A constriction channel analysis of astrocytoma stiffness and disease progression

Abstract

Studies have demonstrated that cancer cells tend to have reduced stiffness (Young's modulus) compared to their healthy counterparts. The mechanical properties of primary brain cancer cells, however, have remained largely unstudied. To investigate whether the stiffness of primary brain cancer cells decreases as malignancy increases, we used a microfluidic constriction channel device to deform healthy astrocytes and astrocytoma cells of grade II, III, and IV and measured the entry time, transit time, and elongation. Calculating cell stiffness directly from the experimental measurements is not possible. To overcome this challenge, finite element simulations of the cell entry into the constriction channel were used to train a neural network to calculate the stiffness of the analyzed cells based on their experimentally measured diameter, entry time, and elongation in the channel. Our study provides the first calculation of stiffness for grades II and III astrocytoma and is the first to apply a neural network analysis to determine cell mechanical properties from a constriction channel device. Our results suggest that the stiffness of astrocytoma cells is not well-correlated with the cell grade. Furthermore, while other non-central-nervous-system cell types typically show reduced stiffness of malignant cells, we found that most astrocytoma cell lines had increased stiffness compared to healthy astrocytes, with lower-grade astrocytoma having higher stiffness values than grade IV glioblastoma. Differences in nucleus-to-cytoplasm ratio only partly explain differences in stiffness values. Although our study does have limitations, our results do not show a strong correlation of stiffness with cell grade, suggesting that other factors may play important roles in determining the invasive capability of astrocytoma. Future studies are warranted to further elucidate the mechanical properties of astrocytoma across various pathological grades.

INTRODUCTION

Tumors derived from glial cells, known as gliomas, account for more than 70% of primary brain tumors.¹ Of the many types of gliomas, the most common form is astrocytoma—a tumor originating from astrocytes and their progenitor cells. Astrocytes are the most abundant cell type in the brain and support neural function.² The World Health Organization classifies astrocytoma into four grades based on histological and molecular characteristics of the tumor: grade I (pilocytic astrocytoma), grade II (diffuse astrocytoma), grade III (anaplastic astrocytoma), and grade IV (glioblastoma, GBM), with increasing grade corresponding to decreased expected outcomes.³ Grade I astrocytoma is slow growing and benign, occurring mostly in children and young adults. Treatment of grade I astrocytoma by complete surgical resection is typically curative, and grade I tumors only rarely progress to higher-grade tumors.^4,5 Grade II astrocytoma tumors do not have well-defined edges, and thus complete surgical resection is generally not possible. Grade II astrocytoma is generally slow growing and considered benign. Grade III astrocytoma is fast-growing and malignant. Glioblastoma (GBM), or grade IV astrocytoma, accounts for 82% of malignant gliomas.⁶ Glioblastoma is highly aggressive and tumors show significant mitotic activity, vascular proliferation, and necrosis; overall survival is only 1–2 years post diagnosis.⁶ Primary (de novo) GBMs account for the majority of astrocytomas, while a minority first begin as grade II or III tumors and typically progress over years.⁷ It is unclear what drives the progress of low-grade astrocytomas.⁸ Primary and secondary GBM are morphologically indistinguishable but arise through different pathways of genetic mutation.⁷

All grades of astrocytoma rarely metastasize (<2% of cases⁹) but typically remain within the central nervous system and are generally unifocal.⁶ Instead, astrocytoma of grades II through IV are characterized by significant infiltration of tumor cells into the brain at the tumor margin. Extensive infiltration of tumor cells makes complete resection impossible and limits radiation therapy efficacy.⁷ For example, 70%–90% of GBM tumors will reoccur within a 2-cm margin of the primary tumor.^10,11 Unlike many tumors that rely on intravascular and lymphatic migration, astrocytoma migration exploits the brain's natural structures and extracellular spaces.¹² Studies reveal that glioma cells actively seek out blood vessels and travel along the basement membrane, displacing astrocyte endfeet and disrupting the blood–brain barrier.¹³ Glioma cells also actively hijack white matter fiber tracks for migration through interstitial spaces of the parenchyma.¹³ To transit the brain's submicrometer spaces, astrocytoma cells employ several migratory behaviors including degradation of the extracellular matrix¹² and volume reduction¹⁴ to squeeze through constrictions. Because glioma cells infiltrate this mechanically challenging environment, it is reasonable to suspect that the intrinsic mechanical properties of the glioma cells change to enhance motility. The mechanical properties of astrocytoma, therefore, may provide insight into migration capability, migration rates, and deviation from the normal mechanical phenotype of healthy astrocytes.

The mechanical properties of primary brain cancers have not yet been well-characterized. Only a few studies have investigated the mechanical properties of primary brain cells and primary brain cancer cells. Most existing studies focused only on healthy astrocytes¹⁵ and/or GBM.^16,17 One study showed that healthy astrocytes from prenatal rat embryos were stiffer than a human GBM (LN229, ATCC) cell line, and that showed a strong correlation between substrate stiffness and cellular stiffness.¹⁷ Another study measured astrocyte mechanical properties and showed that astrocytes are less stiff than neurons, suggesting that astrocytes many not act a structural support but rather a compliant embedding for neurons.¹⁵ One recent study used a microfluidic manometer device to compare the stiffness of healthy human astrocytes to GBM cell lines (A172, 1321N1).¹⁶ This study, although evaluating a small number of cells, suggested that GBM may have increased stiffness compared to healthy normal astrocytes. Studies of non-central nervous system cancers such as breast,^18–22 prostate,²³ liver,²⁴ bladder,^25–27 lung,²⁸ oral,²⁹ thyroid,³⁰ and ovarian³¹ cancer cells have demonstrated reduced stiffness of cancer cells compared to their healthy counterpart cells. Furthermore, studies of breast cancer,^32–34 ovarian cancer,^31,35,36 and colon cancer³⁷ demonstrate that high-grade cancer cells are more compliant than low-grade cancer cells, indicating a link between stiffness and disease progression. Widespread confirmation of the aforementioned trends has led to the paradigm that a reduction in cell stiffness correlates with the metastatic potential in cancer.³⁸ Researchers have hypothesized that the decreased cell stiffness of cancer cells allow the cells to squeeze through the endothelium of blood vessels and enter circulation. However, some studies challenge this paradigm and indicate that increased malignancy and invasion correlate with stiffer cells in some cases.^23,39–44 The unique microenvironmental niche of glioma cells and the low rates of metastasis of glioma suggest that these cells may have mechanical properties and stiffness trends differing from previously studied non-brain cell types. The brain is composed mainly of proteoglycans (lectican family) and associated binders hyaluronan and tenascins, creating a soft gel-like environment that is quite different from the fibrous collagen environments of other tissues.¹³

Conventional methods for testing the mechanical properties of cells include micropipette aspiration⁴⁵ and atomic force microscopy.^35,46 These techniques are commonly used but require expensive equipment and are low-throughput. Microfluidic techniques, alternatively, can be used to investigate the mechanical properties of cells at low cost and high throughput. Microfluidic techniques generally force cells to deform by imposing narrow constrictions,^18,47–51 high fluid shear stress,^52–55 optical stress,^56–62 or electric fields.^63–67 Constrictions channel devices^{16,18,40,47,48–83} are perhaps the simplest microfluidic technique, and force cells to deform as they enter and transit through a narrow channel. Constriction channels mimic pores/capillaries in vivo and can be designed to include multiple constrictions,^84,85 deformable channel walls, or tapered constrictions. In this study, we use a simple constriction channel microfluidic device to deform cells under an applied hydrostatic pressure and measure key parameters of cell stiffness: entry time, transit time, and shape-change parameters.

