1 Introduction

The mechanical properties of materials to which biological cells adhere are now commonly accepted to be a major factor in determining cell function, proliferation, and differentiation [1]. These mechanical properties, usually quantified by elastic modulus, can change either gradually or rapidly during processes such as tissue differentiation, development of fibrotic disease, changes in vascular pressure as the heart beats, sudden impact, or the effects of gravity during activities such as sitting or walking. Many efforts have been directed at developing biologically compatible materials with elastic moduli similar to those of biological tissues [2], and in particular development of methods by which the elastic modulus of a cell substrate can be changed to study how cells respond to this change in substrate stiffness [3]. Most such efforts involve chemical strategies to break network strands or introduce or eliminate network crosslinks, which can change the elastic modulus by several factors over a period of minutes to hours [4, 5]. Such strategies mimic some aspects of normal or pathological stiffness changes but are limited by the relatively slow rate at which the elastic modulus changes and usually by irreversibility of the chemically induced stiffness change. To reproduce the rapid and reversible stiffness changes that occur as the heart beats, blood vessels pulse, or soft tissues are deformed by muscle contraction, chemically induced stiffening or softening materials are inadequate.

An alternative method to change the stiffness of a soft elastomer or a hydrogel is the introduction of ferromagnetic particles into the material and then subjecting the composite material to magnetic fields [6,7,8,9,10]. The most commonly used such materials are magnetoelastomers in which particles such as carbonyl iron spheres are embedded in polydimethylsiloxane (PDMS) or other similar soft materials that have elastic moduli similar to those of some mammalian tissues. These materials have been extensively studied and quantitatively analyzed, with excellent fits of theory to experiment [11, 12]. In part, understanding the effect of magnetic fields on material stiffness is facilitated by the fact that the host rubber-like material, such as PDMS, is linearly elastic, with shear or Youngs moduli that are nearly independent of frequency or strain magnitude over the range of timescales and deformation magnitudes that occur in vivo. Similar considerations also apply to polyacrylamide or other flexible polymer hydrogels embedded with ferromagnetic particles [6, 13, 14], because these hydrogels also have nearly linear elasticity. However the native extracellular matrix, as well as the cytoskeleton in living materials is predominantly formed by relatively stiff fibrous polymer networks that exhibit a rich nonlinear viscoelastic response, with shear moduli that change by orders of magnitude over modest strains, and in some cases with frequency dependent changes in both the shear storage and loss (or elastic and viscous) moduli [15, 16]. Such nonlinear fibrous networks, composed for example of fibrin or collagen, can also be integrated with ferromagnetic particles to allow the elastic modulus of the composite to change by orders of magnitude very rapidly and reversibly by application of magnetic fields that can easily be generated in a laboratory setting [17, 18].

The importance of substrate stiffness and a schematic image of the methods used to study cell response to environmental viscoelasticity is illustrated in Fig. 11.1. Typically, the rigid glass or plastic substrate traditionally used for cell biology in the laboratory is covered by a thin elastomeric or hydrogel material on the surface of which specific adhesion proteins, typically extracellular matrix (ECM) proteins, are covalently attached [19, 20]. The elastic modulus of the deformable elastomer or hydrogel can be varied by altering polymer density, crosslinker concentration, and other features to vary the elastic modulus from less than 100 Pa to kilopascal or megapascal stiffnesses that span the range of most soft tissues, from brain to muscle to cartilage. An illustration of the importance of substrate mechanics is shown by the morphology of cardiac myocytes that are removed from the three-dimensional cardiac tissue and then placed on artificial surfaces with different stiffnesses [21]. Under chemically identical culture conditions, the morphology of these cells can vary from small and round to highly spread and polygonal at the two extremes of stiffness, but only on intermediate stiffnesses of 5–10 kPa, that mimic the stiffness of the native cardiac tissue, the cells acquire the elongated sarcomere-containing structures that allow them to rhythmically contract. In these and similar experiments, substrates of a constant stiffness are used for cell culture, but in living organisms tissue or extracellular matrix stiffness can change due to chemical remodeling or imposition of mechanical stresses. To achieve large, rapid, and reversible changes in substrate stiffness without chemically altering the substrate structure, magnetoelastic materials formed by embedding ferromagnetic particles into elastomers or hydrogels have recently been adopted for a variety of cell biological experiments.

