Defocusing microscopy: An approach for red blood cell opticsLeonardo G. Mesquita, Ubirajara Agero, and Oscar N. Mesquita Citation: Applied Physics Letters 88, 133901 (2006); doi: 10.1063/1.2189010 View online: http://dx.doi.org/10.1063/1.2189010 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/88/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Total three-dimensional imaging of phase objects using defocusing microscopy: Application to red blood cells Appl. Phys. Lett. 104, 251107 (2014); 10.1063/1.4884420 The flow of red blood cells through a narrow spleen-like slit Phys. Fluids 25, 110807 (2013); 10.1063/1.4819341 Optical tweezers assisted quantitative phase imaging led to thickness mapping of red blood cells Appl. Phys. Lett. 103, 013703 (2013); 10.1063/1.4812985 Coarse-grained theory to predict the concentration distribution of red blood cells in wall-bounded Couette flow atzero Reynolds number Phys. Fluids 25, 061901 (2013); 10.1063/1.4810808 Translational velocity measurement for single floating cell based on optical Fourier transform theory Appl. Phys. Lett. 88, 101114 (2006); 10.1063/1.2183747

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Defocusing microscopy: An approach for red blood cell opticsLeonardo G. MesquitaDepartamento de Física, ICEX, Universidade Federal de Minas Gerais, Caixa Postal 702,Belo Horizonte, CEP 30123-970, MG, Brazil

Ubirajara AgeroDepartamento de Bioquímica e Imunologia, ICB, Universidade Federal de Minas Gerais, Belo Horizonte,CEP 30123-970, MG, Brazil and Departamento de Física, ICEX, Universidade Federal de MinasGerais, Caixa Postal 702, Belo Horizonte, CEP 30123-970, MG, Brazil

Oscar N. MesquitaDepartamento de Física, ICEX, Universidade Federal de Minas Gerais, Caixa Postal 702,Belo Horizonte, CEP 30123-970, MG, Brazil

�Received 5 July 2005; accepted 28 February 2006; published online 29 March 2006�

Thin transparent objects �phase objects� can become visible in a bright-field light microscope, if themicroscope is slightly defocused. Thick transparent objects, like red blood cells �RBC�, are seenbecause some of their parts are always out of focus. By applying our recently developed defocusingmicroscopy technique to RBC, we are able to interpret RBC bright-field light microscopy images,an old standing problem. From the average image contrast we obtain RBC shape, size, and refractiveindex. From contrast fluctuations caused by the flicker phenomenon, we obtain RBC bendingmodulus and cytoplasm viscosity. © 2006 American Institute of Physics. �DOI: 10.1063/1.2189010�

It is known that phase objects become visible in aslightly defocused bright-field light microscope. Recently,we proposed a theoretical model for a defocused microscopeand showed that image contrast is proportional to the defocusdistance and to the two-dimensional Laplacian of the phasedifference introduced by the object. For objects with a uni-form refractive index, defocusing microscopy then yields aportrait of the object’s curvature profile.1,2 There is no loss ofthe original microscope lateral resolution, since images areslightly defocused and because of the use of image process-ing techniques. We have recently applied defocusing micros-copy to study motility of a macrophage cytoskeleton and itsrelationship to the macrophage’s phagocytic capacity. Infor-mation about the viscoelastic properties and fluctuation spec-tra of macrophages cytoskeleton has been obtained.1,3 In or-der to explore and test the capabilities of defocusingmicroscopy we study in the present work red blood cells�RBC�, because they are more complex phase objects, anddue to the large amount of data that has been accumulated onshape, the fluctuation surface spectrum caused by the flickerphenomenon, and on optical and mechanical properties.4–8

The theoretical basis for understanding the flicker phenom-enon was published 30 years ago by Brochard and Lennon�BL�,9 where they demonstrated that it is caused by thermalfluctuations of RBC thickness. New interpretation of theRBC fluctuation spectrum and the recognition that nonther-mal �metabolic driven� noise plays an important role in theflicker phenomenon are also discussed.10–12

Applying the defocusing microscopy theory developedin Refs. 1 and 2, we find for the contrast of a defocused RBCthe approximate expression:

C�r� =I�r� − I0

I0

= �n��f − h2�r���2h2�r� − �n��f − h1�r���2h1�r� .