Stiffness, or Young's modulus ($E$), cannot be directly measured from constriction channel experiments, and thus additional analysis, computations, or experiments must be performed to convert experimental results (entry time, transit time, elongation, etc.) to cell stiffness (Young's modulus). For example, if two cells with equal diameters are considered, the cell with a slower entry into the channel (longer entry time) will have a higher stiffness; however, if the two cells have differing diameters, it is not possible to determine which is stiffer by comparing their entry times. There is not a simple relationship between the experimentally measured parameters (entry time, transit time, elongation, cell diameter) and cell stiffness. To obtain stiffness values from experimental parameters, previous studies have used several methods including histogram matching of strain and pressure,⁷⁵ calibration using gel beads,⁷³ analytical relationships,⁷¹ and 2D finite element modeling (FEM).^48,86,87 In this paper, we establish a novel framework to complement existing methods. We construct a computational framework by combining 3D finite element modeling (FEM) with an artificial neural network (ANN) that together are capable of calculating the cell's stiffness based on experimental data such as cell diameter, entry time, and shape-change parameters. Over 150 finite element simulations of cell entry into a constriction channel were performed across a range of cell diameters and stiffness values. In each simulation, the entry time and cell elongation within the constriction were calculated. These FEM results were used to train a neural network to determine cell stiffness from input parameters. We then used the trained ANN to calculate the stiffness of each cell in the experiments by inputting cell diameter, entry time, and cell elongation.

Several studies have used finite element modeling of cell deformation during micropipette aspiration^88–90 or entry into a constriction channel^48,86,87 for mechanical characterization of cells. In almost all the studies, cells were assumed to be incompressible visco-hyperelastic solids. Elastic solid behavior was modeled using the incompressible neo-Hookean model and viscous relaxation was modeled using the Prony series.⁴⁹ In almost all prior studies, cell deformation in three dimensions was approximated using two-dimensional or even two-dimensional axisymmetric plain strain models to reduce the computational cost. Using these computational results, the authors derived numerical or analytical relationships to evaluate mechanical properties of cells. Experimental data were then applied to the derived numerical or analytical relationships to qualify cell mechanical properties. These numerical/analytical models, however, are inconsistent in the literature because there is no exact analytical equation for finding the stiffness (Young's modulus) of a cell because of the assumptions considered while deriving the relations. In this paper, instead of deriving the analytical equations using the results of finite element simulations, we use artificial neural networks for quantifying the mechanical properties of any cell type using the trained network.

An artificial neural network is a nonlinear statistical analysis technique that is suitable for systems hard to describe using the existing physical models. Neural networks provide a way for linking input data to the output data using a set of nonlinear functions. “Generally, choosing the optimum number of layers (i.e., hidden layers) in the network is one of the challenging steps in the neural network modeling. Usually a trial-and-error procedure is applied to get the most accurate neural network. There are some existing studies in which artificial neural networks are used for predicting the mechanical properties of metallic alloys that are difficult to evaluate with the existing models. In these works, computational data are used for training a network, and experimental results are then given as inputs into the trained network and are used to evaluate the mechanical properties of complex systems.^91,92 Additionally, machine learning and artificial neural networks have been used to classify and sort cell populations.^50,74,93,94 According to the authors' knowledge, this is for the first time that artificial neural networks are being used in this field of cellular mechanical property characterization by constriction channels. Prior studies have employed numerical relationships to relate experimental data from constriction experiments with FEM simulations.⁹³ However, numerical relationships become complex when many parameters are considered. Neural networks provide an alternative approach and excel at pattern recognition data interpretation. Relative to numerical relationships, neural networks become comparatively more beneficial as the number of input parameters increases. For this study, we used three input parameters (cell diameter, cell elongation, and entry time), for which numerical relationships for Young's modulus could have been contrived. However, we anticipate that future studies may include additional parameters, making neural networks a preferred method for extracting cell properties. Therefore, our use of neural networks in this paper lays the groundwork for more complex studies in the future.

In this study, we calculate the stiffness of four established astrocyte/astrocytoma cell lines [Healthy: normal human astrocyte (NHA); Grade III: SW1783; Grade IV: U251, U87] and two patient-derived astrocytoma cell lines (Grade II: VTC-018; Grade III: VTC-085). Grade II astrocytoma cell lines are not commercially available, so we investigated a patient-derived astrocytoma grade II cell line. We investigated if a decrease in astrocytoma cell stiffness would correlate with increased pathological grade. We found that, contrary to the prevailing trends with other cancer types, astrocytoma stiffness was not well-correlated with cell grade and in general showed increased stiffness compared to healthy astrocytes.

METHODS

Microfluidic device and experimental setup

The constriction channel device was fabricated from a single layer of polydimethylsiloxane (PDMS) using standard fabrication methods. Briefly, a master mold was negatively patterned by deep reactive ion etching of a silicon wafer and then silanized. PDMS was cast molded on the wafer (10:1 base to cross-linker), cured at 100 °C for 1.5 h, and peeled off. Inlet and outlet holes were punched with a 0.75 mm biopsy punch. The PDMS layers were plasma bonded (Harrick Plasma) to clean glass coverslips and kept under vacuum until experimentation. The assembled devices created a narrow constriction region 8 μm wide, 8.3 μm tall, and 130 μm in length [Fig. 1(a)].

FIG. 1.

The constriction channel device and experimental setup. (a) The microfluidic constriction channel device is composed of a single channel that tapers to a very narrow constriction. The constriction is 8 μm wide, 8.3 μm tall, and 130 μm long. The device is composed of a PDMS layer bonded to a glass slide. Pressure-driven flow through the device is achieved by the vertical offset of two syringe reservoirs aligned on a custom stand to create constant, uniform hydrostatic pressure. A high-speed camera captures the cells passage through the constriction. (b) A schematic of a cell traversing the constriction channel. Entry time ($t_e$) is defined to be the time taken for the cell to completely squeeze into the entrance of the constriction—when the leading edge of the cell reaches the constriction to the time when the trailing edge of the cell is fully within the constriction. Transit time ($t_t)$ is the amount of time taken by the cell to travel the whole length of the channel (130 μm). Cell size is measured before the cell enters the constriction by approximating the cell as an ellipse with a major and minor diameter ($D_m$). Cells elongate within the constriction cells, and elongation is measured as the ratio of the elongated length (L_A) to the cell minor diameter ($D_m$). Cells exit the channel significantly deformed.

