Cancer cell mechanics with altered cytoskeletal behavior and substrate effects: A 3D finite element modeling study

Journal of the Mechanical Behavior of Biomedical Materials xxx (xxxx) xxx–xxx

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Cancer cell mechanics with altered cytoskeletal behavior and substrate effects: A 3D finite element modeling study ⁎

Dinesh R. Katti , Kalpana S. Katti Department of Civil and Environmental Engineering, North Dakota State University, Fargo, ND 58108, USA

A R T I C L E I N F O

A B S T R A C T

Keywords: Finite element modeling Cancer cell Cytoskeleton Substrate

A robust computational model of a cancer cell is presented using finite element modeling. The model accurately captures nuances of the various components of the cellular substructure. The role of degradation of cytoskeleton on overall elastic properties of the cancer cell is reported. The motivation for degraded cancer cellular substructure, the cytoskeleton is the observation that the innate mechanics of cytoskeleton is disrupted by various anti-cancer drugs as therapeutic treatments for the destruction of the cancer tumors. We report a significant influence on the degradation of the cytoskeleton on the mechanics of cancer cell. Further, a simulations based study is reported where we evaluate mechanical properties of the cancer cell attached to a variety of substrates. The loading of the cancer cell is less influenced by nature of the substrate, but low modulus substrates such as osteoblasts and hydrogels indicate a significant change in unloading behavior and also the plastic deformation. Overall, softer substrates such as osteoblasts and other bone cells result in a much altered unloading response as well as significant plastic deformation. These substrates are relevant to metastasis wherein certain type of cancers such as prostate and breast cancer cells migrate to the bone and colonize through mesenchymal to epithelial transition. The modeling study presented here is an important first step in the development of strong predictive methodologies for cancer progression.

1. Introduction Extensive recent studies in the theoretical, computational and experimental evaluation of the mechanical behavior of cellular systems and their relationship to health and disease have been reported. These present an extraordinary opportunity to develop novel detection capabilities for diseases such as cancer and malaria (Suresh, 2007a; Dao et al., 2005; Suresh, 2006; Yallapu et al., 2015). Qualitative and quantitative or semi-quantitative relationships between structure-mechanics-function and disease (Suresh, 2007a; Suresh, 2007b) are explored showing great promise in understanding the connections between underlying cellular and molecular scale mechanisms and disease progression, providing the possibility of novel diagnostic tools and intervention technologies. Cellular shapes and mechanisms of cellular interactions with extracellular environments influence important cellular processes such as proliferation, differentiation, and apoptosis (Keren et al., 2008; McBeath et al., 2004; Zemel et al., 2010; Huang and Ingber, 2000). In particular, the cytoskeleton plays an important role in the cellular response to external stimuli (Fletcher and Mullins, 2010). Microtubules, intermediate filaments, and actin filaments are important structural components of the cytoskeleton. These

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components are also intimately engaged in biological processes. Actin filaments form the basis of filopodial protrusions and support the leading edge in cells which are important for cell motility and are associated with cellular shape changes (Fletcher and Mullins, 2010). The role of intermediate filaments is primarily structural. The diameters of these are in between those of microtubules and actin filaments. Structural details of these components are described in the work of Suresh (2007a). The elastic properties of cancer cells in in vitro studies indicate a much lower value of elastic modulus than healthy cells (Suresh, 2007b). These experiments done using AFM force curves analysis are in agreement with tests done on biopsies of tissue from cancer patients (Cross et al., 2007).Inherently motivated by their potential rewards in enhancing cancer detection, these studies, primarily through AFM force curves have initiated a new era in the possibilities of novel methods of cancer detection and also the characterization of cancer subtypes (Zeng et al., 2016). The anticancer drug and moiety discoveries are also benefitting from their evaluation of anticancer effects of these moieties on cancer cells (Zhao et al., 2013; Saab et al., 2013) The relationship between metastasis and mechanical compliance remains rather unclear (Bastatas et al., 2012). Extracellular biochemical signals translate to mechanical forces in

Corresponding author. E-mail address: [email protected] (D.R. Katti).

http://dx.doi.org/10.1016/j.jmbbm.2017.05.030 Received 24 March 2017; Received in revised form 19 May 2017; Accepted 22 May 2017 1751-6161/ © 2017 Elsevier Ltd. All rights reserved.

