microfluidic chip for electroporation of biological cells

8/2/2019 Microfluidic Chip for Electroporation of Biological Cells

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LabChip 2 project

Microfluidic chip for

electroporation of biological cells

Anil Thilsted (s072154)Nicolai Sanders (s072129)

Nikolaj Feidenhansl (s072168)

Supervisors:Lektor Fridolin Okkels

Lektor Rafael Jozef Taboryski

Technical University of DenmarkDepartment of Micro- and Nanotechnology

2nd

December 2011


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Labchip 2: Project Contents

Contents

1 Introduction 1

2 Literature review 2

3 Electroporation Theory 2

4 Geometry and fabrication 4

5 Simulation setup 5

5.1 Flow simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.2 Electric field simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65.3 Initial simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5.4 Analytical estimations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

6 Building the mesh 8

6.1 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86.2 Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.3 Mesh refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

7 Data processing in Matlab 10

8 Results 11

9 Outlook 13

10 Conclusion 15

Appendix 16

A Matlab script 16

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Labchip 2: Project 1 Introduction

1 Introduction

This report aims to reproduce the results presented by Andresen et al. (2010) [1] and tofurther the investigation the microfluidic design used for electroporation of cells. Electro-poration is a mature technology in which an electric field is used to permeabilize the cellto facilitate the transport of ions and proteins etc. across the cell membrane. One used ofelectroporation is when performing transformation on bacterias, where genetically alteredplasmid-DNA have to be transfered into the cytoplasm.

A literature survey was carried out in which the dierent methods of electroporationwere reviewed, and the advantages of the channel constriction design described. The theoryof electroporation was presented, in which the cell is modelled as a dielectric sphere in anelectric field, and the cell membrane in an external field as a resistor-capacitor circuit.

Simulations using finite element modelling was carried out using Comsol, in which

a quasi-3D geometry was set up for the flow profile and a full 3D model was used forthe electric field distribution. The mesh was optimized until the solution converged to areasonable accuracy. By extracting the flow velocity and electric field, the cell capacitiveresponse was calculated by carrying out a convolution of the electric field and the capacitiveresponse function.

Assuming constant electric potential across the electrodes and a constant microfluidicpressure dierential, the width of the channel constriction was varied in order to find aconfiguration that maximized the throughput of electroporated cells.

All authors contributed equally to the report.

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Labchip 2: Project 2 Literature review

2 Literature review

Electroporation is a method for temporarily increasing the permeability of a cell signifi-cantly by applying an electric field. This process facilitates the transport of ions and othermolecules that cannot otherwise cross the cell membrane, as well as the release of intra-cellular content. Electroporation is a mature technology that is used widely in biomedicalresearch such as genetic manipulation, cloning, cell fusion and antibody production. Acommon method of electroporation is to place a cuvette between two aluminum platesand use a pulsed high voltage source to induce an electric field across the cells held insuspension. Such a device is shown in Figure 1.

Figure 1: Electroporation cuvette withelectrodes and space for the fluid. From [2].

Figure 2: Electroporation using microelec-trodes on single cells. The fluorescent mate-rials entered the cell after applying an elec-tric field of 0.5 V. From [3].

Alternatively, electrode needles can be used to induce an electric field in vivo fortransfection in tissues [4] and microelectrodes can be used to induce a localized electricfield and perform electroporation on single cells [3], shown in Figure 2. Microelectrodeshave the advantage of using low voltage sources, using volts instead of kilovolts, but theyalso apply an inhomogenous electric field and have unwanted eects such as electrodedeterioration, producing potentially harmful products from electrochemical reactions [5]and possibly bubbles [1]. These adverse aects can be minimised by using pulses or ACcurrents and with variations of microchannel and microelectrode design [6].

Andresen et al. demonstrates a method of using an all-polymer injection molded chip,shown in Figure 3 which utilizes a channel restriction to enhance the electrical field. This

avoids the adverse aects from microelectrodes and removes the need for high voltagesources.

