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Braz J Med Biol Res 37(2) 2004

Electro-rotation spectra of particles in liquid solutionsBrazilian Journal of Medical and Biological Research (2004) 37: 173-183ISSN 0100-879X

Mathematical modeling ofelectro-rotation spectra of smallparticles in liquid solutions.Application to human erythrocyteaggregates

1Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónomade Puebla, Puebla, Pue., México2Departamento de Microelectrónica, Instituto de Ciencias, Puebla, Pue., México3Centro de Investigación CENTIA Cholula, Universidad de las Américas,Puebla, Pue., México

A. Zehe1,A. Ramírez2 andO. Starostenko3

Abstract

Electro-rotation can be used to determine the dielectric properties ofcells, as well as to observe dynamic changes in both dielectric andmorphological properties. Suspended biological cells and particlesrespond to alternating-field polarization by moving, deforming orrotating. While in linearly polarized alternating fields the particles areoriented along their axis of highest polarizability, in circularly polar-ized fields the axis of lowest polarizability aligns perpendicular to theplane of field rotation. Ellipsoidal models for cells are frequentlyapplied, which include, beside sphere-shaped cells, also the limitingcases of rods and disks. Human erythrocyte cells, due to their particu-lar shape, hardly resemble an ellipsoid. The additional effect ofrouleaux formation with different numbers of aggregations suggests amodel of circular cylinders of variable length. In the present study, theinduced dipole moment of short cylinders was calculated and appliedto rouleaux of human erythrocytes, which move freely in a suspendingconductive medium under the effect of a rotating external field.Electro-rotation torque spectra are calculated for such aggregations ofdifferent length. Both the maximum rotation speeds and the peakfrequencies of the torque are found to depend clearly on the size of therouleaux. While the rotation speed grows with rouleaux length, thefield frequency νp is lowest for the largest cell aggregations where thetorque shows a maximum.

CorrespondenceA. Zehe

Facultad de Ciencias

Físico-Matemáticas

Benemérita Universidad Autónoma

de Puebla

Apartado Postal 1505

72000 Puebla, Pue.

México

E-mail: azehe@prodigy.net.mx

Research supported by CONACyT,

México.

Received June 14, 2002

Accepted June 24, 2003

Key words• Electro-rotation spectra• Induced dipole moment• Mathematical modeling• Erythrocyte aggregates

Introduction

The development of microdevices for ap-plications in a wide range of biologically andmedically related technologies is increas-ingly directed toward applying the advanced

technologies of microelectronic structuringand fabrication. Such devices permit rapidanalyses of small volume samples and haveapplications in medical and single-cell diag-nostics or chemical detection. The biopar-ticles are suspended in a stationary fluid, and

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tion for the polarization of single-shell ellip-soids has been previously derived for a seem-ingly quite different case, i.e., the meteoro-logical problem of dust particles coveredwith a fluid layer (15), prior to the applica-tion to biological cells (16,17). In order toarrive at explicit solutions of the Laplaceequation, a homogeneous ellipsoid as theonly material body with a constant local(internal) field has to be assumed. Integrat-ing over this field leads to the induced dipolemoment, providing the exposure of the ellip-soid to a homogeneous external field (18,19).The induced dipole moment is directly re-lated to force effects acting on the particle.How precisely the frequency-dependent forceeffects and corresponding spectra can bereproduced depends on the precision of thecalculus for the local field and the dipolemoment.

However, many particles and biologicalcells, including erythrocyte cell aggregates,deviate from the ellipsoidal form, and inorder to account for these specially shapedcells, more adequate models with shapesclose to short cylinders have to be consid-ered (20,21).

Normal human erythrocytes are non-nucleated biconcave disk-shaped cells ofabout 6.6-7.5 µm in diameter with edges thatare thicker than the center part. Neither aspheroid model of rotational symmetry nor aspherical single-shell model approximatesthe cell shape very well. The biconcave shape

Figure 1. A, Approximate shapeof a human erythrocyte with di-ameter 2R and thickness d atthe outer rim of 2.2 µm. Neithera prolate ellipsoid (spheroidal cellmodel of rotational symmetry)nor a spherical single-shell mo-del approaches the shape of thecell very well. B, Rouleaux for-mation of erythrocytes adheringside-by-side forming a cylindricalbody of variable length. Whilethe diameter 2R corresponds tothat of the single erythrocyte,(2R = 7.5 µm), the rouleauxlength is proportional to thestack number s of adhered redblood cells, 2L x s. For s ≥ 4, ashort cylinder with cylinder sym-metry axis of length > 2R isformed.

