Name: finite element modelling of a cellular electric microenvironment
Uploaded: 2021-05-18T12:30:00
Duration: 8 min 23 s
Description: Watch this Scientific Journal Video about Finite Element Modelling of a Cellular Electric Microenvironment at JoVE.com

This work was supported by the 4-year Wellcome Trust PhD Programme in Quantitative &#38; Biophysical Biology

The authors have nothing to disclose.

The proposed protocol suggests a uniform modelling solution for natural and artificial scaffolds and highlights the need to consider the nanostructure of fibrous scaffolds when inspecting the effects of EF on cells seeded onto such materials. Although a coarse approximation for the EF intensity (electrode potential difference divided by the distance between the electrodes) would lead us to expect a field strength of 100 mV/mm, simulations predict stationary field strengths up to 30% higher in different areas of the mat (Figure 5). This result should be of interest in ES experiment design and data interpretation, as cell death can be caused by too strong EFs. Exposing the electrical microenvironment would enable a direct correlation between ES and cellular development. While several studies present detailed morphology analysis of the used scaffolds33,43,59, they do not investigate the interplay between the structure, electrical properties of the materials and the EF. This protocol can enable this link, as parameters such as fiber radius, coating layer thickness, distance between fibers and electrical properties of the component materials can be modified according to each experiment by changing the Global Definitions at steps 1.2 and 1.3. Hence, customized 3D spatially resolved charge and current density predictions can be made for both static and dynamic ES regimes.
Scaffold design optimization can be targeted through the RNC and RNCd models with wide parameter range explorations, scaling the proposed morphologies or parts of them. Alternatively, other scaffold configurations can be investigated with the proposed protocol by changing the Array types from Linear to Three-dimensional in section 1.6.5 and adapting Scaffold Geometry in section 1.6.2. However, scaffold optimization cannot be done without an objective. While for tissue engineering purposes the main focus is cell fate, a clearer picture on what stimuli are its main determinants is essential if its reliable control is desired. Charge and current density are good descriptors of cellular electric microenvironments as they show the interplay between the EF and the electrical properties of the different component materials of complex scaffolds such as ECM. The protocol shows how to compute predictions for those metrics given a nanofibrous scaffold geometry and highlights the importance of the alignment angle of the fibers with the EF. Predictions of charge and current density could then be linked to cellular development and thus scaffold and ES regimes may then be optimized for specific tasks.
Interestingly, a study shows that EF exposure generated mechanical stress more than double in strength in composite films with nanofibers perpendicular to the external EF compared to films with parallel alignment60. The reported mechanical stress could be a result of Coulomb forces acting between charged fibers, predicted by the rough model simulations (RC, RNC, RNCd) (Figure 6). While these simulations could be useful in investigating this hypothesis, it must be noted that the reported experimental results were obtained in a system with capacitive coupling, and the simulation presents direct coupling. 
A limiting factor towards future possible uses of the protocol to estimate a cellular input signal is parameter uncertainty. Geometric uncertain parameters are coating layer thickness and distance between fiber cores. The first one could be inferred by finding the value that leads to a bulk impedance that can be experimentally validated. The second one can be extracted from high resolution material scans. Parameters describing the physical properties of the materials are also affected by uncertainty. However, the electric conductivity and dielectric constant of exemplified materials differ far more than experimental measuring precision (Table 2). Therefore, the reported effects would be maintained despite moderate measurement errors.
The results show how not enough model complexity might hide relevant information. It is important to acknowledge that the protocol simulates a simplified version of the physical phenomenon taking place as it does not account for the different nature of materials involved in the process -conductor (electrodes), semiconductor (coating), dielectric (fiber cores) and electrolytic (surrounding substance) - that are able to influence charge transport. This issue can be accounted for in future model expansions by adding energy transfer delays at the interfaces (i.e., Faradic reactions) and ion transport delays within the electrolyte. Adding complexity should however be guided by experimental validation, as a simple model that reproduces most of what is observed is more useful than a remarkably accurate one that adds little more information but is deeply sensitive to many constituent parameters' uncertainty.