Fig. 11.1
An illustration has a block labeled G = 50 to 50000 Pascal with E C M protein, P D M S or gel, and glass pointing to it. A set of 5 illustrations of cells cultured at 100 Pascal, 300 Pascal, 5 kilopascals, 10 kilopascals, and 30 kilopascals.

Top: Diagram of the use of soft substrates with controlled shear modulus (G) for cell culture. bottom: Effect of substrate stiffness on the morphology of cardiac myocytes. Adapted with permission from Chopra A, et al. J Biomech. 2012;45:824-31. Copyright 2021 Elsevier, Inc.

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Usually, these magnetoelastic substrates, like the example shown in Fig. 11.1, provide a surface on which cells can grow, and the elastic modulus of the substrate is altered by imposition of a magnetic field. A more recent advance has been to employ three-dimensional fibrous networks formed by the same protein filaments that form the extracellular matrices of many soft tissues, such as collagen or fibrin, and add ferromagnetic particles entrapped in the network meshes [17]. This allows cells to be cultured in a three-dimensional matrix that more closely mimics the setting of most cells in the body. Provided that the volume fraction of magnetic particles is sufficiently low, the particles themselves do not alter the structure or the rheology of the fibrous networks and provide the opportunity to change the effective stiffness that cells encounter in a three-dimensional network when a magnetic field is applied.

2 Rheological Properties of Dynamically Stiffening Soft Magnetoelastic Composites

In this study we summarize representative effects of applying uniform magnetic fields to elastomers and hydrogels containing ferromagnetic particles, with an emphasis on the strain dependence of the field-induced stiffening and a comparison of the differences between linear and nonlinear elastic networks. Analysis of the effect of magnetic fields on material stiffening shows that under the magnitudes of field strength and volume fraction of particles used in most such materials suitable for cell biology, the particles are largely immobile and trapped within the surrounding matrix, with the result that the stiffening effect is related to the generation of an array of magnetic dipoles within the network rather than application of local stress to the network. We illustrate the utility of magnetic stiffening in soft fibrous networks formed by collagen and fibrin, within which cells are embedded in a three-dimensional matrix. The rapid and reversible change in stiffness generated as the field is applied, without imposition of a local force on the cell, enables studies of both acute and chronic responses of cells to substrate stiffening. The most rapid response of the cell to a stiffened environment occurs within seconds and appears to involve activation of ion channels, that later lead to cell remodeling and changes in cell fate.

2.1 Magnitude of Magnetic Stiffening of Polydimethylsiloxane Containing Carbonyl Iron Particles

The magnitude of the stiffening in an elastomer substrate caused by a uniform magnetic field is seen in the examples shown in Fig. 11.2. In this study polydimethylsiloxane (PDMS) elastomers with a shear modulus of approximately 5 kPa were formed with 10% weight fraction of randomly distributed carbonyl iron spheres with a diameter of approximately 3 microns (Sigma-Aldrich C3518). As seen in Fig. 11.2a the shear modulus increases from its initial value of 5 kPa to approximately 20 kPa in the presence of the 400 mTesla magnetic field. A theoretical model that computes the additional resistance to shear deformation provided by a random array of magnetic dipoles that mimic those that would be formed by the carbonyl iron particles predicts that the shear modulus should rise with a square of the magnetic field magnitude [18]. The fit of this theory to the experimental data shows very close agreement, from which the magnetic susceptibility of the particles can be computed. This theory also predicts that along with the resistance to shear deformation, increasing magnetic fields will generate a normal force within the material, and Fig. 11.2b shows that this normal force also rises with the square of the field strength, as predicted by the theoretical model (blue arrows). In these experiments the elastomer was placed between two rigid plates within a rheometer, but in settings in which a magnetic field is generated by a permanent magnet placed beneath it in a cell culture dish, the resulting normal force can lead to wrinkling of the upper surface of the elastomer to which the cells adhere [11]. The magnitude of this wrinkling effect, which could perturb cell adhesion to the surface, depends on the relative magnitudes of the normal force and the shear modulus of the elastomer. Direct measurements of elastomers suitable for cell culture show that the surface roughness caused by high fields is on the order of 10–100 s of nm [11], but does not seem to perturb the cell morphology.

