Dp

ID

a

ARRA

KPDCFF

1

ivapfl(drabTaawt

(

0d

BioSystems 102 (2010) 134–147

Contents lists available at ScienceDirect

BioSystems

journa l homepage: www.e lsev ier .com/ locate /b iosystems

iffusion–convection effects on drug distribution at the cell membrane level in aatch-clamp setup

rina Baran ∗, Adrian Iftime1, Anca Popescu1

ept. of Biophysics, “Carol Davila” University of Medicine and Pharmacy, 8 Eroilor Sanitari Blvd., Bucharest 050474, Romania

r t i c l e i n f o

rticle history:eceived 2 July 2010eceived in revised form 2 September 2010ccepted 4 September 2010

eywords:atch-clampiffusiononvection

a b s t r a c t

We present a model-based method for estimating the effective concentration of the active drug appliedby a pressure pulse to an individual cell in a patch-clamp setup, which could be of practical use in the anal-ysis of ligand-induced whole-cell currents recorded in patch-clamp experiments. Our modelling resultsoutline several important factors which may be involved in the high variability of the electric responseof the cells, and indicate that with a pressure pulse duration of 1 s and diameter of the perfusion tip of600 �m, elevated amounts of drug can accumulate locally between the pipette tip and the cell. Hence,the effective agonist concentration at the cell membrane level can be consistently higher than the initialconcentration inside the perfusion tubes. We performed finite-difference and finite-element simulationsto investigate the diffusion/convection effects on the agonist distribution on the cell membrane. Our

inite difference simulations

inite element simulations model can explain the delay between the commencement of acetylcholine application and the onsetof the whole-cell current that we recorded on human rhabdomyosarcoma TE671 cells, and reproducequantitatively the decrease of signal latency with the concentration of agonist in the pipette. Results alsoshow that not only the geometry of the bath chamber and pipette tip, but also the transport parametersof the diffusive and convective phenomena in the bath solution are determinant for the amplitude andkinetics of the recorded currents and have to be accounted for when analyzing patch-clamp data.

. Introduction

Diffusive–convective bio-transport can play an important rolen a number of processes with physiological and clinical rele-ance. For instance, exocytotic release of neurotransmitters suchs adrenaline or serotonin from granules or vesicles can be accom-anied by convection associated with both the extracellular bulkow and the dilution of the granule matrix due to fast swellingFan and Fedorov, 2004). New methods of convection-enhancedrug delivery (Sawyer et al., 2006; Oh et al., 2007) were developedecently to deliver compounds throughout the brain by applyingn external pressure gradient to induce fluid convection in therain via a small catheter with an internal diameter of about 1 mm.his flexible technique which allows dosing of large areas of tissue

nd concentrating the infusate in situ can be used in chemother-py for intratumoral drug administration (e.g., inside gliomas), asell as in gene therapy or immune therapy. However, although,

he method has been proven to be generally reproducible and clin-

∗ Corresponding author. Tel.: +40 21 3180765; fax: +40 21 3125955.E-mail addresses: baran@ifin.nipne.ro, baran@lns.infn.it, virbaran@yahoo.com

I. Baran), ai@biofizica-umfcd.ro (A. Iftime), ap@biofizica-umfcd.ro (A. Popescu).1 Tel.: +40 21 3180765; fax: +40 21 3125955.

303-2647/$ – see front matter © 2010 Elsevier Ireland Ltd. All rights reserved.oi:10.1016/j.biosystems.2010.09.003

© 2010 Elsevier Ireland Ltd. All rights reserved.

ically safe, unpredictable fluid flow may arise due to the complexanatomy of the brain which can lead to collection of the drug inthe perivascular spaces and cause incidences of edema (Sawyer etal., 2006), which points to the need for detailed quantitative mod-els of drug transport via diffusive-convective processes inside thebrain tissue (Sarntinoranont et al., 2006; Linninger et al., 2008;Smith and García, 2009). Several complications of the problemarise from the variable contribution at a microscopic scale of geo-metric and viscous factors that may affect mass transport, suchas the molecular composition of the extracellular medium, tis-sue porosity and connectivity between fluid-filled spaces, or thepresence of macromolecular obstacles which can retard diffusionand contribute to the characteristic tortuosity of the nervous tis-sue (Rusakov and Kullmann, 1998; Syková and Nicholson, 2008).Nevertheless, the biophysical issues are complex and not fullyresolved at the moment. A similar principle is involved in the tech-nique of microdyalisis, which has been widely used to measureacetylcholine (ACh) release in vivo and has revealed the effectsof psychoactive and therapeutic drugs on cholinergic transmis-

sion (Bruno et al., 2006). In this procedure a micropipette withan inner radius of ∼10 �m is used to deliver ACh or other prob-ing solutions via pressure ejection directly into the brain tissue,and specific microelectrode devices are used to probe the ensuingresponse at a distance of several hundreds of micrometers away

I. Baran et al. / BioSystems

f2

ipan(itsgsTrdiaasttLpueftstuu

ostftctouva2atfidtI

In an initial set of simulations, we considered a spherical simu-

Fig. 1. Schematic representation of the perfusion system in a patch-clamp setup.

rom the micropipette (Bruno et al., 2006; Syková and Nicholson,008).

Pressure-controlled ejection through the tip of a micropipettes also a common method for agonist delivery to individual cells inatch-clamp measurements. The patch-clamp method (Hamill etl., 1981) is one of the most widely used electrophysiological tech-iques for studying both voltage-activated and ligand-activatedsuch as ionotropic receptors) membrane ionic channels. Whenonotropic membrane receptors are studied by the patch-clampechnique, the agonist is applied by pressure through a perfusionystem where the flow of the active solution is controlled via aroup of pinch valves, and the flow velocity depends on the pres-ure of the air communicating with the drug containers (Fig. 1).he effective amount of the drug interacting with the membraneeceptors of a studied cell depends, under these conditions, on therug concentration in the application containers (which is read-

ly controllable by the experimenter), but also depends on otherdditional parameters of the patch-clamp setup. Such parametersre: the perfusion pressure, the mobility of the drug in the batholution, as well as geometrical parameters such as the diame-er of the perfusion tubes, the distance between the cell and theip of the perfusion system, or the size of the cell. The studies ofisk and Desay (2006) on red blood cells have indicated that theerfusion parameters critically affect the quality of the seal, andnderlined that perfusion artefacts can impair the quality of thelectrophysiological recordings. Di Angelantonio and Nistri (2001)ound that short pressure pulses (10–50 ms) and agonist concen-rations in the range 20–100 �M produce linear responses in theame cells. Using diffusion equations for a continuous point source,hey calculated that the amount of agonist (nicotine in their study)ndergoes limited dilution in the extracellular microenvironmentnder their experimental conditions.

To the best of our knowledge, there are no quantitative modelsn the kinetics of drug distribution at the cell membrane level inpecified patch-clamp setups based on pressure-controlled ejec-ion of the agonist in the extracellular bath solution. Our studyocuses on the particular case of acetylcholine (ACh) applicationo evoke whole-cell currents on human rhabdomyosarcoma TE671ells, which express nicotinic acetylcholine receptors (nAChR) onhe plasma membrane. The influence of geometric parametersf the environment on the amount of acetylcholine that stim-lates postsynaptic nicotinic receptors has been studied for inivo situations (Whatey et al., 1979; Khanin et al., 1994; Smartnd McCammon, 1998; Tai et al., 2003; Popescu and Morega,004). However, in most patch-clamp studies the authors gener-lly present the particular conditions of their own setups, makinghe comparison between the results of different studies quite dif-

cult. Taking into account the different perfusion conditions andifferent shapes/dimensions of the perfusion chambers and pipetteips could provide a consistent background for such a comparison.n this paper, we present a modelling study on the effects of the

102 (2010) 134–147 135

geometry and flow parameters on the actual amount of agonist thatreaches the cell membrane, with the aim of determining the opti-mal experimental conditions required for studying ligand–receptorinteractions by the patch-clamp technique, as well as to providepossible corrections to be accounted for when interpreting thepatch-clamp data.

In our model, we consider that the transport of the neuroactivedrug (acetylcholine here) to a relatively close cell is determined bydiffusive and convective processes. The release source is realisti-cally modelled as being distributed over the surface of the pipettetip (with a diameter of 600 �m), and the duration of the pressurepulse (1 s) is longer than in previous studies (Di Angelantonio andNistri, 2001). We used two different numerical methods (a finitedifference and a finite element method, respectively) to solve thetransport equations in the computation domain, and the agreementbetween the results obtained with the two methods was good. Toincrease the computation speed, we modelled the decay of the flowvelocity with the distance from the tip as an exponential function,which is supported by classical theory of the “submerged jet” offluid ejected from a point source or from a cylindrical tube intoan unconfined space filled with the same fluid. Complex computa-tional models incorporating data provided by magnetic resonancemicroscopy and diffusion tensor imaging scans also indicate that invarious convection-enhanced drug delivery protocols the intersti-tial fluid velocity decreases exponentially with the distance fromthe injection site (Sarntinoranont et al., 2006; Smith and García,2009), with a decay rate which appears to depend on the type oftissue (i.e., white or gray matter). Our simulations indicate that highamounts of the drug can accumulate locally between the pipettetip and the cell and hence the effective concentration at the mem-brane level is consistently larger than the initial concentration inthe pipette. The modelling results in terms of the delay between thebeginning of acetylcholine application and the onset of the whole-cell current have been compared to a set of experimental data weobtained on TE671 cells. Some relevant predictions provide us withan estimate of the threshold level of nAChR activation required forthe initiation of a measurable cell signal.