3 Electroporation Theory

The equations for electroporation are found by approximating the cell periphery as a non-conducting membrane, which is placed in an external electric field E0, and solving theLaplace equation for the potential:

r2U(r) = 0 (1)

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Labchip 2: Project 3 Electroporation Theory

With the usual continuity boundary conditions applied for the electric field, the solution,in spherical coordinates (r,,) becomes [7]:

U = E0r cos() (2)

The cell is modelled as a dielectric sphere with dielectric constant i placed in a dielectricfluid medium with dielectric constant e. The electric field polarizes the cell, which inturn perturbs the external electric field. The solution for the potential within the cell isexpressed as:

Uc(r) =3e

i + 2eE0r cos() , for r < a (3)

For a time-dependent electric field E(t), the cell membrane can be modelled as an RC-

circuit, sketched in Figure 4.

(A) (B)

Figure 3: Micrograph of flow channelwith the constriction. The blue areas arethe electrodes. From [1].

Uext(t)!

Rm!

Cm! Um(t)!

Figure 4: Equivalent electrical circuit ofa cell membrane.

The response function for such a circuit is:

hc(t) =1

RmCmexp

t

RmCm

H(t) (4)

in which Rm [m2] and Cm [Fm

2] are the specific membrane resistance and the specific

membrane capacitance, respectively, and H(t) is the Heaviside function. The Heavisidefunction ensures that causality is maintained. Multiplying the resistance and capacitancegives the membrane charging time m. Typical values for the variables are [1], [8]:

Description Variable Value

Cell diameter 2a 5 30 mRelative permittivity of water e 80

Relative permittivity inside cell i 50Membrane charging time m = RmCm 1 ms

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Labchip 2: Project 4 Geometry and fabrication

For a pulsed electric field the convolution of an external field E(t) = E0 H(t) andthe response function hc(t) gives an exact solution for the potential U(t)m across the cell

membrane [9]:

Um(t) =3ei + e

E0a cos()

1 exp

t

RmCm

(5)

The cross-membrane potential U(t)m that is required for dielectric breakdown andelectroporation of the membrane is temperature dependent, ranging from 1 V at 4C to0.64 V at 30C [10]. If the electric field is pulsed, the membrane reseals on the order of10 s [11], allowing for the diusion of ions and other molecules into the cell. This diusionis expressed by the NernstPlanck equation [8]:

@c

@t = r

Drc +

Dze

kBTcr(Um)

(6)

in which D is the diusivity of the species, c is the concentration, z is the valence of thespecies, e is the elementary charge, kB is Boltzmanns constant and T is the temperature.The NernstPlanck equation takes into account both diusion and the electrostatic forces.

This project focuses on the device fabricated and tested by Andresen et al. and simpli-fies the electroporation to assume that it occurs when the capacative response of the cellmembrane due to a time-dependent electric field results in a cross-membrane potential ofat least 1 V [10].

4 Geometry and fabricationThe structure of the electroporation chip is a microchannel with a constriction. Twoelectrodes are connected to the fluid at the inlet and outlet, to apply a potential dierencethrough the fluid. The channel is made of the polymer TOPAS 8007, which has a dielectricconstant of 2.35 [12]. Only the constriction part of the channel will be simulated, since theflow before and after this area is Poiseuille flow. The basic structure including dimensionsfor the Comsol simulation is shown in Figure 5, with the height of the channel being80 m. This structure is based on the work by Andresen et al. and will be the standardstructure for the simulations. The constriction width will be varied to investigate theeects on the flow rate and electric field intensity in the constriction.

A method for mass production of this device is injection moulding. With this method,two moulds are joined together to define one half of the chip with the channels, inletsand outlets included. The polymer is then heated above the glass transition tempera-ture, pumped into the mould and cooled before separating the two moulds to release thecomponent. This component, containing the channels and inlet/outlets is then positionedon a flat chip where the conductors are structured. The conductors could be defined byphotolithography and metal deposition or etch of a conducting polymer applied by spincoating. The electrodes should either be submerged in the substrate polymer or thinenough to avoid leaking. The two parts are bonded together e.g. by plasma-assistedthermal bonding as in [1]. The finished chip is shown in Figure 6.

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Labchip 2: Project 5 Simulation setup

50 m1 mm

1.1 mm

Fluid channel

TOPASx!

y!50 m

Curvature:50 m

Figure 5: Sketch of the Comsol structureincluding dimensions, the hight of the chan-nel is 80 m. The sketch is not to scale.

