study¹⁰ the modulus of VSMCs was not found to vary much over the central part of the cell and the nucleus did not seem to have a significant effect on the cell mechanical properties in these experiments. The modulus did increase significantly at the edge of the cell (within 1–2 µm from edge) when the height of the cell is < 1 µm. In order to estimate the contribution of actin stress fibers to the cells mechanical properties, VSMCs were treated in separate experiments with 1 µM cytochalasin D at 37 °C for one hour prior to testing to depolymerize f-actin.¹¹

FIGURE 2.

Cell B as viewed from below through AFM camera with 60X oil objective (a) prior to indentation with arrow pointing to the nucleus and (b) during indentation with arrow pointing to 5 µm diameter spherical probe at right-hand side of AFM cantilever above the cell.

Confocal Imaging of Cells

Following mechanical characterization, the cells were fixed using 4% paraformaldehyde and permeabolized using a solution of 0.1% Triton-X and 0.01 M glycine (to quench the excess aldehyde). They were then treated with solutions of 130 nM AlexaFluor 488 phalloidin and 350 nM DAPI to fluorescently label the actin stress fiber network and nucleus, respectively (Fig. 1). The stained cells were imaged using an Olympus PLAPON60XO 60 × oil objective (NA = 1.42) on an Olympus IX81 inverted microscope equipped with a DSU spinning disc confocal unit and a Hamamatsu ImagEM CCD camera (Hamamatsu Photonics K.K., Hamamatsu City, Japan). The lateral resolution of this imaging technique is 230 nm, and the axial resolution is 840 nm.⁹ It would admittedly be ideal to use an imaging technique with higher resolution due to the small diameter of the actin stress fibers and the thinness of the spread VSMCs (Fig. 1). However, the authors chose to utilize confocal microscopy for this study because of its ability to image cells in an aqueous environment at the same temperature as mechanical characterization, thereby allowing for minimal disturbance of subcellular features relative to their orientation during mechanical characterization. In addition, the use of an inverted optical microscope allowed for easy combination with an AFM, which further allowed the authors to image the exact same cells that were characterized mechanically within minutes of said characterization.

Analysis of Mechanical Characterization

The Hertz contact model is used initially to estimate the apparent elastic modulus of the cell. The equation governing the relation of force of indentation to the elastic modulus of an indented sample using a spherical indenter according to the Hertz model is given by:

$$F=\frac{4}{3}\frac{E}{(1-{\nu }^2)}{R}^{\frac{1}{2}}{\delta }^{\frac{3}{2}}$$ (1)

where E and ν are the elastic modulus and Poisson’s ratio of the sample, respectively, δ is the depth of indentation, and R is the radius of the spherical indenter. The Hertz model shown in Eq. (1) is calculated using a depth of 200 nm indentation, or approximately 10% of cell height, as the value for δ at this depth is within the range that the Hertz model remains accurate.¹⁰

Finite-Element Model of System

Modeling Assumptions

The microtubule network of VSMCs is highly complex, with no common, easily discernible patterns typically present. Therefore, in order to reduce the complexity of this model the microtubule network is assumed to be distributed homogeneously throughout the cell and can therefore be modeled as a continuum. Additionally, it is highly difficult to discern from confocal microscopy images the number of actin microfilaments comprising a single actin stress fiber. It is therefore assumed that each stress fiber has a radius of 100 nm based on previously reported values.²²