2. Physical Model

In a patch-clamp setup, the cell under study is placed in a bathsolution which is relatively similar in composition to the extracellu-lar environment. After reaching the desired recording configuration(whole-cell, in our case), the cell is positioned in front of the tip ofthe perfusion system, from where the activating drug (in the exper-iments presented here, acetylcholine) is applied. According to thestandard pressure-controlled delivery method (Di Angelantonioand Nistri, 2001; Oancea et al., 2006; Sawyer et al., 2006), the solu-tion containing the agonist is placed in reservoirs which deliver thesolution through a thin tube ending with the perfusion tip. The solu-tion flow is controlled through a computer-controlled pinch valve.The solution reservoirs have another opening, situated above thefluid level, where air under pressure is applied (in absence of thepressure there will be no solution flow through the perfusion tip,even with an open valve, due to the very small diameter of the per-fusion tubes). The use of fluid reservoirs downstream the controlledpressure source prevents the pressure spikes that may appear afterthe opening of the valve and thus allows the delivery system tooperate at constant pressure and hence constant flow rate (Sawyeret al., 2006). Fig. 1 shows the general schematic of the setup.

lation volume of 1.2–1.5 cm radius, with the cell placed at its centre(Fig. 2). The cell is considered spherical, with a radius ≈30 �m, andthe centre of the cell was considered as the origin of the coordinatesystem, O. For simplicity, the simulation domain was assumed to be

136 I. Baran et al. / BioSystems

Extracellular solution

Rmax = 1.5 cm

rp= 500 µm

60º

Pipette tipACh

AChCell

Fsd

stpooodeFiatv

3

3

≈do

˚

wpicSapaolpupstoawu

ig. 2. Geometrical configuration of the cell–pipette system used in the model withpherical geometry. A section through the equatorial plane of the computationomain is depicted. For display reasons, the draw is not to scale.

pherical, with a total volume of about 7–14 ml, a typical value forhe volume of the bath solution used in experiments. The recordingipette attached to the cell was ignored in the simulations. The tipf the perfusion system was considered to have the basal surfacef ≈600 �m in diameter and situated at ≈500 �m from the centref the cell. First we studied the effects of diffusion alone on AChelivery to the cell in this geometrical arrangement and then wextended the model to include the convective transport of the drug.urther on, as will be detailed in Section 5, we studied the changesn the distribution of the drug at the cell membrane level in situ-tions with different geometrical parameters (different surfaces ofhe perfusion tip, different shapes of the bath chamber, differentolumes of the bath solution).

. Mathematical Model

.1. General Equations

The experimental perfusion debit measured in our setup is Q1 ml/(8 min) ≈1.938 �l/s. The corresponding flux of acetylcholineelivered through the pipette tip, which is assumed to be uniformver the entire perfusion surface, is then written as:

={

Q × [ACh]p/Sp, 0 ≤ t ≤ �0, t > �

(1)

here [ACh]p represents the concentration of acetylcholine in theipette solution, � = 1 s is the duration of the pressure pulse, and Sp

s the perfusion surface. In our simulations with various geometri-al arrangements, we modelled the pipette tip in such a manner thatp gets as close as possible to the value 2.86 × 10−7 m2 calculated forcircular area of 600 �m in diameter (the computational details areresented in the next sections and in Appendix A). Zero-flux bound-ry conditions were imposed on all boundaries with the exceptionf the perfusion surface, where the flux was given by Eq. (1). Theocal direction of the ACh flow into the bath solution was alwayserpendicular to the perfusion surface in the respective point. Theniformity of the acetylcholine flux considered at the exit from theipette reflects the diffusion effect of suppressing hydrodynamictretching inside the perfusion tube by reducing the mean varia-

ion in the particle speeds, hence producing a rather uniform profilef particle velocities along the lateral direction of the flow (Du etl., 2010). Moreover, simulations of nozzle microjets impinging onall-tube electrodes have shown that for low-viscosity solvents aniform velocity across the inlet, corresponding to plug flow imme-

102 (2010) 134–147

diately before and after the nozzle exit, provides a much betterexplanation for the experimental data than a parabolic inlet flowprofile (Rees et al., 2003). We should note that theoretical calcula-tions have also been done for a round jet of fluid ejected with anuniform flux at the exit (Revuelta et al., 2002), as discussed below.

In our experiments, the perfusion rate is very low and the totalvolume of the injected fluid into the bath solution is negligible(<0.03% of the bath volume). Therefore, in order to reasonablyreduce the excessive computational time, we made a number ofsimplifying assumptions, as will be described below. We consid-ered that the agonist flow inside the bath solution is driven bydiffusional-convective transport of the ACh molecules into theaqueous solution, according to the equation:

∂c(�r, t)∂t

= D∇2c(�r, t) − ∇ · [c(�r, t) × �v(�r, t)] (2)

where c represents the concentration of acetylcholine, �r – thevector of position, t – time, D – diffusion coefficient of the drugin the bath solution, and �v – the convection velocity. We usedD = 4 × 10−10 m2/s (Smart and McCammon, 1998; Tai et al., 2003).

If the drug transport is exclusively diffusive (v = 0), Eq. (2)becomes:

∂c(�r, t)∂t

= D∇2c(�r, t) (3)

In order to estimate the general effects of convection on theeffective agonist concentration on the membrane, we consideredtwo simplified scenarios for the convective flow in the bath solu-tion. In the first one, the convection velocity is assumed to beconstant and so we considered that the velocity v0 (in absolutevalue) at the exit from the pipette tip into the bath solution is givenby:

v0 = Q

Sp(4)

For simplicity, in most of the simulations with spherical geom-etry we assumed that the injected ACh molecules advance radiallyinto the solution (Fig. 2), and once reaching the cell membrane theycease convective motion. From there, they can move further awayby simple diffusion. This idealized scenario can provide us usefulclues about the fastest achievable rate at which ACh can reach thecell and the corresponding distribution of the agonist that yields amaximal concentration at the cell membrane level.

Within the second scenario, we considered an exponential decayof the velocity with the distance from the perfusion tip, accordingto the equation:

v(r) = −v0 exp[− rp − r

d0

](5)

where v(r) is the convection velocity at radial coordinate r, v0 > 0,rp is the radial coordinate of the perfusion tip at its inferior edge, rp

– r is the distance from the perfusion tip along the radial direction(denoted d), and d0 is a constant depending on the intermolecu-lar interactions between acetylcholine and the molecules in thebath solution, which we refer to as the specific convection length. Asbefore, we assume that convective movement stops when the AChmolecules reach the cell.

Because the intermolecular interactions and the collisionfrequency inside the solution depend on its composition and tem-perature, we used different values for d0, in the range 50–500 �m.A semi-quantitative comparison with the classical case of the “sub-

merged jet” emerging from a point source into an infinite volumeof fluid (Landau and Lifshitz, 1987) justifies the relevance of usingthese orders of magnitude for the decay constant d0. Along thepipette axis, the radial velocity of the fluid flow should decreaseas 1/d if the jet were originating from a point source (Landau

I. Baran et al. / BioSystems

/0

c/0

x

0

0.5

1

1.5

2

0.20 0.4 0.6 0.8 1

Point source Round jet (theory) Round jet (experiment) This work

1

1.5

2B

A

Point source

Round jet

This work

d (μμμμm)

0

0.5

10 00 200 300 40 50 00

Fig. 3. Decay of the radial velocity with the distance from the jet exit. (A) The cen-treline velocity vc scaled with the exit velocity v0 as a function of the normalizedaxial distance x = d/Re/a, predicted for a point source (Landau and Lifshitz, 1987) –dashed line – or for a round jet (Revuelta et al., 2002) – squares. Experimental data(diamonds) are from (Kwon and Seo, 2005). The best exponential fit to the data isshown (continuous line). (B) The decay of the average radial velocity v scaled withv as a function of the radial distance from the perfusion surface (d) in our workingcms

atamwtnarwti(�iTwtvtst

the diffusion equation is to be solved is reduced to one quarter.

0

onditions. The predictions for a point source (dashed line) and a round jet (dia-onds) are shown together with the exponential decrease assumed in the present

imulations (continuous line).

nd Lifshitz, 1987; Revuelta et al., 2002) located at the symme-ry centre of the perfusion surface. We calculated a numericalpproximation of the average radial velocity of the flow, withore computational details presented in Appendix B. In Fig. 3e compare the velocity decay obtained with a point jet and

he corresponding profile obtained with a round jet. The expo-ential decrease of the average radial velocity (in absolute value)ssumed in our diffusion–convection simulations (d0 = 500 �m cor-esponding to our experimental conditions) is also shown in Fig. 3B,hich is in good agreement with the theoretical solution for

he round axisymmetric jet ejected into a saline solution hav-ng the density � = 1010 kg/m3, dynamic viscosity � = 10−3 kg/msequal to the viscosity of water at 20 ◦C) and kinematic viscosity= �/� = 0.99 × 10−6 m2/s. As expected, the behaviour at increas-

ng distances approaches the far-field solution for the point source.hus, our simulations with different convection lengths supply usith a general picture of the kinetic variations in the drug dis-

ribution on the cell membrane in environments with different

iscosities. As discussed in Appendix B, the ratio of the convec-ion lengths characterizing two different solutions under otherwiseimilar conditions (that is, for the same injection rate Q) is equalo the ratio of their kinematic viscosities. Hence, d0 = 50 �m would

102 (2010) 134–147 137

correspond to an extracellular environment with kinematic viscos-ity of ∼=10−5 m2/s (more comments are given in Section 6).