Figure 6: Illustration of the final chip,showing the inlet/outlet ports, the con-stricting and the electrodes. From [1].

h!

L!

w!Qreal!

w!Q

sim!

u(y)!

u0!

y!x!z!

pin = p0 +p!

pout = p0!

Figure 7: Sketch of the flow in the real microchannel and in the quasi-3D simulation, bothapproximated as infinite parallel-plates.

5 Simulation setup

An important aspect of numerical simulations is that the physical setup should be as closeas possible to the real structure. In this section, the simulation setup in Comsol is pre-

sented, including some initial results and comparison with simplified analytical estimates.

5.1 Flow simulation

The flow simulation in Comsol is simulated with the Laminar Flow physics in theFluid Flow module. It solves for the stationary flow solutions by applying the timeindependent NavierStokes equation and the conservation of mass:

(u r) u = rpI + r h

ru +(ru)T

23(r u) I

i+ f (7)

r (u) = 0 (8)

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Labchip 2: Project 5.2 Electric field simulation

in which is the density, u is the flow field, I is the identity matrix, is the dynamicviscosity and f is the body forces.

As the fluid is incompressible, the NavierStokes equation simplifies to:

(u r) u = rp + r

ru +(ru)T

+ f (9)

The values used for the fluid are: = 1000kg/m3 and = 1.002mPa s, with a pressuredierence of 38 Pa applied along the channel, yielding a flow rate of 0.178 L/s for thestandard geometry. In the following simulations, the pressure dierence is held constantand the flow rate will change due to changes in the hydraulic resistance of the dierentgeometries. The boundary conditions are no-slip boundaries for the channel sidewalls.

To shorten the computation time, the flow simulation is performed in 2D with a quasi-3D adjustment in which the no-slip condition on the top and bottom of the channel is taken

into account by applying an opposing body force. The force is determined by consideringtwo cases of flow in infinite parallel-plates, one which corresponds to the real flow in themicrochannel with no-slip boundaries and hence a parabolic flow profile, and one whichcorresponds to the quasi-3D simulation with a constant flow value and an opposing bodyforce, f= u, see Figure 7. The factor can then be determined as the flow rates in thetwo channels must be equal1:

Qreal =h3w

12

p

L(10)

Qsim =

Zw0

Zh0

u0 dx dy = hwu0 = hwp

L(11)

Qreal = Qsim , =12h2

(12)

in which the steady state velocity u0 is determined frompL

u0 = 0, in which p is thepressure dierence along the channel. The body force is then found as:

f= 12

h2u (13)

By including this body force in the 2D flow simulation, the flow rate and mean velocitywill be the same as for a full 3D simulation.

5.2 Electric field simulation

The electric field simulation is performed with the Electric Currents physics in theAC/DC module, in which the stationary solver is applied. Comsol then finds thestatic solution for the current from the equation of continuity with Ohms law, includinga term for an externally generated current density, Je:

r J = 0 , r J = r (E + Je) (14)

The material parameters were:

1The solution to the flow in the no-slip parallel-plate channel, Qreal, is from [7]

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Labchip 2: Project 5.3 Initial simulations

Domain Conductivity [ S/m] Permittivity

Fluid 1.7 80

TOPAS 110

10 2.35

The value for the conductivity of TOPAS should have been zero, since it is an isolatingpolymer, but as Comsol could not perform the calculation, it was set to a very smallnumber instead. This imposes a slight error in the simulation, but a value of 11010

should be significantly lower than the other numerical noise sources.The electric field simulation was carried out in 3D, as the eects from the top and bot-

tom of the channel could not otherwise be taken into account. Two 80 m slabs of TOPASwere therefore added on both sides of the channel, and a potential dierence of 50 V ap-plied through the fluid. The remaining boundary conditions were electric insulation on allouter surfaces of the TOPAS.

5.3 Initial simulations

To check whether the simulation functioned correctly, some initial simulations were carriedout on the standard geometry. These are shown in Figure 8 in which the flow velocity andelectric field distribution is plotted. These plots show that the maximum velocities andE-fields are in the constriction, as expected,.

a b

Figure 8: Initial simulations on the standard geometry showing a) the velocity in the channel

and b) the distribution of the electric field.