Geometric Model and Mesh Characteristics

Three-dimensional confocal microscopy image stacks are analyzed in MATLAB R2010a (MathWorks, Natick, MA) using an algorithm capable of generating representative model geometries consisting of the cell body, nucleus, and actin stress fiber network. The algorithm uses standard image segmentation and thresholding techniques that are similar to commercially available software techniques and are well reviewed in the literature^{16,19,21,22,25,33,34,37,41} to generate the cell body and nucleus geometries. We first select an image of a single cell. The cell and nucleus are segmented separately by isolating the images corresponding to the appropriate fluorescence channels. For each channel, we normalize the images to the maximum intensity so that all the values are between 0 and 1. We then select all voxels having an intensity above an empirically determined threshold (0.25 for the nucleus, and 0.05 for the cell body). The unselected voxels are set to 0 and the selected voxels are set to 1 to create a black and white image. The gaps in the interior of the white voxels are filled. All objects lying along the border of the image are removed. The image is smoothed with a diamond shaped erosion element. Finally, since the initial image contained only one cell and nucleus, we select the connected component in the processed image which contains the most voxels. For the cell body, we also fill downward to ensure that there is no empty space underneath the cell. This assumption is made because the cells used in this study were cultured in 2D conditions on rigid substrates. Following segmentation of the cell body and nucleus, a 2D frequency analysis combined with linear programming approach is used to generate the actin stress fiber network. This approach relies primarily on the directionality of the actin stress fibers within the image to generate a statistically representative synthetic actin stress fiber network for incorporation into the finite element model. The directionality of the actin network is calculated using a Fourier transform on the local neighborhood surrounding each voxel in the confocal image stack. The linear program uses the directionality throughout the image to compute the extent to which each prospective fiber should be present in such a way that it minimizes the discrepancy between the superposition of these fibers and the actual confocal image.⁴⁰ Fibers are chosen for inclusion if the linear program selects them with a weight above a specific threshold (40% in this study), empirically chosen to generate an FEM model of reasonable computational complexity; it can easily be adjusted to vary the total number of fibers used. It is important to emphasize that this approach does not attempt to faithfully reproduce the exact actin stress fiber network observed in the confocal images, but rather to generate a statistical representation of it.

In order to replicate physical conditions and ensure that the model can be solved successfully, both ends of all fibers are anchored at nodes along the cell periphery and fibers are not allowed to pass beyond either the cell or nucleus boundaries (i.e., fibers are not allowed to exist in whole or in part either outside of the cell or inside of the nucleus). The fibers are therefore contained completely within the cytoplasmic volume of the model cell. While this means that the fibers are not made explicitly to terminate at focal adhesion points, the fibers in the model will terminate very close to focal adhesions with high probability because the fibers are chosen by the linear program based on the cell’s image data and because the mesh density used is high (~1.5 nodes per µm²). Actin stress fibers in live VSMCs are known to anchor to the nucleus of the cell.²⁹ However, stress fibers in the current model were not anchored directly to the nucleus, but rather rely on the cytoplasm to transfer forces between the two. This lack of connectivity was not explicitly built into our actin fiber generation algorithm, but rather a byproduct of limiting the number of synthetic actin stress fibers used in the model and the fact that there were few if any actin stress fibers that terminated at the nucleus in the confocal images of the cells utilized for this study. However, the model building algorithm can be easily modified in the future to include fibers that anchor to the nucleus.

The resulting geometries are shown in Fig. 3, with the grey surface representing the cell periphery, the blue surface representing the nucleus periphery, and the green lines representing the actin stress fibers for each cell. Cell A was generated with 291 fibers, Cell B was generated with 361 fibers, and Cell C was generated with 615 fibers. Geometries were imported from MATLAB to MSC Patran 2010.2.3 64-bit (MSC Software Corporation, Santa Ana, CA) for mesh generation before being submitted to MSC Marc 2010.2 (MSC Software Corporation) for solving.

FIGURE 3.

Geometry of (a) Cell A (291 fibers), (b) Cell B (361 fibers), and (c) Cell C (615 fibers) as generated by MATLAB image processing: grey represents the cytoplasm, blue represents the nucleus, and green represents the actin fibers; units on all axes are in micrometers.

The geometry created in MATLAB is composed of two-dimensional 3-noded triangle (Tria3) and one-dimensional 2-noded bar (“Bar2”) elements in 3D space and is imported into Patran via a session file written in MATLAB. The session file builds the mesh into Patran from the ground up by specifying the coordinates of each individual node and element. It automatically creates meshes constructed of 10-noded tetrahedral (“Tet10”, MSC Marc Element Type 127) elements for the nucleus and cytoplasm using the “TetHybrid” mesher, meshes the actin stress fibers with Bar2 (MSC Marc Element Type 9) elements, and constructs the spherical cap representative of the AFM probe directly above the highest node of the cell (Fig. 4). Positioning the AFM probe in such a manner typically ensures proper probe placement to match the experimental condition of indenting the cell at the approximate center of the cell.

FIGURE 4.