3.2. Finite Difference Method

3.2.1. Diffusion Modelling in Spherical CoordinatesEq. (3) has been solved numerically using an explicit formula

with finite differences in spherical coordinates. Previously we useda similar method of diffusion modelling in cylindrical coordinates,in a series of simulations of calcium release from the endoplasmicreticulum (Baran, 2007; Baran, 2008; Baran and Popescu, 2009).In this study we perform space discretization in spherical geome-try with variable radial step (�r) and constant angular incrementfor both angular coordinates, �� = �ϕ = /18. An individual vol-ume element (voxel) is defined by three integer numbers (i, j, k),which represent the iteration numbers for r, � and ϕ, respectively.The first actual iteration for all the spherical coordinates, whichwas employed in integration of Eq. (3), corresponds in fact to thevalue i = 2, j = 2, and k = 2, respectively, because iteration “1” is onlyused for defining the boundary conditions in corresponding virtualelements.

The radial coordinate (ri) at iteration “i” represents the distancefrom the centre of the current voxel to the origin O. At each iteration,the radial step �ri = ri − ri−1 is increased with a fixed value ı, whichcan be calculated in function of the size of the simulation space(Rmax = 1.5 cm, measured from the origin O), the radius of the firstsphere surrounding the origin (R0 = 2 �m), and the total number ofradial iterations (nr = 451), according to the relation:

ı = 2R0

nr − 2× [Rmax/R0/(nr − 1) − 1] (6)

We obtain that the radius of the outer sphere delimiting a voxelV(i, j, k) is:

Ri = (i − 1)

[1 + ı

R0× (i − 2)

2

]× R0 (7)

the radial step at radial iteration “i” is

�ri = R0 + ı (i − 1.5) (8)

and the radial coordinate of the voxel centre is

ri = Ri − �ri

2(9)

From the values given above, we obtain ı = 139.6 nm, with min-imal and maximal values of the radial step �r1 = 1.93 �m (near theorigin) and �rnr = 64.7 �m (at the outer boundary). The radius ofthe simulated cell domain is then Rc = 29.68 �m, corresponding tothe radial iteration ic = 12.

The angular increment is constant for both � and ϕ.Since � ∈ [0,) and ϕ ∈ [0,2), the total numbers of angu-lar iterations performed in the actual integration aren� − 1 = /�� = 18 and nϕ − 1 = 2/�ϕ = 36, respectively. There-fore, the simulation space, which is a sphere of radius1.5 cm with a total volume Vtot = 14.14 ml, is divided in(nr − 1) × (n� − 1) × (nϕ − 1) = 450 × 18 × 36 = 291,600 voxels.Given the geometrical symmetry of the problem, we can usesymmetric boundary conditions on two particular surfaces (thesymmetry planes of the pipette, corresponding to the equatorialplane � = /2, and to the vertical plane ϕ = /2, respectively; thesymmetry axis of the pipette is given by the intersection of thesetwo orthogonal planes). In this way, the number of elements where

The running code was written in FORTRAN and implementedon either UNIX workstations or LINUX PCs. Numerical integration(detailed in Appendix A) was performed with variable time step(�t). At each time step, integration begins with the maximal value

1 stems

�scrfic�

ctspsckisrdtSsvd

dbttveds

3

cd

sb

bgvpE

c

wr(

w

v

“

c

38 I. Baran et al. / BioSy

t = 10 �s and the diffusion equation is solved in every voxel, byuccessively scanning the discretized values of the three sphericaloordinates and increasing their respective iteration rank in theange i = 2,. . .,nr; j = 2,. . .,n�; k = 2,. . .,nϕ. If in one single voxel wend negative values of the solution, we reset all values to the initialonditions at that time, and repeat the integration procedure witht reduced to half of its previous value.The pipette tip is located at a distance dcp ≈ 470 �m from the

ell surface. The perfusion surface is assumed to be a patch ofhe spherical shell having the radius closest to 500 �m, and theurface area closest to the experimental Sp value. The simulatedipette tip is delimited laterally by four planar surfaces, corre-ponding to iterations 8 and 13, respectively, for both angularoordinates (with corresponding notations jp1 = 8, jp2 = 13, kp1 = 8,p2 = 13), yielding a 60◦-angular aperture of the pipette in bothts symmetry planes. The perfusion surface belongs to the outerpherical shell at the radial iteration ip = 73, corresponding to aadial distance Rp = 500.74 �m from the origin O and a distancecp = 471.06 �m from the cell surface, measured along the symme-ry axis. With these values, the area of simulated perfusion surface isp = 2.63 × 10−7 m2, which is calculated by summing the areas of theurface elements dS (ip, j, k) with jp1 ≤ j ≤ jp2 and kp1 ≤ k ≤ kp2. Theolume of the bath solution is Vsol = 12.96 ml, whereas the volumeisplaced by the pipette in the bath solution is Vtot − Vsol = 1.18 ml.

In Appendix A we present more details for the integration proce-ure and boundary conditions. The ACh concentration is calculatedy integration of Eq. (3) in the bath domain only, after the concen-ration in a number of virtual elements has been preset accordingo Eqs. (A.22)–(A.26). At time t = 0, the concentration c = 0 in all theoxels of the bath domain. We averaged the ACh concentration oververy 0.05 s time interval in all the elements located in the imme-iate vicinity of the cell (162 voxels covering one quarter of the cellurface).

.2.2. Diffusion–Convection Model in Spherical CoordinatesWhen molecular transport is achieved through diffusive and

onvective motion, the global transport equation in spherical coor-inates reads as:

∂c(r, �, ϕ, t)∂t

= D∇2c(r, �, ϕ, t) − ∂

∂r[v(r, t) × c(r, �, ϕ, t)] (10)

We approximate that convection takes places during the perfu-ion only, for 1 s, and is restricted to the spatial domain Dc definedy ic+1 ≤ i ≤ ip, jp1 ≤ j ≤ jp2 and kp1 ≤ k ≤ kp2.

In the first scenario, the velocity v of the molecules is assumed toe constant all over the convection domain (v0 = −v0, with v0 > 0iven by Eq. (4); �v is oriented radially, toward the origin O), and= 0 in the remaining of the computation domain. From the valuesresented before, we obtain v0 = 7.92 mm/s. Then, the solution ofq. (2) is:

(i, j, k, t) = cdif(i, j, k, t) + v0 × �t × f ′r (i) (11)

here f ′r (i) is the first-order derivative of the concentration with

espect to r, and cdif(i, j, k, t) is the solution of the diffusion equationEq. (3)) calculated in the voxel V(i, j, k).

In the second scenario, the velocity decreases exponentiallyith the path length:

(i) = −v0 exp[− rip − ri

d0

](12)

where v(i) is the velocity of the ACh molecules at radial iteration

i”.

The numerical solution of the transport equation (Eq. (2)) is

(i, j, k, t) = cdif(i, j, k, t) − v(i) × �t × [f ′r (i) − cold(i, j, k, t) × d0]

(13)

102 (2010) 134–147

The convection–diffusion relation (either Eq. (11) or Eq. (13)) isapplied in the convection domain only, whereas in all the remainingelements the solution satisfies for the diffusion equation (Eq. (3))so that c (i, j, k, t) = cdif (i, j, k, t).

3.3. Finite Element Model

We used a Galerkin finite element method as a second numer-ical method for solving the mathematical model. The model wasimplemented in the software package Comsol 3.2 (Comsol AB) in 2Dgeometry with axial symmetry. We used a direct solver (UMFPACK)implemented in the software, with a variable time-step. The simu-lation mesh was refined selectively and consisted of 8000 to 10,000elements (16,000 to 20,000 degrees of liberty). In order to choosean appropriate mesh size we performed initial simulations withincreasingly more elements, until no more differences between thesimulation results on different meshes were noticeable. The fore-last size of the mesh was the one used to solve the model, providingthe best balance between the solution accuracy (which is higher atfiner meshes) and the required computational power. The aver-age ACh concentration on the cell was calculated as the integralconcentration in a 1 �m thick shell surrounding the cell.

There are three main factors that can produce slight differencesbetween the results obtained with the two simulation methods,namely: (1) the differences in mesh generation (the finite ele-ment method allows a more flexible partitioning of the simulationdomain), (2) the refinement of time discretization, and (3) the spacescale used for averaging the ACh concentration in the immediatevicinity of the cell (1 �m vs. 3.5 �m in the finite-element vs. thefinite-difference method, respectively). However, by controllingthat the total number of ACh molecules at each time step remainswithin 2% error, we could ensure a good accuracy of the numericalsolution with both methods.