5.4 Analytical estimations

In order to determine whether the simulation model yields reasonable results, simpleanalytical estimations were carried out and compared with the simulations.

For the flow simulation, the flow rate is estimated from the pressure dierence andthe hydraulic resistance of the constriction. This estimation then neglects the hydraulicresistance for the wider parts of the channel, but as the hydraulic resistance for a rectan-gular channel decays with 1

w, the contribution from these areas is small. The flow for a

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Labchip 2: Project 6.2 Gradients

boundary condition must be applied. For the current simulation, the symmetry line willhave equal current densities in both directions resulting in zero net current. This can be

modelled with an electrically insulating boundary. In the flow simulation Comsol hasthe boundary condition Symmetry plane, which is a boundary that has no penetrationand vanishing shear stress. These symmetry lines were not used in the simulations, sinceit was preferred to visualize the full structure, as this gave a more intuitive view of thevelocity and electric field.

6.2 Gradients

From the initial simulations it was discovered that the highest gradients were located in theconstriction, and therefore these areas needed a much denser mesh than the wider part ofthe channel and the TOPAS. The meshing was done by specifying the maximum element

size and growth factor for the boundaries. For the 2D fluid flow simulation a boundarylayer mesh is applied which consists of rectangular elements along the boundary, yieldinga high density mesh perpendicular to the boundary while keeping it coarse along theboundary.

200 m

25 m

a

25 m

b

Figure 9: The mesh used for the standard structure in Comsol in a) for the 2D flowsimulation and b) the 3D mesh for the electric simulation both including a zoom-in on thehigh density area. The box in the 3D constriction is a volume with increased mesh density.The simulation times were 6 s and 200 s for 2D and 3D respectively.

The final element sizes were determined from the analysis in the following section. The

final mesh consists of 5760 mesh elements, as shown in Figure 9a which includes a zoom-inon the high density constriction area. The mesh for the 3D electric simulation is seen inFigure 9b. Note the box in the constriction, which is used to increase the mesh density.The mesh is most dense in the constriction while the wide parts of the channel have muchlarger elements. The flow simulation time for the 2D mesh was around 6 s with a theboundary mesh on all boundaries of two layers, a stretch factor of 1.2 and the values:

Domain/boundary Max element size Growth factor

Fluid domain 80 m 1.1Constriction walls 2 m 1.1

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Labchip 2: Project 6.3 Mesh refinement

6.3 Mesh refinement

The trade-o

between mesh density and computation time was investigated by simulatingthe structures with decreasing mesh dimensions and comparing the maximum velocityfrom each simulation to determine when the value converged. The distribution of meshdimensions were based on the previous work of identifying high gradient areas, with theinitial values for the domain and boundary set to:

Domain/boundary Max element size Growth factor

Fluid domain 320 m 1.1Constriction walls 8 m 1.1

while the boundary mesh was kept constant with two layers and a stretch factor of 1.2.When decreasing the mesh dimensions, this distribution is kept constant. Both values

were divided by the same factor to evenly reduce the mesh dimensions. Seven simulationswere done with scale-down factors from 1 to 7, resulting in the graph seen in Figure 10.This figure reveals that the first refinements alter the velocity significantly, after which itconverges rapidly with increasing mesh elements. The mesh consisting of 5760 elementswas chosen as the best trade-o, since the maximal velocity changes insignificantly in thefiner meshes. The parameters for this mesh were the ones shown in section 6.2.

Figure 10: Plot of the maximum flow velocity from simulations with varying numbers ofmesh elements, the simulation time is stated for each datapoint.

7 Data processing in MatlabThe data output from Comsol consists of line plots showing the flow velocity u(x) andelectric field strength E(x) along the center line of the geometry, as functions of theposition x. The center line is chosen because the field strength in the constraint is lowestat this point, and because the flow is fastest there. Thus, it is both the worst case scenarioand the most relevant case for a majority of the cells. In order to evaluate whetherthe cells experience a sucient cross-membrane potential to be electroporated, the datais converted to the electric field E(t) and cell position x(t) as functions of time. The

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Labchip 2: Project 8 Results

conversion is illustrated below:

u(x) =dx

dt ) (18)

dt =1

u(x)dx ) (19)

Zt0

1 dt =

Zx0

1

u(x)dx ) (20)

t(x) =

Zx0

1

u(x)dx (21)

f1(t(x)) = x(t) (22)

where f1 is the function inversion operator.