As the end goal of tissue engineering is to create bioreactors that not only mimic one or two aspects of in vivo environments, but replicate and control all cellular developmental cues61, electromagnetic and mechanical in silico models as well as models of heat transfer between bioreactor components will need to be combined. In a subsequent modelling phase, coupling phenomena between those interactions such as ohmic heating, electrolytic fluid flow, morphological scaffold deformations in response to electrical stimulation60 and piezoelectricity62 can also be added. However, models should be merged only after each one has been experimentally validated. This way, we can gain a better understanding of each component's influence in the cellular microenvironment, and how stimuli can be optimized. 
If the proposed model is experimentally validated, it can be combined with models of biological cells - Figure 1. Charge density patterns and modulations could asymmetrically influence specific ion pumps' activity, impact attachment to the fiber of proteins driving membrane adhesion63 and hence guide migration, proliferation patterns and morphogenesis64. Exploring those hypotheses is the way forward in understanding the mechanisms underpinning tissue and cell responses to ES.

ES is the use of EFs with the aim of controlling biological cells and tissues. Its mechanism is based on the physical stimulus transduced to the cell when the biomolecules inside and surrounding it are exposed to an externally generated voltage gradient. Charged particles are engaged in an organized motion governed by Coulomb&#39;s law, generating drag forces upon the uncharged particles. The resulting fluid flow and charge distribution alter cell activities and functions such as adhesion, contraction, migration, orientation, differentiation and proliferation1 as the cell attempts to adapt to the change in the microenvironmental conditions.
As EFs are controllable, non-invasive, non-pharmacological and shown to have an effective impact on essential cell behavior, ES is a valuable tool for tissue engineering and regenerative medicine. It has been successfully used to guide neural2, skeletal3, cardiac muscle4, bone5 and skin6 development. Moreover, as it enhances iontophoresis7, it is used as an alternative or complementary treatment to conventional pharmacological ones. Its efficiency in pain management is still debated as higher quality clinical trials are awaited8,9,10. Nevertheless, no adverse effects were reported and it has the potential to improve patient welfare11,12,13,14,15.
While only clinical trials can give a definitive verdict for the efficacy of a procedure, in vitro and in silico models are required to inform the design of predictable ES treatment as they offer stronger control over a wider range of experimental conditions. The investigated clinical uses of ES are bone regeneration16,17, recovery of denervated muscles18,19, axonal regeneration after surgery20,21, pain relief22, wound healing23,24,25 and iontophoretic drug delivery26. For ES devices to be widely introduced on all possible target applications, clinical trials have yet to establish stronger evidence for efficient treatment. Even in domains where both in vivo animal and human studies consistently report positive outcomes, the great number of reported methods coupled with too little guidance on how to choose between them and high acquisition price deters clinicians from investing in ES devices27. To overcome this, the target tissue can no longer be treated as a black box (limit of in vivo experiments) but must be seen as a complex synergy of multiple subsystems (Figure 1).
Multiple ES experiments have been carried out in vitro over the years28,29,30,31,32,33,34. Most of these only characterize the ES through the voltage drop between the electrodes divided by the distance between them - a rough approximation of the electric field magnitude. However, the electric field itself only influences charged particles, not cells directly. Also, when multiple materials are interposed between the device and the cells, the rough approximation may not hold.
A better characterization of the input signal requires a clear view on how the stimulus is transduced to the cell. Main methods of delivering ES are direct, capacitive and inductive coupling35,36. Devices for each method differ with electrode type (rod, planar or winding) and placement relative to the target tissue (in contact or isolated)35. Devices used in vivo for longer treatments need to be wearable, thus the electrodes and most times the energy source are either implanted or attached to the skin as wound dressings or electroactive patches. The generated voltage gradient displaces charged particles in the treatment area.