Mesh of (a) Cell A, (b) Cell B, and (c) Cell C with geometry of AFM probe shown in zero load state as displayed in Patran after importing from MATLAB; AFM probe diameter = 5 µm in all instances.

The numbers of elements and nodes used to construct each component of each of the cells in this study are presented in Table 1. All Bar2 elements were generated with average global edge length approximately consistent with the persistence length of actin filaments (15 µm).²⁷ Global edge lengths for all Tet10 elements were set to approximately 0.2 based on data from mesh convergence tests (not shown) through down-sampling of photomicrographs prior to geometry construction.

TABLE 1.

Number of elements and nodes in each cell component.

	Actin fibers		Nucleus		Cytoplasm
Cell name	Elements	Nodes	Elements	Nodes	Elements	Nodes
Cell A	2909	3200	1210	1211	8261	7695
Cell B	3609	3970	3407	3772	17,220	18,376
Cell C	6149	6764	1007	970	14,778	14,810

Material Models

As discussed above, three cells (labeled A–C) were modeled in this study. For each cell, the model was run using three different types of simulations (labeled I–III) (Fig. 5). Previous studies have shown VSMCs to exhibit viscoelastic behavior.¹¹ However, all materials were modeled as linearly elastic in this study for two reasons. The viscoelastic behavior of VSMCs has been shown to exhibit a linear behavior at two time scales (fast and slow).¹¹ We therefore chose to incorporate actin stress fibers with material properties that differed greatly from their surrounding materials to investigate whether or not this system would recreate the nonlinear mechanical response observed in previous studies. Linear elastic materials were also chosen for this study in order to minimize the computational complexity for this initial study of the model. All three simulation types for this model were constructed with a Poisson’s ratio of 0.49¹⁰ and a nucleus with an elastic modulus of 3.30 kPa.¹

FIGURE 5.

Material models of the three different simulation types used in this study. The no actin FEM (ST I) models a cell treated that has been with cytochalasin D and therefore has depolymerized actin. The discrete actin FEM (ST II) and the bulk material FEM (ST III) both model untreated cells. The discrete actin FEM (ST II) models actin fibers explicitly inside the cytoplasm. The bulk material FEM (ST III) models the cell cytoplasm including the actin as a single continuum.

For simulation type (ST) I, each modeled cell consists of a nucleus surrounded by soft cytoplasm devoid of actin stress fibers. This represents a cell treated with an actin depolymerizing agent such as cytochalasin D. Actin stress fiber depolymerization was accomplished by treating the cells with cytochalasin D using previously reported methods¹¹ prior to testing. An elastic modulus of 2.25 kPa is used for the cytoplasm in ST I. This value was obtained from Hertzian analysis (Eq. (1)) of AFM indentation data of live VSMCs with depolymerized actin stress fibers which were found to have an elastic modulus of 2.25 ± 0.80 kPa.¹¹ This type of simulation is similar in structure to those of earlier models^7,31,36 and is used here for verification that our model behaves in the expected manner upon the addition of actin stress fibers (Fig. 6). Of course, one of the limitations of using these experimental values as the “actin-free” cytoplasm elastic modulus is that when the cell is treated with cytochalasin D, there may be some remodeling of other non-actin cellular components, which may also affect the elastic modulus

FIGURE 6.

Effect of actin fibers on force-indentation curves of representative (a) live VSMCs with intact (Control) and depolymerized (CytoD) actin stress fibers and (b) simulated VSMCs with (ST II) and without (ST I) actin fibers.

The discrete actin FEM (ST II) is the primary focus of our study. For the discrete actin FEM (ST II), each modeled cell consists of a nucleus surrounded by a soft cytoplasm with individual actin stress fibers explicitly modeled inside the cytosolic space as shown in Figs. 3 and 4. An elastic modulus of 1.90 GPa is used for the actin stress fibers in the discrete actin FEM (ST II).²⁷ The elastic modulus for the cytoplasm in the discrete actin FEM (ST II) is identical to that in the no actin FEM (ST I) as it represents the cytoplasm with no polymerized actin. This type of simulation is similar in structure to the model developed by Slomka and Gefen.³⁵ The novel feature of the discrete actin FEM (ST II) that distinguishes it from its predecessors is that the actin stress fibers in the model are arranged so that they are representative of those observed in the confocal image of the particular modeled cell.

