accurate dielectric modelling of shelled particles and cells

Journal of Electrostatics 57 (2003) 143–156

Accurate dielectric modelling of shelled particlesand cells

Miguel Sancho*, Genoveva Mart!ınez, Carla Mart!ın

Departamento de F!ısica Aplicada III, Universidad de Madrid, Complutense, 28040 Madrid, Spain

Received 15 March 2002; received in revised form 22 July 2002; accepted 2 August 2002

Abstract

Dielectric modelling of shelled particles is usually based on analytical solutions of Laplace’s

equation, assuming a simplified spherical or ellipsoidal geometry. We propose a boundary

element method, derived from an integral equation for the charge density at the interfaces, to

overcome the limitations of the analytical approach. It can be applied to a variety of situations

involving stress effects or dielectric response of individual particles and colloidal suspensions,

with particular reference to biological cells. Effects of particle shape and membrane structure,

are analyzed. Polarizabilities of a spheroid with a layer defined by confocal surfaces and

different elongated particles with a layer of uniform thickness, are compared. Numerical

stability and efficiency of the proposed method are shown to be excellent.

r 2002 Elsevier Science B.V. All rights reserved.

Keywords: Polarizability; Single-shell model; Dielectric cell properties; Boundary element method;

Dielectrophoresis; Dielectric shells; Biological cell

1. Introduction

The study of the response of polarizable particles, particularly biologicalcells, to alternating electric fields has gained considerable attention in recent years,because of several important reasons. First, different applications of moderntechnology in electronic, food, pharmaceutical or metallurgical industries, involvethe electric processing of particles [1–3]. In particular, the introduction ofmicroelectrode technology has facilitated the development of sophisticated methods

*Corresponding author. Tel.: +34-3944445; fax: +34-3944688.

E-mail address: [email protected] (M. Sancho).

0304-3886/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 3 0 4 - 3 8 8 6 ( 0 2 ) 0 0 1 2 3 - 7

for manipulating, trapping and separating bio-particles, from bacteria to viruses andmacromolecules such as nucleic acids and proteins [4–7]. Second, exposure ofbiological cells to electric fields can produce different biophysical and biochemicalresponses, leading in the extreme case of very high field to membrane electroporationor fusion. But even at relatively low levels that do not involve detectable heating orcellular damage, some physiological alterations have been reported [8]. Publicconcern about the possible health effects of these weak RF fields has becomewidespread. Third, measurement of the dielectric permittivity of a colloidalsuspension has frequently been used for elucidating electrical properties of individualparticles or cells [9–11]. Recently, observations of the response of a particle to anonuniform (DEP) or rotating (ER) AC fields, have constituted single particlealternatives to the dielectric spectroscopy [1]. This electrokinetic response has beenused to characterize the type of cell, its physiological state as well as possiblefunctional alterations caused by physical or chemical agents [12–17].For all these applications, the theoretical modelling of the electrical response

of a homogeneous or compartmentalized particle to an external field is essential.Many relevant types of particles consist of a layered structure, either intrinsic oracquired during its handling. Important examples are biological cells that canbe approximated by rigid particles with a shell representing the membraneand coated particles with modified conductive properties. Classical models of cellsbased on shelled spheres or ellipsoids, initiated by Fricke [18] have been successfullyused to interpret many experiments involving single cells or colloidal suspensions[19,20]. They provide relatively simple analytical solutions that make it possible torationalize observed impedance or force effects in terms of a few structuraland electrical parameters. However, they have severe limitations: cell shapemay differ considerably from spherical or ellipsoidal geometry. There is no possibleanalytical solution for more realistic cell shapes, e.g. cylinders or rods, since anexplicit solution of the Laplace equation requires a geometry consisting of one orseveral uniform media separated by interfaces which coincide with a surface ofconstant coordinate, within a certain set of coordinate types [21]. Even for ellipsoids,this constraint requires the surface of the membrane to be confocal with thecore, producing a shell of nonuniform thickness. As the membrane is a site of a highfield amplification it is uncertain how this detail of modelling can affect the accuracyof the predictions in the electric behavior. Gheorghiu and coworkers [22–24],using an approach similar to ours, have examined shape effects on the permittivity ofcell suspensions and shown the importance of an adequate representation of the cellgeometry. Gimsa [25] has warned that the use of confocal shells may lead tolarge errors in the calculated polarizability of ellipsoids with large excentricity.For thin, little conductive shells, he has proposed a new RC element model.Whilst it has the great advantage of simplicity and can be very useful to analyzethe physical origin of different dielectric relaxations shown by cells and forqualitative analysis of fields and forces, this model, in which a prismatic piece ofdielectric medium is approximated by a simple RC circuit, seems to be valid onlyunder the assumption of constant field and consequently, only for shells of vanishingthickness.