Fig. 11.2
A scatterplot of G prime versus H has plots along an increasing curve with an equation for G prime. A scatterplot of normal stress versus H has plots along an increasing curve with an equation for p.

Effect of magnetic field on the shear storage modulus (a) and normal stress (b) when a magnetic field is applied to crosslinked polydimethylsiloxane containing 10% by weight carbonyl iron beads

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Both theory and experiment show that the contribution of the ferromagnetic particle array in the magnetic field to the shear modulus is additive above the modulus of the elastomer in the absence of a field, or in the absence of ferromagnetic particles. The relative contributions from the magnetic particles and the underlying elastomer depend on the magnitude of the shear deformation. Figure 11.3 shows that an elastomeric material like PDMS, which exhibits nearly perfectly linear elastic response up to strains of at least 50%, becomes significantly strain softened after its stiffening by the magnetic field. In the presence of the particles but no magnetic field the shear modulus is nearly constant over the entire range of shear strains. After stiffening by the magnetic field, however, the shear modulus decreases significantly as shear strain magnitude is increased up to 100%. This softening is not the result of plastic deformation or damage to the network, because decreasing the shear strain magnitude to low values immediately leads to a higher value of measured shear modulus.

Fig. 11.3
A scatterplot of G prime versus strain has decreasing plots. From the top to the bottom, the plots are for 1 T, 0.6, 0.6, 0.4, 0.2, and 0. The label 10% carbonyl iron in P D M S is in the graph.

Shear modulus of magneto-elastomeric PDMS decreases with increasing strain in the presence of magnetic fields

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2.2 Theoretical Model for Stiffening of a Linearly Elastic Materials Containing Ferromagnetic Particles

The experimental results on magnetorheological properties of ferrogels can be rationalized by considering the dependence of their magnetic permeability tensor \(\mu_{ik}\) on the deformation \(u\) [22]:

$$\mu_{ij} = \mu_{0} \delta_{ij} + a_{1} u_{ij} + a_{2} u_{kk} \delta_{ij} .$$
(11.1)

The magnetostriction coefficients \(a_{1} ,a_{2}\) are calculated in [23] using a self-consistent mean field approach (the magnetic susceptibility dependence on the particle concentration is taken into account in \(a_{2}\)) and are given as

$$a_{1} = - \frac{2}{5}\left( {\mu_{0} - 1} \right)^{2} ;a_{2} = - \frac{1}{5}\left( {\mu_{0} - 1} \right)^{2}$$
(11.2)

The coefficient \(a_{2}\) plays a role only in compressible ferromagnetic media and will not be further considered.

Equations (11.1) and (11.2) allow calculation of the shear modulus dependence on the magnetic field. The volume density of magnetic torque determines the antisymmetric stress.

\(\sigma_{ij}^{a} = \frac{1}{2}e_{ijk} \left[ {\vec{M} \times \overrightarrow {{H_{0} }} } \right]_{k}\) [24].

In the magnetic field \(\overrightarrow {{H_{0} }} = \left( {0,0,H_{0} } \right)\) at a shear deformation \(u_{xz} = \frac{1}{2}\frac{{\partial u_{x} }}{\partial z}\) a transversal component of the magnetization, Mx, arises:

$$M_{x} = \frac{{a_{1} H_{0} }}{{2\mu_{0} }}\frac{{\partial u_{x} }}{\partial z}.$$

As a result, additional shear stress in the gel appears

$$\sigma_{xz}^{a} = \frac{{a_{1} H_{0}^{2} }}{{16\pi \mu_{0} }}\frac{{\partial u_{x} }}{\partial z}$$

Thus, the effective shear modulus of the gel increases by the magnitude

$${\Delta }G^{\prime} = \frac{{\left( {\mu_{0} - 1} \right)^{2} H_{0}^{2} }}{{40\pi \mu_{0} }}$$