For the bulk material FEM (ST III), each modeled cell consists of a nucleus surrounded by a bulk material which represents both cytoplasm and actin stress fibers. The cytoplasm and actin stress fibers in the bulk material FEM (ST III) are modeled together as a single continuum material. The elastic modulus values used for the cytoplasm in the bulk material FEM (ST III) vary from cell-to-cell, and range from 2.21 to 3.31 kPa. The elastic modulus for each cell is based on the apparent elastic modulus estimate obtained via Eq. (1) from the AFM indentation data of that particular cell in vitro [Eq. (1) fit with an R² = 0.99 for all three cells for indentation of 0–200 nm]. This type of simulation is similar in structure to those of earlier models^7,31,36 and is used here for purposes of comparison.

The actin stress fibers in the discrete actin FEM (ST II) are represented in the modeling software as embedded truss elements (MSC Marc Element Type 9) which can be thought of as similar to rebar in reinforced concrete, and the cell body and nucleus are both constructed of Tet10 (10-noded tetrahedral) elements (MSC Marc Element Type 127). General beam elements were initially desired for this model, however, truss elements were chosen instead of general beam elements for actin stress fibers based on advice from engineers at MSC Software. Truss elements were recommended to mitigate issues with computational intractability using general beams, since truss elements are a more stable 1D embeddable element type. It should be noted that truss elements do have two primary limitations relative to general beam elements for this type of model. First, they only transmit forces axially and therefore do not exhibit any bending characteristics. Secondly, they are incapable of supporting prestresses. However, prestress values in actin stress fibers vary throughout the cell.^6,30 Accurately determining the exact prestress value for each fiber that would be needed for incorporation into the model would therefore be difficult even if prestress of the elements was supported by the software.

All pre- and post-processing of the model is performed in Patran 2010. However, Patran 2010 is not capable of embedding 1D elements into 3D solids as needed for a model of this nature. Therefore, the model was submitted to the Analysis Deck in Patran for the discrete actin FEM (ST II) simulations, and an INSERT card is manually written into the resulting.dat file in order to define the host and embedded elements before submission to MSC Marc for analysis.

Loads and Boundary Conditions

Two sets of boundary conditions are applied to the model to match experimental conditions. First, the bottom-most layer of nodes is fixed in position to represent the physical attachment of the cell to its substrate. Second, a rigid spherical cap is plunged into the deformable cell body in the negative 3 direction using contact parameters to represent the indentation of the AFM probe into the cell, as described in the Analytical Parameters section below.

Analytical Parameters

Analysis is performed using MSC Marc 2010.2. In order to simulate contact between the AFM probe and the cell, each component is defined as a separate frictionless contact body with the former being rigid and the latter being deformable. Global remeshing is utilized for the analysis, with a strain of 0.25 used as the threshold to trigger remeshing; however it should be noted that the 0.25 strain threshold is not reached in any of the simulations performed in this study. To match experimental conditions, the probe is prescribed a displacement of 1 µm into the cell in 25 nm increments at 0.5 µm/s. For all simulations, an implicit static solution strategy is employed, with non-linear and large displacement/large strains with no follower loads solution parameters utilized. The default bandwidth optimized multifrontal sparse solver is used for this study.

RESULTS

Sensitivity Analyses

Effect of Actin Stress Fibers

In order to determine if the addition of actin stress fibers to the model has a similar effect on the mechanical properties as the addition of actin stress fibers does to a live cell, the force-indentation curve of a control VSMC was compared against the force-indentation curve of a VSMC treated with cytochalasin D (Fig. 6a). A similar curve was generated for a representative cell using the no actin FEM (ST I) and the discrete actin FEM (ST II) as shown in Fig. 6b. Although the values of the two sets of data are different, the data and model curves show similar decreases in force with the removal of actin. In fact, when Eq. (1) is used to determine the relative elastic modulus of each data set, the untreated live VSMC is found to be 2.7 times stiffer than the VSMC treated with cytochalasin D at 200 nm indentation using Eq. (1) whereas the discrete actin FEM (ST II) cell is found to be 2.5 times stiffer than the no actin FEM (ST I) cell at the same indentation depth.