Please cite this article as: Katti, D.R., Journal of the Mechanical Behavior of Biomedical Materials (2017), http://dx.doi.org/10.1016/j.jmbbm.2017.05.030

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living tissues and cells are constantly faced with a variety of mechanical loading conditions. The interactions of cells with the extracellular components influences cellular morphology resulting in changes to cellular structure (Ingber and Tensegrity, 2003). Cellular components have been known to contribute to cellular mechanics. In particular, the cytoskeleton plays a key role in cell mechanics, function, differentiation, locomotion, etc. (Suresh, 2007a; Suresh 2007b; Fife et al., 2014; McKayed and Simpson 2013). The cytoskeleton is an important structural member contributing to mechanics of cell. The cytoskeleton is often described as the response unit responsible for altering cellular mechanical behavior in response to events of extracellular nature and undergoes remodeling through a systematic remodeling of molecular characteristics of the cytoskeleton. Hence, cytoskeleton disruption is often targeted by drugs (Jordan and Wilson, 1998; Haga et al., 2000; Rotsch and Radmacher, 2000). The changes to cytoskeleton are often captured as changes to genes through gene expression studies (Janmey, 1998). Two of the cytoskeleton filaments. microtubules and actin filaments, participate in important cellular functions such as cell division, cell signaling, and motility. These are often considered the most important contributors to the elastic response of the cell (Sato et al., 1990; Wang, 1998; Tseng and Wirtz, 2001). Reorganization of these filaments is often the targeting activity of anti-cancer drugs (TerHaar et al., 1997; Jonnalagadda et al., 1997; Kowalski et al., 1997). In light of this fact, several recent studies have attempted to build computational models of human cells incorporating various elements of cellular substructure and its properties (Barreto et al., 2013; Xue et al., 2015). Computational studies using finite element modeling have been used to model a single cell (Barreto et al., 2013; Vaziri and Gopinath, 2008; Mijailovich et al., 2002; Viens and Brodland, 2007). These studies often investigate the role of external mechanical stimuli. Membrane thickness, cytoskeletal density, and extent of loading is varied parametrically. Biological behaviors such as cellular ageing are also investigated through the finite element models (Xue et al., 2015). Many of the finite element models built for the cells are simplistic and make large approximations of cellular substructures and properties (Fallqvist et al., 2016).Recent studies also dissect the specific contribution of some of the individual cytoskeletal components such as actin and vimentin intermediate filaments (Gladilin et al., 2014). Experimental evidence indicates that the quantitative contribution of these elements to elastic behavior of the cell is often cell type dependent (Grady et al., 2016) which further increases the complexity of evaluation of cell mechanics during disease progression. Thus this work attempts to include the various evaluations of mechanical properties of cancer cellular substructure from literature into the development of a robust finite element model that accurately captures nuances of the various components of the cellular substructure. In addition, the mechanical behavior of cells is often influenced by the substrate and hence an evaluation of mechanics of cancer cell on various substrates is also attempted. In particular, during cancer metastasis (Hanahan and Weinberg, 2011) cells migrate to environments remote to the original location, such as breast cancer cells migrating to the bone. Hence mechanics of cancer cells during this stage of metastasis is influenced by their behavior on osteoblastic cells. Similarly, recent attempts at developing cancer tumor models in vitro using 3D scaffolds (Katti et al., 2016; Adjei and Blanka, 2015; Subia et al., 2015; Zhu et al., 2015) suggest a need for evaluating the mechanical behavior of cancer cells on nanocomposite biomaterials. Nanoindentation experiments are important tools to evaluate cell mechanics. In this work, the results of simulations of nanoindentation on cancer cell using flat tip nanoindentor to evaluate the effect of cytoskeletal degradation and substrate stiffness on the force displacement behavior of the cell are described.

Fig. 1. Scanning electron micrograph of breast cancer cell seeded on the nanoclayhydroxyapatite-PCL composite scaffold.

2. Materials and methods 2.1. Cancer cell model construction A three dimensional finite element model of a cell having dimensions similar to that of breast cancer cells observed in the laboratory (Fig. 1) is constructed. The 5 µm diameter cell consists of all major components identified as contributing to the mechanics and cell conformation including the cytoskeleton (actin filaments, intermediate filaments, and microtubules), cytoplasm, cell membrane, and nucleus (Fig. 2). The three dimensional geometry of the cell membrane, cytoplasm and nucleus are created in Solid Works™ software and exported to finite element pre-processing software MSC/Mentat™. The geometry of the cytoskeleton elements is created in the Mentat software. Since the cytoskeleton plays a critical role in defining the shape of the cell, the actin filaments and the intermediate filaments have circular conformations and symmetric to mimic the initial 3D spherical shape of the cell. The microtubules are hollow tube strut like structures having a diameter of 25 nm and wall thickness of 7 nm emerging from the cytoskeleton surrounding the nucleus and extending through the cytoplasm to the cytoplasm-cell membrane interface. The microtubules are connected to the other cytoskeletal subunits that prevent buckling and allow for the microtubules to carry relatively larger compressive forces. The intermediate filaments play a tension bearing role. The intermediate filaments, having a diameter of 10 nm form a ring around the nucleus and extend to the cytoplasm-cell membrane interface and support the plasma membrane. At the cytoplasm-cell membrane interface, the intermediate filaments have a circular conformation to mimic

Fig. 2. 3D Finite element model of the 5-micron diameter spherical cancer cell. The cutout shows the various components of the cell.