As it impacts the resulting charged particle flow in the vicinity of the cells, scaffold structure is of utmost importance in the design of ES protocols. Different charge transport configurations arise if the platform material, synthesis technique, structure or orientation relative to the voltage gradient change. In vivo, the availability and movement of charged particles is impacted not only by cells but also by the collagen network and interstitial fluid composing the supporting ECM. Engineered scaffolds are increasingly used to better recreate natural cell microenvironments in vitro1,35. Concurrently, the ECM is a complex natural scaffold.
Artificial scaffolds are based on metals, conducting polymers and carbon, engineered with a focus on balancing biocompatibility with electrochemical performance and long-term stability36. One versatile scaffold type is the electrospun fibrous mat that offers a controllable nanoscale topography. This can be engineered to resemble the ECM, thus deliver similar mechanical cues that aid regeneration of a wide range of tissues37. To significantly impact ES, the mats need to be conductive to some degree. However, conductive polymers are difficult to electrospin and blending with insulating carriers limits the conductivity of the resulting fibers38. One solution is polymerizing a conductive monomer on the surface of a dielectric fiber, resulting in good mechanical strength and electrical properties of the end product38. An example is coating silk electrospun fibers with the semi conductive PEDOT-PSS39. The combination of mechanical and electromagnetic cues significantly accelerates neurite growth40,41,42. Neurites follow scaffolds fibers alignment, and elongate more after exposure to an EF parallel to the fibers than to a vertical one43. Similarly, alignment of fibrous scaffolds to the EF also promotes myogenic maturation33.
The ECM is mainly composed of fibrous-forming proteins44, out of those collagen type I being the major constituent in all animal tissues apart from cartilage (rich in collagen type II)44. Tropocollagen (TC), triple helical conformation of polypeptide strands, is the structural motif of collagen fibrils45. Transmission electron microscopy and atomic force microscopy images of collagen fibrils show a D-periodic banded pattern46 explained by the Hodge &#38; Petruska model47 as regular arrays of TC gaps and overlaps45. Tendons are composed of an aligned collagenous fibrillar matrix shielded by a non-collagenous highly hydrophilic proteoglycan matrix48,49. Decorin is a small leucine-rich proteoglycan (SLRP) able to bind the gap regions of collagen fibrils and connect with other SLRPs through their glycosaminoglycan (GAG) side chains49. Studies done on tendons show that their electrical properties change significantly when hydrated50,51, charge transport mechanism changing from protonic to ionic as hydration level increases51. This suggests that electric conduction along a collagen type I fibril could be enabled by a Decorin-water coat, with gap and overlap regions having different electrical conductivities and dielectric constants.
As identical recreation of the ECM by artificial scaffolds is improbable, the knowledge producing synergy between in vivo and in vitro enabled by translatable results seems to be at a dead end. In silico modelling not only re-enables translation between the two, but also adds important benefits in characterizing the unknown processes involved in ES. Comparing the in vivo observations with the in vitro can bring information on the coupling strength between the target tissue and the rest of the organism but does not uncover current knowledge limits. The unknown can be exposed by observing the difference between what is expected to happen based on the current knowledge and what happens. In silico experiments based on mathematical modelling allow splitting the process into known and unknown subprocesses. This way, phenomena not accounted for in the model come to light when in silico predictions are compared to in vitro and in vivo experiments.
Forming and testing hypotheses regarding the underlying mechanism(s) of how cells and tissues are affected by electrical fields is hindered by the great number of parameters52 that need to be tested separately. To define representative experimental conditions, the ES process must be split in subprocesses (Figure 1) and dominant input signals affecting cell behavior must be identified. Models representing fundamental physical effects of ES on cells describe the domain that couples the EF with the cell - that of charged particles53. The behavior of particles exterior to the cell depends on the microenvironment and can be investigated separately from the cell. The dominant input signal for the cell is the subset of ES device outputs that causes the greatest degree of variability in the cell response. The smallest subset of the full experimental parameters that can generate variations in all the dominant cell input signals can be used to decrease the parameter space dimension and the number of test cases.