A

B

2R

= 2L.s

various force effects are imparted onthem by the application of alternating cur-rent electric fields for dielectric character-ization, manipulation, trapping or separation(1-12).

Calculations of the frequency depend-ence of force effects are primarily based onspherical or ellipsoidal models, the standardapproach to biological cells. In particular,single-shell models with Maxwell’s stresstensor (13) or the Laplace equation (14) areused. Interestingly, the general Laplace solu-

Figure 2. Schematic representa-tion of the electric field distribu-tion for prolate ellipsoids (a > b= c) in field direction E

→→→→→0 (A). An

oblate ellipsoid corresponds to b> a = c. B, Dielectric cylinders oflength 2L and diameter 2R.

dF→

(r)

FE→

0(r→)

b r→12

a

R

L

FE→

0(r→)

xr→12

A B

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Electro-rotation spectra of particles in liquid solutions

of erythrocytes is usually flatter than theb/a-axis ratio of an ellipsoid would indicate,since the curvature at the equatorial zone ofthe cell is more critical for the polarizationthan the actual axis ratio (see Figures 1A and2). The determination of the induced dipolemoment for such a structure in the directionof the rotation axis will be possible only witha certain approximation, say, of a very shortcircular cylinder (disk) of radius R and halflength L with L/R<<1.

A controlled formation of erythrocytedoublets has been shown recently by electro-fusion (22), leading to an axis relation of2L/R = 0.534. Rouleaux formation (Figure1B) dominates the dielectric behavior of cellsuspensions (23). Their size has been shownto be of clinical relevance (24).

The aggregation of disk-shaped objectsto columns has a clear effect on the localfield and the induced dipole moment. Whilethe depolarizing factor in the direction of thecylinder axis (parallel to the external electricfield) has a value close to 0 for L/R>>1, forthe short cylinder or disk, this value is closerto 1.

It is important to visualize that the aggre-gation of individual erythrocytes generatescolumns of length l = s × d with d = 2L thethickness of a single disk (d = 2.2 µm) and s= 1, 2, 3, etc. Depending on the number s, theaxis ratio of the columns L/R, the depolariz-ing factors f (q) and the induced dipole mo-ments change correspondingly, and so doesthe electro-rotation torque acting on the cellaggregates.

Numerical methods most commonly usedfor field characterization within a biologicalstructure are based on Mie theory (25-27) orthe finite element and the finite differencetime domain techniques. However, exten-sive computer calculations are required(7,28) and a simpler approach is desirableand often sufficient.

The rouleaux formation of erythrocytesdue to side-by-side adherence of several in-dividual cells translates into a cylindric cell

model of variable but discrete cylinder height.Characterized by an integer factor of singlecell heights, the aggregation of two, threeor more cells changes the cell geometryfrom oblate to prolate. Thus, the local fieldand the polarization are affected consider-ably.

In the present study we calculated theelectro-rotation spectra of erythrocytes un-der the aspect of cell aggregates of differentstacking size. The multiplicity of these cellaggregates can be monitored by electro-rota-tion forces, and size-selective spectra ap-pear, given a suitable suspending mediumconductivity and frequency of the appliedcircular polarized electric field. The approxi-mation procedure for dielectrics of generalshape, departing from the simple ellipsoidapproach, is included and applied to rou-leaux of human erythrocytes of differentlength.

The problem is of practical importancesince erythrocyte cell aggregation stems fromthe fact that rouleaux formation is caused byan increased blood concentration of fibrino-gen, globulin or paraproteins. Associatedclinical disorders include acute and chronicinflammatory disorders, macroglobulinemia,and multiple myeloma.

Electro-rotation

A circularly polarized rotating electricalfield induces a circulating dipole moment ina dielectric object. Due to ever-present dis-persion processes in the exposed sample, aspatial shift between the external field vectorand the induced dipole moment occurs. Theinteraction of the out-of-phase part of thisdipole moment and the electrical field causesa torque on the object.