4. Materials and Experimental Methods

4.1. Cell Cultures

TE671 cells of the human Caucasian rhabdomyosarcoma cell line (ECACCNo. 89071904) were cultured in DMEM medium (Dulbecco’s modified Eaglemedium with GlutaMAXTMI and glucose 4.5 g/l, without sodium piruvate, GIBCO-Invitrogen), supplemented with 10% (v/v) fetal bovine serum, 50 units/ml penicillinand 50 �g/ml streptomycin. The cultures were grown at 37 ◦C in a humidified incu-bator with a 5% CO2 atmosphere. Subculturing was performed by trypsinizationevery 4-5 days, when the cultures reached ∼50–60% confluence.

4.2. Electrophysiology

Patch-clamp measurements (Axon Guide, 2003) were performed 24–72 hafter subculturing, using an Axopatch 200A amplifier (Axon Instruments) undervoltage-clamp conditions, in the whole-cell configuration (Hamill et al., 1981).The membrane potential was held at −70 mV during measurements. The record-ing electrodes were made from capillaries of borosilicate glass (Science productsGB150T-8P), gravitationally elongated with the use of a vertical pipette puller PB-7(Narishige). Pipette resistances typically ranged between 2 and 4 M�. The perfusionsolution containing the agonist (acetylcholine) was introduced into a reservoir con-nected to a perfusion tip with diameter ≈600 �m, which was submerged into thebath solution at ≈500 �m away from the cell. Application of the perfusion solutionwas controlled with the use of a valve system (General Valve Corporation, Fairfield,USA) automatically controlled (pClamp8 software, Axon Instruments, Foster City,CA, USA) by using a computer connected to a home-made trigger system. Transientmacroscopic currents evoked by acetylcholine were recorded in the whole-cell con-figuration. The recorded data were digitized at 100 kHz using the Digidata 1320Asystem (Axon Instruments) which was controlled with the pCLAMp8 software (AxonInstruments). The signals were filtered at 5 kHz with a low-pass filter. All experi-ments were done at room temperature (∼22 ◦C).

4.3. Reagents

Cell currents were measured in a bath solution containing (in mM): 135 NaCl, 3KCl, 2 CaCl2, 2 MgCl2, 30 HEPES (pH 7.4/NaOH). The pipette solution contained (inmM): 140 CsCl, 1 MgCl2, 1 CaCl2, 5 HEPES, 11 EGTA (pH 7.2/CsOH). The solutions ofacetylcholine which were added to the pipette solution were freshly prepared before

I. Baran et al. / BioSystems 102 (2010) 134–147 139

F(oi

tp

5

5

uvpmtprdacaAetmodtdsttm

macmldtrscat(v

Fig. 5. Spatial distribution of acetylcholine established at t = 50 s with 10 �M AChin the perfusion solution (A) and the concentration profile on the cell–pipette axis,computed at t = 10, 20, 30, 40 and 50 s, respectively (B). Pressure is applied for 1 s,

ig. 4. Comparison between the diffusion profiles computed by the finite-differenceFD) and the finite-element method (FE), respectively. The maximal concentrationf acetylcholine on the cell membrane is represented as a function of time. Pressures applied for 1 s, starting with t = 0.

he experiment from a stock solution (1 mM) maintained frozen. All chemicals wereurchased from Sigma.

. Results

.1. Diffusional Mass Transport

We first computed the maximal ACh concentration that buildsp on the cell membrane as a result of simple diffusion (the con-ection term in Eq. (2) was set to zero) in a spherical domain. Fig. 4resents the model results obtained with the two methods in theain geometry and conditions described in Section 3.2 (radius of

he simulation domain: 1.5 cm) but with a different shape of theipette tip (a truncated cone in the finite-element method, vs. theegular shape with four lateral flat faces described in the finite-ifference method). In both situations, the perfusion surface wasssumed to be rounded and belonged to the corresponding spheri-al shell at the level of the inferior edge of the pipette tip. We noticefairly close similarity between the kinetic profiles of the maximalCh concentration reached on the cell membrane in the two differ-nt situations, which display similar rising rates and times (≈100 so the plateau phase). Nevertheless, the overall agreement of both

ethods under similar conditions is generally good and supportsur further investigations of mass transport in several systems withifferent geometries and perfusion parameters. In addition, simula-ions with the bath volume reduced to half (radius of the simulationomain 1.2 cm) have shown that the variations in the numericalolution at the cell level are small (<10%, not shown), indicatinghat the local geometry of the cell–pipette tip configuration playshe dominant role in shaping the distribution of agonist on the

embrane.Perfusion with 10 �M ACh in the pipette produces 33–37 �M

aximal ACh concentration in the immediate vicinity of the cell,nd a much higher concentration near the pipette tip (736 �M AChalculated at 45 �m away from the edge of the tip). It is to beentioned that the maximal ACh concentration at the membrane

evel (∼35 �M), which is reached within 90 s in the diffusion model,oes not represent the equilibrium value. Our simulations indicatehat diffusion is slow, and there appears to be an initial, transientegime in which the acetylcholine molecules accumulate in thepace between the pipette tip and the cell. We obtained numeri-

ally that homogenization within the bath solution is achieved afterlong time (>10,000 s) after the perfusion. The equilibrium concen-

ration can be calculated as [ACh]ech = Q × [ACh]p × �/Vsol ≈ 1.4 nMwhich was reproduced by numerical simulation), a much loweralue than that obtained at the plateau phase. However, the asymp-

starting with t = 0.

totic behaviour of the [ACh] distribution is not of interest, sincetypical experimental times are of the order of 10 s.

The pseudocolor image in Fig. 5A illustrates the spatial profileof the ACh distribution in a domain of interest, obtained at t = 50 sunder similar conditions with the exception of the perfusion sur-face, which now is planar. Fig. 5B presents several examples of theACh distribution along the symmetry axis at various times. One cannotice that a high level of ACh persists locally for a long time afterthe perfusion pulse, and hence the ACh front reaches the cell withconsiderable delay.

All these findings indicate that diffusion determines a consider-able accumulation of the active drug in the vicinity of the perfusiontip and show clearly that diffusion alone cannot explain by itselfthe rapid onset of the electric signal observed in our patch-clamprecordings. The predictions of the diffusion model indicate thatwith 10 �M acetylcholine in the pipette there is a delay of about10–20 s between the beginning of ACh application and the devel-opment of a sufficiently high level of agonist on the cell membraneto activate the ACh receptors and subsequently trigger the macro-scopic current. However, the experimental value of this lag time

is <0.5 s (discussed in Section 5.4), which indicates that the modelmust take into account the convective phenomena, too.

140 I. Baran et al. / BioSystems 102 (2010) 134–147

F [ACh]c

s nt moo ectiont is pre

5

aattcdivccvFoic

F[d

ig. 6. Time variation of the maximal ACh concentration on the cell membrane,olution contains 10 �M ACh. The perfusion pulse lasts 1 s, starting with t = 0. Differef the ACh specific convection length (d0) used as indicated or with a constant convhe entire simulation period (100 s) is shown. In (B) a time interval of interest (10 s)

.2. Diffusive–Convective Mass Transport

We next investigated the effects of convection on the spatialnd temporal variation of ACh distribution on the cell membranend in the bath solution for an applied drug level of 10 �M inhe pipette. As we expected, if very low d0 values are used inhe diffusion–convection model, the obtained numerical results arelose to the predictions of the diffusion model, whereas with higher0 the transport gets faster and the time lag becomes increas-ngly shorter (Fig. 6A). In the limit case of constant convectionelocity, the most rapid and highest increase of the average AChoncentration on the cell surface is obtained. The maximal ACh con-entration sensed by the cell increases with the average convectionelocity and consequently with the specific convection length d0.

ig. 6B presents the time variation of the maximal concentrationf acetylcholine at the membrane level, [ACh]cell-max, over a timenterval of 10 s. In Fig. 7 we present the kinetics of the ACh con-entration averaged over the entire surface of the cell in the same

ig. 7. Time course of the average ACh concentration on the cell membrane,ACh]cell-av, obtained with the diffusion and diffusion–convection models. Otheretails are as in Fig. 6.

ell-max, obtained with the diffusion and diffusion–convection models. The pipettedes of mass transport were considered (see details in the text) with different valuesvelocity (v0 = 7.92 mm/s). For the diffusion case, the velocity was set to zero. In (A)sented for several convection parameters as indicated.

scenarios as before. In the case of simple diffusion, the average AChconcentration begins to increase significantly after about 20 s. Asthe convection velocity increases, the ACh concentration on thecell rises at a faster rate. The increase is monotonous in all cases,except in the situation of convection with constant velocity, wherea transient, sharp peak (up to 79.1 �M) is obtained during the per-fusion interval. The actual case of our working bath solution used inexperiments is d0 = 500 �m, for which the maximal and the averageconcentration of agonist reached on the cell membrane at the endof perfusion are 980 and 27 �M, respectively.

We performed similar simulations for different ACh concentra-tions of the perfusion solution, namely 20, 30, 40, 50 and 100 �M,and we found that both the maximal and the average concentrationon the cell surface are proportional with the concentration of AChin the perfusion solution.