The form of the integral implies that time and cell position start at t = 0 and x = 0.t(x), the time it takes for the cell to reach position x, is then easily inverted to x(t),since they are bijective functions. E(t) is then found by taking E(x(t)). To find the crossmembrane potential Um(t), E(t) is convoluted with the capacitive response function, hc(t),described in Equation (4).

Um(t) =3ei + e

a

Z1

1

E(t )hc() d (23)

The initial data is not evenly spaced in x, with relatively large spaces at the edges, anda much higher density of points at the constriction. Furthermore, the number of points forthe E(x) and u(x) data are not the same. In order to evaluate and manipulate these data

in Matlab, they are interpolated onto a new x-axis with the interp1 function, using thepchip3 method. The new x-axis has the same endpoints as the old one, but is uniformlyspaced and contains 40000 points. The integral in Equation (21) is then evaluated with the

cumtrapz4 function. The inversion in Equation (22) is done by defining a new uniformlyspaced time vector, and then interpolating the old x-vector onto the new one. The E(t)vector is also found by interpolating the old E(x) onto the new t-vector. The convolutionin Equation (23) is done by using the conv function. An example of the code used forprocessing and plotting the data can be seen in appendix A.

8 Results

To optimize the structure to yield a high flow rate, simulations were carried out for a rangeof constriction widths. For each geometry, the electric field and flow was evaluated to de-termine whether it was sucient to electroporate the cells, and how much the constrictionhindered the flow of water.

Using the convulution method described in section 7, the cross-membrane potentialwas calculated for the standard configuration, that is, 50 V applied across the electrodesand a fluid pressure dierential of 38 Pa. The result is plotted in Figure 11. This figureshows that the electric field is strongest when the cell passes the the center of the channel

3Piecewise Cubic Hermite Interpolating Polynomial4Cumulative trapezoidal numerical integration

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Labchip 2: Project 8 Results

constriction (t = 0) and is significantly lower before and after with a full width at halfmaximum (FWHM) of 2.77 ms. This is somewhat in agreement with the FWHM of

1 4 ms for the relative resistance change obtained by Andresen et al. Resultantly, thecross-membrane potential peaks but with a time delay of 0.55 ms with a maximum valueof 1.22 V and a FWHM of 4.90 ms. As the cross-membrane potential has been greaterthan 1 V for longer than the membrane charging time m = 1 ms, it is assumed that thecells will electroporate reliably [1].

Figure 11: E-field and cross-membranepotential as functions of time. Note thetime delay in the cross membrane potential.

Figure 12: Flow as function of constric-tion width. Note the diminishing returnsfor higher constriction widths.

The challenge is then to maximise the flow such that a large number of cells can

be electroporated in a short time span while maintaining a high enough electric fieldto electroporate successfully. By increasing the width of the channel constriction, onereduces the hydraulic resistance and thus increases the flow rate. The flow rate wasfound by integrating across the fluid outlet boundary for a range of constriction widthsand constant pressure dierential of 38 Pa. The results are plotted in Figure 12. Asexpected, the flow rate increases for larger channel constriction widths. By increasing theconstriction width from 50 m to 185 m, there is a three fold increase in flow rate andhence the number of cells passing through the constriction in a given time span.

In order to assess for which constriction widths electroporation still occurs, the widthwas varied and, by extracting the flow velocity and electric field data, the cross-membranepotential as a function of time was calculated for a range of constriction widths. The

results are shown in Figures 13 and 14.Figure 13 shows that, for increasing constriction widths, the FWHM of the cross-

membrane potential increases. There are two contributions to this eect. First, the fluidvelocity at the site of the constriction is reduced, such that the cells are in the constrictionfor a longer period of time. Secondly, the electric field is less confined such that the cellsexperience a relatively larger potential before and after the channel constriction. Thus, thecharging time for this range of channel constriction width never falls below the membranecharging time m 1 ms and this therefore is not a limiting factor for this design.