Very often only frequency dependenciesof a physical measure are of interest. Thus, itis sufficient to consider the Clausius-Mossottifactor K(ω), which is the frequency-depend-ent part of the induced dipole moment. Thetorque is proportional to the imaginary part

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A. Zehe et al.

of this factor and describes the frequencydependence of the rotation of an individualcell under study in or against the rotationsense of the external field.

The time-averaged rotational torque incircularly polarized external fields, excertedon a particle, is given by the vector productof induced dipole moment and conjugatefield

(Eq. 1)

The induced dipole moment m → is propor-tional to the external field E, the suspendingmedium permittivity ε*

m and the volume V ofthe object. Let the principal axes of the (el-lipsoidal) object be oriented parallel to thevectors of the base system, and then m → willbe given by

(Eq. 2)

(i, r - imaginary and real part, respectively,and j = (-1)1/2 - imaginary unit). With theexternal alternating current field written E =E0 × exp(jωt), and assuming that its compo-nents Ex, Ey, Ez are parallel to the sameorthonormal base system, it follows fromEquation 1 that

(Eq. 3)

An electric field circulating at constantamplitude in the x-y plane can be written as

(Eq. 4)

For Ez considered to be zero, and as

established in practical cases, where oneof the principal axes of the ellipsoid is al-ways perpendicular to the field plane, thek-th component of the acting torque resultsin

(Eq. 5)

For a homogeneous ellipsoidal particle,the Clausius-Mossotti factor (19) in the xdirection is given by

(Eq. 6)

where ε* p, ε*

m are the complex permittivity ofthe sample and the suspending medium, re-spectively, and fx(q) is the depolarizing fac-tor along this direction with q being a num-ber describing the axis relation (29-31). TheClausius-Mossotti factor is a measure of theeffective polarizability of the particle anddepends for fx(q) strongly on the geometricalshape of the ellipsoidal object.

With ε the permittivity and σ the conduc-tivity of a dielectric medium, the complexpermittivity is defined as

(Eq. 7)

Consequently, the Clausius-Mossotti fac-tor depends on the frequency of the appliedfield in addition to the dielectric propertiesof particle and medium. When only frequencydependencies are important in the study, it issufficient to consider K(ω) as the only fre-quency-dependent part of the induced dipolemoment. Variations of this factor give rise tothe electro-rotation forces described in Ref.5, which are unique to a special particle type.This concerns not only intrinsic dielectricproperties, but also the geometrical shapevia the depolarizing factors fx. It is this latter

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aspect that plays a particular role in thepresent study.

Shape variation of the particles affectsK(ω), leading to readily achievable electro-rotation spectra. The design and geometry ofthe microelectrodes used to generate the ro-tating electrical fields are of course impor-tant factors to be considered. To determinethe orientation of particle movement, onlythe directions (signs) of the torque compo-nents corresponding to Equation 3 areneeded. The sign of the torque about thez-axis, as considered in this report, is equiva-lent to that of the frequency-dependent K(ω)-term:

(Eq. 8)

Thus, if the imaginary component of m → ispositive, corresponding to Equation 1, theexerted torque will be negative and cause theparticle to rotate in a sense that opposes thatof the rotating field.

Calculation of the inducedpolarization in linear rouleaux

In the Laplace model, a homogeneousellipsoid always exhibits a constant localfield. Integration over this field leads to in-duced polarization and thus to expressionsrelated to the actions of force on the specialobject. On the other hand, in such importantcases as a cube or a short cylinder it isdifficult to calculate the depolarization fac-tors without accepting an ellipsoid to substi-tute the shape of the object. But even thenthe best shape to be used and the next ap-proximation step are not a straightforwardchoice. This chapter deals with an approxi-mation procedure for the calculation of theinternal depolarization field E

→→→→→i (r → ) in a mate-

rial body of general shape and a complexdielectric constant ε*

p, which is brought intoan external field E

→→→→→0 (r

→ ), acting inside a medi-

um of dielectric permittivity ε* m.