M. Sancho et al. / Journal of Electrostatics 57 (2003) 143–156144

It turns out that only numerical methods can give a sufficiently precise estimationof field values in realistic cell structures. This type of studies [26–30] has to face thedifficulty of handling regions of very different size scales: microns for the celldiameter and nanometers for the membrane thickness. As the numerical solution ofthe Laplace equation in the form of finite differences involves a kind of polynomialapproximation in nodes of a convenient grid, the existence of very small domainsmakes it necessary to use a very dense grid in traditional finite difference (FDM) orfinite element methods (FEM). Except by using sophisticated adaptive meshingtechniques and extensive computer resources, the resulting number of equations tobe handled is prohibitively large. Using a boundary element method (BEM), onlybounding surfaces of the geometry need be discretized which reduces the memoryspace required and makes this approach especially appropriate for treating openproblems, as it is the case of a dielectric particle immersed in an electric field. But inany of FEM, FDM or standard BEM, the unknowns to be determined are the valuesof the electric potential at the nodes of a 3D grid or at the subareas of a 2D surfacemesh. Having computed the potential distribution, the dielectric properties areobtained only after a suitable treatment. For example, Brusseau and Beroual [27],using FEM and BEM, compute the permittivity of dielectric heterostructures, firstobtaining the potential and its derivatives at the nodes of a mesh and then computingthe electrostatic energy by integration. The electrostatic energy stored in the mediumgives a measure of the equivalent permittivity of the composite. Sekine [28] employsa similar standard BE approach and as a second step, performs a kind of fittingprocedure by equating the calculated potential to that produced by a dipole withprincipal components of polarizability to be determined. In this way, he is able tocalculate the cell polarizability and the permittivity of a suspension of cells. Thesesimulation methods for obtaining dielectric characteristics are indirect and sufferfrom supplementary errors besides those intrinsic to the numerical solution of theelectric potential.In this article we propose a new technique pertaining to the class of BEM. As

distinguished from the standard BEM for the electric potential solution, it is basedon an integral equation for the polarization charge density induced on the dielectricinterfaces, in quasi-static approximation. Since the numerical discretization of thisintegral equation does not involve any type of polynomial approximation, thesurface mesh may consist of a reasonably small number of elements, producing alinear system of equations easily solvable. Contrary to the similar integral equationformulated in terms of the electric potential for a current source [31], our approach isfree of numerical instabilities. As additional advantages, we can mention that,besides the equivalent polarizability of the particle, the method provides a directcomputation of the stresses produced by the electric field [32], even for nonuniformfields. It also gives physical insight into the induced charges and interfacial relaxationmechanisms that take place along the frequency spectrum. In this paper we focus onthe efficiency and accuracy of the method by examining several cases ofhomogeneous and layered particles with rotational symmetry around the fielddirection, but the method can be generalized to arbitrary particle shapes andorientations with respect to the field.

M. Sancho et al. / Journal of Electrostatics 57 (2003) 143–156 145

2. Integral equation for the induced charge density

Consider a dielectric object immersed in a fluid and subjected to a uniform,linearly polarized, AC electric field of magnitude E0 at angular frequency o: Usingphasor notation, we define a complex charge density at the interface as

*t ¼ ð*e1 � e0Þ *E12 þ ð*e2 � e0Þ *E21 ¼ e0*e1 � *e2

*e2*E12 ð1Þ

which is a generalization of the corresponding expression for the polarization chargedensity at the interface between media without losses. *e1 and *e2 are complexpermittivities of object and medium, respectively and e0 the permittivity of vacuum.*E12 denotes the normal component of the field directed from medium 1 to 2 andsimilarly *E21: The real part of the complex density *t comprises both polarizationcharges produced by the different dipole densities induced by the field at both sidesof the interface ðtpÞ and free charges built up at the surface as a consequence of thedifferent conductivities of both media ðtf ): Reð*tÞ ¼ tp þ tf :The quasi-static electric potential is obtained by superposition of the contributions

of external sources and that of free and polarization charges (assumed to be in thevacuum),