The input of the biological ES target model must be a subset of the output signals produced by the ES device that are useful in describing the physical effects of ES on cells. A simple bioreactor with direct coupling has the same structure as electrolytic electrochemical cells. Models of those show the primary (accounting for solution resistance), secondary (also accounting for faradic reactions) or tertiary (also accounting for ion diffusion) current density distribution. As complexity translates into computational cost, the simplest model is most suitable for parameter space explorations. Simulations of fibrous composites motivated by material properties54 focus on bulk material properties as a result of complex micro-architecture, hence cannot describe local effects of EF exposure. Existing in silico models, motivated by ES, focus on the biological sample, be it a single cell immersed in a homogenous medium55,56,57, or complex tissues with homogenous extracellular space58. Charge and current density (Figure 2) can act as interface signals between models of the ES device and the biological sample, or between different components of the ES device. The proposed FEM based protocol uses the equations described in Figure 2 and was used to study how scaffold dependent parameters can be used to modulate those two signals, independent of the EF generated by a direct coupling setup. Results stress that it is necessary to account for scaffold or ECM electrical properties when investigating how ES impacts target cells.

<table border="1" fo:keep-together.within-page="1" fo:keep-with-next.within-page="always">
	<tbody>
		<tr>
			<td>Comsol multiphysics 5.2 AC/DC module</td>
			<td>COMSOL</td>
			<td>-</td>
			<td>FEM modelling software</td>
		</tr>
	</tbody>
</table>

finite element modelling of a cellular electric microenvironment

Clinical studies show electrical stimulation (ES) to be a potential therapy for the healing and regeneration of various tissues. Understanding the mechanisms of cell response when exposed to electrical fields can therefore guide the optimization of clinical applications. In vitro experiments aim to help uncover those, offering the advantage of wider input and output ranges that can be ethically and effectively assessed. However, the advancements in in vitro experiments are difficult to reproduce directly in clinical settings. Mainly, that is because the ES devices used in vitro differ significantly from the ones suitable for patient use, and the path from the electrodes to the targeted cells is different. Translating the in vitro results into in vivo procedures is therefore not straightforward. We emphasize that the cellular microenvironment's structure and physical properties play a determining role in the actual experimental testing conditions and suggest that measures of charge distribution can be used to bridge the gap between in vitro and in vivo. Considering this, we show how in silico finite element modelling (FEM) can be used to describe the cellular microenvironment and the changes generated by electric field (EF) exposure. We highlight how the EF couples with geometric structure to determine charge distribution. We then show the impact of time dependent inputs on charge movement. Finally, we demonstrate the relevance of our new in silico model methodology using two case studies: (i) in vitro fibrous Poly(3,4-ethylenedioxythiophene) poly(styrenesulfonate) (PEDOT-PSS) scaffolds and (ii) in vivo collagen in extracellular matrix (ECM).

The proposed model describes features of a composite mat with parallel fibers, immersed in a conductive substance and exposed to an externally generated electric potential gradient. Simulations show that accounting for the different components of a scaffold is important on a microscale and explore how change in alignment angle (input signal) of the fibers to the EF can generate variability in the current and charge density (output signals) in the vicinity of the fibers.
Five different geometrical complexity stages are presented, each having an effect upon the simulation result:&#160;smooth conductive slab (SC), smooth slab with non-conductive embedded fibers (SNC), rough conductive composite (RC), rough composite with non-conductive embedded fibers (RNC), rough composite with non-conductive embedded fibers and two types of periodic coating (RNCd) (Figure 3). Section 1.5 of the protocol presents the steps to importing the geometries in a project and section 1.6 shows how to build those step by step. The first two models do not account for surface morphology. SC and RC do not account for the fiber core dielectric properties. The RNC is the proposed model for nanofibrous artificial scaffolds, while RNCd is the proposed model for an ECM segment.