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the interface. The actin filaments have a smaller diameter of 6 nm and also have a circular conformation. Focal adhesion of the actin to the cell membrane and the cell is created using the links feature of the Finite Element Method that allows for creating rigid or deformable connections between adjacent components via the nodes. The cytoskeleton itself is modeled as a contact unit that is “glued” to the cytoplasm with adhesion values that can be input based on the interface properties of cytoskeleton-cytoplasm. The cell membrane is made of hex8 elements (eight-node, isoparametric, arbitrary hexahedral elements) 10 nm thick. These elements are excellent for contact analysis. The cytoplasm and nucleus are composed of isoparametric three-dimensional tetrahedron (tetra4) elements. The actin filaments and intermediate filaments are composed of solid cylindrical beam elements having a diameter of 6 nm and 10 nm respectively. The microtubules are made of cylindrical hollow tube elements with a diameter of 25 nm and wall thickness of 7 nm (Suresh, 2007a). The cell model consists of 82,024 elements and 23,497 nodes. The flat punch nanoindenter tip is modeled as a flat rigid solid geometric entity with a node attached to the center for application of force and evaluation of displacement. In the cases of rigid substrate simulation, a flat rigid solid geometric entity is used beneath the cell. Both entities are modeled as contact bodies. To model the substrate with different stiffness, a deformable contact body, 5 µm thick is constructed with fixed nodal displacements imposed at the bottom of the substrate. The mesh of the deformable substrate consists of 29,570 3D hex8 elements (eight-node, isoparametric, arbitrary hexahedral elements). The cytoskeleton is dynamic during the lifecycle of the cell. In this paper, the structural features are maintained and cytoskeletal degradation is simulated by changes to the mechanical properties of the cytoskeleton elements.

culture (Katti et al., 2016; Yoshii et al., 2011; Lin and Chang, 2008; Elliott and Yuan, 2011). Further, 3D in vitro models allow potentially cost effective and relatively less time-consuming studies of cancer tumors without the complexity of in vivo models. Recent studies also show advantages of bone-mimetic 3D scaffolds for evaluation of metastasis (Katti et al., 2016). These bone mimetic systems provide an added advantage of evaluation of cancer in its metastatic stage. Hence many model substrate materials are thus considered. The parameters used in the deformable substrate analysis are shown in Table 2. A value of 0.30 is used for the Poisson's ration (ν) unless otherwise noted in the table. 2.3. Simulation of mechanics of cancer cell The simulations were conducted using the general purpose finite element software MSC/MARC™. The software has robust contact analysis capability and scales well for conducting simulations on massively parallel platforms. The simulations were carried out as quasi static loading analysis in force control mode. In the case of continuously increasing loading, the loading rate was kept constant. In the case of loading-unloading paths, a trapezoidal loading function was used. In all cases, the load applied by the flat indenter is applied on the top of the cancer cell by a rigid contact body mimicking the flat indenter tip. Contacts between all contact bodies are established during the first increment, followed by application of the load increments. An adaptive stepping criterion is used for applying load increments. The solution control is done by the full Newton-Raphson scheme. All computations are conducted in extended precision mode. Simulations were conducted on 4-core Intel Xeon 1.6 GHz processor with a computation time of between 10 and 12 cpu hours per simulation. Force, displacement and rotation values are computed at all nodes at each load increment, and the stresses and strains are also calculated at each load increment. The maximum force application is limited so at to not exceed about 120 nm of indentation depth on the top of the 5 µm diameter cell. The vertical deformation of the cell does not exceed 2.5 percent of the cell diameter. Three sets of simulations were conducted, 1) continuously increasing load (monotonic loading) applied on the top of the cancer cell and three levels of cytoskeleton mechanical degradation, 2) loading-unloading load path applied on the top of the cancer cell and three levels of cytoskeleton mechanical degradation and 3) loading-unloading load path applied on the top of the cancer cell with undegraded cytoskeleton and placed on a deformable substrate of various stiffness values.

2.2. Material properties The magnitudes of material parameters for various components of the cell are based on parameters reported in the literature and are shown in Table 1. In the case of simulations with the mechanically degraded cytoskeleton (50 and 90 percent), the elastic moduli of the cytoskeleton elements, actin filaments, intermediate filaments and the microtubules are reduced by 50 percent and 90 percent of the values shown in Table 1. Various substrate materials are relevant for the evaluation of mechanics of the cancer cell. Cancer metastasis involves migration of the cancer cells to a remote location (Hanahan and Weinberg, 2011) such as bone and hence we have considered a range of substrates that have elastic properties over a wide range such as those of bone, osteoblasts, and hydrogels. Further, recent studies evaluate the use of 3D scaffolds for evaluation of cancer progression. These scaffold-tumors present a unique 3D in vitro method to evaluate cancer. The recent introduction of 3D in vitro tumor model has brought considerable improvements in cancer research to overcome the limitations of 2D