*fðrÞ ¼ *f0ðrÞ þ1

4pe0

ZS

*tðr0ÞRdS0; ð2Þ

where *f0 is the external potential, R ¼ jr � r0j and the integral extends to the particle-medium interface S.From Eq. (2) we obtain the field at a point r inside the particle, close to the

interface. When r tends to the interface, the integral on the right-hand side shows anintegrable singularity of the solid angle type [33], giving for the normal component ofthe field at the interface

*E12ðrÞ ¼ *E0nðrÞ �1

4pe0

ZS

*tðr0Þqqn

1

R

� �dS0 � 2p*tðrÞ

� �: ð3Þ

*E0n is the normal component of the field produced by the external sources. UsingEq. (1) we obtain

*tðrÞ ¼ 2*e1 � *e2*e1 þ *e2

*t0ðrÞ �1

2p*e1 � *e2*e1 þ *e2

ZS

*tðr0Þqqn

1

R

� �dS0; ð4Þ

where *t0ðrÞ ¼ e0 *E0nðrÞ:Eq. (4) can be easily generalized to the case of several dielectric interfaces

separating media of different permittivity. In such a case,

*tðrÞ ¼ 2*ei � *ej

*ei þ *ej*t0ðrÞ �

1

2p*ei � *ej

*ei þ *ej

XSk

ZSk

*tðr0Þqqn

1

R

� �dS0; ð5Þ

where r represents a point at the interface between media i and j and the sum in thelast term extends to all the interfaces Sk: This expression is a Fredholm integralequation of the second kind for *tðrÞ:


For the numerical solution of Eq. (5), each surface Sk is divided into nk suitablesmall elements D1;y;Dnk

: On each of these elements we assume *tðrÞ approximatelyconstant, thus resulting in the following set of linear equations:

*ti ¼XN

j¼1

Aij *tj þ Bi; i ¼ 1;y;N ð6Þ

with

Bi ¼ 2*ek � *el

*ek þ *el

*t0ðriÞ; ð7Þ

where ri is the position vector of the representative point (usually the center) of theelement Di: *ek and *elare the permittivities corresponding to both media in contact atri: The complex matrix coefficients Aij are given by

Aij ¼ �1

2p*ek � *el

*ek þ *el

ZDj

qqn

1

R

� �dS0 ð8Þ

with R ¼ jri � r0j: Aij are proportional to the normal component of the electric fieldcreated at ri by a uniform distribution of charge on the element Dj ; with unit density.The permittivities ei and ej are those corresponding to both media in contact at ri:Wewill restrict the analysis to cases of rotational symmetry around the external fielddirection and consequently, the elements Dj will be zones with this symmetry. Forexample, for the case of an spheroidal particle with its rotation axis coinciding withthe z-axis, the elements are chosen as thin strips between z constant planes. The fieldthat one of these elements, with unit charge density, produce at any point, iscalculated by numerical integration of the elementary field created by a chargedcircumference. In the spheroid case, the integration is performed along the arc of theellipse, between the extreme values of z:

3. Numerical tests

Numerically, our integral equation (5) produces a dense coefficient matrix. Thesolution of the linear system can be performed by either a direct method or aniterative solver [34]. In our study we employ a routine based on LU factorization ofthe complex matrix and posterior iterative refinement. We use increasing numbers ofelements to see how the numerical solution converges. We find that for a value of N

as small asB100 we already obtain the definitive solution. Having solved the linearsystem and computed the polarization charges *ti; the dipole moment induced by theexternal field is calculated as

p ¼XN

i¼1

*tiDiri

and hence, the corresponding polarizability is obtained. Some results given belowshow examples of computed and analytical polarizabilities of homogeneous andshelled particles. In these and other checked examples, including the most


unfavorable case of needle-shaped particle, the agreement is very good. We concludethat when the coefficients are calculated accurately, the solution is also very accurate,even using a small number of elements.An important issue in BEM is the stability of the numerical solution. Eq. (5) is