Two 1-ml syringes without plungers were used as reservoirs for the hydrostatic pressure system and connected to the constriction device with 30 AWG tubing (Cole Palmer). Because of the extremely low fluid flow rate through the narrow constriction, the reservoir levels remained constant for the duration of experiments. The fluid interfaces were at atmospheric pressure and were precisely aligned using a custom laser-cut acrylic stand [Fig. 1(a) and Fig. S1 in the supplementary material] to ensure a consistent pressure drop across experiments. The reservoir liquid levels were separated by exactly 30 cm, producing a pressure drop across the system of 3.0 kPa. Nearly all the pressure drop occurs across the narrow constriction due the very high hydraulic resistance of this channel. A pressure of 3.0 kPa is within the upper pressure range for intracranial hypertension,⁹⁵ but is a bit high for physiological relevance. The high pressure was required for cell deformation and entry into the narrow constriction.

Prior to an experiment, a constriction channel device was placed on the stage of an inverted microscope (Leica DMI6000). A high-speed EM-CCD camera (Hamamatsu 9100-02) and MicroManager software were used to capture high-speed videos of cells passing through the constriction. A 20× objective (NA 0.4) was used to capture a field of view of approximately 400 × 40 μm². Images were captured at 200 fps.

Cell culture

Six cell lines (healthy astrocytes and astrocytoma grades II–IV) were cultured according to standard culture procedures. Flasks were routinely passaged at 70%–90% confluency, and only low passage number cells (passages 3–11) were included in the experiments. Normal human astrocytes (NHA, Lonza) were cultured with an AGM astrocyte growth media bullet kit (Lonza) consisting of 500 ml ABM basal medium (CC-3187) supplemented with 15.0 ml fetal bovine serum (FBS), 5.00 ml L-glutamine, 0.50 ml GA-1000, 0.50 ml ascorbic acid, 0.50 ml HEGF, and 1.25 ml insulin. Grade II astrocytoma cell lines are not commercially available, so we established a grade II astrocytoma cell line (VTC-018) from excised patient tissue. Likewise, we also established a grade III (VTC-085) astrocytoma cell line from excised patient tissue. Both patient-derived cell lines were cultured in Dulbecco's modified Eagle's media (DMEM) with supplemented with 15% FBS and 1% penicillin streptomycin (PS). The SW1783 cell line (ATCC) is a commercially available grade III astrocytoma and was cultured in Leibovitz's L-15 media supplemented with 10% FBS and 1% PS. We cultured two glioblastoma (grade IV astrocytoma) cell lines, U251 cells (Sigma Aldrich) and U87 (ATCC) cells. U251 cells were cultured in DMEM supplemented with 10% FBS and 1% PS. U87 cells were cultured in Eagle's minimum essential media (EMEM) media supplemented with 10% FBS and 1% PS. All cell lines except SW1783 were cultured at 37 °C under 5% CO₂. SW1783 cells were cultured at 37 °C under 100% atmosphere.

To prepare a suspension of cells for experimentation, the following protocol was used. Cells cultured in T75 flasks were washed twice with 5 ml phosphate buffered saline (PBS), trypsinized with 1.5 ml of trypsin, and incubated for 1.5–2 min at 37 °C. A 2 ml pipette was used to break apart cell clumps before 4 ml of cell culture media was added to deactivate the trypsin in the flask. The cell suspension was transferred to a centrifuge tube, and 1 ml was transferred to an automatic cell counter (Beckman Coulter, Vi-CELL) to determine the cell density and viability. The cell suspension was centrifuged at 150 × g for 5 min. The supernatant was decanted and 5 ml of our experimental buffer solution was added to resuspend the cells. For our experiments, we used a buffer solution of Dulbecco's phosphate-buffered saline (Ca/Mg⁺²-free) supplemented with 1% bovine serum albumin (BSA), 5 mM EDTA, and DNase I (25 μg/ml). This buffer was formulated to minimize cell–cell and cell–surface adhesion, both of which adversely affect constriction channel experiments. Devices were not pre-coated with BSA prior to experiments. After resuspension in the buffer, the cells were recentrifuged at 150 × g for 5 min. The supernatant was decanted and the pellet was resuspended at 1 × 10⁶–3 × 10⁶ cells/ml. A cell strainer (Fisher, 40 μm filter) was used to filter out cell clumps. A 1 ml syringe was connected to a 15-in. 30 AWG tube with a plastic dispensing tip (Jensen Globlal), and the syringe was filled with 500 μl of the cell solution. The syringe and tubing were then transferred to the hydrostatic pressure stand. During the experiment, the plunger of the syringe was removed and the fluid within the syringe was at atmospheric pressure.

Measured parameters

This study measured several parameters as cells traversed the constriction channel including cell size ($D_m$), entry time ($t_e$), transit time ($t_t$), cell elongation ($L_e$), and cell circularity. Image data were processed semi-manually in ImageJ according to the following procedures.

Cell size: Before cells were deformed by the constriction, cells generally had a circular or slightly elliptical shape when viewed from the microscope. To quantify cell size, we used ImageJ software to trace a cell's perimeter using a spline and then fit an ellipse to the spline using ImageJ's Fit-Ellipse function. This analysis yielded the major diameter ($D_M$) and the minor diameter ($D_m$) for each cell. Most cells were highly circular and yielded very similar values for $D_M$ and $D_m$. During experiments, we found that cells typically rotated such that their minor diameter was oriented parallel with the constriction width as they approached the entrance to the constriction channel (data not shown). For this reason, we used minor diameter as the metric for cell size [Fig. 1(b)]. For brevity, we refer to the minor diameter as simply cell diameter.

Entry Time ($t_e$): Entry time was defined to be the time taken by the cell to enter the constriction. Entry time was calculated from the time the leading edge of the cell crossed into the constriction (8-μm wide region) to the time when the cell's trailing edge had completely entered the constriction. Entry time is illustrated in Fig. 1(b).

Transit Time ($t_t$): Transit time was defined as the time taken by the cell to travel the entire length of the constriction (130 μm). Transit time was calculated from the time the trailing edge of the cell entered the constriction to the time when the trailing edge of the cell exited the constriction channel as shown in Fig. 1(b).

Elongation ($L_E$). Cell elongation ($L_E$) was defined as the ratio of the length of a deformed cell (fully within the constriction channel) to the diameter of the cell prior to entering the constriction region,

$$L_E={\displaystyle \frac{L_A}{D_m},}$$ (1)

where $L_E$ is the cell elongation, $L_A$ is the deformed cell length while in the constriction, and $D_m$ is the minor diameter of the cell prior to entering the constriction [Fig. 1(b)].