We also investigated the potential effects of uniform convec-tive transport from a planar perfusion surface by performing finiteelement simulations in spherical geometry. It was considered thatafter leaving the pipette tip, the ACh molecules move uniformlyinside the bath solution along a direction parallel to the cell–pipetteaxis. Fig. 8A shows the distribution of ACh along the cell–pipetteaxis, obtained at different times with 10 �M ACh in the appliedsolution and a constant convection velocity of 1 mm/s (we usedthis value as a measure of the average velocity of ACh molecules inthe bath solution that could explain the experimental delay timeobtained with 10 �M ACh in the pipette). In this case, the maximalconcentration on the cell reaches 2.3 mM at t = 1 s and subsequentlydecreases rapidly to 1.4 mM at t = 1.1 s and further on to 85 �Mat t = 2 s, which is followed then by a very slow, continuous decay(Fig. 8A and B). The kinetic profiles of the maximal concentrationof ACh on the cell membrane obtained with different ACh concen-trations in the pipette are shown in Fig. 8B. As before, we obtainedproportionality between the agonist concentration on the mem-brane and the concentration of the applied solution.

5.3. Effects of the Perfusion Rate in a More Realistic Geometry

In this section we present more results regarding the depen-dence of the transport kinetics on the bath volume, the geometryof the bath chamber and the shape of the perfusion surface. Weperformed finite element simulations and compared the results

I. Baran et al. / BioSystems 102 (2010) 134–147 141

Fig. 8. Distribution of acetylcholine on the cell–pipette axis computed at varioustimes as indicated on the plot, with 10 �M ACh in the perfusion solution and con-vection velocity of 1 mm/s (A). Kinetic profiles of the maximal ACh concentrationon the cell membrane, obtained with different ACh concentrations in the perfusionpipette, as indicated (B).

Fig. 9. Variation of the average ACh concentration on the cell membrane with theconvection velocity, obtained from finite element simulations for different geomet-rical configurations, at t = 10 s. Pressure is applied for 1 s, starting with t = 0. Datalabelled as “spherical” were obtained in the spherical geometry of the simulation

Fig. 10. Time course of the maximal ACh concentration on the cell membrane, computedand different convection velocities. Other details are as in Fig. 9.

domain (total volume 14 ml), with a rounded perfusion surface. Data labelled as“cylindrical” were obtained with a cylindrical geometry of the bath chamber (totalvolume 3.5 ml) and a flat perfusion surface.

obtained in the usual spherical geometry (radius of the simula-tion domain: 1.5 cm) with a rounded perfusion surface, to thoseobtained in a non-spherical geometry with axial symmetry, whichis closely similar to our experimental arrangement. In the lattercase, we modelled the computation domain as a cylinder withradius of 1.5 cm and height of 5 mm, providing a total volume of3.53 ml. The cell was positioned at the centre of the bath domain andthe pipette tip was placed vertically, with the lower edge situated470 �m above the cell. The pipette tip was modelled as a truncatedcone, the perfusion surface was considered flat and horizontal, andthe ACh flow was oriented vertically. In addition, it was assumedthat the convective motion ceases once the molecules reach theinferior boundary of the bath chamber.

The convection velocity was varied from 0.5 to 4 mm/s betweenthe simulations. Interestingly, our data indicate that the aver-age ACh concentration on the cell is affected by the geometrical

arrangement only at low convection velocities (0.5 mm/s here),whereas at velocities ≥1 mm/s there are no significant differencesbetween the results (Fig. 9), which appear to be insensitive to the3.7-fold variation in the bath volume, as well as to the consistent

for different shapes of the perfusion surface, different volumes of the bath solution

142 I. Baran et al. / BioSystemsS

igna

l lat

ency

(ms)

ACh concentration (µM)

0

100

200

300

400

500

120100806040200

Fig. 11. The delay time between application of pressure and the onset of themacroscopic cell current decreases with the ACh concentration in the pipette. Theedt

dOetiiesa

5E

eotoagdbtaanAitcotmwbi(f1wc

main geometrical and transport parameters can greatly increase

xperimental data, obtained from 10 to 20 cells, are presented as mean ± standardeviation. The best fit of the diffusion–convection model to the data is shown (con-inuous curve).

ifference in the pattern of the molecular flow in the bath solution.n the contrary, the maximal concentration on the cell membranexhibits a large variation (up to one order of magnitude) betweenhe two situations at all convection velocities (Fig. 10), suggest-ng that the local distribution of the agonist on the cell membranes highly dependent on the geometry of the system, whereas theffects of geometrical differences between various experimentalystems can be strongly attenuated when the concentration is aver-ged over the whole membrane.

.4. Estimation of nAChRs Activation Threshold fromxperimental Data

In Fig. 11 we present a series of delay times measured in ourxperimental setup between application of pressure and the onsetf the whole-cell current evoked by acetylcholine, for ACh concen-rations in the pipette ranging from 10 to 100 �M. By comparingur model results to these experimental values, we could providen estimate for the threshold of nAChR activation that could trig-er a measurable cell response. The equilibrium constant of AChissociation from the nicotinic receptor has been determined toe 106–160 �M in the closed state and of the order of 10 nM inhe open state of the receptor (Purohit and Grosman, 2006; Akknd Auerbach, 1999; Grosman and Auerbach, 2001), which yieldsn apparent Kd ≈20–28 �M and an open probability of the chan-el ≈0.1 in the presence of 10 �M ACh (Purohit and Grosman, 2006;kk and Auerbach, 1999). By using a standard scheme for ACh bind-

ng to the receptor together with known functional dependences ofhe open probability and the effective opening rate on the ACh con-entration (Akk and Auerbach, 1999; Auerbach and Akk, 1998), webtained from the best fit of the model to the experimental data thathe threshold for receptor activation (averaged over the entire cell

embrane) is Ptho = 0.067 ± 0.006 (the central value of this interval

as determined by the best least-square fit to the data, whereas theoundary values were calculated from the restriction that the max-

mal deviation from the data remains within 25%). Our calculationsdetailed in Appendix C) indicate that the effective opening time

rom the unbound closed state of the receptors decreases from 20 to0 ms in average at applied levels of 10–100 �M ACh in the pipette,hich is much lower than the observed latencies of the whole-cell

urrent, whereas the computed time needed for the ACh perfusion

102 (2010) 134–147

flow to reach the cell membrane (minimal 0.5 �M ACh in the firstmembrane voxel) is ∼=100 ms for all application levels, which leavesanother additional 60–210 ms of delay that can be attributed tothe time needed for ACh distribution over the cell membrane andconsequent activation of a minimal number of receptors. Consis-tent with this interpretation, the patch-clamp data presented inFig. 11 could only be explained by assuming that the latency ofthe whole-cell current is determined by the activation of nAChRsover the entire membrane, since more localized increases of AChconcentration at the “impact” cell region exposed to the perfusionflow could only account for a 10–20% difference in the latenciesrecorded over the entire range of the ACh concentration in thepipette (10–100 �M), which is much lower than the variation (42%)observed in our experiments.

6. Discussion

We performed finite difference and finite element simulationsof drug transport in a patch-clamp setup where the agonist (acetyl-choline) is delivered by pressure application though a fine tippedpipette to an individual cell placed 500 �m away from the pipette.Our results indicate that with a release source distributed over aregion of 600 �m in diameter and a pressure pulse duration of 1 s,the concentration of the drug reaching the cell depends not only onthe initial concentration in the perfusion system, but is also criti-cally affected by the geometry of the bath chamber and the shapeof the pipette tip, as well as by the velocity and direction of theperfusion jet.

We found that the timescale of ACh diffusion in this cell–pipetteconfiguration is slow and therefore cannot account for the experi-mental time delay that we recorded between pressure applicationand the onset of the macroscopic current evoked by acetylcholine(≈150–350 ms obtained with 10–100 �M ACh in the pipette andperfusion velocity ≈8 mm/s). However, incorporation into themodel of the convective mode allowed us to obtain a good descrip-tion of the experimental data. Moreover, simulation results, withor without convection included, show that elevated amounts ofdrug can accumulate locally between the pipette tip and the celland hence the effective agonist concentration at the cell membranelevel can be consistently higher than the initial concentration insidethe perfusion tubes. Our results are different from the those of DiAngelantonio and Nistri (2001) who calculated that diffusion aloneproduces maximal levels of agonist (nicotine therein) on the cellmembrane which are about 10% lower than the concentration in thepipette, with perfusion achieved via short pressure pulses (<50 ms)applied through a fine-tipped pipette (3 �m in diameter, approxi-mated to a point source) with 0.1 �l/s flow rate. However, in theirarrangement the cell was positioned much closer to the pipette tip(at 20 �m distance) and the calculated exit velocity of the fluid jetwould be 14 m/s, which is 2000-fold higher than in our set-up. Inour experiments we used rather large pipette tips in order to oper-ate under slow flow conditions and thus reduce the shear stress onthe cell membrane. Similar conditions were applied in a previouspatch-clamp study (Oancea et al., 2006) of the response of TRPM7channels to fluid mechanical stimulation of vascular smooth mus-cle cells. Thus, with a flow rate of 2–20 �l/s, diameter of the pipettetip 250 �m, distance between the cell and the tip 150 �m, andfluid velocities of 29–290 mm/s, significant stress on the cell wallcould be induced. So, in patch-clamp experiments where the ago-nist is applied by pressure ejection, an accurate knowledge of the

the relevance of the information that is extracted from the currentrecording data.