similar to the integral equation for the electric potential caused by a current source,used in the interpretation of neuromagnetic data. In that case however, the derivativeunder the integral symbol is performed with respect to the normal at r0 while inEq. (5) it is taken with respect to the normal at r: It is known that this problem hasno unique solution, corresponding to the physical fact that the electric potential isdefined only up to an additive constant. Besides this ambiguity that needs a deflationprocedure to be removed, the matrix system is not well conditioned for the case of anobject coated with an insulating shell. For this reason, and considering that abiological cell is just the case of a particle covered with an insulating membrane, wehave carefully examined the numerical behavior of our system of equations. Fig. 1shows an estimation of the condition number, cond ðAÞ ¼ norm ðAÞ: norm ðA�1Þ;which is a good indicator of the stability of the matrix solution, for different numberof elements used in the numerical solution. We see that the linear system solution isstable even when N is large.In this work we have restricted ourselves to cases with rotational symmetry, so

that the elements are strips with a common axis of symmetry and coefficients arecomputed by numerical Gaussian quadrature of the fields produced by a uniformlycharged circumference. However the method has general validity and can be appliedto an arbitrary geometry, using a convenient division of the interfaces into smallpatches. This generalization will be the subject of future work.

0 200 400 600 800 1000

103

104

105

106

107

cond

ition

num

ber

N

Fig. 1. Condition number of the coefficient matrix versus number of elements. The condition number has

been estimated using the approach described in Ref. [35] (The studied linear system corresponds to one of

the examples of shelled particles presented in Section 5.).


4. Homogeneous dielectric particles

We first consider the case of a homogeneous particle immersed in a uniform field.Eq. (5) gives the distribution of free and polarization charges on the surface of theparticle. A parameter of interest is the effective polarizability a or relation betweenthe effective dipole moment and the applied field. This parameter determines, for asmall particle, its dielectrophoretic response to a nonuniform field [32]:

F ¼ �1=2Re½ðpeff rÞEn� ¼ �1=2ReðaÞrE2: ð9Þ

Thus, according to our approach, we study the longitudinal polarizability of theparticle. In order to check the efficiency and accuracy of the method, we have studiedthe case of a spheroid and compared our results with the analytical solution. Fig. 2shows analytical and computed values of polarizability for a prolate spheroid witheccentricity e ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� b2=a2

p¼ 0:87: An excellent agreement is found for all tested

values of eccentricity and electrical parameters.Dielectric objects and bioparticles with different shapes have been usually

modelled by ellipsoids. We have investigated the accuracy of this approximation bysolving Eq. (5) for three different geometric shapes of particles with axial symmetry:spheroid, circular cylinder and rod ended by hemispheric caps.Fig. 3 shows the real and imaginary part of a for an elongated spheroid with

relative semiaxis lengths a ¼ 1; b ¼ c ¼ 0:2; a cylinder and a rod-like particle withthe same length and radius l ¼ 2a; r ¼ b ¼ c; as a function of frequency. Rod andcylinder have close spectra, but they differ from that of spheroid at low frequency.The analytical approximation in this case results in an inaccuracy of up to 30%. Abetter approximation to elongated particles consists of using a spheroid of the same

104 105 106 107 108 109 1010

-50

0

50

100

150

200

f (Hz)

Re

(α)/

ε 0,

Im (

α)/ε

0

Fig. 2. Real and imaginary parts of the effective longitudinal polarizability of a prolate spheroid with

relative values of semiaxes a ¼ 1; b ¼ c ¼ 0:5: Solid lines represent the analytical solution and dots arecomputed values. Polarizability scales as the particle volume. Assumed particle and medium permittivities

are e1 ¼ 50 e0; e2 ¼ 80 e0; and conductivities s1 ¼ 0:1 S m�1; s2 ¼ 0:5 S m�1; respectively.


volume rather than of the same length and radius. In Fig. 4, the correspondingspectra for particles of equal volume are compared, showing a closer agreement thanin the previous case.