3. Results 3.1. Monotonic loading of the cancer cell The force versus displacement responses of the cancer cell subject to monotonic loading by a flat rigid contact are plotted in Fig. 3. The three curves plotted are for the three levels of cytoskeleton mechanical degradation, undegraded, elastic modulus of the cytoskeleton degraded by 50 percent and elastic modulus of the cytoskeleton degraded by 90%. In all the three cases, the cell is placed on a rigid substrate. The plots show a nonlinear response, with the plot showing a concave shape and with significant deformations under the relatively small magnitude of external force during the early stage of loading. The deformation appears to be mainly the deformation of the cell membrane which is 10 nm thick. However, the mechanical integrity of the cytoskeleton also influences the early deformation which was found to be 8.39 nm for the case where cytoskeleton is intact, 9.13 nm for elastic modulus of the cytoskeleton degraded by 50 percent and 11.76 nm elastic modulus of the cytoskeleton degraded by 90% for a load of 0.002nN. As shown in Fig. 3, significant differences are observed in the force-displacement responses of the cell with various levels of degradation. The F-D response of the cell with increasing degradation is significantly softer. For example, for a displacement of about 109 nm, the corresponding

Table 1 Properties of cell components used in the finite element model of breast cancer cell. Elastic Modulus

Poisson's Ratio

Limiting Strain

Reference

Cell Membrane

1.8 KPa

0.30

*

Cytoplasm

0.25 KPa

0.49

*

Nucleus

1.0 KPa

0.30

*

Actin Filaments Intermediate Filaments Microtubules

1.0 GPa 1.0 GPa

0.30 0.30

20% 20%

(Yokokawa et al., 2008) (Barreto et al., 2013) (Barreto et al., 2013) (Suresh, 2007a) (Suresh, 2007a)

1.9 GPa

0.30

60%

(Suresh, 2007a)

*Unknown.

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Table 2 Elastic moduli of substrate materials used. Substrate material

Elastic Modulus

Reference

Elastic Modulus Parameters used in Simulations

Human Osteoblasts

1–12 MPa

(Khanna et al., 2011)

Nanocomposite Bone tissue engineering scaffold (ChiPgaHAP) substrate Polycapralactone Nanocomposite of polycaprolactone with HAP and nanoclay Polymer modified HAP nanoparticles Hydroxyapatite nanoparticles untreated Human bone

4–8 GPa 0.4 GPa 2.8 to 1.7 GPa 10–40 GPa 147 GPa 13–190 GPa

(Khanna et al., 2010) (Ambre et al., 2015) (Ambre et al., 2015) (Khanna et al., 2009) (Snyders et al., 2007) (Gu et al., 2016)

1 MPa (ν= 0.45) 4 GPa 0.4 GPa 1.7 GPa 10 GPa 147 GPa 190 GPa

Cartilage and soft tissue engineering hydrogel scaffolds PVA with alginate microspheres

0.14-0.04 MPa (Poisson's ratio(ν): 0.29-0.22)

(Scholten et al., 2011)

10 kPa (ν = 0.4)

Cartilage and soft tissue engineering hydrogel scaffolds of Chitosan modified PVA

0.6524 kPa – 69.26 kPa (Poisson's ratio 0.12–1.33)

(Lee et al., 2009)

10 kPa (ν = 0.4)

degradation of the elastic modulus of the cytoskeleton and 90% degradation of the elastic modulus of the cytoskeleton. As in the monotonic condition, the maximum load for a similar magnitude of deformation decreases with increased cytoskeleton degradation. The unloading portions of the F-D curves show noticeable differences. In the case of the intact cytoskeleton, the unloading curve does not coincide with the loading portion of the curve and meets the displacement axis when completely unloaded indicating the irrecoverable displacement of 10.87 nm. However, in the case of cytoskeleton elastic modulus degraded by 90%, the unloading curve, and the loading curves, although not coinciding, are significantly closer indicating a more elastic response compared to the cell with intact cytoskeleton F-D response. Snapshots of the cell cytoskeleton before loading and after complete unloading are shown in Fig. 7. The colors of the symbols on the cytoskeleton elements represent magnitudes of displacement indicated in the legend in the figure. Before the load is applied, as expected, the various locations on the cytoskeleton have an identical displacement of zero as indicated by all red symbols in Fig. 7a. Fig. 7b shows a snapshot of the cytoskeleton for intact cytoskeleton cell. The purple and blue colored symbols on the cytoskeleton indicate residual displacements of the cytoskeleton after complete unloading of the cell. Portions of the cytoskeleton have no residual displacements. Thus, the residual displacements in the cytoskeleton are not uniform. Fig. 7c shows a snapshot of the cytoskeleton where the elastic modulus of the cytoskeleton has been degraded by 90 percent. The snapshot also indicates residual displacements of the cytoskeleton. However, the displacements appear to be more uniform over the entire cytoskeleton.