A significant conclusion of our investigations is that while localnon-uniformities in drug distribution may be less important in the

tems

dcmainouflaitaoadrpdct(

omdttoeocaotjracfbpomptibpowMrlaitaa

ctiwst

I. Baran et al. / BioSys

evelopment of the macroscopic current recorded in a whole-cellonfiguration, they can lead to different patterns of activation of theembrane receptors which in turn may interfere with the receptor

ctivation/desensitization kinetics and therefore could affect thenterpretation of the experimental results. Thus, we obtained byumerical simulations that in different geometrical configurationsf the patch-clamp setup, with consistent differences in the vol-me of the bath solution and in the spatial pattern of the molecularow imprinted by the specific shape of the pipette tip, the aver-ge concentration of acetylcholine on the cell membrane is fairlynsensitive to such parameters, whereas the maximal ACh concen-ration on the membrane can display large variations in differentrrangements. These findings were valid for convection velocitiesf 1–5 mm/s, but not for 0.5 mm/s. However, perfusion velocitiess low as 0.5 mm/s should not be used in whole-cell experimentsue to the inherent difficulties in interpreting the kinetic data ofeceptor activation, as discussed above. Nevertheless, the modelresented here can be further developed by incorporating hydro-ynamic effects of the perfusion jet or restricted diffusion at theell surface, where a slow-flowing boundary layer may form dueo the presence of extracellular elements that can slow diffusionShepherd and McDonough, 1998).

By fitting the model to a set of experimental data we obtainedn TE671 cells, we could derive a quantitative criterion for theacroscopic activation of nicotinic ACh receptors by acetylcholine

elivery from an extended source located at 500 �m away fromhe cell. The best fit suggests that opening of nAChR receptors dis-ributed over the cell membrane with an average open probabilityf 0.067 is sufficient to initiate a macroscopic cell current and alsoxplains the decrease of the signal latency at increasing quantitiesf agonist delivered through the pipette. The computed averageoncentration of acetylcholine on the cell membrane needed tochieve this level of activation is about 7–12 �M for applicationf 10–100 �M in the pipette, whereas the receptors located withinhe membrane area directly exposed to the perfusion flow are sub-ected to high levels of ACh (about 150–350 �M). Hence, theseeceptors can remain active for a sufficient time, ≈300 ms in aver-ge (Auerbach and Akk, 1998), to sustain the initiated whole-cellurrent until the redistribution of the agonist over the membrane isully developed. In our recordings the whole-cell currents evokedy acetylcholine generally displayed an abrupt and consistent risehase (not shown), suggesting that activation of the ACh receptorsver a wide membrane domain is rather unitary and rapid. Ourodel results are consistent with this interpretation, as the open

robability of the receptors located in areas immediately exposedo the fluid stream appears to be high at the moment of currentnitiation. Subsequently, ACh spreads progressively over the mem-rane and activates more receptors, thus contributing to the risinghase of the whole-cell current. However, the abrupt and unitarynset of the cell current also may suggest that restricted diffusionithin an unstirred layer adjacent to the membrane (Shepherd andcDonough, 1998; Endeward and Gros, 2009) can play a certain

ole in the delayed receptor activation and thus could contribute, ateast to a certain extent, to agonist equilibration on the membrane,nd hence to a more uniform cell response. Therefore, numer-cal simulations that include convection and diffusional effectsogether with detailed microscopic schemes of receptor activationnd desensitization should be performed in order to evaluate thectual contributions of the different delay sources.

We investigated by numerical simulations the effects of the vis-osity of the extracellular medium on the kinetic distribution of

he drug on the cell membrane. In general a significant increasen viscosity can be induced by the presence of high molecular-

eight molecules such as dextran or hyaluronan in the aqueousolution (Rusakov and Kullmann, 1998; Vercruysse et al., 1995). Inhis case, the viscosity increases with the volume fraction occu-

102 (2010) 134–147 143

pied by the large particles (Rusakov and Kullmann, 1998), so thataccording to the Stokes–Einstein equation the global viscosity ofthe extracellular fluid in the brain should be about 5 times higherthan the corresponding viscosity manifested in free aqueous phase(Rusakov and Kullmann, 1998; Syková and Nicholson, 2008). It isknown that hyaluronan, a polysaccharide found in all tissues andbody fluids, is present in notable amounts in brain tissue and is animportant component of the interstitial fluid (Fraser et al., 1997;Syková and Nicholson, 2008) thus contributing to its increasedviscosity. Interestingly, interstitial fluid velocity in convective-enhanced drug delivery also decreases in an exponential mannerwith the distance from the injection site (Sarntinoranont et al.,2006; Smith and García, 2009) within a given tissue region (i.e.,white or gray matter region), and the velocities attained in the graymatter appear to be about 30 times lower than in the white matter(Sarntinoranont et al., 2006) due to increased hydraulic resistanceof the tissue. Anisotropic variation in the specific convection lengthd0 can also manifest along different directions inside the brain tis-sue (Sarntinoranont et al., 2006). Our estimations indicate that thefluid velocity decreases exponentially with the distance from theinjection site with a decay constant d0 = Q//� (Q – volumetric infu-sion rate, � – kinematic viscosity). Therefore, for a given infusionrate, a simple preliminary analysis can be made in order to predictthe spatial range of drug dispersion into the brain tissue on variousdirections. The data presented in Figs. 7 and 9 of Sarntinoranontet al. (2006) indicate that for infusion with Q = 0.1 �l/min into thewhite matter dorsal column of the rat spinal cord, d0 is about170–200 �m along the axis of the cord and roughly 50 �m in thetransverse plane on the horizontal direction toward the pial mem-brane, which would correspond to an apparent kinematic viscosityrelative to water of ∼3 and ∼10 along the two axes. Similar valuesderive from the modelling data of Smith and García (2009), whered0 ≈ 4–5 mm for Q = 6 �l/min, indicating a relative kinematic vis-cosity of ∼6–8. Thus, our diffusion–convection model of transportcould provide a simple way to estimate the drug distribution intoa given tissue, which in a first approximation could be consideredas a medium of high viscosity.

Acknowledgements

This work was partially supported by the Romanian Ministry ofEducation and Research under CNCSIS-UEFISCSU Grant PNII-IDEIno. 1138/2009, code 1449/2008, and Grant PNII-IDEI no. 326/2007,code 251/2007. The finite element simulations have been per-formed at the Department of Bioengineering and Biotechnology ofthe Politehnica University Bucharest. We would like to thank Prof.Alexandru Morega for access to the laboratory and for his precioussupport and advice in the preparation of this paper.

Appendix A. Finite difference simulation method

A.1. Numerical solving of the diffusion equation with variableradial step

The surface element in spherical coordinates has the area:

dS = r2 sin � d� dϕ (A.1)

and the elementary volume is

dV = r2 sin � dr d� dϕ (A.2)

A voxel V(i, j, k) is confined within the volume defined by:

Ri−1 ≤ r ≤ Ri (A.3)

�j−1 + ��

2≤ � ≤ �j + ��

2(A.4)

1 stems

ϕ

w

�

ϕ

a

a�i

d

Rt

d

∇

MLir

c

wvstr

d

d

d

d

f

f

f

t

t

44 I. Baran et al. / BioSy

k−1 + �ϕ

2≤ ϕ ≤ ϕk + �ϕ

2(A.5)

here

j = (j − 1.5) × �� (A.6)

k−1 = (k − 1.5) × �ϕ (A.7)

re the angular coordinates of the voxel (associated to its centre).The area of the outer face of the voxel, having radius Ri, angular

perture �� from � = (j − 2)�� to � = (j − 1)��, and angular apertureϕ from ϕ = (k − 2)�ϕ to ϕ = (k − 1)�ϕ, and calculated by integrat-

ng Eq. (A.1), is obtained as:

S(i, j, k) = R2i {cos[(j − 1)��] − cos[(j − 2)��]}�ϕ (A.8)

The volume of the voxel delimited by two spherical shells of radiii−1 and Ri, and angular aperture as above, is calculated, accordingo Eq. (A.2), as:

V(i, j, k) =R3

i− R3

i−1

3{cos[(j − 1)��] − cos[(j − 2)��]}�ϕ (A.9)

The Laplacean operator in spherical coordinates is:

2 = ∂2

∂r2+ 2

r

∂

∂r+ 1

r2 sin2 �

∂2

∂ϕ2+ cos �

r2 sin �∂

∂�+ 1

r2

∂2

∂�2(A.10)

We used a finite difference formula in 4 points (Koonin anderedith, 1990) to obtain a numerical approximation of the

aplacean and then the solution of the diffusion equation (Eq. (3))n the volume element V(i, j, k) at a certain time, according to theelation:

dif(i, j, k) = cold(i, j, k) + D �t

{2

f ′r

d1(i)+ f ′′

r

+ [cold(i, j + 1, k) − 2cold(i, j, k) + cold(i, j − 1, k)]d2(i)

+f ′�

d3(i, j)+ [cold(i, j, k + 1) − 2cold(i, j, k) + cold(i, j, k − 1)]

d4(i, j)

}(A.11)

here cold(i, j, k) is the concentration value calculated at the pre-ious time step in the voxel V(i, j, k), f ′

r and f ′′r are first- and

econd-order partial derivatives of the concentration with respecto r, f ′

�is the first-order partial derivative of the concentration with

espect to �, and

1(i) = ri (A.12)

2(i) = (ri × ��)2 (A.13)

3(i, j) = r2i × tg[(j − 1.5)��] (A.14)

4(i, j) = {ri × �ϕ × sin[(j − 1.5)��]}2 (A.15)

According to the four-point formula, we obtain the derivatives′�

and f ′ϕ as:

′�

= [−2cold(i, j + 1, k) − 3cold(i, j, k) + 6cold(i, j − 1, k) − cold(i, j − 2, k)]/6��

(A.16)

′ϕ = [−2cold(i, j, k + 1) − 3cold(i, j, k) + 6cold(i, j, k − 1) − cold(i, j, k − 2)]/6

�ϕ

(A.17)

Nevertheless, in the vicinity of various boundaries, the choice ofhe four voxels must take into account the boundary conditions.