Figure 14 shows how the maximum electric field, and the maximum cross-membranepotential decreases for larger constriction widths. For constriction widths of 180 m,

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Labchip 2: Project 9 Outlook

Figure 13: Contour plot of Cell membrane

potential as a function of time and constric-tion width. The thick line indicates thelimit to which the cells are electroporated.

Figure 14: Maximum E-field and cross

membrane potential as function of constric-tion width.

the maximum potential is approximately 1 V, and this is therefore the approximate limitbefore the microchannel geometry is not capable of electroporating cells.

9 Outlook

As an outlook of the microfluidic design fabricated and tested by Andresen et al., analternate design was proposed and simulated in Comsol. The alternate design is one inwhich two additional side channels connect to the original channel immediately after theconstriction. This design ensures that the cells are first exposed to the ions or proteins afterthe cells are electroporated. This allows for use of proteins which denaturate in the highelectric field in the constriction, and for more accurate control of the timing and diusionflux into the cell. The geometry has two side channels of 300 m situated approximately140 m after the constriction. Electric field simulations showed no significant change inthe electric field profile in the constriction and the electric field strength at the site of theadditional channels was negligible. The two side channels had a pressure of 38 Pa appliedlike the original channel. Diusion of particles from the side channels was simulated bysolving the convection-diusion equation

@c@t

+ r (2Drc) + u rc = 0 (24)

in which c is the concentration, D is the diusion coecient and u is the convection whichuses the velocity field calculated in the flow simulation. The concentration was normalizedto a value of 1 at the entrance of both side channels and 0 on the original channel.The channel was extended such that the diusion and flow could be visualized furtherdownstream of the constriction. The results from the flow and the diusion simulation isshown in Figure 15.

For the figure, the flow velocity is seen to be homogenous downstream of the threeinlets with a value of 0.36 cm/s, and the concentration gradually increases in the center

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Labchip 2: Project 9 Outlook

Figure 15: Contour plots of, top: the velocity in the channel system, bottom: the concen-tration distribution due to convection and diusion.

flow containing the cells, due to diusion. In this simulation, the diusion constant is setto the value for potassium ions K+ = 1.957 109 m2/s [13], resulting in a Peclet numberof:

Pe =V0L

DK+=

0.36 cm/s 0.1 cm

1.957 109 m2/s= 1840 (25)

in which V0 = 0.36 cm/s is the characteristic flow velocity, and L = 0.1 cm is the charac-

teristic length.The concentration is half of the input concentration at a distance of 4.63 mm down-stream of the channel constriction. With a flow velocity of 0.36cm/s, the cell reaches thisposition 1.29 s after being electroporated, which is inside the 10 s time frame where themembrane is permeabilized [1]. Hence, this design is suitable for controlled diusion ofions into cells.

However, for larger molecules with lower diusivity, the distance and time before su-cient mixing has occurred will be larger. For the losyzyme protein, the diusion coecientis D = 1.17 106m2/s yielding a Peclet number of 30800. The diusion simulation showsthat the center concentration in the channel reaches half of the initial concentration 9.61s downstream of the constriction, assuming a flow velocity of 0.36 cm/s. Thus, for electro-

poration of larger molecules like proteins or DNA, a passive mixer such as a herringbonestructure could be required.

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Labchip 2: Project 10 Conclusion

10 Conclusion

This project presents finite element modeling carried out in Comsol to analyze the flowvelocity and electric field magnitude in a microfluidic device capable of electroporation.Areas with high gradients were identified and the mesh was optimized for these regions.By considering simplified models, analytical estimations were carried out to determinewhether the Comsol results were reasonable.

To determine the cross-membrane potential on the cells passing the constriction, theelectric field data from Comsol and the cells response function was convuluted in Mat-lab. It was found that by increasing the constriction width to 180 m a flow rate of0.5 L/s could be realized while maintaing a sucient electric field for electroporation inthe constriction.

A possible redesign of the electroporation channel was presented which, by adding

inlet channels after the electroporating constriction, allowed for diusion of biomaterialinto the cells which otherwise would be denatured by the high electric field.

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Labchip 2: Project A Matlab script

Appendix

A Matlab script

An example of the code used for processing and plotting the data obtained from Comsol.