Another effect which is possible to measure by rheometry is the normal force acting on the plates holding the sample under the action of the normal field. The general expression for the magnetic energy of the body \(- \frac{1}{2}\smallint \vec{M} \cdot \overrightarrow {{H_{0} }} dV\) [22] in the case of the gel layer with area \(S\) and thickness \(h\) gives

$$E = - \frac{1}{8\pi }H_{0}^{2} Sh\frac{{\mu_{zz} - 1}}{{\mu_{zz} }}$$

Its variation from the isotropic case at deformation \(u_{zz} = \frac{\xi }{h}\) (\(\xi\) is the displacement of the upper plate of the rheometer) gives

$$\delta E = - \frac{{a_{1} H_{0}^{2} }}{{8\pi \mu_{0}^{2} }}S\xi$$

Therefore, the force per unit area of the plate \(F\) is

$$F = - \frac{{\left( {\mu_{0} - 1} \right)^{2} H_{0}^{2} }}{{20\pi \mu_{0}^{2} }}$$

It may be noted that accounting for the elastic energy of an incompressible gel with volume V, \(V\frac{{3Gu_{zz}^{2} }}{2}\) for shrinking deformation of the sample in the form of a disk of large radius we obtain

$$\left( {M_{0} = \frac{{\left( {\mu_{0} - 1} \right)H_{0} }}{{4\pi \mu_{0} }}} \right)$$
$$u_{zz} = - \frac{{4\pi M_{0}^{2} }}{15G},$$

This expression coincides with that derived in [25] for the description of the Procrustes effect. The quadratic dependencies of shear modulus increase and the normal force on the applied field correspond well with the experimental data.

2.3 Magnetoelastic Materials Formed by Fibrous Biopolymer Networks

Magnetoelastic materials formed by incorporation of ferromagnetic particles into polymeric elastomers have now been extensively used to study the effects of stiffness changes on induction of differentiation pathways in precursor cells, phenotypic changes in muscle cells, and other applications [6, 7, 9, 18, 26, 27]. Some limitations of solid elastomers are that cells can only be cultured on their surfaces, and the shear modulus of the elastomer before application of the field is generally above a few kPa, and stiffer than some of the softest tissues such as embryos, bone marrow, brain, or fat.

To circumvent the limitations of magnetoelastic elastomers, similar materials have also been formed by adding ferromagnetic particles to hydrogels. Some of the first examples were hydrogels formed by polyacrylamide or carrageenan, with initial elastic moduli below a kilopascal [6, 28]. Since the stiffening effect of the magnetic field on the ferromagnetic particles is additive to the initial elastic modulus of the host material, the fractional change produced by the same volume fraction of particles is much greater when the initial elastic modulus is low. Additionally, a change in elastic modulus that is adequate to alter cell phenotype can be produced by a lower volume fraction of ferromagnetic particles when the initial substrate stiffness is low.

To take a step closer to the native extracellular matrix environment of cells in three-dimensional cultures, magnetoelastic materials have also recently been made using the native biopolymer networks formed by fibrin or collagen that constitute the material into which cells infiltrate during wound healing or that surround the cell in homeostasis [17, 18]. In addition to providing a more native environment for the cell, the large mesh size and biocompatibility of fibrin and collagen enable cell culture in three-dimensional environments. An example of the formation of an optically translucent magnetoelastic material from a biopolymer is shown in Fig. 11.4. Carbonyl iron particles are suspended within culture medium prior to mixing with a solution of fibrinogen, the protein that polymerizes to form a blood clot after its activation by thrombin. Before fibrinogen is activated by the thrombin, the mixture can be poured into a mold or microfluidic chamber. Figure 11.4a shows a cylindrical fibrin gel with 0.5 percent carbonyl iron particles by weight. The sample is grey but partly transparent. The resulting fiber network entraps the carbonyl iron particles within it, as shown by the scanning electron micrograph in Fig. 11.4b.