The problem can be formulated asfollows (see Figure 2): the local field E

→→→→→i (r → )

causes a polarization P→→→→→

= ε0(ε* p − ε*

m)E→→→→→

i . Thispolarization generates on the surface ele-ment ∆F

→→→→→ of the dielectric body a polarization

charge ∆q =σpol.∆F =P

→→→→→ .∆F→→→→→

, which by vir-tue of the Coulomb law, together with theunperturbed field E

→→→→→0 (r

→ ), produces the localfield of such a strength that

(Eq. 9)

The integration is carried out on the sur-face of the dielectric body; ∆F

→→→→→ points out-

ward, and r →

12 combines the origin r →

1with theintegration element at r →

2. We assume therelation between E

→→→→→0 (r

→ ) and E→→→→→

i (r → ) to be linear:

(Eq. 10)

In general, α (r→ ) is a tensor since thedirections of E

→→→→→i and E

→→→→→0 are not necessarily

parallel. It further depends on the place in-side the sample due to the locally differentaction of the polarization charges.

In order to calculate P→→→→→

(r → ) or E→→→→→

(r → ) fromEquations 9 and 10, we assume that α doesnot depend on r → .

The polarization established inside thedielectric particle is due to the displacementof electrical charges enforced by the fieldE→→→→→

0(r→ ). Surface charges are built up and coun-

teract the complete displacement whichwould correspond to the field E

→→→→→0(r

→ ). We willassume here that the whole set of chargesexperiences the same displacement, whichmeans that in Equation 10, α = constant. Wefurther assume that the polarization vectorP→→→→→

points more or less towards the directionof E

→→→→→0 (r

→ ), i.e., we will consider the projectionof the field generated by the polarizationcharges in the direction of E

→→→→→0(r → ):

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This α1 value allows us to consider a firstapproximation of the polarization P

→→→→→1, which

generates charges on the surface of the di-electric and thus an additional field insidethe dielectric. The problem would be com-pletely solved if the total field at any placefulfilled the condition

(Eq. 12)

but, in general, the polarization P→→→→→

1 of thefirst approximation step will not be suffi-cient to describe the real situation, and afield E

→→→→→1(r → ) will keep acting on the dielectric

with the effect of the additional polarizationP→→→→→

2(r → ),

(Eq. 13)

P→→→→→

2(r → ) can be calculated with E→→→→→

1(r → ) inthe same way as P

→→→→→1(r → ) was calculated with

E→→→→→

0(r → ). The number of approximation stepsneeded to achieve the best result depends onthe complexity of the shape of the dielectricbody, as well as on the allowed error of theresult.

We will apply this procedure now to acylinder-shaped object of the rouleau kind.Exact solutions proposed by Fuhr et al. (11)with a constant α are known for the sphere,as well as the infinitesimal thin wire (needle),and the infinitesimal extended disk (sheet).

When our approach is applied here, the firstapproximation step (11) gives the exact so-lution, as it should, when α (r → ) = α1 is aconstant.

The polarization of a prolate spheroid(Figure 2A), considered together with Equa-tion 10 and the axis relation q2 = b2/(a2-b2),yields

(Eq. 14)

An oblate ellipsoid yields

(Eq. 15)

and consequently with q → ∞ (or a = b = c)we obtain α1 = 3/(ε*

p + 2ε* m) for the sphere-

shaped dielectric body, and thus the knownresult for the polarization vector

(Eq. 16)

which would be a rouleau of length = 2Lswith s ≥ 4.

Not so straightforward is the situation inthe case of a short cylinder which would bea rouleau of length = 2Ls with s ≤ 4 (seeFigure 2B). As a first approximation, weobtain

(Eq. 11)

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Electro-rotation spectra of particles in liquid solutions

Such a homogeneous polarization is thefirst approximation and good enough onlyfor long cylinders with L>>R or for diskswith R>>L. Due to the choice of the centerpoint at z = 0, the P

→→→→→ generated field will be

too weak in the transversal plane at z = 0, butalong the z-axis at the limiting surfaces ofthe cylinder it is too strong. A field E

→→→→→1(r → )

remains as given in Equation 13, which de-livers the depolarization at the cylinder topand bottom surfaces in a second approxima-tion.

The integrations involved in this step arequite tedious and will not be presented here.