Fig. 3. Force-displacement response of the cancer cell obtained from FEM simulations under increasing load mimicking nanoindentation tests using rigid flat punch indenter for different levels of mechanical property degradation of the cell cytoskeleton.

force for the cancer cell nanoindentation simulation for intact cytoskeleton, cytoskeleton degraded by 50 percent and cytoskeleton degraded by 90 percent are about 0.1 nN, 0.072 nN and 0.041 nN respectively. Fig. 4a is a snapshot of the cytoskeleton of the cell before deformation. Snapshots showing changes to the cytoskeleton are presented in Fig. 4b and c and represent cytoskeleton structures at maximum loads for intact cytoskeleton and cytoskeleton mechanical properties degraded by 90 percent respectively. Visual observations of the structures indicate more cytoskeleton conformation changes in the case of intact cytoskeleton compared to the degraded cytoskeleton. In Figs. 4d and e, contour bands of displacements through a cross-section of the cell are shown for the cases of the intact cytoskeleton and cytoskeleton mechanical properties degraded by 90 percent respectively. These snapshots are captured at maximum applied force. The blue to yellow color range represents lower to higher displacements. As expected, the larger magnitude of displacements are observed in the cytoplasm in the case of the degraded cytoskeleton. However, in the degraded cytoskeleton case, the displacements are more uniformly distributed across the cytoplasm than in the case of the intact cytoskeleton.

3.3. Effect of substrate stiffness Live cell indentation is conducted on cells growing on a variety of substrates. The mechanical behavior of the substrates can vary from stiff to soft. FEM simulations of nanoindentation experiments on cells placed on substrates with a wide range of elastic modulus values are performed. The substrates considered in the simulations are listed in Table 2 along with the corresponding elastic modulus values used in the simulations. The substrates listed have been used by researchers to grow cells on them for tissue engineering applications. The purpose of these simulations is to understand the effect of the elastic modulus of the substrate on the F-D response of the cancer cell placed on the substrate. The elastic modulus values for the substrates used in the simulations vary from 190 GPa to 10 KPa, a six order spread in the elastic modulus values. The simulations were conducted in force control mode using a trapezoidal loading function described earlier with a maximum load of 0.07 nN for all the simulations. Simulation with the rigid substrate is conducted for comparison. The F-D responses of the cancer cell on various substrates are plotted in Fig. 8. The shape of the loading portions of the F-D curves of

3.2. Loading-unloading path The loading-unloading path used in these simulations have a trapezoidal function (Fig. 5) similar to commonly used paths in nanoindentation experiments. The loading is applied by a rigid contact solid, and the cell is placed on a rigid contact body mimicking a rigid substrate. The F-D curves are shown in Fig. 6 for the three levels of mechanical degradation of the cytoskeleton, no degradation, 50% 4

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Fig. 4. (a-c) Snapshots of cell cytoskeleton, (a) before application of indentation load, (b) for undegraded cytoskeleton after approximately 100 nm of indentation showing significant deformation and (c) for cytoskeleton after approximately 100 nm of indentation where mechanical properties are degraded by 90%, showing less deformation compared to the undegraded case; (d-e) displacement contour bands inside the cell at peak loads for, (d) undegraded cytoskeleton and (e) cytoskeleton degraded by 90%.

Fig. 5. Force-time function used in the FEM simulations to mimic force path typically used in nanoindentation experiments.

Fig. 6. Force-displacement response from FEM simulation of the cell subjected to loading and unloading path for various magnitudes of mechanical degradation of the cytoskeleton. The simulations indicate more elastic response with increasing cytoskeleton degradation.

the cancer cell for all underlying substrates are concave. However, differences are observed for the magnitudes of displacements at the peak load, the shape of the curve at the beginning of unloading and the residual displacements at the end of unloading. Based on the observations of the shapes, displacements of the peak load and the residual displacements of the F-D curves, the curves can be classified as being part of four “clusters.” The first cluster comprises of the F-D curve of the cancer cell on a rigid substrate. The peak displacement value at a load of 0.07 nN is 77.35 nm, and the residual displacement at the end of unloading is 10.87 nm. The second cluster of F-D curves is for cancer

cell on substrates with elastic modulus values ranging from 190 GPa to 0.40 GPa. The peak displacements values for the F-D curves in this cluster fall in a narrow range of displacements between 81.51 nm to 82.7 nm. The residual displacements at the end of unloading are once again in the narrow range between 17.81 nm and 19.35 nm. The loading and unloading curves in this cluster have a similar shape and to a large extent coincide. The third cluster contains the F-D curve for a substrate with an elastic modulus of 1 MPa. The differences between 5