It is worth noting, however, that the indentation curve of the discrete actin FEM (ST II) cell is more similar in slope to its live cell counterpart than is the no actin FEM (ST I) cell. The most likely explanation for this phenomenon is that the nuclear mechanical properties were kept constant in the model and set to literature values. The literature value for the elastic modulus of the nucleus used in the no actin FEM (ST I) cell was almost 50% stiffer than the surrounding cytoplasm. This difference coupled with the placement of the probe above the approximate center of the nucleus could lead to the mechanics of the nucleus dominating the response of the model.

Fiber Number Sensitivity Analysis

Fiber density sensitivity analysis for the discrete actin FEM (ST II) was performed, with results of the model compared when varying numbers of synthetic actin stress fibers are utilized (Fig. 7). A diameter of 200 nm was used for all models in this analysis. The apparent elastic modulus was calculated using the Hertz Analytical Model (Eq. (1)) at an indentation depth of 200 nm, and values were normalized to the elastic modulus of the cytochalasin D analog model. As shown in Fig. 7, the elastic modulus of the cells did increase with addition of fibers. In fact, the force calculated at any specific indentation depth for a particular cell is always higher in the model which included fibers compared to that with no fibers. However, the increase in calculated elastic modulus with addition of fibers varied widely between the cells. For instance, Cell 2 had only a 2% increase in elastic modulus with the addition of 500 fibers. In contrast, Cell 1 had a 27% increase and Cell 3 had a 60% increase in elastic modulus with the addition of 500 fibers. There is a qualitative trend towards a higher elastic modulus with increasing numbers of fibers. However, while the initial addition of 100 fibers increased the elastic modulus of all the cells, the further addition of fibers did not always lead to an additional increase in elastic modulus (e.g., the elastic modulus for Cells 2 and 3 reached a plateau at 300 and 400 fibers, respectively, while the modulus for Cell 1 continued to increase with increasing fiber number for the range tested). This indicates that while addition of fibers does increase the elastic modulus, this effect may reach a plateau. This data suggests the hypothesis that this effect may due to the image-based sampling method used to construct the model, which results in the final geometric arrangement of the added fibers in the model. Our image analysis algorithm picks fibers to include in the model based on the calculated confidence that a particular fiber is observed in the image. The more fibers we force the model to include the more likely we will include fibers which have lower calculated image confidence. Our results indicate that these fibers are also less load-bearing then their higher confidence counterparts. While the initial 100 fibers selected for inclusion tend to bear significant loads (up to ~120 pN), fibers added beyond a certain threshold are less load bearing and do not contribute as strongly (~8 pN loads per fiber).

FIGURE 7.

Fiber number sensitivity analysis for the discrete actin FEM (ST II) using fibers of 200 nm diameter, (a) average elastic modulus results with values in normalized to the results for cytochalasin d analog models; error bars show 99% confidence intervals (n = 3) and (b) indentation curves from a single representative cell with varying numbers of fibers.

Fiber Diameter Sensitivity Analysis

Fiber diameter sensitivity analysis for the discrete actin FEM (ST II) was performed, with results of the model compared when varying diameters of synthetic actin stress fibers are utilized (Fig. 8). The number of fibers utilized was 500 for all models in this analysis. The apparent elastic modulus was calculated using the Hertz Analytical Model (Eq. (1)) at an indentation depth of 200 nm, and values were normalized to the elastic modulus of the model incorporating 7 nm diameter fibers (the diameter of a single f-actin filament²⁷). As shown in Fig. 8, no clear correlation between the diameter of fibers used in the model and the resulting elastic modulus of the cell was observed within the range of 7–250 nm diameter fibers. This suggests that the diameter of fibers used in the model is not a primary determinant of the elastic modulus of the model within the range of parameters tested here, which encompasses the 100 nm diameter of the fibers used in the discrete actin FEM (ST II) indentation studies.

FIGURE 8.

Fiber diameter sensitivity analysis for the discrete actin FEM (ST II) using 500 fibers, (a) average elastic modulus with values normalized to the results for models with fibers the diameter of a single actin filament; error bars show 99% confidence intervals (n = 3) and (b) indentation curves from a single representative cell with varying fiber diameters.