Cell shape change: We measured cell circularity before and after the constriction channel to measure cell shape change. Circularity was calculated as

$$\mathrm{Circularity}=4\pi \left({\displaystyle \frac{A}{P^2}}\right),$$ (2)

where A is the z-projected area of the cell and P is the z-projected perimeter of the cell. Circularity values range from 0.0 to 1.0, with a value of 1.0 representing a perfect circle. ImageJ was used to trace a cell's edges with a spline in ImageJ and circularity was automatically calculated using ImageJ's Analyze Menu. When measuring circularity after exiting the constriction channel, contents of ruptured cells were not included in this calculation as they were difficult to resolve due to low contrast.

Numerical simulations

Finite element simulations of the cell entry in to the constriction channel were performed using the finite element package ABAQUS.⁹⁶

Geometrical and material modeling

Three-dimensional cell deformation was conducted in the ABAQUS/Explicit solver. Cells were modeled as a continuum sphere across a range of cell diameters to collect as broad a data set of numerical results as possible. The constriction channel was modeled as a rigid body since the stiffness of PDMS (∼1 MPa) is much higher than that of the cell stiffness. The model of the constriction channel had the same dimensions as in experiments but incorporated an initial entrance region with increased channel height (20 μm), such that cells could be initially modeled as spheres before being flattened by the low channel height (8.3 μm) just prior to the constriction channel entrance. The material model for cells was taken to be incompressible neo-Hookean with visco-hyperelastic behavior.⁴⁵ The constitutive equation for the neo-Hookean material is

$${\sigma }_s=C_{10}\left(\overline{B}-{\displaystyle \frac{1}{3}tr(\overline{B})I}\right)+{\displaystyle \frac{2}{D_1}(J-1)I,}$$ (3)

where ${\sigma }_s$ is the Cauchy stress tensor, $\overline{B}$ is defined as $\overline{F}.{\overline{F}}^T$ with $\overline{F}$ being the distortion gradient which is equal to $J^{1/3}F$ (where J is elastic volume ratio), $D_1$ is the temperature dependent material parameter, and I is the identity tensor. $C_{10}$ and $D_1$ are given as

$$C_{10}={\displaystyle \frac{G_0}{2},}$$ (4)

$$G_0={\displaystyle \frac{E_0}{2(1+\nu )},}$$ (5)

$$D_1={\displaystyle \frac{3(1-2\nu )}{G_0(1+\nu )},}$$ (6)

where $G_0$ is the initial shear modulus and ν is the Poisson's ratio. Stiffness corresponds to Young's modulus, $E_0$, and is referred to elsewhere in this study as simply E. Combining Eqs. (4) and (5), we find that

$$C_{10}={\displaystyle \frac{E_0}{4(1+\nu )}.}$$ (7)

Viscoelastic behavior is defined using the Prony series expansion of time dependent shear and volumetric behavior formulated:⁸⁹

$$g_R(t)=1-\sum _{i=1}^N{\overline{g}}_i^p\left(1-e^{-\frac{t}{{\tau }_i^G}}\right),$$ (8)

where $g_R(t)$ is a dimensionless shear relaxation modulus, ${\overline{g}}_i^p$ is the dimensionless Prony series parameter for shear modulus, and ${\tau }_i^G$ is the relaxation characteristic time.

$$k_R(t)=1-\sum _{i=1}^N{\overline{k}}_i^p\left(1-e^{-\frac{t}{{\tau }_i^K}}\right),$$ (9)

where $k_R(t)$ is a dimensionless bulk relaxation modulus, ${\overline{k}}_i^p$ is the dimensionless Prony series parameter for bulk modulus, and ${\tau }_i^K$ is the relaxation characteristic time. The bulk relaxation term will be equal to zero because the material is considered to be incompressible. The density of the cell was considered 1000 kg/m³ and the elastic behavior was modeled using the term C₁₀.⁴⁸ For incompressible material D₁ is equal to zero. The viscoelastic behavior was modeled using the Prony series where we specified the dimensionless shear modulus and relaxation characteristic time.

Simulation parameters are presented in Table I. In this study, we performed multiple simulations by varying the stiffness between 3.6 and 21.6 kPa for a set of cell diameters varying from 11 μm to 19 μm in an interval of ∼1 μm, while keeping all the remaining parameters constant in all the finite element simulations. The entry time of a cell depends on the stiffness ($E$) and the relaxation parameters ($g$, $\tau$); however, throughout our analysis, we keep the viscoelastic properties as constants.⁴⁷ Thus, the stiffness of a cell is assumed to be the only parameter that affects the entry time.

TABLE I.

Simulation parameters used in FEM models.

Simulation parameter	Value(s)
Density (ρ kg/m3)	100048
Poisson's ratio (ν)	0.548
Dimensionless shear modulus (g)	0.948
Relaxation characteristic time (τ s)	2048
Cell stiffness (E) investigated in FE simulations (kPa)	3.6, 4.2, 4.8, 5.4, 6.0, 6.6,7.2, 7.8, 8.4, 9.0, 9.6, 10.2, 10.8, 11.4, 12.0, 13.2, 14.4, 15.6, 16.8, 18.0, 20.0, 21.6.
Cell minor diameters investigated in FE simulations (μm)	11.58, 12.64, 13.45, 14.54, 15.4, 16.4, 17.2, 18.5, 19.1

Boundary conditions and meshing

As the spherical cell enters the constriction channel, the cell is significantly deformed due to smaller cross-sectional area of the channel compared to that of the cell. To model these large deformations, geometrical nonlinearity was considered in the analysis. Friction between the channel and cell was modeled using a simple Coulomb friction model, in which critical shear stress on the sliding surfaces was directly proportional to the contact pressure between the surfaces with a constant frictional coefficient $f_c$. The rigid channel was kept fixed throughout the simulation while the spherical cell was acted upon by a uniform pressure loading of 3.0 kPa, equal to the pressure that was used in the experiments. As a cell deforms and enters into the constriction channel, the surface area that is exposed to the pressure keeps changing. Due to this change, a uniform pressure could not be used on a pre-selected region of the cell surface. To overcome this issue, a user subroutine VDLOAD was used for applying the pressure loading on the outer boundary of the cell that was outside of the constriction channel. This area was updated in each time step during the simulation. Cells were discretized with C3D10M quadratic elements, which are very efficient in modeling large deformations and complex contact situations. The constriction channel was meshed with a default discrete rigid element mesh.

Artificial neural network model

We used an artificial neural network designed by the available MATLAB neural network toolbox. Generally, the number of neurons in the input and output layers of a neural network are determined by the number of input and output variables. Here we used three input nodes (entry time, cell diameter, and elongation) and only one output node for Young's modulus. For training and validating the network, we used the results collected from the finite element simulations. A total data set of 155 simulations was used to train and validate the network. Among that data set, 150 data points were used for training, which was performed using a batch training process. The remaining five data points were used for testing the network. The network was trained using the fastest and standard Levenberg–Marquardt algorithm. The optimum number of neurons in the hidden layer of the network was determined by checking the error convergence rate between the simulations and predicted stiffness data. After checking results from networks with various numbers of neurons, we found that eight neurons formed an efficient network. Our network with eight neurons in the hidden layer was trained in cycles until the simulation and predicted stiffness values were very close. Our validation process demonstrated that our trained network could predict stiffness with less than seven percent error.