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Fig. 7. Snapshots of the cytoskeleton showing the magnitude of residual displacement in the cytoskeleton for (a) initial condition before indentation force is applied to the cell, (b) at the end of unloading for undegraded cytoskeleton and (c) at the end of unloading for mechanical properties of the cytoskeleton degraded by 90%. (b and c) show residual displacements in the cytoskeleton as indicated by colored symbols on the skeleton representing displacements different from the initial condition where the displacements are zero. Figure b shows residual displacements are less uniform and larger in magnitude compared to displacements in figure c.

loading and unloading portions of the F-D curve are close to one another. Fig. 9 shows the plots of F-D curves on rigid, and substrates with elastic modulus of 190 GPa and 10 KPa to clearly delineate the differences. In Fig. 10a, the F-D curve for the force exerted by the cancer cell on the surface of the substrate immediately below the cell as a consequence of loads applied by the indenter on the top of the cell (Fig. 8) is shown. The elastic modulus of the substrate is 1 MPa. Since the substrate is modeled as a deformable contact body that is elastic, the force-deformation response is linear during loading and unloading. In order to compare the substrate F-D responses to cell loading, the forcedisplacement curves are plotted with displacement in the log scale (Fig. 10b). The curves shown contain both loading and unloading paths, and for each substrate type, the loading and unloading portions of the F-D curves coincide. The maximum surface displacements are six orders of magnitude different, mimicking the range of elastic moduli of the substrate. Fig. 11 shows snapshots from the simulation. Fig. 11a and b show the cell on substrate model before and immediately after the maximum load is applied by the indenter on the top of the cell. Fig. 11c is the view from above the cell placed on the substrate at maximum load applied on the top of the cell. Fig. 11d shows the vertical displacement contour bands on the top of the substrate as a result of maximum load applied to the cell by the indenter. The maximum displacement occurs immediately beneath the tip of the cell, and the displacements become smaller in magnitude moving away from the contact. Fig. 11e and f show cross-section through the cell and the substrate beneath the cell when the cell is subject to maximum load by the indenter. The contour bands of vertical displacements in the figures show the extent of displacements in the substrate. In addition, the flattening of the cell at the cell-substrate interface can also be observed. Snapshots of the cytoskeleton on different substrates after the cell has been loaded to 0.07nN and completely unloaded are shown in Fig. 12. The colored arrows superimposed on the cytoskeleton represent displacement vectors with colors defining the magnitudes indicated in the legend and the length of the arrows providing a visual representation of relative magnitudes of the displacement vectors. The displacements shown are the residual displacements at various points in the cytoskeleton. Fig. 12a,b and c are snapshots of the cytoskeleton of cells placed on rigid, E=190 GPa and E=10 KPa substrates respectively. The F-D curves for the three cases are shown in Fig. 9, and the residual displacements of for the cells placed on rigid, E=190 GPa and E=10 KPa substrates are 10.87 nm, 19.35 nm and 5.89 nm respectively. The comparison of Fig. 12a,b and c show that the residual displacements of the cytoskeleton for the cell placed on the substrate with E=190 GPa are the largest, followed by the residual displacements in the cytoskeleton for the cell on a rigid substrate with the

Fig. 8. Force-displacement response from FEM simulation of rigid flat punch nanoindentation of a cell placed on a substrate of various stiffness.

Fig. 9. Force-displacement response from FEM simulation of rigid flat punch nanoindentation of a cell placed on rigid, stiff and soft substrates.

the curve in this cluster and the second cluster are, a larger displacement at the maximum loading which is found to be 87.67 nm and the shape at the beginning of unloading. The fourth cluster comprises of the F-D curve of cancer cell on a substrate with an elastic modulus of 10 KPa. The displacement for the maximum force of 0.07 nN is found to be 91.26 nm, significantly higher than F-D curves in clusters 1 and 2. The residual displacement at the end of unloading is 5.89 nm, a significantly smaller magnitude of residual displacement compared to F-D curves in clusters 1, 2 and 3. Also, the shape of the loadingunloading curve is different from the cluster 2, and 3 curves since the

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Fig. 10. Force – surface displacement of the substrate beneath the cell during nanoindentation of the cell, (a) Force – surface displacement of osteoblast cell beneath the cancer cell, (b) comparison of Force – surface displacement of the substrate beneath the cancer cell with the varying stiffness of the substrate. Note the displacements are in logarithmic scale for ease in comparison.

Fig. 11. (a) Snapshot of 3D FEM model of cell on a substrate, (b) deformed at a load of 0.07nN on the cell placed on hydrogel substrate, (c) top view of cell on substrate, (d) contour bands of vertical deformation of the hydrogel substrate beneath the cell, (e-f) cross-sections showing displacement contour bands in the cell and hydrogel substrate at a load of 0.07 nN on the cell.