Minimization of computational cost was accomplished by reducing the ES device geometry to a model unit volume representing the microenvironment. While an ES device and scaffold&#39;s width and length can easily be at the order of a few centimeters, the containing fibers&#39; diameter is usually lower than a micron. Here, we use a scaffold cut comparable to the fiber diameter to reduce the computational cost induced by the aspect ratio and highlight the effect of the scaffold&#39;s fibrous nature on the electric microenvironment. The rest of the ES device is replaced with electric potential boundary conditions chosen so that a rough approximation for the magnitude of the electric field is 100 V/m, a frequently reported stimulation parameter. Moreover, a unit volume with five parallel fibers - as the one used in simulations, presented in Figure 3 - is assumed to be representative of a whole planar fibrous mat. Three types of fibers can be distinguished in a 1D array: interior central (with the longitudinal symmetry plane of the scaffold splitting them in half), interior transitory (with lateral surface surrounded by other fibers but with asymmetrical sides), and exterior (at the edge of the scaffold). Five is the minimum number of fibers required in order to include all the three types defined.
The model mesh element size requires special attention as it may impact simulation results and thus fail to expose important effects&#160;(Figure 4). This is a general rule of the finite element method and an implication of the Nyquist-Shannon sampling theorem. The faster the essential simulation signals fluctuate in space the smaller the mesh elements need to be to produce a loyal representation of the phenomenon. On the other hand, the smaller the element, the greater the total number of model building blocks and the computational cost. The adaptive mesh refinement set up in section 2.1 is a good and facile method to balance those opposing objectives by decreasing the element size only where and as long as this operation produces a significant change.
A model that is too simplistic can fail presenting important effects (Figure 5,6).&#160;Simulations show that accounting for surface morphology and scaffold component electrical properties is not redundant in predicting electric microenvironments. While surface morphology has a direct impact on the stationary EF (compare SC and SNC with RC, RNC and RNCd), a comparison between RC and RNC predictions shows that nonconductive fiber cores amplify this effect. From the point of view of modelling cellular electric microenvironments on nanofibrous scaffolds, the SC, SNC and RC models are thus sub-optimal. However, it is good practice to incrementally add complexity as comparisons between the different stages help indicate what features give rise to specific effects.
Model complexity impacts current and charge density change with fiber alignment to the EF. The proposed protocol helps highlight the effect (Figure 5,6). While the SC model shows no variation in the proposed metrics when its alignment to the electric potential gradient is changed, the RNC model simulations predict a powerful contrast between the mat unit with fibers aligned to the EF and the one with fibers perpendicular (Figure 7). When the non-conductive cores come in the way of the current flow, they form periodic dams that lead to alternating regions of high and low charge density.
Dynamic ES regimes can be simulated with time dependent studies. Videos in supplementary files show predictions made for a sinusoidal input voltage on a full artificial scaffold model (RNC), with fibers parallel or perpendicular to the electric potential gradient. Small currents along the fibers perpendicular to the EF appear when charge is released from the scaffold as the EF magnitude decreases. This shows that stimulation could occur not only while the external EF is present, but also right after it is disconnected - See supplementary files for videos.