104 105 106 107 108 109 1010

-10

0

10

20

30

40

50

60

70

f (Hz)

Re(

α)/ε

0, I

m(α

)/ε 0

Fig. 3. Computed longitudinal polarizability of a prolate spheroid, a circular cylinder and a rod with

hemispheric caps, of the same length and radius. Solid line: spheroid; dash-dotted line: cylinder; dashed

line: rod. Relative values of spheroid semiaxes a ¼ 1; b ¼ c ¼ 0:2: Electrical parameters are the same as inFig. 2.

104 105 106 107 108 109 1010-10

0

10

20

30

40

50

f (Hz)

Re(

α)/ε

0, I

m(α

)/ε 0

Fig. 4. Computed longitudinal polarizability of a prolate spheroid, a circular cylinder and a rod with

hemispheric caps, with characteristics identical to those in Fig. 3, but cylinder and rod have the same

radius and the same volume as the spheroid. Solid line: spheroid; dash-dotted line: cylinder; dashed line:

rod.


5. Shelled particles and cells

Modelling of layered particles with loss is of special interest because of the manyindustrial and biomedical applications. The dispersive behavior of these particles ismuch more complex than that of homogeneous ones, due to the combined Maxwell–Wagner polarizations at the different interfaces. Dielectrophoretic and electrorota-tional spectra have been used to characterize microorganisms. The unique solutionof the inverse problem, i.e., obtaining the dielectric parameters of a cell from theobserved frequency response, requires the relationship between effective polariz-ability and dielectric model to be accurately established.Variations in the assumed characteristics of the membrane may produce

significant alterations to the calculated polarizability. For example, the cellmembrane is known to have a high electrical resistance and is usually representedby a very thin layer with dielectric constant between 2 and 10 and low conductivity,with values that range between 0 and 10�2 S m�1: It is important to observe thateven values of membrane conductivity that are small compared to those ofcytoplasm and medium lead to spectra which are very different to that obtained withnull conductivity, as shown in Fig. 5, obtained by using the analytical solution for aconfocal model of the cell.Therefore, given this sensitivity of the results to modelling details, it can be

expected that removal of the restriction of confocal surfaces of the layer and the useof more realistic descriptions of membrane and cell shapes would lead to significantlybetter descriptions of the electrical behavior of cells. In order to check the limitationsof the current analytical model which assume ellipsoidal and confocal surfaces, we

104 105 106 107 108 109

-100

-50

0

50

100

150

f (Hz)

Re(

α)/ε

0

Fig. 5. Spectra of the effective longitudinal polarizability of a shelled spheroid, for different values of shell

conductivity. Solid line: s2 ¼ 0; dashed line: s2 ¼ 10�3 S m�1; dash-dotted line: s2 ¼ 10�2 S m�1:Parameters used in the calculations are a ¼ 1; b ¼ c ¼ 0:5; da ¼ 0:01; e1 ¼ 50 e0; e2 ¼ 10 e0; e3 ¼ 80 e0;s1 ¼ 0:1 S m�1; s3 ¼ 0:5 S m�1: Indices 1, 2, 3, refer to core, shell and external medium, respectively.


have applied our numerical BEM to different elongated shelled particles. Thegeometrical and electrical parameters used in all the calculations of this section are:ellipsoid semiaxes a ¼ 1; b ¼ c ¼ 0:2; layer thickness in the major axis directionda ¼ 0:01; the resulting layer thickness in the minor axis direction being db ¼ 0:058;permittivity and conductivity of core, shell and external medium e1 ¼ 50e0; s1 ¼0:1 S m�1; e2 ¼ 10 e0; s2 ¼ 1 mS m�1; e3 ¼ 80 e0; s3 ¼ 0:5 S m�1; respectively.In Fig. 6, the cases of ellipsoids with confocal surfaces and membrane of uniform

thickness are compared. For the last case, two possibilities are studied: membranethickness equal to da; and membrane thickness equal to db: In the same figure,analytical and numerical values for the confocal model are plotted, showing the highaccuracy of the BEM approach. It can be realized that the results obtained using theconfocal approximation differ greatly from those predicted by a more realistic modelof the membrane. At each frequency, the analytical value for the confocal model isintermediate between the results obtained for spheroids with membranes of uniformthickness da and db; respectively. This suggests that, besides shape effects, volumeeffects due to the contribution of the polarization of the different dielectric phases,are important. These results contrast to those reported by Fear and Stuchly [29] whostudied the transmembrane potential at low frequency, using a FE approach. Theyfound that small cell configurations could be approximated with the analyticalsolutions for similarly shaped ellipsoidal cells to within 10%. A possible explanationfor this discrepancy with our findings is that the overall dielectric response of the cellis much more sensitive to shape details than a local property as the transmembranepotential. In this connection, it is worth to note that, due to the boundary condition,the field at the outermost end of the cell is related to the external field by the quotientof permittivities of membrane and medium, independently of the cell shape.With the aim of examining the influence of shape factors on the polarizability, we