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Fig. 12. Snapshots of the cytoskeleton of cells on three substrates loaded to 0.07 nN and completely unloading. Displacement vectors are superimposed. The substrate (a) rigid, (b) E=190 GPa and (c) E=10 KPa. Significant differences are observed in the residual displacements in the cytoskeleton of the cell on the three substrates after unloading.

substrate condition. In the case of much softer substrates, E=1 MPa and E=10 KPa, the maximum displacements were 10 nm and 14 nm respectively more than the displacements at the maximum load for the cell on a rigid substrate. Also, for the case of a substrate with E=10 KPa, the residual deformation of the cell is about half the residual deformation observed for the case of the rigid substrate. The effect of mechanical properties of the substrate beneath the cell do influence the F-D response of the cell and would be an important consideration when comparing nanoindentation test results of a cell on substrates with disparate mechanical properties. This is in addition to the consideration that substrate mechanical properties also affect cell adhesion on the surface and the cell mechanics. The maximum substrate deformation is quite small compared to the maximum deformation of the cell for all cases with the exception of the substrate with E=10 KPa.The magnitude of substrate deformation does not seem to directly contribute to the differences in the maximum cell deformation for stiffer substrates used in the simulations where the elastic modulus values ranged from E=190 GPa to E=0.40 GPa, and the observed maximum deformations were 7.58E-7nm and 3.6E-4nm respectively and the difference in the maximum cell displacements with respect to the rigid substrate were 4 to 5 nm, about four to seven orders of magnitude higher. In the case of a substrate with E=1 MPa, the maximum substrate deformation is 0.132 nm while the difference in the maximum cell displacement with respect to the rigid substrate is about 10 nm. In addition, the residual displacements in the cell are 64 to 78% larger than the residual displacements for the cell on a rigid surface, although there are no residual displacements in the rigid substrate at the end of unloading. However, in the case of the soft substrate where E=10 KPa, the maximum substrate displacement is 13.12 nm while the difference in the maximum cell displacement with respect to the cell on a rigid substrate is about 14 nm which are comparable values. However, the residual displacements in the cell are about 50% of the cell residual displacements for the rigid substrate. The observations made in the paper are for the relatively small magnitude of vertical displacement, less than 120 nm or less than 2.5% of the cell height. It appears that the deformations in the substrate beneath the cell during nanoindentation alter the arrangement of cytoskeleton elements during loading leading to the differences in the F-D response of the cell placed on substrates with different elastic moduli.

smallest magnitude of residual cytoskeleton displacements exhibited in the cell on a substrate with E=10 KPa. The trend of the magnitudes of the residual cytoskeleton displacements compares well with the trend of the residual displacements from the F-D curves, indicating the important role of rearrangement of the cytoskeleton elements during loading of the cell on cell deformation on substrates with various magnitudes of elastic modulus. It should be noted that all cells were loaded to 0.07 nN and then fully unloaded. 4. Discussion The significant changes in the F-D response of the cell are contributed by the relatively small amount of the cytoskeleton elements, about 0.24% of the cell volume in the present cell model. The differences would be significantly greater if the volume fraction of the cytoskeletal elements is larger. Depending on the cell type and other biological factors, the volume fraction of cytoskeletal elements of 20 percent is reported in the literature. The amount of degradation of individual elements such as actin, intermediate filaments and microtubules would not be uniform and would impact the F-D response. In the case of the mechanically degraded cytoskeleton, the cytoskeleton is more compliant and does not distort to the extent observed in the intact cytoskeleton case. In the degraded cytoskeleton case, the displacements are more uniformly distributed throughout the cytoplasm than in the case of the intact cytoskeleton. The residual displacements in the cytoskeleton after complete unloading of the cell are indicative of movement and rearrangement of the cytoskeleton in the cytoplasm matrix and contributes to the observed inelastic F-D response. The differences in the cytoskeleton displacements between the intact cytoskeleton and mechanically degraded cytoskeleton indicate that the mechanically degraded cytoskeleton because of the lower elastic modulus value is more compliant. It appears that although the material responses of the cell constituents are elastic, the structural rearrangement of the cytoskeleton elements introduces the inelastic response observed in the F-D curves and the behavior of mechanically degraded cytoskeleton is observed to be more elastic. The results also show that the mechanical properties of the substrate beneath the cell affect the F-D responses of the cell by varying amounts and are significantly different from the response exhibited by the rigid substrate. In the current simulations, the residual displacements of the cell with substrate elastic modulus ranging from 190 GPa to 1 MPa were 64 to 78% higher than the rigid substrate case. As shown in Fig. 8, for the same magnitude of the load, the maximum displacements observed for cells on deformable substrates were larger than the deformation of the cell over the rigid substrate. In the case of substrate elastic modulus range of 190 GPa to 0.40 GPa, the maximum displacements were four to five nanometers larger than the cell over rigid

5. Conclusions A 3D-finite element model of a cell is constructed that includes discrete cytoskeleton architecture, cytoskeletal elements and other constituents including the cell membrane, cytoplasm, and the nucleus. Flat punch tip nanoindentation experiments in force control mode are simulated to evaluate the force-displacement response of the cell. The 8