<img alt="Figure 1" class="xfigimg" src="/files/ftp_upload/61928/61928fig01.jpg" /> 
Figure 1: Hierarchical block diagram of modelling - advantages and limitations of modelling with in vivo and in silico models. Block color marks blocks on the same hierarchical level. Lower rank blocks are included in higher rank ones. Block stroke color marks possibility to include the block into a certain type of model - coupling with other system blocks do not have yellow in their stroke, as they are not components for in vitro models. Bullets act like valves and signify controllability of the block. When a valve is ON, signal can pass through all arrow paths in the subordinate subsystems that have the color of the valve in their stroke. Interpretation of the diagram: the ES process is composed of the stimulation device and biological target, each with several inter-connected deterministic or stochastic sub-processes that cannot be separated in vivo or in vitro, thus they have no red or yellow valve. Stochastic processes also intervene on the interface between the simulation device and biological sample when they are both stimulated. An in vitro model decouples the system of interest (i.e., skin segment) from the rest of the organism. Thus, only intrinsic processes of the system of interest topped by stochastic processes of different nature can be observed. However, the different intrinsic processes involved cannot be stimulated and identified separately. The in silico models are parametric for known components - their behavior is expected to be of a certain shape - and non-parametric for the unknown - as there is no mechanistic reason to give credence to a certain extrapolation. All the in silico components can be simulated separately or in different combinations, allowing the portrayal of different hypothesis. <a href="https://www.jove.com/files/ftp_upload/61928/61928fig01large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<img alt="Figure 2" class="xfigimg" src="/files/ftp_upload/61928/61928fig02.jpg" style="margin-bottom: -10px;" /> 
Figure 2: (A) Coulomb&#39;s Law (B) Electric potential field and mobile probe charge (C) Electric current (D) Charge density (E) Current density (F) Equation of continuity (G) Charge conservation law. (A) Electrically charged stationary particles q and Q interact electrostatically through Coulomb&#39;s force <img alt="Equation 1" src="/files/ftp_upload/61928/61928eq01.jpg" style="margin-bottom: -10px;" />. (B1) Each charged particle Q generates a scalar field called electric potential at all positions <img alt="Equation 2" src="/files/ftp_upload/61928/61928eq02.jpg" style="margin-bottom: -5px;" /> in space: <img alt="Equation 3" src="/files/ftp_upload/61928/61928eq03.jpg" style="margin-bottom: -10px;" />. The maximum work required to move another charged particle q from its position <img alt="Equation 4" src="/files/ftp_upload/61928/61928eq04.jpg" style="margin-bottom: -5px;" /> is the product between the charge q and the electric potential generated by Q at position <img alt="Equation 4" src="/files/ftp_upload/61928/61928eq04.jpg" style="margin-bottom: -5px;" />. The electric potential field generated by multiple particles is the sum of the fields generated by each individual particle. (B2) A stationary field with fixed generator particles&#160;q and Q, acts with a <img alt="Equation 5" src="/files/ftp_upload/61928/61928eq05.jpg" style="margin-bottom: -10px;" /> upon a probe particle with positive charge qp. In response, qp moves to minimize its position&#39;s electric potential. To describe the motion of qp, one can derive <img alt="Equation 5" src="/files/ftp_upload/61928/61928eq05.jpg" style="margin-bottom: -10px;" /> and the electric field <img alt="Equation 6" src="/files/ftp_upload/61928/61928eq06.jpg" style="margin-bottom: -10px;" /> from the electric potential field: <img alt="Equation 7" src="/files/ftp_upload/61928/61928eq07.jpg" style="margin-bottom: -15px;" />. (C) Multiple mobile positively charged probe particles uniformly released in a stationary electric field follow an organized motion. To track