In order to calculate the partial derivatives f ′r and f ′′

r in a cer-ain voxel V(i, j, k), we have used the Taylor series expansion up

102 (2010) 134–147

to the three-order derivative and used the function values in foursuccessive voxel on the same radial direction (j, k = const.):

ci+1 − ci = �rif′r +

(�r2

i

2

)f ′′r +

(�r3

i

6

)f ′′′r (A.18)

ci−1 − ci = �ri−1 f ′r +

(�r2

i−1

2

)f ′′r +

(�r3

i−1

6

)f ′′′r (A.19)

ci+2 − ci = (�ri + �ri+1)f ′r +

[(�ri + �ri+1)2

2

]f ′′r

+[

(�ri + �ri+1)2

6

]f ′′′r (A.20)

ci−2 − ci = −(�ri−1 + �ri−2)f ′r +

[(�ri−1 + �ri−2)2

2

]f ′′r

−[

(�ri−1 + �ri−2)2

6

]f ′′′r (A.21)

where f ′′′r is the third-order partial derivative of the concentra-

tion with respect to r, in the voxel V(i, j, k). For each volumeelement, at each time step, we selected, in the most convenientway, three equations from Eqs. (A.18)–(A.21) and solved the sys-tem with unknown f ′

r , f ′′r and f ′′′

r , by choosing four voxels where theconcentration is already known and taking into account the vari-ous boundary conditions. However, sufficiently far away from theboundaries, one can choose from Eqs. (A.18)–(A.21) any combina-tion of three equations. After solving this three-equation system inthe voxel V(i, j, k), we use the obtained f ′

r and f ′′r values to determine

cdif(i, j, k) according to Eq. (A.11).

A.2. Boundary conditions

The zero-flux conditions for the lateral faces of the pipette readas:

f ′ϕ = 0 for i > ip, kp1 ≤ k ≤ kp2 and j = jp1 − 1 or j = jp2 − 1

(A.22)

f ′� = 0 for i > ip, jp1 ≤ j ≤ jp2 and k ≤ kp1 − 1 or k = kp2 − 1

(A.23)

Other boundary conditions are

f ′r = 0 for i = ic+1 (at the cell surface) (A.24)

f ′r = 0 for i = nr (at the outer boundary) (A.25)

f ′r = for i = ip, jp1 ≤ j ≤ jp2 and kp1 ≤ k ≤ kp2

(at the perfusion surface) (A.26)

Eq. (A.26) ensures that there is no flux from the bath solutionback into the pipette.

For the bath solution elements facing the perfusion surface of thepipette tip, we imposed the condition that the ACh concentrationincreases, at every time step, with a term given by the perfusion

flux:

c(ip, j, k, t) = cdif(ip, j, k, t) + ˚ × dS(ip, j, k)dV(ip, j, k)

× �t (A.27)

where jp1 ≤ j ≤ jp2 and kp1 ≤ k ≤ kp2.

tems 102 (2010) 134–147 145

A

l

[

wo

i

d

N

wmve

A

(L

v

w

f

F

ˇ

˛

b

cf

J

w1pt(

Fig. B.1. Geometrical framework for calculating the radial velocity in a given pointP (A). O′ represents the intersection of the pipette axis with the perfusion surface.The radial velocity (v) of the fluid in point P depends on the perfusion velocity (v0),

I. Baran et al. / BioSys

.3. Other relations

The average concentration on the cell surface has been calcu-ated, at each time step, as:

ACh]cell-av =

n�∑j=2

nϕ∑k=2

c(ic+1, j, k, t)dV(ic+1, j, k)

n�∑j=2

nϕ∑k=2

dV(ic+1, j, k)

(A.28)

hich was stored as 0.05 s time average values. The total numberf moles of acetylcholine in the bath solution is calculated at each

nstant t as N = 109 × 4∑i,j,k

c(i, j, k, t)dV(i, j, k) over the simulation

omain, and then compared to the theoretical value:

th ={

Q × [ACh]p × t, 0 ≤ t ≤ 1s

Nmax, t > 1s(A.29)

here Nmax = 109 × Q × [Ach]p × 1 s is the total number of ACholes introduced into the bath solution during the perfusion inter-

al. The integration time step is decreased sufficiently to keep therror of N within 2%.

ppendix B. The submerged jet

For a laminar jet emerging from a point source located in O′

Fig. B.1) into an infinite volume of the same fluid (Landau andifshitz, 1987), the radial velocity in the point P can be written as:

rps(d, �, ϕ) = [F(˛) cos(˛ + ˇ) − f (˛) sin(˛ + ˇ)]

cos(˛ + ˇ/2)d cos(ˇ/2)

(B.1)

here

(˛) = −2� sin ˛/(A − cos ˛)�

(B.2)

(˛) = −2�[(A2 − 1)/(A − cos ˛)2 − 1]�

(B.3)

= arccos

(1 − b2/2

rP2

)(B.4)

= arctan

{b cos (ˇ/2)/d − sin ˇ

cos ˇ + b sin (ˇ/2)/d

}(B.5)

= RP[(sin �1 cos ϕ1 − sin � cos ϕ)2 + (sin �1 sin ϕ1 − sin � sin ϕ)2

+ (cos �1 − cos �)2]1/2

(B.6)

In Eq. (B.6), (�1 = /2, ϕ1 = /2) and (�, ϕ) represent the angularoordinates of O′ and P, respectively. The constant A can be foundrom the relation:

= 16�2 A{1 + 4/3/(A2 − 1) − A/2 ln [(A + 1)/(A − 1)]}�

(B.7)

here J = �v20Sp represents the jet momentum (Landau and Lifshitz,

987; Revuelta et al., 2002). The numerical solution of Eq. (B.7) forarameter values given before is A = 3.326. In Fig. 3 we representedhe decay of the centreline velocity (Fig. 3A) along the pipette axisobtained with ˛ = ˇ = 0 and � = ϕ = /2), as well as the decay of the

on the radial distance from the perfusion surface (d) and on the position of P. Thedecay of the radial velocity with d predicted for a jet emerging from a point sourceis shown for three different angles ˇ (B).

average radial velocity vps (Fig. 3B), which was computed as thenumerical approximation of

vps(d) =∫∫

vrps(d, �, ϕ)sin �d�dϕ∫∫

sin �d�dϕ(B.8)

within the domain of interest Dc.For a round laminar jet emerging from a circular surface, the

velocity field can be generally expressed (Revuelta et al., 2002) interms of the normalized transverse and longitudinal coordinatesy = yt/a and x = d/Re/a where a is the exit radius at the tube end,d is the longitudinal distance from the jet exit, yt the transver-sal distance measured from the axis, and Re = �v0a/� = v0Rp/2/� isthe Reynolds number, which for our parameter values becomesRe = 2.00. When expressed in this way, the axial and transversalvelocity profiles scaled with v0 and �/�/a, respectively, becomeindependent of the Reynolds number (Revuelta et al., 2002). InFig. 3A, we present a set of theoretical (Revuelta et al., 2002) andexperimental data obtained with Re = 177 (Kwon and Seo, 2005)of the centreline velocity for a round jet (we limit the analysis toour range of interest x ≤ 1), which were fitted to an exponential

function, yielding:

vc(x) = 1.18v0 exp(

− x

x0

)(B.9)

1 stems

wt

d

wmofltd

v

wca

�

vi

tse

A

AA

C

t

�

wA2

q

q

q

awAdsatlria

P

wcPa

46 I. Baran et al. / BioSy

here x0 = d0/Re/a, with d0 = 500 �m in our working conditions, sohat x0 = 0.997. A general relation can be then obtained as:

0 ∼= Re × a = Q//v (B.10)

here Q is the volumetric injection rate and � = �/� is the kine-atic viscosity of the solution. Here we use the theoretical results

f Revuelta et al. (2002) made under the assumption that the fluidux is uniform at the jet exit. The longitudinal velocity (parallelo the cell–pipette axis) presents a Gaussian dependence on theeparture from the axis (Kwon and Seo, 2005) of the form:

l(x, y) = vc(x) exp[−y2/2/�(x)2] (B.11)

here � increases linearly with x (Kwon and Seo, 2005). From thealculations presented in Fig. 1b of Revuelta et al. (2002), we couldpproximate from the best linear fit to their numerical data that

(x) = 0.75 + 1.95 x (B.12)

Finally, for our geometry, we computed the radial componentr = vl cos ˇ and then calculated v as the average of vr weighted byts volumetric fraction (similarly to Eq. (B.8)) within the domain Dc.