have next studied the dielectric behavior of shelled cylinder and rod like particles.

104 105 106 107 108 109

-10

-5

0

5

10

15

20

25

30

35

Re(

α) /

ε 0

f (Hz) (a)

104 105 106 107 108 109

-10

-5

0

5

10

15

20

25

Im(α

) / ε

0

f (Hz) (b)

Fig. 6. Real (a) and imaginary (b) parts of the effective longitudinal polarizability of a shelled cell-type

spheroid. Solid line: spheroid with confocal layer, analytical solution; dots: numerical solution; dashed

line: spheroid with layer of uniform thickness d ¼ da ¼ 0:01; dash-dotted line: spheroid with layerthickness d ¼ db ¼ 0:058:


Fig. 7 shows results of polarizability corresponding to particles of equal externallength and radius. In Fig. 8, we have also compared particles having the same radiusbut adjusting their lengths so that they have the same volume. In this case thegreatest polarizability corresponds to spheroidal shape.

6. Conclusions

Dielectric modelling is an important tool in the development of techniques forselective handling and characterization of particles and biological cells. In this paper

104 105 106 107 108 109-20

-10

0

10

20

30

40

Re(

α) /

ε 0

f (Hz) (a)

104 105 106 107 108 109

-20

-10

0

10

20

30

Im(α

) / ε

0

f (Hz) (b)

Fig. 7. Real (a) and imaginary (b) parts of the effective longitudinal polarizability of elongated shelled

particles with different shapes and the same length and radius. Solid line: circular cylinder; dashed line: rod

ended by hemispheric caps; dash-dotted line: spheroid with shell of uniform thickness d ¼ da ¼ 0:01:

104 105 106 107 108 109-10

-5

0

5

10

15

20

25

30

35

Re(

α) /

ε 0

f (Hz) (a)104 105 106 107 108 109

-15

-10

-5

0

5

10

15

20

25

Im(α

) / ε

0

f (Hz) (b)

Fig. 8. Idem as in Fig. 7, but particles have the same radius and volume.


we have shown the importance of a numerical approach to obtain an accuratesolution of the dielectric response using realistic models. Our boundary elementapproach has the advantage of not involving any linear expansion of potentials at apoint in terms of the potentials in neighboring nodes. The only approximationinvolved in the numerical solution of the exact integral equation (5) is that the chargedensity *t is approximately constant over each subarea Dk: Commercial programmes,using FEM with adaptive meshing procedure or standard BEM have been previouslyapplied to obtain field values at the different regions in a layered structure, but thecomputation of polarizability, surface charges or electric forces involves additionaltreatment of data presumably producing a considerable error. Our method providesa direct calculation of surface charges giving the particle complex polarizability aswell as electric stresses on the surface. As it has been shown the accuracy andreliability of the technique are high. The technique is useful for analysing a variety ofphenomena, i.e., DEP, ER, pearl chaining, including mutipole contributions notconsidered in the usual approximation or the deformation of cells by an electric field.When applied to different elongated particles uniformly shelled, it proves that theinfluence of shape and membrane structure cannot be ignored. It also allowsinterpretation of impedance results for a colloidal suspension in terms of thecharacteristics of individual particles. Finally, it may be used to predict levels offields and deposited energy in a model of cells exposed to electromagnetic radiation.Extension of this work to treat force and torque effects on arbitrary shaped andoriented particles and cells will be the subject of future research.

Acknowledgements

The authors would like to thank W.B. Betts and A.P. Brown for valuablediscussion and advice. Financial support from a EU grant (QLK2-CT-2001-70561RASTUD) is gratefully acknowledged.

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accurate dielectric modelling of shelled particles and cells

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