the charge configuration without tracking every particle, one can specify at every instant: (D) how space is occupied by particles, assigning a charge density <img alt="Equation 8" src="/files/ftp_upload/61928/61928eq08.jpg" style="margin-bottom: -10px;" /> to each infinitesimal volume, according to Gauss&#39;s Law, and (E) how particles pass through the boundary surfaces between neighbouring infinitesimal volumes, assigning a current density <img alt="Equation 9" src="/files/ftp_upload/61928/61928eq09.jpg" style="margin-bottom: -10px;" /> to each boundary according to Ohm&#39;s Law. (F) Charge and current density evolve co-dependently according to the Equation of continuity, as non-uniform particle displacement leads to either accumulation or loss of particles in a certain volume. (G) Within an isolated system, the Charge conservation law prevails and there is no inflow or outflow of charged particles. Notations used:- q,Q,qp charge and name of the charged particle; <img alt="Equation 10" src="/files/ftp_upload/61928/61928eq10.jpg" style="margin-bottom: -10px;" /> - Euclidian norm of the position vector; k - Coulomb&#39;s constant; <img alt="Equation 11" src="/files/ftp_upload/61928/61928eq11.jpg" style="margin-bottom: -5px;" />- gradient operator, &#949;a - absolute permittivity of medium; &#963; - conductivity of medium. <a href="https://www.jove.com/files/ftp_upload/61928/61928fig02large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<img alt="Figure 3" class="xfigimg" src="/files/ftp_upload/61928/61928fig03.jpg" /> 
Figure 3: Five different levels of complexity for a fibrous mat. SC- smooth with conductive embedded fibers, the simplest model, not accounting for surface morphology or different properties of the constituent components; SNC- smooth with non-conductive embedded fibers; RC- rough with conductive embedded fibers,&#160;accounting for surface morphology but not for different component properties; RNC- rough with non-conductive embedded fibers, full proposed model of nanofibrous artificial scaffolds; RNCd- rough with non-conductive embedded fibers coated with two different materials, full proposed model for a sheet of collagen fibers. Length unit used: nanometers. <a href="https://www.jove.com/files/ftp_upload/61928/61928fig03large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<img alt="Figure 4" class="xfigimg" src="/files/ftp_upload/61928/61928fig04.jpg" /> 
Figure 4: Example results of the adaptive mesh refinement and the resulting charge density following the simulation. (Left) Automatically generated mesh with extra coarse tetrahedral elements; (Right) Initial mesh adaptively refined during stationary study; smaller elements are required for an accurate result in the areas where simulated signals have abrupt spatial changes. <a href="https://www.jove.com/files/ftp_upload/61928/61928fig04large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<img alt="Figure 5" class="xfigimg" src="/files/ftp_upload/61928/61928fig05.jpg" /> 
Figure 5: Fiber alignment angle to electric potential gradient impacts EF strength in surrounding cell culture media when enough complexity is accounted for. SC, SNC, RC, RNC and RNCd are the different levels of complexity for the fibrous mat model presented in Figure 3. Vertical axis marks the alignment angle of the fibers to the electric potential gradient. Abstract electrodes featured - bottom side with high electric potential and top side with low electric potential. <a href="https://www.jove.com/files/ftp_upload/61928/61928fig05large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<img alt="Figure 6" class="xfigimg" src="/files/ftp_upload/61928/61928fig06.jpg" /> 
Figure 6:&#160;Fiber alignment angle to electric potential gradient impacts space charge density in surrounding cell culture media when enough complexity is accounted for. SC, SNC, RC, RNC and RNCd are the different levels of complexity for the fibrous mat model presented in Figure 3. Vertical axis marks the alignment angle of the fibers to the electric potential gradient.&#160;Abstract electrodes featured-bottom side with high electric potential and top side with low electric potential. <a href="https://www.jove.com/files/ftp_upload/61928/61928fig06large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<img alt="Figure 7" class="xfigimg" src="/files/ftp_upload/61928/61928fig07.jpg" /> 