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vertical displacements are limited to less than 120 nm or less than 2.5% of the cell height. The overall conclusions from this study are: (a) The mechanical response of the cell is significantly influenced by the cytoskeleton, although it constitutes a small portion of the overall cell volume. (b) The mechanical degradation of the cytoskeleton dramatically affects the mechanical response of the cell. (c) The F-D response of the cell shows inelastic behavior even at a low magnitude of vertical displacement and can be attributed to movement and rearrangement of cytoskeleton elements during loading. (d) The F-D response of the cell with intact cytoskeleton has a larger inelastic response than the cell with the degraded cytoskeleton. (e) The deformations in the substrate beneath the cell during nanoindentation alter the arrangement of cytoskeleton elements during loading leading to the differences in the F-D response of the cell placed on substrates with different elastic moduli. The results confirm the important role of the cytoskeleton in cell mechanics and provide an insight into the mechanisms that influence the F-D response of cancer cell in nanoindentation experiments. The role of the nanomechanical behavior of cancer cells in the progression of the disease or rather the implications of nanomechanics in cancer progression is an emerging area of research that can be largely benefitted through development of mechanics based modeling of cancer cells. Acknowledgements Authors would like to acknowledge support from NDSU Grand Challenge Program for funding for Center for Engineered Cancer TestBeds. Authors would also like to acknowledge Computationally Assisted Science and Technology (CCAST) Center for providing computational resources at North Dakota State University. References Adjei, I.M., Blanka, S., 2015. Modulation of the tumor microenvironment for cancer treatment: a biomaterials approach. J. Funct. Biomater. 6 (1), 81–103. http://dx.doi. org/10.3390/jfb6010081. [PubMed PMID: MEDLINE:25695337]. Ambre, A.H., Katti, D.R., Katti, K.S., 2015. Biomineralized hydroxyapatite nanoclay composite scaffolds with polycaprolactone for stem cell-based bone tissue engineering. J. Biomed. Mater. Res. Part A 103 (6), 2077–2101. http://dx.doi.org/10. 1002/jbm.a.35342. [PubMed PMID: WOS:000354024100019]. Barreto, S., Clausen, C.H., Perrault, C.M., Fletcher, D.A., Lacroix, D., 2013. A multistructural single cell model of force-induced interactions of cytoskeletal components. Biomaterials. 34 (26), 6119–6126. http://dx.doi.org/10.1016/j.biomaterials.2013. 04.022. [PubMed PMID: WOS:000321079600003]. Bastatas, L., Martinez-Marin, D., Matthews, J., Hashem, J., Lee, Y.J., Sennoune, S., et al., 2012. AFM nano-mechanics and calcium dynamics of prostate cancer cells with distinct metastatic potential. Biochim. Biophys. Acta-General. Subj. 1820 (7), 1111–1120. http://dx.doi.org/10.1016/j.bbagen.2012.02.006. [PubMed PMID: WOS:000305366100038]. Cross, S.E., Jin, Y.S., Rao, J., Gimzewski, J.K., 2007. Nanomechanical analysis of cells from cancer patients. Nat. Nanotechnol. 2 (12), 780–783. http://dx.doi.org/10.1038/ nnano.2007.388. [PubMed PMID: WOS:000251456500015]. Dao, M., Lim, C.T., Suresh, S., 2005. Mechanics of the human red blood cell deformed by optical tweezers (vol 51, pg 2259, 2003). J. Mech. Phys. Solids. 53 (2), 493–494. http://dx.doi.org/10.1016/j.jmps.2004.10.003. [PubMed PMID: ISI:000226555700010]. Elliott, N.T., Yuan, F., 2011. A review of three-dimensional in vitro tissue models for drug discovery and transport studies. J. Pharm. Sci. 100 (1), 59–74. http://dx.doi.org/10. 1002/jps.22257. [Epub 2010/06/10], [PubMed PMID: 20533556]. Fallqvist, B., Fielden, M.L., Pettersson, T., Nordgren, N., Kroon, M., Gad, A.K.B., 2016. Experimental and computational assessment of F-actin influence in regulating cellular stiffness and relaxation behaviour of fibroblasts. J. Mech. Behav. Biomed. Mater. 59, 168–184. http://dx.doi.org/10.1016/j.jmbbm.2015.11.039. [PubMed PMID: WOS:000376693900015]. Fife, C.M., McCarroll, J.A., Kavallaris, M., 2014. Movers and shakers: cell cytoskeleton in cancer metastasis. Br. J. Pharmacol. 171 (24), 5507–5523. http://dx.doi.org/10. 1111/bph.12704. [PubMed PMID: WOS:000345965500004]. Fletcher, D.A., Mullins, D., 2010. Cell mechanics and the cytoskeleton. Nature 463 (7280), 485–492. http://dx.doi.org/10.1038/nature08908. [PubMed PMID: WOS:000273981100040].

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