Nucleus-to-cytoplasm (NC) ratio calculation

The nucleus-to-cytoplasm (NC) ratio is the ratio of nuclear volume to cytoplasmic volume. To calculate the NC ratio, we stained cell nuclei with NucBlue (two drops per ml) and the cytoplasm with calcium AM (4 μl/ml). Cells were stained in suspension, and then transferred to a well-plate and imaged at 10× before cell adhesion occurred. Thus, cells were assumed to be spherical. To calculate the NC ratio, nuclear and cytoplasmic channels were individually thresholded and binarized to black and white. ImageJ's Analyze Particle function was used to calculate the z-projected area, the major diameter (of a fitted ellipse), and the minor diameter (of a fitted ellipse) of the nuclei and cells. Only cells (and nuclei) that were relatively circular (circularity >0.75) were analyzed. The average of the major and minor diameters was assumed for the out-of-plane direction, and volumes were calculated with these diameters. To accurately compare NC data to the constriction channel data, we adjusted cell diameters (originally measured on spherical cells) to be equivalent with the diameters calculated for cells just prior to entering the constriction channel (flattened due to low channel height). See the supplementary material for derivation.

Statistics

Statistics were performed in JMP Pro 15. Statistical differences were determined by ANOVA tests at $\alpha =0.05$. Multiple comparison Student t-tests were used to compare statistical significance between multiple groups ($\alpha =0.05$).

RESULTS

Healthy astrocytes and astrocytoma grade II, III, and IV cells were investigated with constriction experiments, and over 75 cells from each cell type were analyzed as presented in Table II. The median minor diameter of cells analyzed ranged from 16 to 21 μm depending on the cell type (Table II). Cells with diameters larger than 25 μm often did not reach the constriction channel because of the relatively low channel height (8.3 μm) of the device. Furthermore, many large cells (>25 μm) that reached the constriction channel could not enter the constriction at the experimental pressure (3.0 kPa) due to the large deformation required and thus became stuck prior to entering the channel. For these reasons, we focused our analysis on cells with diameters between 10 and 25 μm.

TABLE II.

Cell grades, origin, and number analyzed.

Grade	Cell line	Cell type	Number analyzed	Cell diameter (μm)
				Median	Mean ± STD
Healthy	NHA	Primary	n = 270	16.5	16.9 ± 3.4
Grade II	VTC-018	Primarya	n = 110	18.8	19.4 ± 3.3
Grade III	VTC-085 SW1783	Primarya Immortalized	n = 81 n = 130	18.9 21.1	18.9 ± 3.5 21.4 ± 4.3
Grade IV	U251 U87	Immortalized Immortalized	n = 129 n = 113	20.5 15.9	20.9 ± 3.9 16.0 ± 2.1

^aPatient-derived cell line

Figures 2(b)–2(c) show two representative cells traversing the constriction. Prior to entering the constriction channel, cells display a high degree of circularity and high velocity. Upon encountering the constriction, cell velocity is dramatically reduced as significant deformation is required for a cell to enter the constriction channel. Deformation usually occurred as an elongation of the leading (aspirated) edge of a cell and the lateral compression of the minor diameter of a cell to the width of the channel. In many cases, especially larger cells, the leading (aspirated) surface of the membrane of the cell and the trailing edges (at both contact point with channel sidewalls) showed blebbing, which indicates membrane rupture and leakage of cytoplasmic contents [Fig. 2(c)]. While confined to the channel, cells elongated in the direction of the channel. Upon exiting the constriction, cells had highly deformed shapes with significantly decreased circularity (circularity decrease = 0.34 ± 0.13, average for all cell types).

FIG. 2.

Constriction channel results. (a) Brightfield images of each cell line showing cell morphology of adherent cells. Scale bars 100 μm. (b) and (c) Time series of two representative cells traversing the constriction channel. (c) A small diameter cell (SW1873, diameter 15.8 μm) entering and transiting the constriction. (d) A larger diameter cell (VTC-085, diameter 20.3 μm) entering and transiting the constriction channel. Scale bars 25 μm. (d) Entry time, transit time, and elongation index for all cell types plotted against cell diameter. Entry time and transit time are roughly linearized on a log scale when plotted against cell diameter.

Entry time, transit time, and elongation

The entry times ($t_e$) for all cells analyzed are plotted in Fig. 2(d) against the cell diameter. The relationship between entry time and diameter is nonlinear but can be roughly approximated as a log relationship as shown by the best-fit lines on the semi-log plots in Fig. 2(d). Interestingly, while the slopes of the best-fit semi-log relationships for healthy cells through grade III cells are quite similar, the slopes for grade IV cells appear quite different—with U251 cells showing a low slope and U87 cells showing a high slope. We used cell entry time as an input to the neural network for predicting the stiffness of cells.

The transit times ($t_t$) for all analyzed cells are show shown Fig. 2(d). As with the entry time, transit time is roughly linearized when plotted on a log scale against diameter. Best-fit log relationships show similar trends in their slopes as with entry time. Because our finite element simulations could not recapitulate cell transit through the constriction (see the Analysis of cellular entry process and numerical simulations section), transit time was not included in the analysis of cell stiffness.

Cells elongated in the direction of the channel when confined within the constriction [Figs. 2(b) and 2(c)]. We measured cell elongation ($L_E$, see Methods) and found that the elongation was approximately linearly correlated with cell diameter, as shown in Fig. 2(d). We used elongation as an input to the neural network for predicting the stiffness of cells.

Analysis of cellular entry process and numerical simulations

Using finite element modeling, we simulated cell entry into the constriction channel. Figure 3 shows a sequence of images illustrating the characteristics of cell entry. Cell deformation from our numerical simulations agreed well with the sequential cell entry process into the channel measured in the experiments as shown in Fig. 2. Aspiration length, defined as the distance traveled by the leading edge of the cell past the entrance of the constriction, plotted against time revealed that the process of cell entry is divided into three stages as in Fig. 3(f). In the first stage, aspiration length shows a rapid, nonlinear increase as the cell's leading edge enters the constriction. However, the forward progress of the leading edge is quickly impeded in the second stage as the cell becomes deformed by the constriction [see Fig. 3(b)]. This deformation causes the second stage to have a linear relationship between time and the aspiration length, with a small slope. The slow movement of the cell into the channel at this stage is illustrated in Figs. 3(c)–3(d). In third stage of entry, the trailing edge of the cell completely enters the constriction, leading to an increased velocity as the cell begins transiting the constriction [Fig. 3(e)].

FIG. 3.

Numerical simulations of cell entry into the constriction channel. (a)–(e) Snapshots showing cell deformation during cell entry into the constriction channel. (a) Cells approach the constriction with some deformation present due to the low channel height. (b)–(d) The leading edge of the cell enters the channel (stage I), and the cell undergoes slow deformation [(c) and (d)] as the entire cell enters the constriction (stage II). (e) After entry, the cell transients through the channel (stage III). (f) Aspiration length vs time illustrates the three stages of the entry process.