1 %%%%%%%%%%%%%%%%%%%%%%

2 % This program reads in the Flow and Efield data from COMSOL line plots,

3 % and calculates the xposition, Efield and cell membrane voltage of a ...

cell as function of time

4 % Written by Nicolai Sanders, 25/112011

5 %%%%%%%%%%%%%%%%%%%%%%

6 %%

7 clc

8 clear all9 format short

10

11 Wopen=[30,40,50,60,90,120,150,175,185,200];

12 %%

13 Nt=40000; % Number of points in final time vector

14 % Preallocation of the output for increased speed

15 tout=zeros(length(Wopen),Nt);

16 Eout=zeros(length(Wopen),Nt);

17 CMRout=zeros(length(Wopen),Nt);

18 xout=zeros(length(Wopen),Nt);

19 Vout=zeros(length(Wopen),Nt);

20 %%

21 for n=1:length(Wopen)

22 % Read in the data

23 data=dlmread(['Wopen ' num2str(Wopen(n)) ...

'mu 200sSolutionTime.txt'],'',8,0);

24 xE=data(:,1).*1e6; % x vector corresponding to Efield data in cm

25 E=data(:,2); % Efield as function of x, in kV/cm

26 data=dlmread(['Wopen ' num2str(Wopen(n)) 'mu Flow.txt'],'',8,0);

27 xV=data(:,1).*1e6; % flow Velocity as function of x

28 V=data(:,2); % x vector corresponding to Velocity data

29 clear data

30 % Interpolation to make x linearly spaced with sufficient resolution

31 Nx=min([floor((xV(end)xV(1))/min(diff(xE))),100000]); % Choose number ...

of x points for sufficient resolution

32 % Nx=10000;

33 x=linspace(xV(1),xV(end),Nx); % define linear x vector34 dx=mean(diff(x));

35 E=interp1(xE,E,x,'pchip'); % interpolate new E(x)

36 V=interp1(xV,V,x,'pchip'); % interpolate new V(x)

37 t=cumtrapz(x,1./V); % integrate to find t(x)

38 % Interpolate to find the things as function of time

39 tl=linspace(t(1),t(end),Nt); % Define linear time vector

40 dt=mean(diff(tl));

41 xl=interp1(t,x,tl,'pchip'); % interpolate new x(t)

42 El=interp1(t,E,tl,'pchip'); % interpolate new E(t)

43 Vl=interp1(t,V,tl,'pchip'); % interpolate new V(t)

44 [C,I]=max(El); % Find the time with maximum Efield

45 tl=tltl(I); % Normalize the time vector

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46 % Calculate the cell membrane response

47 tau1=1e3; % Cell membrane response time

48

epsilone=80; % Constants49 epsiloni=50;

50 a=1e5;

51 CR=(3*epsilone/(epsiloni+2*epsilone))*a/tau1*exp((tl1*mean(tl))/tau1) ...

.*heaviside((tl1*mean(tl))); % Cell membrane capacitive response ...

function

52 CMR=conv(El,CR,'same')*dt; % Cell membrane voltage response

53

54 tout(n,:)=tl; % Output time matrix

55 Eout(n,:)=El; % Output Efield matrix

56 xout(n,:)=xlmean(xl); % Output x matrix

57 Vout(n,:)=Vl; % Output flow velocity matrix

58 CMRout(n,:)=CMR; % Output Cross membrane potential matrix

59 end

60 %% Write output to files61 dlmwrite('standard CMR.txt',CMRout(3,:).','\t');62 dlmwrite('standard t.txt',tout(3,:).','\t');63 dlmwrite('standard E.txt',Eout(3,:).','\t');64 dlmwrite('Wopen.txt',Wopen.','\t');65 dlmwrite('Max CMR.txt', max(CMRout,[],2),'\t');66 dlmwrite('Max E.txt', max(Eout,[],2),'\t');67 %% Plot the cross membrane potential as functions of time for different Wopen's

68 figure(1)

69 clf

70 hold all

71 for n=1:2:length(Wopen)

72 plot(tout(n,:),CMRout(n,:));

73 end

74 set(gca,'FontSize',20)

75 xlabel('Time [s]','fontsize',20);

76 ylabel('Voltage [V/m]','fontsize',20);

77 xlim([0.01, 0.01]);

78 % ylim([0,max(max(Eout))]);