Fig. 11.4
A photograph of a fibrin gel with a scale at the back. An electron micrograph with particles observed at 5 micrometers. A graph of G prime versus H has a concave up increasing curve and scattered plots along an increasing curve. A scatterplot of G prime versus strain has increasing plots for H = 0.4 T and H = 0.2 T.

a Photograph of a 10 mg/ml fibrin gel with 0.5% carbonyl iron beads. Scale: 1 mm between lines. b scanning electron micrograph of fibrin gel with ferromagnetic particle (arrow) at right. Scale bar: 5 µm. c effect of magnetic field on shear modulus of magnetoelastic fibrin gel. D. Strain dependence of magnetoelastic fibrin in presence of magnetic fields

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Figure 11.4c shows that the shear modulus of a fibrin gel can be increased greatly by relatively modest magnetic fields. A fibrin gel with 10% carbonyl iron particles that has an initial shear modulus of 200 Pa is stiffened by a factor of 40 to over 8 kPa for the same magnitude of field that increased the stiffness of the initially stiffer PDMS elastomer by only a factor of 4 (Fig. 11.2). In addition, the nonlinear rheology of the semiflexible fibrin gel is also evident in the response of the magnetically stiffened fibrin gel to increasing shear strain magnitudes. Figure 11.4d shows that the initial decrease in shear modulus of the magnetically stiffened fibrin/bead composite switches to shear strain stiffening, at strains above 10%, consistent with the strong increase in shear modulus of the fiber network. Several aspects of the elastic response of magnetically stiffened fibrin gels are not yet explained by theoretical models. For example, the rise in shear modulus of the fibrin/bead composite does not follow a quadratic relation to the magnetic field strength, as seen for the magnetoelastic PDMS elastomer (Fig. 11.2). Similarly, the strain dependence of the magnetoelastic effect is also not evident from current theoretical models.

3 Effects of Magnetoelastic Substrate Stiffening on Live Cells in 3D

The open fibrous meshwork of soft fibrin or collagen gels combined with the use of low volume fractions of carbonyl iron particles that permit imaging by light microscopy within the 3D network/cell composite creates new opportunities to study mechanobiology over a range of time scales in 3D environments that are close to the physiological setting. A schematic diagram of the method is shown in Fig. 11.5. The example shown here is for fibrin gels, but the same method can be used by adding cold acidic collagen in place of fibrinogen and initiating its polymerization by neutralization and warming to 37 °C as the cells are added. The utility of this system is demonstrated by effect of magnetic stiffening on the rapid change in intracellular Ca2+ flux when stiffness is suddenly changed.

Fig. 11.5
A flowchart illustration is as follows. A set of sample tubes has Iron beads, add D M E M, add fibrinogen, add thrombin and cells, mold or microfluidic chamber.

3-D magnetoelastic cell culture system. Carbonyl iron particles are suspended in cell culture medium such as DMEM and then mixed with fibrinogen. The enzyme thrombin is then added at the same time that cells are added, and the polymerizing fibrin network can be poured into molds or microfluidic chambers before the fibrin gels

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3.1 Magnetic Stiffening of Magnetoelastic Fibrous Networks Occurs in the Absence of Network Deformation by the Field

Figure 11.6a shows that fluorescently labeled live cells can be clearly imaged within a magnetoelastic collagen gel, and displacements of the beads can be used to measure a strain field caused by cells contracting the matrix. Figure 11.6b shows that the shear modulus of the magnetoelastic collagen gel containing only 0.5% carbonyl iron increases from ~ 0.5 to 1.6 kPa when a 400 mT field is applied and returns to its baseline level with the field is removed. The carbonyl iron microparticles can also be visualized within the hydrogel using fluorescent labeling to assess whether application of the magnetic fields causes displacement independent of cell contractility. As suggested by the theoretical model for stiffening in linear elastic materials, the contribution of the microparticles to the shear modulus of the fibrous network is predicated on the dipoles being entrapped in the network. Figure 11.6c demonstrates that application of a 250 mT field induces displacement of the particles prior to hydrogel polymerization, but there is minimal particle displacement once the collagen has polymerized. This result verifies that the field does not displace the particles once they are entrapped in the biopolymer network. In stiffer materials like PDMS, there is less potential for microparticles to move in the presence of a field. However, in biological materials like collagen and fibrin, the shear moduli may be too low to constrain the particles, mitigating the effect of the microparticles on network stiffness. These results indicate that at least in the case of 2 mg/mL collagen (storage modulus ~ 30–50 Pa), a magnetic field that is large enough to substantially increase hydrogel stiffness does not cause microparticle displacement. However, future studies using softer hydrogels or larger fields should verify that the microparticles are not displaced by application of the field.