In this study, we performed multiple simulations by varying the stiffness between 3.6 and 21.6 kPa for a set of cell diameters varying from 11 to 19 μm in an interval of ∼1 μm, while keeping all the remaining parameters constant in all the finite element simulations. From these simulations, we calculated the entry time ($t_e$) and elongation ($L_E$) for each simulation of cell diameter ($D_m$) and Young's modulus. We optimized our simulations to capture the largest range of cell diameters and stiffness values as possible. However, simulations with cell diameters greater than 19 μm were not possible due to the very large deformations, and entry times of greater than 2 s were excluded from analysis to the limited accuracy of model predictions.

Artificial neural network training and validation

We trained a neural network to predict the stiffness of a cell using our collected data from the finite element simulations. A schematic of the network is shown in Fig. 4. We found that a neural network with one hidden layer composed of eight neurons was able to predict the stiffness for the finite element data with less than 7% error. The network, once trained, could estimate stiffness for any cell when time entry ($t_e$), entry diameter ($D_m$), and elongation index ($L_E$) values were given as input into the network. For example, Figs. 4(b)–4(d) show the contour plots of cell stiffness for multiple combinations of the input parameters ($t_e$, $D_m$, and $L_E$).

FIG. 4.

(a) Schematic representation of the neural network architecture. The network predicts stiffness (E) based on three inputs: entry time ($t_e$), diameter ($D_m$), and elongation ($L_E$). (b)–(d) Contour plots show stiffness values for different combinations of cell diameter and entry time ($t_e$) when elongation values are (b) 1.6, (c) 1.9, and (d) 2.2.

Stiffness values of astrocytes and astrocytoma cell lines

Using the trained neural network, we calculated the stiffness values for our experimental data for cells between 10 and 25 μm in diameter, as shown in Fig. 5. Table III presents the average and median values of the stiffness for each cell type. We found that average stiffness values varied between 7 and 13 kPa depending on the cell type, and an ANOVA test revealed significant differences between cell types. Table III also includes the groupings from multiple comparison Student t-tests; each cell type not connected by the same letter has significantly different stiffness. We found that cell grade did not show a strong correlation with average (or median) cell stiffness. Interestingly, NHA cells (healthy astrocytes) and U87 cells (grade IV) had significantly lower average (and median) stiffness values compared to the other cell types. Contrarily, the patient-derived grade III astrocytoma cell line (VTC-085) showed the highest average (and median) stiffness, with significantly higher stiffness than all other cell types. To better understand the origin of stiffness differences, we calculated the nucleus-to-cytoplasm (NC) ratio for all cell types.

FIG. 5.

Neural network calculation of stiffness. (a) Stiffness for each cell analyzed. Horizontal black lines show median values. (b) Average stiffness values with 95% confidence intervals for all cell types. Dashed black lines separate cell grade (healthy through grade IV from left to right). Bars show significant differences between astrocytoma cell lines and healthy astrocytes (Student t-tests). Analysis performed on cells with diameters between 10 and 25 μm.

TABLE III.

Stiffness (E) of all cell types.

Grade	Cell	Number analyzed	Stiffness (E) kPa			Student t-test groupings
			Mean	STD	Median
Healthy	NHA	234	7.6	5.4	5.0	A
Grade II	VTC-018	91	9.5	6.0	7.2		B
Grade III	VTC-085	52	12.3	7.2	12.3			C
Grade III	SW1783	86	9.9	6.1	7.1		B
Grade IV	U251	98	8.8	5.8	6.1	A	B
Grade IV	U87	111	7.5	4.9	5.3	A

Nucleus-to-cytoplasm (NC) ratio

To better understand the results of our stiffness measurements, we calculated the nucleus-to-cytoplasm (NC) ratio for all cell types. Figure 6 shows the NC ratio for each cell type for cells 10–25 μm in diameter (cell diameters adjusted for equivalence with constriction experiments—see Methods). As can been seen from our results, the NC ratio differs significantly between cell types. Most notably, SW1783 (grade III) and U87 (grade IV) cells show different NC characteristics. Both SW1783 and U87 cells have significantly lower NC ratios compared to the healthy astrocytes (NHA). Furthermore, we found that U251 cells had a significantly higher NC ratio than the healthy astrocytes (HNA). Figure S2(a) in the supplementary material shows the density plots of NC ratio vs cell diameter for all cell sizes. From the distribution, it is clear that SW1783 and U87 cells show different NC ratio characteristics compared to other cell lines.

FIG. 6.

Nucleus-to-cytoplasm (NC) ratio for all cell types. Compared to healthy astrocytes, SW1783 cells (Grade III) and U87 cells (Grade IV) have significantly lower NC ratios, while U251 cells (Grade IV) have significantly higher NC ratios. NC ratios calculated for cells with diameters between 10 and 25 μm (see Methods) for consistency with constriction channel analyses.

DISCUSSION

The prevailing paradigm in cancer progression and metastasis is that increased metastatic potential of cancer cells correlates with decreased cell stiffness. Due to the significant deformations that must occur for a cell to migrate through confined spaces, decreased stiffness is thought to increase a cells ability to traverse these mechanically challenging spaces. In this study, we investigate whether the malignancy of astrocytoma correlates with cell stiffness. We used a microfluidic constriction channel device to deform healthy astrocytes and astrocytoma cells of grades II–IV to investigate the stiffness of these cells and were able to analyze the passage of over 75 cells for each cell type through a microfluidic device consisting of a single constriction channel (8 × 8.3 × 130 μm). Using finite element simulations of cell entry into the constriction, we trained a neural network to predict stiffness (E) values based on cell diameter, cell entry time, and cell elongation within the channel.

Several studies have applied machine learning to classify cells based on their physical properties.^50,74,93,94 These studies use experimental data to train neural networks (or other algorithms) to differentiate cell populations by cell type^50,74,93 or phenotype.⁹⁴ In contrast to these studies, our study is the first to use an artificial neural network, trained with finite element models, to compute the stiffness (E) of cells. Furthermore, our study represents the first investigation of cell stiffness for grade II and grade III astrocytoma. Unfortunately, research on low-grade gliomas (grades II and III) has been limited. Glioma research is generally directed toward grade IV astrocytoma (glioblastoma), as these grade IV gliomas present the most severe pathology and many grade IV cell lines are readily available and easy to work with. Unfortunately, at this time, there are no grade II astrocytoma cells commercially available, further limiting research efforts. Here, we investigated a patient-derived grade II and grade III astrocytoma cell line and measured cell stiffness. Additionally, we investigated another grade III astrocytoma cell line that is commercially available.