79 title('Voltage across cell membranes','fontsize',20);

80 saveas(gcf,'Wopen sweep Voltage across cell membranes.png');

81 % %% Plot the Efield as functions of position

82 % figure(2)

83 % clf

84 % hold all

85 % for n=1:length(Wopen)

86 % % plot(xout(n,:)+xout(n,end),Eout(n,:));

87 % plot(tout(n,:),Eout(n,:));88 % end

89 % set(gca,'FontSize',20)

90 % xlabel('Time [s]','fontsize',20);

91 % % xlabel('xposition [m]','fontsize',20);

92 % ylabel('Voltage [V/m]','fontsize',20);

93 % % ylim([0,max(max(Eout))]);

94 % xlim([0.01, 0.01]);

95 % title('Voltage experienced by cells','fontsize',20);

96 % saveas(gcf,'Wopen sweep Voltage experienced by cells.png');

97 %% Plot the flow velocity as functions of time

98 figure(3)

99 clf

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100 hold all

101 for n=1:length(Wopen)

102

plot(tout(n,:),Vout(n,:));103 end

104 set(gca,'FontSize',20)

105 xlabel('Time [s]','fontsize',20);

106 ylabel('Flow [m/s]','fontsize',20);

107 xlim([0.01, 0.01]);

108 % ylim([0,max(max(Eout))]);

109 title('Flow experienced by cells','fontsize',20);

110 saveas(gcf,'Wopen sweep Flow experienced by cells.png');

111 %% Contour plot of cross membrane potentials

112 figure(5)

113 clf

114 % colormap(jet(512))

115 % pcolor(tout,Wopen,CMRout)

116 contourf(tout*1e3,repmat(Wopen.',[1,40000]),

CMRout,linspace(0,2.2,12))117 % shading interp

118 set(gca,'FontSize',20,'XTick',linspace(10,10,9),'YTick',[50,75,100,125,150,175,200])

119 ylabel('W {open} [\mum]','fontsize',20);120 xlabel('Time [ms]','fontsize',20);

121 xlim([10, 10]);

122 colorbar('fontsize',20,'YTick', linspace(0,2.2,12));

123 title('Cross membrane potential [V/m]','fontsize',20);

124 grid on

125 saveas(gcf,'Wopen color Voltage across cell membranes.png');

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Labchip 2: Project References

References

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[2] Eppendorf. http://www.eppendorf.com, Retrieved 25.11.2011.

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[4] Jun-ichi Miyazaki Hiroyuki Aihara. Gene transfer into muscle by electroporation invivo. Nature Biotechnology, 16:867870, 1998.

[5] Jessica Olofsson, Kerstin Nolkrantz, Frida Ryttsen, Bradley A Lambie, Stephen GWeber, and Owe Orwar. Single-cell electroporation. Current Opinion in Biotechnol-ogy, 14(1):29 34, 2003.

[6] M. Fox, D. Esveld, A. Valero, R. Luttge, H. Mastwijk, P. Bartels, A. van den Berg,and R. Boom. Electroporation of cells in microfluidic devices: a review. Analyticaland Bioanalytical Chemistry, 385:474485, 2006. 10.1007/s00216-006-0327-3.

[7] Henrik Bruus. Theoretical Microfluidics. Oxford University Press, 2008.

[8] Malmivuo J and Plonsey R. Bioelectromagnetism. New York: Oxford UniversityPress, 1995.

[9] Lecture slides 2011-09-30 from course 33336.

[10] Hans G. L. Coster and Ulrich Zimmermann. The mechanism of electrical breakdownin the membranes of valonia utricularis. Journal of Membrane Biology, 22:7390,1975. 10.1007/BF01868164.

[11] Gorazd Pucihar, Tadej Kotnik, Damijan Miklavcic, and Justin Teissie. Kinetics of

transmembrane transport of small molecules into electropermeabilized cells.Biophys-

ical Journal, 95(6):2837 2848, 2008.

[12] MatWeb Material property data. http://www.matweb.com, Retrieved 25.11.2011.

[13] E. Samson, J. Marchand, and K. Snyder. Calculation of ionic diusion coecientson the basis of migration test results. Materials and Structures, 36:156165, 2003.10.1007/BF02479554.

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