Indentation

The force-indentation curves of AFM nanoindentation and both the discrete actin FEM (ST II) and the bulk material FEM(ST III) cases for all 3 cells are shown in Fig. 9. The AFM curve is presented with error bars displaying the99%confidence interval (n = 5). Both the discrete actin FEM (ST II) and the bulk material FEM (ST III) cases match the experimental data very well (average R² > 0.99) up to an indentation depth of 250 nm and still match the experimental data reasonably well (average R² > 0.9) at an indentation depth of 500 nm, however there is a precipitous drop-off in the correlation between the experimental data and the model at or beyond 750 nm of indentation (average R² < 0.5). As such, it is not recommended that the current model be used to predict AFM nanoindentation data beyond 500 nm of indentation. As mentioned above, analyses were performed on the sensitivity of the model to the number of fibers and fiber diameter and neither was found to be a primary determinant of the results of the model, thus reinforcing the importance of the arrangement of the cytoskeleton.

FIGURE 9.

AFM and FEM force-indentation curves for (a) Cell A, (b) Cell B, and (c) Cell C; error bars display 99% confidence interval (n = 5). The discrete actin FEM (ST II), which models the actin specifically as separate elements, fit the data more closely than the bulk material FEM (ST III), which represented the cytoplasm and actin together as one bulk material. The discrete actin FEM (ST II) fit well (R² > 0.9) for indentation depths of up to 500 nm.

On average, the discrete actin FEM (ST II) matches the experimental data more closely (R² = 0.961 ± 0.040 up to 500 nm indentation) than the bulk material FEM (ST III) does (R² = 0.920 ± 0.089 up to 500 nm indentation). This difference is statistically significant (p = 2.70 × 10⁻¹⁵ ± 4.39 × 10⁻¹⁵) using two-tailed Student’s t test. This demonstrates that the cytoskeletal geometries generated by our novel image processing techniques are successful at replicating AFM nanoindentation experiments more accurately than traditional finite element model techniques. The apparent elastic modulus estimates of each cell based on the indentation curve obtained in the discrete actin FEM (ST II) are shown in Fig. 5.

The distribution of von Mises stresses throughout Cell B for the discrete actin FEM (ST II) is shown in Fig. 10 at a probe indentation depth of 1 µm. Forces for actin fibers were calculated using the stress reported by Patran and the cross-sectional area of the individual fiber itself (based on a 200 nm circular fiber diameter). For the cross-sectional area of nodes in the cytoplasm and nucleus, we calculated the distance to the nearest node in all elements sharing the node of interest. We then assumed the stress reported by Patran for that element to be distributed over a circle of that diameter. The maximum amount of force experienced by the cytoplasm and nucleus of the cell is 4.48 nN and the maximum amount of force experienced by the actin stress fiber network is 9.77 nN. At five randomly chosen points far from the indentation site (21.7 ± 6.1 µm), the average amount of force experienced by the cytoplasm of the cell is 4.62 ± 3.50 pN and the average amount of force experienced by the nearest actin nodes (432 ± 62 nm away from their respective cytoplasm nodes) is 52.0 ± 42.9 pN. While these values are hard to assess on their own since there are no experimental methods to measure intracellular stresses with which to compare them against, they do provide a promising demonstration of two different principles.

FIGURE 10.

Representative von Mises stress distribution (shown in Cell B for the discrete actin FEM [ST II]) at a probe indentation depth of 1 µm. More stress is exerted on the model actin fibers than the cell cytoplasm and nucleus. In addition, stresses are distributed throughout the cell away from the indentation site through the model actin fibers.

First, the fact that the average amount of force experienced by the actin stress fibers is 12.7 ± 10.5 times higher than in the average amount of force experience by the rest of the cell body suggests that the load exerted upon the cell is primarily distributed through the actin stress fibers, which matches physical expectations. Secondly, the forces are carried to many points within the cell that are far from the indentation site via the actin stress fiber network, matching observations of cell behavior obtained experimentally.²⁴ Of note, Fig. 10 shows an anisotropic distribution of stresses throughout the cell, emphasizing the importance of the geometric arrangement of the actin stress fibers within the cell on the mechanical characteristics of the cell.