Several prior studies using constriction channels have used cell entry time directly to infer cell stiffness rather than calculating stiffness.^16,77 However, entry time values alone have limited usefulness since they cannot be compared across studies due to the dependence on channel geometry and pressure drop across the channel. Our calculation of stiffness via finite element simulations enables our results to be easily compared with other studies. In relatively good agreement with previously calculated values of stiffness for astrocytes and GBM cells,^15,17 we found that average stiffness values varied between 7 and 13 kPa depending on the cell type. Interestingly, we found that the average stiffness of the cell lines tested did not have a strong correlation with cell grade. Compared to the healthy astrocytes (NHA cells), most astrocytoma cell types were found to have higher stiffnesses than their healthy counterpart. U87 cells and U251 cells (both grade IV) did not show a statistically significant increase in stiffness; however the median and average stiffness of U251 cells was higher than healthy (NHA) cells. The patient-derived grade III cell line showed the highest average stiffness. To calculate these stiffness values, we included three input values (cell diameter, entry time, and elongation) into our neural network.

The nucleus is the stiffest organelle of the cell, and thus we calculated the nucleus-to-cytoplasm (NC) ratio for each cell type to understand the impact of nuclear size on our results. NC ratios revealed that U87 and SW1783 cells had significantly smaller nuclei than other cell types. Furthermore, U251 cells showed higher NC ratios compared to other cell types. Low stiffness values of U87 seem to be explainable by their smaller-than-normal nuclei. Since the nucleus is generally 5–10 times stiffer than the surrounding cytoskeleton,⁹⁷ smaller nuclei require less deformation to enter the constriction channel, leading to decreased entry times and decreased stiffness values for these cells. Likewise, the large nuclei of U251 cells may explain their higher-than-normal stiffness values. NC ratios do not explain all trends in cell stiffness, however, as both grade II and III patient-derived cells had high stiffness values but showed nearly equivalent or lower average NC ratios than healthy astrocytes (NHA) (Fig. 6 and Fig. S2(b) in the supplementary material).

Several additional factors may contribute to the trends in cell stiffness we observed. Patient-derived primary cells, used at low passage number, may be mechanically different than established or immortalized cell lines that have been adapted for cell culture. Studies have shown that astrocytes change their stiffness and behavior depending on the stiffness of their culture platform;^17,98 thus, patient-derived cells may differ from established cell lines. Low-grade cells may be innately stiffer than healthy astrocytes due to their pathology. Astrocytes are generally quiescent and non-migratory but can become highly migratory when activated by brain injury.⁹⁹ Healthy astrocytes, therefore, may be less stiff than low-grade gliomas due to this migratory capability. Our culture conditions for the astrocytes may have promoted a migratory phenotype due to the inclusion of insulin in the culture media, which has been shown to induce migration in other cell types.^100–102 Given the migration potential of healthy astrocytes, it may prove insightful for future studies to include grade I astrocytoma, which typically does not show cell migration in vivo. Finally, patient-derived cells were more difficult to work with (due to cell clumping and clogging in the channels), and therefore their stiffness values may have more error. Clumping may indicate altered surface properties in these cancer cells.^103,104 Cell surface properties have been previously shown to affect cell transit in constriction channels,¹⁰⁵ and altered surface properties may have contributed to the higher stiffness values we observed for these cell types. Interestingly, low-grade astrocytoma cells showed equal or higher stiffness values compared to glioblastoma. Future studies are warranted to better understand the nature of high stiffness values we found in low-grade astrocytoma.

The increased invasion and malignancy of high-grade astrocytoma is well documented.^6,7 However, grade IV gliomas showed lower stiffness than low-grade gliomas, and healthy astrocytes had approximately equal stiffness with grade IV gliomas. Our finding that astrocyte and astrocytoma stiffness is not well-correlated with malignancy goes against the prevailing paradigm in cancer mechanics and warrants future studies to investigate this further. Along with our results, several prior studies have similarly reported higher stiffness values for malignant, late-stage, or invasive cancer cells compared to healthy or less malignant/invasive cells. Bastatas et al. found that highly metastatic prostate cells were stiffer than lowly metastatic cells,³⁹ and Faria et al. similarly showed that a highly invasive prostate line had higher stiffness than a non-invasive prostate line.²³ Zhang et al. found that hepatocellular carcinoma cells were more stiff than healthy hepatocytes.²⁴ Darling et al. found no correlation between malignancy and stiffness in chondrosarcoma cell lines, with the cells of greatest malignancy showing the lowest stiffness.¹⁰⁶ Deregulation of gene expression in diseased cells can alter protein expression and lead to both changes in stiffness and changes in cell migratory capability. For example, Nguyen et al. demonstrated that more invasive pancreatic ductal adenocarcinoma cells had increased stiffness and showed that increased lamin A expression increased both invasive potential and Young's modulus.^40,41 Kim et al demonstrated that β-adrenergic receptor signaling increased cell stiffness and invasion in vitro through altered actin dynamics and myosin II activity.⁴² Rathje et al. showed that oncogenes induced alterations of vimentin intermediate filaments and microtubules, leading to increased cell stiffness that was shown in vitro to increase the invasion capabilities of the cells.⁴³ Likewise, Yu et al. found that lung cancer cells had altered actin and focal adhesion dynamics (driven by β-PIX signaling) that resulted in greater mobility of the cells and increased cellular stiffness.^42,44 Our results add to the findings of these studies that suggest that malignancy and invasiveness are not always correlated with stiffness.

Limited correlation between astrocytoma grade and cell stiffness may suggest that stiffness is not the driving (or rate-limiting) factor in astrocytoma migration. Cell migration is an active process, requiring the coordination of many cytoskeletal, adhesion, and motor proteins. During cell migration, cells reach forward with protrusions, anchor to the extracellular matrix (ECM), contract with myosin II motors to pull themselves (and their nucleus, the stiffest organelle) forward, and detach at their trailing end.^107,108 Thus, alteration of proteins involved with the cell cytoskeleton and nucleus, such as actin and lamins, can impact cell migration via changes to cell deformability or by altering the dynamics of the molecular mechanisms of cell motility.^109–111 Actin filaments, for example, form tension-bearing stress fibers anchored at focal adhesions and regulate cell stiffness, especially for adherent cells.^112–114 Deregulation of actin-related molecules, however, can also increase cancer cell motility.^101,115 Likewise, the intermediate filament proteins lamins, which provide structure to the nuclear membrane, directly impact nuclear deformability^116–118 but also can affect migration through mechanosensing and gene regulation.¹¹⁹ Therefore, cell mechanics and migration are deeply interrelated. Additionally, several studies show that astrocytoma cells use active cellular processes during invasion. First, glioma cells are able to degrade the extracellular matrix by secreting proteases and other pro-migratory molecules.¹² Degradation of ECM enables glioma cells to enlarge and transit narrow constrictions.¹²⁰ Myosin II activity is likewise critical for glioma cells to pull themselves through narrow spaces within the brain.¹²¹ Additionally, ligand-mediated cascades allow glioma cells to exchange ions and water and decrease their volume up to 30% to squeeze through narrow passages.¹⁴ Other studies, however, have shown evidence that suggests nuclear stiffness may be the rate-limiting step in glioma invasion.^120–123 For example, knockdown of lamin A, an important component of nuclear lamina, decreases nuclear stiffness in U251 cells and increased transmigration across a transwell membrane.¹²⁴ Lamin A/C has been linked with nuclear stiffness and rate-limiting migration.¹²² Future studies are warranted to investigate if there are differences of nuclear stiffness between grades. Since we forced cell deformation in our experiment, we cannot capture active cell processes such as ECM degradation, myosin II activity, and volume regulation that are involved in migration. Additionally, our experiments were performed on cells in suspension and thus were not able to measure the effect of substrate/ECM stiffness on cell properties. Substrate/ECM stiffness impacts cytoskeletal dynamics and cell stiffness,^125–127 and thus future studies may yield more information by measuring the rate of active cell migration through constrictions for various grades of astrocytoma.