�1�

In Eq. �1�, C�r� is the image contrast profile, I�r� is theintensity at the position r on the RBC image, I0 is the back-ground intensity nearby the RBC, h2 and h1 are, respectively,the heights of the RBC upper and lower surfaces, in relationto the microscope coverslip; �f is the position of the micro-scope objective focal plane in relation to the coverslip; �n isthe difference in refractive index with �n=nRBC−nmedium, forlight coming from the medium and entering the RBC. In thesecond term of Eq. �1�, the sign of �n is negative, since lightgoes from the RBC toward the medium. For a discocyteRBC, in addition to the circular symmetry, there is a reflec-tion symmetry such that �2h1�r�=−�2h2�r�, where �2h2�r�= �1/r��� /�r��r��h2 /�r�� is the radial part of the Laplacian ofh2 in circular coordinates. Then, the resulting expression forthe defocused contrast of a discocyte RBC is greatly simpli-fied to read as

C�r� = 2�n��f −h2�r� + h1�r�

2��2h2�r� . �2�

It is important to notice that h2�r�+h1�r� has a constantvalue that corresponds to the height of the thickest part of theRBC. Equation �2� shows that the image contrast of a defo-cused discocyte RBC is proportional to its curvature profile�2h2�r�. Figure 1�a� shows a RBC defocused image andFig. 1�b� shows the corresponding image contrast along theRBC horizontal diameter. A high-pass red filter, which trans-mits light with wavelengths larger than 590 nm, is used toavoid absorption, such that only defocusing effects areobserved.

Our microscope is a Nikon inverted microscope �EclipseTE300� equipped with an objective of 100X and numericalaperture NA=1.4. The images are captured by a CCD cam-era �CCD-72 Dage-MTI� and recorded in a DVCAM�DSR-20 Sony� for further analysis. The images are digitizedby a firewire card and converted to 8-bits format for imageanalysis with the public domain program ImageJ. Since thecamera and image acquisition system give image intensities

APPLIED PHYSICS LETTERS 88, 133901 �2006�

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on a gray-level scale �from 0 to 255�, one needs a calibrationto obtain the light intensity contrast C�r� from the gray-levelcontrast CG�r�. The calibration procedure is described inRef. 2. For the particular camera settings used in this workwe have C�r�=0.75CG�r�. RBC samples are prepared at thetime of the experimental runs. Fresh blood is diluted in asolution of PBS �pH 7.4� with BSA �1 mg/ml� in order toobtain discocyte RBC, following the procedure of Ref. 12.We use a Newport PZT translator to move the sample verti-cally for scanning �f .

From Eq. �2� one sees that for �f = �h2+h1� /2, i.e., whenthe focal plane is just at the midplane of the RBC, imagecontrast vanishes. Therefore, we have an accurate method todetermine the midplane position of RBC in relation to thecoverslip, and, consequently, the height �h2+h1� of the thick-est part of the RBC. Contrast profiles C�r� as a function of�f were obtained and confirmed the validity of Eq. �2�. Con-trast inversion is clearly seen around the midplane, wherecontrast is zero. The linear dependence between C�r� and �foccurs for defocus distances in the range 2 �m in relation tothe RBC midplane. Within this range, one can choose anycontrast profile for shape profile determination. The contrastprofile of Fig. 1�b� is numerically integrated twice to obtain2�n��f − �h2�r�+h1�r�� /2�h2�r�. This can be simply done byusing the integrate-curve option of the Kaleidagraph �Syn-ergy� plot program. Since we have previously determined�f − �h1+h2� /2 and �h1+h2�, we obtain the RBC profile �Fig.2� and also the difference in the refractive index,�n=0.042±0.005, for this RBC, which is within the range ofvalues measured by other techniques.4 The error is due touncertainties in the determination of the actual RBC height.

For the double integration, the constant for the first integra-tion is zero; otherwise one would have a logarithmic diver-gence for r=0; the second constant is determined after theintegration by setting the value of h2 �in relation to the mid-plane� equal to zero at the edge of the RBC. The smoothnessof the RBC profile �Fig. 2� is remarkable, because the inte-grations performed on the experimental data filter small-scale fluctuations.

RBC shape profiles can be obtained in a much simplerand practical way if one uses the known value of �n. In thiscase it is enough to have two image contrasts of a RBC,C1�r� and C2�r�, measured at two different defocus distances�within the linear range of �f�, �f1 and �f2. From Eq. �2�we have that

�2h2�r� =1

2 �n�C2�r� − C1�r�

�f2 − �f1� . �3�

No special reference position is required to obtain �2h2since �n is known. We use the procedure described previ-ously to integrate �2h2 to obtain h2. There is no need of aspecial translator, just a careful adjustment of the microscopefocusing knob.