DISCUSSION

The approach taken here represents a new method for the construction of a cellular mechanics model. Previous attempts to model the mechanical behavior of cells have focused primarily on the material parameters of the model rather than the arrangement of those materials to match experimental data. This model, however, exclusively utilizes material parameters taken from experimental data (either directly from the cell modeled or from literature values) and relies on the geometric arrangement of the structural components of the cell to provide the model with realistic results. It is important to note that the results above could be made to better match the experimental data by varying the various material parameters, as all material properties from the literature were presented in range form, or by fitting one or more of the material parameters to each cell’s experimental data instead of keeping all parameters fixed.

The major advantage of this new method of model construction is that it is, in theory, equally capable of modeling a wide variety of cells because the parameters for the material properties are of secondary importance to the arrangement of those materials. This potentially allows for results that may be predictive across samples and even across cell types. Additional advantages of this method of model construction over previous models are that it is entirely automated and that it is capable of incorporating cytoskeletal elements in a non-arbitrary manner.

Although this approach has some advantages over previous models, it does have some limitations due to the methods that the material parameters from the literature were obtained. The first of these factors is that the material parameters for actin stress fibers taken from the literature were based on experiments from single filaments of f-actin in dilute solution. Therefore, three assumptions were made for the actin stress fibers in our model. The first is that actin stress fibers (which are composed of many actin filaments) have the same tensile modulus as a single filament (~2 GPa observed,^8,15,27 1.9 GPa in our model), which is highly unlikely. Since the stress fiber is composed of several filaments, it is likely that the tensile modulus value is actually higher. Secondly, because the fibers in the model are all independent of one another, the model relies on the assumption that the cross-linking of actin stress fibers observed in VSMCs does not alter the mechanical properties observed from actin filaments in dilute solution. Finally, prestress has been shown to play a critical role in the mechanical behavior of actin stress fibers,¹⁷ however due to limitations of MSC Marc we were unable to both account for prestress and embed the actin stress fibers into the cells of our model. However, while these assumptions do pose limitations, this is an important first step towards models of cells that can use observed cellular structures from images. Even with these limitations, our model has shown reasonable agreement with experimental data. In addition, it predicts known cell mechanical behaviors such as anisotropic stress distribution and the transfer of forces to points away from the indentation site via the actin stress fiber network, which is hard to predict without the inclusion of fibers in geometries that are representative of the native cell.

It is also noteworthy that the model proposed herein is similar in nature to the multiphasic models commonly utilized to characterize the mechanical responses of biological tissues. However, these are not appropriate for this model because, in the case of mixture models they assume that the material is a superimposed continua of two materials where each point within the material is occupied simultaneously by a material point of each phase and in the case of biphasic models such as poroelasticity, they assume that the mechanical properties are dominated by factors such as fluid flux and permeability. Mixture models could potentially be incorporated into a model of the type proposed here to describe the mechanics of the cytoplasm. However, to use these mixture models to describe the mechanics of the entire system would neglect the geometric arrangements that we have shown to play a crucial role the mechanics of cells.

Finally, the most important limitation of a finite element model of this type is that cells are living systems capable of active responses, which this type of model cannot replicate. We may be able to use this passive model to better distinguish the difference between the active and inactive responses of cells. Active responses could be incorporated in the future as the cell’s cytoskeletal arrangement could be made to change with time and in response to external stimuli.

Future models could improve upon the current study by expanding the model to different cell types and different physical experiments, incorporating appropriate models of viscoelasticity (e.g., the Generalized Maxwell model), incorporating techniques to visualize changes in the actin stress fiber network of live cells during physical experimentation, and/or incorporating additional structural components within the modeled cells.

The work presented in the current study represents an important step toward the ability to use finite element models to accurately predict the mechanical response of biological cells to AFM nanoindentation and provides a solid foundation from which to build even more representative models. It is capable of both construction of highly representative geometries produced in a completely automated manner and validation of its results using experimental data taken from the same cell that was modeled on the same day that the images of the cell were obtained. Such models could potentially be utilized to elucidate the mechanisms of mechanotransduction or even increase the speed and decrease the cost of drug development, tissue engineering, and regenerative medicine therapies, thereby possibly increasing the quality and longevity of life for medical patients in the future.

ABBREVIATIONS

AFM

Atomic force microscope

CytoD

Cytochalasin D

FEM

Finite element model

ST

Simulation type

VSMC

Vascular smooth muscle cell