(Eq. 20)

We obtain

(Eq. 19)

The surface charge density σ2 = ε0(ε* p -

ε* m)α2 E

→→→→→1⊥ when added to σ1 has the effect of

reducing the charge density of the side sur-faces, but increasing it on the cover areaclose to the side surfaces.

The dipole moment of the cylinder oflength 2L in a second approximation yieldsthen Equation 20:

By comparing this equation with Equa-tion 10, an analytical expression for α isfound after two approximation steps and canbe used for rouleaux of different length.

Electro-rotation torque calculationand discussion

The frequency response of the electro-

(Eq. 17)

(Eq. 18)

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A. Zehe et al.

Figure 3. A, Electro-rotationspectra of erythrocyte rouleauxof different lengths = s x 2L, atmedium conductivity σm = 100mS/m. 2L = 2.2 µm is the thick-ness of a single erythrocyte. Thevertical axis displays a measureof the acting torque. Only thespectral range of positive torqueis shown, where no membraneeffects are involved. Cytoplasmparameters are εp = 50 and σp =0.535 S/m. B, Same as A, butthe medium conductivity is re-duced to σm = 1 mS/m. Themaximum torque on the par-ticles is increased up to a factorof three.

rotation torque on linear erythrocyte cellarrangements in rouleaux is governed by thecorresponding Clausius-Mossotti factor,which is related to α (Equations 10 and 20) by

(Eq. 21)

In the limiting cases of a long cylinder withα = 1/ε*

m, a sphere with α = 3/( ε* p + 2 ε*

m) , and adisk with α = 1/ε*

p, respectively, the relationsknown for ellipsoids are of course reproduced.In the present case, the rouleaux are approxi-mated to solid homogeneous cylinders of length

= 2L × s (Figure 1C) with a relative permit-tivity εp = 50 and conductivity σp = 0.5 S m-1

(32). For s ≥ 4, the aggregation length corre-sponds to short cylinders whose symmetryaxis lines up with the external electric fieldvector, while for s ≤ 3 a reorientation willoccur with the erythrocyte radius vector nowin parallel to the external field. By applying thegiven electrical parameters, a plot of the fre-quency variation of the polarization factorestimated from Equation 20 was calculated asa function of frequency and rouleaux-lengthparameter s in medium conductivities of σm =1 mS/m and 0.1 S/m, respectively.

60

50

40

30

20

10

00.1 1 10 100 1000 10000

Frequency (MHz)

0.1 1 10 100 1000

Frequency (MHz)

45

40

35

30

25

20

s ×

ImK

(ω)

15

10

5

s ×

ImK

(ω)

s = 30s = 30s = 30s = 30s = 30

s = 15s = 15s = 15s = 15s = 15

s = 3s = 3s = 3s = 3s = 3s = 2s = 2s = 2s = 2s = 2s = 1s = 1s = 1s = 1s = 1

s = 10s = 10s = 10s = 10s = 10

s = 9s = 9s = 9s = 9s = 9s = 8s = 8s = 8s = 8s = 8s = 7s = 7s = 7s = 7s = 7s = 6s = 6s = 6s = 6s = 6s = 5s = 5s = 5s = 5s = 5s = 4s = 4s = 4s = 4s = 4

s = 10s = 10s = 10s = 10s = 10

s = 9s = 9s = 9s = 9s = 9

s = 8s = 8s = 8s = 8s = 8

s = 7s = 7s = 7s = 7s = 7

s = 6s = 6s = 6s = 6s = 6

s = 5s = 5s = 5s = 5s = 5s = 4s = 4s = 4s = 4s = 4

s = 1s = 1s = 1s = 1s = 1

s = 3s = 3s = 3s = 3s = 3

s = 2s = 2s = 2s = 2s = 2

A

B

0

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Electro-rotation spectra of particles in liquid solutions

Only the high frequency (positive torque)part is shown, which reproduces the cellbehavior without accounting for the electri-cal properties of the cell envelope, but dis-plays the effect of different rouleaux lengths.

Figure 4. Peak frequency νp ofthe maximum torque on a rou-leaux of length = s x 2L, wheres = 1 to 20 characterizes thelength of the cell aggregation.At s ≤ 3 the particle rotation axisis the rouleaux cylinder axis,while for s ≥ 4 the erythrocyteradius vector is the rotation axis.Medium conductivities are intro-duced as parameters: σm = 0.1S/m (filled circles); σm = 0.05 S/m(open squares); σm = 0.01 S/m(filled triangles); σm = 0.001 S/m(open circles).