Fig. 11.6
A microscopic scan has dotted particles with a few stains. A graph of storage modulus versus time has a fluctuating curve along a column for 400 m T and label 0.5 w t % M P. Two fluorescent scans labeled pre-polymerization and post-polymerization have slightly large particles in the first and similar, round particles in the second.

Copyright 2021 American Chemical Society

a ifeACT-transfected hCASMC in hydrogels consisting of 5 mg/mL collagen, 1 mg/mL HA, and 0.5 wt% carbonyl iron particles. hCASMC are labeled with the live molecular probe, LifeACT (red), and green denotes the local strain caused by cell contractility. b change in shear modulus of hydrogel as uniform 400 mT field is applied and removed. c fluorescently labeled microparticles in a 2 mg/mL collagen hydrogel before and after polymerization in the presence of a 250 mT field. Scale = 50 µm. Adapted with permission from K.A. Tran, et al., ACS Appl. Mater. Interfaces 13, 20,947–2094759 (2021).

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3.2 Rapid Response of Cells to Sudden Stiffness Changes Involves Calcium Ion Fluxes

The ability to image cells within biopolymers containing carbonyl iron microparticles and the rapid and reversible changes to network mechanical properties made possible by the application of a magnetic field provide a unique glimpse into the dynamic cell response to extracellular matrix stiffness. As Fig. 11.6b indicates, the hydrogel storage modulus is increased to a steady value within milliseconds of the application of the magnetic field. Previous studies have used this rapid change to characterize how quickly cells respond to a shift in extracellular matrix mechanics [18]. Figure 11.7a shows averaged calcium transients from cells treated with Fluo-4, a calcium sensitive fluorescent dye, in the presence of an intermittent magnetic field. The transient is significantly different during the time when the field is applied (red) compared to when there is no field (blue): both in the overall rate as well as the initial slope of the transient (Fig. 11.7b, c). This result indicates the near instantaneous response of the cells to the field, as well as the reversibility of this response. There does not appear to be any inertia in the calcium flux in the cells: once the field is removed and the hydrogel stiffness decreases, the transients return to baseline levels.

Fig. 11.7
A graph of F minus F 0 over F max minus F 0 has two right-skewed curves. A scatterplot of calcium influx rate versus time has scattered plots. A column chart with error bars illustrates the initial slope of the transient with a taller column for w over M F and a shorter column for w over O M F.

Copyright 2021 American Chemical Society

Effects of dynamically altered hydrogel mechanics on cellular calcium transients. a Average calcium transients in carbonyl iron–seeded collagen hydrogels with field on (red) and off (blue). b calcium influx rate (slope of calcium transients) as magnetic field is turned on and off every 150 s. c average initial slope of calcium transients in matrices with or without stiffening by magnetic field. *P < 0.01. (n = 5 cells per condition, 10 transients for MF on and 5 transients for MF off per condition). Adapted with permission from K.A. Tran, et al., ACS Appl. Mater. Interfaces 13, 20947–20959 (2021).

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4 Conclusion

These results validate the ability of magnetically active biopolymer hydrogels to interrogate the dynamics of cell mechanics. The effect of the magnetic field on calcium handling demonstrates that cells respond rapidly to changes in hydrogel stiffness, but a host of questions remain about whether the acute response is limited to cytoskeletal-mediated changes or whether gene transcription or translation is also affected in the following seconds and minutes. Future work can also include studies to determine the effect of the magnetic field on the viscoelastic properties of biopolymers like collagen and fibrin. In contrast to linear elastic materials like PDMS and polyacrylamide, collagen and fibrin have substantial viscous dissipation that is also affected by the application of a magnetic field. Using magnetic fields to tune the viscoelastic properties would provide new avenues to understand cell response to changes in their physical surroundings.