Having determined �n, and knowing �f and the positionof the RBC mid-plane, one can convert contrast fluctuationsinto curvature fluctuations. A normalized curvature spatialcorrelation function is shown in Fig. 3 and a curvature time-autocorrelation function is shown in Fig. 4. These correla-tions are obtained along a RBC diameter; they correspond toaverages over the RBC thickness.

We adapted a Brochard-Lennon �BL� model9 to calculatecurvature instead of thickness correlation functions of fluc-

FIG. 1. �a� A RBC defocused image. �b� The corre-sponding RBC contrast profile along the horizontal di-ameter. The two peaks are due to edge effects that canbe taken into account if one uses the integral represen-tation for the defocused electric field of the light, fromwhich the model used in this Letter is a first orderexpansion.

FIG. 2. RBC profile obtained from the integration of the experimental con-trast data of Fig. 1�b�. The integration is performed up to the data pointclosest to zero contrast at the increasing part of the peak. This procedureintroduces an error of about 0.1 �m in the determination of the height of theRBC thickest part, and an overestimation of the RBC radius by about 3%.The difference in refractive index obtained is �n=0.042±0.005.

FIG. 3. Normalized spatial-correlation function of RBC curvature fluctua-tions. Dots are the experimental points and the continuous curve is the fitusing the theoretical prediction from the Brochard-Lennon model for curva-ture fluctuations. The characteristic length obtained is �=0.45 �m forf =30 Hz.

133901-2 Mesquita, Agero, and Mesquita Appl. Phys. Lett. 88, 133901 �2006�

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tuations. The BL model replaces the cell shape by parallelplates, which is a reasonable approximation in the shortwavelength limit. The bending fluctuations of a RBC aredetermined by spherical harmonics and the constraints offixed volume and area lead to truncation of the longest wave-length modes, as shown by Peterson.13 From the experimen-tal spatial correlation function and fit to BL model, we obtainthe characteristic length ��30 Hz�=0.45 �m �Fig. 3�. Spatialcorrelation functions are determined for different positionsalong a horizontal radial direction, for consecutive frames,such that f =30 Hz. Using a BL model we find that the cur-vature time-autocorrelation function is a power law given by

��0���t� �kbT

4�KC� ��1/3�t−1/3

3�2�2�f�1/3 − q02� , �4�

where � is curvature fluctuation, ��1/3��2.68, kB is Boltz-mann’s constant, T is absolute temperature, KC is the RBCbending modulus, � is the characteristic length, f =30 Hz isthe sampling frequency, and q0 is the wave vector of thefundamental vibration mode of the RBC. In Fig. 4 we showthe data and the fit using Eq. �4�, with KC and q0 as fittingparameters, and with fixed �=0.45 �m obtained from thespatial correlation function. From the fit we obtainKC=2.2x10−19 J for this RBC, a value within the rangeobtained by other techniques,5–8 and q0=0.585 �m−1, a

value surprisingly close to the one of the fundamental modeof a drumhead of radius 4 �m �0.6 �m−1�, which should besuppressed in the case of RBC.13 Inserting the values abovefor �, KC and the average thickness of this RBCd�1.3 �m in the BL theoretical expression for �, we obtainthe cytoplasm viscosity ��10 cP, which is higher than 6 cP,the value considered by BL.9 A more detailed account of thisexperiment and calculations will appear in a futurepublication.

As conclusions, we showed that defocusing microscopyis a powerful technique to visualize and obtain quantitativestatic and dynamic information from transparent curved ob-jects, therefore well suited to work with biological samples.We applied it to a red blood cell and from the average con-trast we determined its shape, size, and index of refraction.From the contrast fluctuations due to the flicker phenom-enon, we obtain its bending modulus and cytoplasm viscos-ity. We expect that defocusing microscopy becomes a popu-lar technique, since it provides accurate data and it isrelatively simple to use.

The authors thank N. Richmond and R. Dickman for acritical reading of this manuscript. This work was supportedby the following Brazilian agencies: CNPq, FINEP-PRONEX, and Instituto do Milênio de Nanociência-MCT.

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FIG. 4. Time-auto-correlation function of RBC curvature fluctuations. Dotsare the experimental points and the continuous curve is the fit using Eq. �4�.From this fit we obtain q0=0.585 �m−1 and KC=2.2�10−19 J. Inset: linear-log plot of the same curve.

133901-3 Mesquita, Agero, and Mesquita Appl. Phys. Lett. 88, 133901 �2006�

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