We should note here that, for simplicity, in this section onlyhe absolute value of the velocity has been taken into considered,o that the sign “−” should be incorporated when the velocity isxpressed as a function of r, in accord with Eq. (5).

ppendix C. Opening rate and open probability of nAChR

According to the microscopic scheme of sequential binding ofCh and opening of the nAChR receptor (Akk and Auerbach, 1999;uerbach and Akk, 1998):

↔ AC ↔ A2C ↔ A2O ↔ A2D ↔ . . .

he minimal opening time � of the nAChR receptor is:

= �1 + �2 + �13 = 1q1

+ 1q2

+ 1q3

(C.1)

here the state-transition rates of the first three steps are (Akk anduerbach, 1999; Auerbach and Akk, 1998; Grosman and Auerbach,001):

1 = kon[ACh] (C.2)

2 = q1

2(C.3)

3 = q3M

1 + (EC50/[ACh])H(C.4)

q1 and q2 represent the rate of ACh binding to the unliganded (C)nd monoliganded (AC) closed state of the receptor, respectively,ith kon = 111 �M−1 s−1 (Akk and Auerbach, 1999; Grosman anduerbach, 2001). q3 represent the effective rate of opening from theiliganded closed (A2C) state of the receptor to the diliganded opentate A2O, with q3M = 60,000 s−1, EC50 = 532 �M and H = 1.6 (Akknd Auerbach, 1999). In the transition scheme presented above,he transition to the doubly liganded desensitized state A2D is fol-owed (not shown) by consecutive unbinding steps such that theeceptor may finally recover from desensitization and return to thenitial unbound closed state C (Akk and Auerbach, 1999; Auerbachnd Akk, 1998).

The open probability is expressed as

o = Pmaxo

h(C.5)

1 + (Kd/[ACh])

here Kd = 28 �M and h = 1.8 represent the apparent dissociationonstant and Hill coefficient of ACh binding to the receptor, andmaxo = 0.96 (Akk and Auerbach, 1999). The average open probabilitynd opening time are computed by weighting the corresponding

102 (2010) 134–147

value in each voxel at the cell membrane level by its surface fraction(similarly to Eq. (B.8)).

References

Akk, G., Auerbach, A., 1999. Activation of muscle nicotinic acetylcholine receptorchannels by nicotinic and muscarinic agonists. Br. J. Pharmacol. 128, 1467–1476.

Auerbach, A., Akk, G., 1998. Desensitization of mouse nicotinic acetylcholine recep-tor channels: a two-gate mechanism. J. Gen. Physiol. 112, 181–197.

Baran, I., 2007. Characterization of local calcium signals in tubular networks ofendoplasmic reticulum. Cell Calcium 42, 245–260.

Baran, I., 2008. Modulation of calcium signals by fluorescent dyes in the pres-ence of tubular endoplasmic reticulum: a modelling approach. BioSystems 92,259–269.

Baran, I., Popescu, A., 2009. A model-based method for estimating Ca2+ release fluxesfrom linescan images in Xenopus oocytes. Chaos 19, 037106.

Bruno, J.P., Gash, C., Martin, B., Zmarowski, A., Pomerleau, F., Burmeister, J., Huettl, P.,Gerhardt, G.A., 2006. Second-by-second measurement of acetylcholine releasein prefrontal cortex. Eur. J. Neurosci. 24, 2749–2757.

Di Angelantonio, S., Nistri, A., 2001. Calibration of agonist concentrations appliedby pressure pulses or via rapid solution exchanger. J. Neurosci. Methods 110,155–161.

Du, Y., Hancock, M.J., He, J., Villa-Uribe, J.L., Wang, B., Cropek, D.M., Khademhos-seini, A., 2010. Convection-driven generation of long-range material gradients.Biomaterials 31, 2686–2694.

Endeward, V., Gros, G., 2009. Extra- and intracellular unstirred layer effects inmeasurements of CO2 diffusion across membranes—a novel approach appliedto the mass spectrometric 18O technique for red blood cells. J. Physiol. 587,1153–1167.

Fan, T.H., Fedorov, A.G., 2004. Transport model of chemical secretion processfor tracking exocytotic event dynamics using electroanalysis. Anal. Chem. 76,4395–4405.

Fraser, J.R., Laurent, T.C., Laurent, U.B., 1997. Hyaluronan: its nature, distribution,functions and turnover. J. Intern. Med. 242, 27–33.

Grosman, C., Auerbach, A., 2001. The dissociation of acetylcholine from open nico-tinic receptor channels. Proc. Natl. Acad. Sci. U.S.A. 98, 14102–14107.

Hamill, O.P., Marty, A., Neher, E., Sakmann, B., Sigworth, F.J., 1981. Improved patch-clamp techniques for high-resolution current recording from cells and cell-freemembrane patches. Pflugers Arch. 391, 85–100.

Khanin, R., Parnas, H., Segel, L., 1994. Diffusion cannot govern the discharge of neu-rotransmitter in fast synapses. Biophys. J. 67, 966–972.

Koonin, S.E., Meredith, D.C., 1990. Computational Physics (FORTRAN Version).Addison-Wesley Publishing Company, Redwood City, California.

Kwon, S.J., Seo, I.W., 2005. Reynolds number effects on the behavior of a non-buoyantround jet. Exp. Fluids 38, 801–812.

Landau, L.D., Lifshitz, E.M., 1987. Fluid Mechanics, second ed. Pergamon Press,Oxford.

Linninger, A.A., Somayaji, M.R., Erickson, T., Guo, X., Penn, R.D., 2008. Computationalmethods for predicting drug transport in anisotropic and heterogeneous braintissue. J. Biomech. 41, 2176–2187.

Lisk, G., Desay, S.A., 2006. Improved perfusion conditions for patch-clamp recordingson human erythrocytes. Biochem. Biophys. Res. Commun. 347, 158–165.

Oancea, E., Wolfe, J.T., Clapham, D.E., 2006. Functional TRPM7 channels accumu-late at the plasma membrane in response to fluid flow. Circ. Res. 98, 245–253.

Oh, S., Odland, R., Wilson, S.R., Kroeger, K.M., Liu, C., Lowenstein, P.R., Castro, M.G.,Hall, W.A., Ohlfest, J.R., 2007. Improved distribution of small molecules and viralvectors in the murine brain using a hollow fiber catheter. J. Neurosurg. 107,568–577.

Popescu, A., Morega, A., 2004. Acetylcholine diffusion within the synaptic cleft: amodeling study. In: Roman, E., et al. (Eds.), Mathematical Modeling of Environ-mental and Life Sciences Problems. Proceedings of Workshops 2003, PublishingHouse of the Romanian Academy. Bucharest, pp. 242–253.

Purohit, Y., Grosman, C., 2006. Estimating binding affinities of the nicotinic recep-tor for low-efficacy ligands using mixtures of agonists and two-dimensionalconcentration–response relationships. J. Gen. Physiol. 127, 719–735.

Rees, N.V., Klymenko, O.V., Coles, B.A., Compton, R.G., 2003. Hydrodynamics andmass transport in wall-tube and microjet electrodes: an experimental evalua-tion of current theory. J. Phys. Chem. B 107, 13649–13660.

Revuelta, A., Sánchez, A.L., Linan, A., 2002. The virtual origin as a first-order cor-rection for the far-field description of laminar jets. Phys. Fluids 14, 1821–1824.

Rusakov, D.A., Kullmann, D.M., 1998. Geometric and viscous components of the tor-tuosity of the extracellular space in the brain. Proc. Natl. Acad. Sci. U.S.A. 95,8975–8980.

Sarntinoranont, M., Chen, X., Zhao, J., Mareci, T.H., 2006. Computational model ofinterstitial transport in the spinal cord using diffusion tensor imaging. Ann.Biomed. Eng. 34, 1304–1321.

Sawyer, A.J., Piepmeier, J.M., Saltzman, W.M., 2006. New methods for direct delivery

of chemotherapy for treating brain tumors. Yale J. Biol. Med. 79, 141–152.

Shepherd, N., McDonough, H.B., 1998. Ionic diffusion in transverse tubules of cardiacventricular myocytes. Am. J. Physiol. 275, H852–H860.

Smart, J.L., McCammon, J.A., 1998. Analysis of synaptic transmission in the neu-romuscular junction using a continuum finite element model. Biophys. J. 75,1679–1688.

tems

S

S

T

Vercruysse, K.P., Lauwers, A.R., Demeester, J.M., 1995. Absolute and empiri-cal determination of the enzymic activity and kinetic investigation of the

I. Baran et al. / BioSys

mith, J.H., García, J.J., 2009. A nonlinear biphasic model of flow-controlled infu-sion in brain: fluid transport and tissue deformation analyses. J. Biomech. 42,

2017–2025.

yková, E., Nicholson, C., 2008. Diffusion in brain extracellular space. Physiol. Rev.88, 1277–1340.

ai, K., Bond, S.D., MacMillan, H.R., Baker, N.A., Holst, M.J., McCammon, J.A., 2003.Finite element simulations of acetylcholine diffusion in neuromuscular junction.Biophys. J. 84, 2234–2241.

102 (2010) 134–147 147

action of hyaluronidase on hyaluronan using viscosimetry. Biochem. J. 306,153–160.

Whatey, J.C., Nass, M.M., Lester, H.A., 1979. Numerical reconstruction of the quantalevent at nicotinic synapses. Biophys. J. 27, 145–164.