60

50

40

30

20

10

0

ν p (

MH

z)

0 5 10 15 20Aggregation number s

and depends of course on the dielectric dataof cell and medium, as well as on the cellshape via the depolarization factor f (q). Mem-brane data do not interfere with the high-frequency part of the spectrum.

The fitting of theoretical electro-rotationspectra to experimentally determined datamight require the consideration of hydrody-namic friction. Indeed, the counteracting fric-tional force will increase with cell volumetoo, and smaller rotation speeds are meas-ured.

The peak frequency νp of the rotation

spectra for different rouleaux lengths is pre-sented in Figure 4, showing a shift to lowervalues with increasing length parameter s.While these shifts are more pronounced be-tween short rouleaux s ≥ 4, a computer dif-ferentiated presentation [dT(ω)/dω] wouldindicate with high precision the frequencymaxima and their correlation with the lengthparameter s. In order to understand the posi-tive peak frequency shift in νp for values s =1 to 4, one has to consider the orientation ofthe main (longest) particle axis with respectto the rotating electric field vector. Since the

The results are shown in Figure 3. The de-pendence of the peak rotation speeds on therouleaux size for a certain medium conduc-tivity is given by the maximum value of eachcurve in this figure. Its theoretical height is

(Eq. 22)

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Cells rotate around their small semi-axis.A reorientation of rouleaux occurs at thetransition from s = 2 to 4. Rouleaux with s =3 show an axis relation of approximately 1:1and are not well defined in terms of reorien-tation.

The magnitude and polarity of the torqueare related to both the geometrical structureand the dielectric properties of the rouleaux.

The experimental determination ofelectro-rotation spectra on the one hand, andthe fitting to theoretically determined electro-rotation curves, particularly to the character-ized frequencies νp, on the other, allows usto determine the set of dielectric and geo-

metric properties of the cells in correspond-ing frequency ranges.

The rotation can be observed using eitherdigital image processing techniques or directCCD analysis. The typical induced rotationspeed is in the range of one revolution persecond, and can of course be controlled bythe driving voltage.

Acknowledgments

We are grateful to Ms. Guadalupe LealSantos and Mr. Eduardo Ramírez Solís forhelp with computer programming and manu-script typing.

rouleaux length is defined by = 2L × s, andthe erythrocyte thickness at the rim by 2L =2.2 µm, only for values s ≥ 4 will the mainparticle axis be the rouleaux cylinder axis.The erythrocyte diameter 2R = 7.5 µm iswider for s ≤ 3 than the rouleaux length andcorrespondingly the radius vector aligns with

the external field, i.e., the shorter rouleauxcylinder axis is the rotation axis.

The peak frequency is related to the celland medium properties, as well as to therouleaux size via the depolarization factorf (q) as

(Eq. 23)

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Electro-rotation spectra of particles in liquid solutions

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The Program is partially supported byAFIP (Associação Fundo de Incentivo à Psicofarmacologia),

a non-profit organization. This advertisement is financed by AFIP.

The Department of Psychobiology, Escola Paulista de Medicina, is aWHO Collaborating Center for research and training in MentalHealth, and has laboratories for biochemical, physiological andbehavioral research.It also has a Clinical Psychobiology Research Center, with a 16-bedinpatient unit, sleep and biofeedback laboratories, a clinical analysislaboratory, and an outpatient clinic. This Center provides consultingservices to the federal and state governments.

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The Department of Psychobiology, EPM, offers basic courses toundergraduate students of Medicine and Biological Sciences, aclinical psychopharmacology course to residents in Psychiatry, anda PSYCHOBIOLOGY GRADUATE PROGRAM designed to trainstudents for academic, research, and clinical activities.

For more information please contact:DEPARTAMENTO DE PSICOBIOLOGIA - ESCOLA PAULISTA DE MEDICINA. A/c Secretaria do Curso de Pós-Graduação,

Rua Botucatu, 862, 1º andar, Caixa Postal 20399.8, CEP 04023-062 São Paulo, SP, Brasil. Tel. (011)576-4550/(011)576-4504