Figure 7: Charge movement is influenced by scaffold fiber alignment relative to the EF. Both panels illustrate steady state RNC model predictions. On the left side the fibers are parallel to the EF, while on the right side they are perpendicular.&#160;The light red to blue color volume marks charge density, while the arrow volume marks current density orientation. The color of the arrows corresponds to the current density norm. <a href="https://www.jove.com/files/ftp_upload/61928/61928fig07large.jpg" target="_blank">Please click here to view a larger version of this figure.</a>
<table border="1" fo:keep-together.within-page="1" fo:keep-with-next.within-page="always">
	<tbody>
		<tr>
			<td>Name</td>
			<td>Expression</td>
			<td>Description</td>
		</tr>
		<tr>
			<td>Ws</td>
			<td>10*Rc*med_ratio</td>
			<td>Scaffold width</td>
		</tr>
		<tr>
			<td>Ls</td>
			<td>10*Rc*med_ratio</td>
			<td>Scaffold length</td>
		</tr>
		<tr>
			<td>Hs</td>
			<td>2*Rf</td>
			<td>Scaffold height</td>
		</tr>
		<tr>
			<td>med_ratio</td>
			<td>1.5</td>
			<td>Ratio cell culture media to scaffold</td>
		</tr>
		<tr>
			<td>Rc</td>
			<td>278.5[nm]</td>
			<td>Fiber core radius</td>
		</tr>
		<tr>
			<td>r</td>
			<td>1.5</td>
			<td>Fiber core to coat ratio</td>
		</tr>
		<tr>
			<td>Rf</td>
			<td>Rc*r</td>
			<td>Fiber with coat radius</td>
		</tr>
		<tr>
			<td>theta</td>
			<td>90[deg]</td>
			<td>Fiber orientation angle</td>
		</tr>
		<tr>
			<td>Lf</td>
			<td>1.3*(Ls*cos(theta)+Ws*sin(theta))</td>
			<td>Fiber length</td>
		</tr>
		<tr>
			<td>tes</td>
			<td>1</td>
			<td>Ratio fibre core radius to distance between fibres</td>
		</tr>
		<tr>
			<td>n_1</td>
			<td>2*(fix((Ws/(2*cos(theta))-Rf)/(2*tes*Rc))+3)*(cos(theta)!=0)+1*(cos(theta)==0)</td>
			<td>Max number of fibers if theta&#60;=45</td>
		</tr>
		<tr>
			<td>n_2</td>
			<td>2*(fix((Ls/(2*sin(theta))-Rf)/(2*tes*Rc))+3)*(sin(theta)!=0)+1*(sin(theta)==0)</td>
			<td>Max number of fibers if theta&#62;45</td>
		</tr>
		<tr>
			<td>excess</td>
			<td>1.2+0.3*abs(sin(2*theta))</td>
			<td>First fiber relative offset from scaffold</td>
		</tr>
		<tr>
			<td>D</td>
			<td>Lf/5</td>
			<td>Coat periodicity</td>
		</tr>
		<tr>
			<td>prop</td>
			<td>0.46</td>
			<td>Length of first coat relative to periodicity D</td>
		</tr>
		<tr>
			<td>E</td>
			<td>100[mV/mm]</td>
			<td>Electric field magnitude</td>
		</tr>
		<tr>
			<td>V0</td>
			<td>E*Ls*med_ratio</td>
			<td>Terminal Voltage</td>
		</tr>
		<tr>
			<td>omega</td>
			<td>500[Hz]</td>
			<td>Time dependent study Voltage frequency</td>
		</tr>
		<tr>
			<td>p_sigma</td>
			<td>0.5</td>
			<td>Second coating relative conductivity</td>
		</tr>
		<tr>
			<td>p_eps</td>
			<td>1.5</td>
			<td>Second coating relative dielectric constant</td>
		</tr>
	</tbody>
</table>
Table 1: Parameters used for simulation
<table border="1" fo:keep-together.within-page="1" fo:keep-with-next.within-page="always">
	<tbody>
		<tr>
			<td></td>
			<td>Culture Media</td>
			<td>PEDOT:PSS 1</td>
			<td>PEDOT:PSS 2</td>
			<td>Collagen Hydrated 1</td>
			<td>Collagen Hydrated 2</td>
			<td>Silk Fibroin</td>
			<td>Collagen Dry</td>
		</tr>
		<tr>
			<td>Electrical Conductivity (S/m)</td>
			<td>1.7014</td>
			<td>1.00E-01</td>
			<td>p_sigma * 0.1</td>
			<td>2.00E-05</td>
			<td>p_sigma * 2e-5</td>
			<td>1.00E-08</td>
			<td>2.50E-08</td>
		</tr>
		<tr>
			<td>Relative Permittivity</td>
			<td>80.1</td>
			<td>2.2</td>
			<td>p_eps * 2.2</td>
			<td>9.89</td>
			<td>p_eps * 9.89</td>
			<td>7.81E+00</td>
			<td>4.97</td>
		</tr>
	</tbody>
</table>
Table 2: Material properties used in simulation
Supplemental Files. <a href="https://www.jove.com/files/ftp_upload/61928/supplemental files.zip" target="_blank">Please click here to download this File.</a>

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