In our experiments, pressure was held constant by a hydrostatic pressure difference maintained by vertically displaced syringe pumps. Due to the low fluid flow, the fluid level in the syringe pumps remained constant. However, our device did not enable real-time monitoring of pressure levels, as more complex microfluidics devices¹⁶ or pressure-based flow controllers^71,76 can enable. During experiments, large cells (or clumps of cells) progressively became lodged upstream of the constriction channel, potentially decreasing the pressure drop across the constriction channel. To minimize unwanted pressure drops upstream of the channel, we performed our experiments on multiple constriction channel devices and only performed experiments on constriction channel devices for short durations (around 5 min) before replacing the device. Additionally, we created a buffer solution to minimize cell–cell and cell–wall interactions. Devices were not pre-coated with BSA, however, and pre-coating the devices may have further reduced cell–wall interactions.⁷⁷

The geometry of our microfluidic channel (particularly the low channel height, 8.3 μm) limited the size of cells that reached and transited through the constriction at the hydrostatic pressures used in these experiments. Constriction channel devices are known to be sensitive to cell size, since constrictions too narrow do not allow cell passage but channels too wide do not cause significant deformation. Unfortunately, the astrocytoma cell lines in this study showed a very wide range of sizes, making it difficult or impossible to collect data for all cell diameters. With the exception of U87 cells, most cell types had a significant percentage of cells too large for analysis with our device. Consequently, future work is warranted to confirm that stiffness trends from this experiment apply to cells with larger diameters. Smaller cells, however, tend to have less cytoplasm and thus larger NC ratios. Therefore, contributions from the nuclear stiffness to the overall cell stiffness may be more pronounced in our experiments. This hypothesis seems reasonable, since NC ratios show some correlation with cell stiffness [Fig. S2(b) in the supplementary material]. A higher sensitivity to nuclear stiffness may be desirable, since previous studies have demonstrated that the nucleus, the stiffest part of the cell, may play an invasion-limiting role by slowing the rate that cells can migrate.^120–123 Our brightfield constriction experiments did not enable the measurement of nuclear size/shape, as has been done previously via fluorescence microscopy.⁷⁶ Measurement of nuclear size and deformation during constriction experiments could further reveal contributions of the nucleus to cell mechanics. To improve the throughput and reduce clogging events in future studies, a multi-constriction channel device with a higher channel height could be implemented. Increasing the device channel height and constriction cross-sectional area would have enabled the entire distribution of cell diameters to enter the channel and may have improved our results. Improvements of our device could also include a method for real-time pressure measuring to ensure any small pressure changes during and between experiments are known. Additionally, studies have shown transient cellular and nuclear softening after multiple constrictions,^68,69,84,85 and thus a device that includes multiple sequential constrictions could investigate the effects of repetitive deformation.

Due to the high deformability required by cells to enter the constriction channel, finite element modeling of cell entry into the constriction was very difficult due to large mesh deformation leading to model convergence errors. We maximized the parameter space possible for modeling cells of various diameters. However, at cell diameters greater than 19 μm, cell deformation was quite large and convergence of simulations was not possible. Consequently, our neural network estimated stiffness for cell diameters greater than 19 μm by extrapolating data from smaller diameters. Therefore, our calculation of stiffness for cells larger than 19 μm may have some error. Limiting our experimental data set to cells with diameters below 19 μm was not practical due to the low number cells within this range for some cell types. This limitation of our analysis could be mitigated in future studies by increasing the dimensions of the constriction to reduce deformation of large-diameter cells or by implementing improved FEM techniques able to handle larger mesh deformations. Additionally, the large deformations required for larger cells to enter the constriction channel led to cell blebbing/rupturing. Blebbing/rupturing is a result of the lipid bilayer detaching from cortical actin that supports the membrane and provides mechanical strength. This phenomenon cannot be replicated in simulations as it involves discrete cell components with different mechanical properties and may also alter cell volume and thus violate the incompressibility assumption of our models. Our simulations treat cells as homogenous visco-hyperelastic bodies with uniform properties. Thus, significant cell blebbing and rupturing may lead to inaccuracies of our simulations. Furthermore, we did not model the cell nucleus or other organelles; thus, our stiffness parameter combines contribution from the cytoskeleton, cytoplasm, nucleus, and other organelles. As the nucleus is the stiffest component of the cell, FEM simulations including nuclear size and stiffness would provide additional insight. We leave this to a future study, however, since nuclear size and deformation were not measured concurrently during our constriction experiments. Modeling of cytoskeletal dynamics could likewise provide further insight of cell properties.^128–130 In future works, more experimental parameters (transit time, nuclear size, nuclear deformation/elongation, etc.) could be used to train a neural network for even more reliable prediction of stiffness ($E$) from constriction channel data. Furthermore, in this study, we held the dimensionless shear modulus ($g$) and relaxation characteristic time (τ) constant; however, an investigation of the creep response during cell entry into the constriction through higher frame-rate data could enable these viscoelastic properties to be predicted using a neural network as well.

CONCLUSIONS

In this study, we used a constriction channel microdevice to deform cells for the purpose of calculating cell stiffness of healthy astrocytes, and grade II, III, and IV astrocytoma cells. Using measured values of cell diameter, entry time, and elongation, we calculated the stiffness of cells using neural network trained with finite element simulations. Our study provides the first calculation of stiffness for grade II and grade III astrocytoma and is the first to apply a neural network analysis to a constriction channel device for determining cell stiffness. Our findings suggest that low-grade gliomas may have elevated stiffness compared to healthy and high-grade gliomas and warrants future studies to investigate this further. Differences in NC ratios between the cell types only partly explain the stiffness values. While most prior studies show reduced stiffness of cancer cells compared to their healthy counterparts, we found that most astrocytoma cell lines had higher stiffness values than the healthy astrocytes. Increased migratory potential of higher-grade astrocytoma cells may be driven by factors other than cell stiffness, such as active cell processes like volume regulation, ECM degradation, or altered dynamics of the molecular mechanisms regulating cell motility.

SUPPLEMENTARY MATERIAL

See the supplementary material for additional figures: Photos of experimental setup (Fig. S1), supplemental NC ratio plots (Fig. S2), and calculation of NC ratio details (Fig. S3).

DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.