Light scattering from cells: finite-differencetime-domain simulations and goniometric measurements

Rebekah Drezek, Andrew Dunn, and Rebekah Richards-Kortum

We have examined the light-scattering properties of inhomogeneous biological cells through a combina-tion of theoretical simulations and goniometric measurements. A finite-difference time-domain ~FDTD!technique was used to compute intensity as a function of scattering angle for cells containing multipleorganelles and spatially varying index of refraction profiles. An automated goniometer was constructedto measure the scattering properties of dilute cell suspensions. Measurements compared favorably withFDTD predictions. FDTD and experimental results indicate that scattering properties are stronglyinfluenced by cellular biochemical and morphological structure. © 1999 Optical Society of America

OCIS codes: 170.3660, 290.0290.

1. Introduction

A. Background

Increasing the present understanding of the interac-tion of light with tissue at a cellular level will promotethe development of optical diagnostic techniques po-tentially capable of rapid, noninvasive assessment oftissue pathology. In the past few years, a number oftechniques have been developed that rely in somemanner on measuring the scattering properties oftissue to determine physiological state. These tech-niques range from direct imaging methods such asconfocal microscopy1 and optical coherence tomogra-phy,2 to more indirect approaches including elastic-scattering spectroscopy3 and photon migration.4

It is known that light scattering from tissue is de-pendent on the tissue’s morphological and biochemicalstructure. However, the details of this dependenceare not fully understood. As a result, although manyelastic-scattering methods are able to measure differ-ences between normal and diseased tissue, it is diffi-cult to meaningfully relate measurements to specificunderlying chemical and physical changes. Further-more, without a sufficient quantitative model capable

R. Drezek and R. Richards-Kortum are with the BiomedicalEngineering Program, University of Texas at Austin, EngineeringScience Building, Room 610, Austin, Texas 78712. A. Dunn iswith the Beckman Laser Institute and Medical Clinic, Universityof California, Irvine, Irvine, California 92697-1875.

Received 2 October 1998; revised manuscript received 16 Feb-ruary 1999.

0003-6935y99y163651-11$15.00y0© 1999 Optical Society of America

of predicting elastic scattering based on changes intissue on a microscopic level, it is challenging to makea priori predictions of how effectively a particular op-tical diagnostic technique will be able to detect diseasein a specific tissue. In addition, lack of accurate,quantitative models makes it difficult to assess howvariations in optical system design might be able toenhance differences between normal and diseased tis-sue, improving the diagnostic performance of the in-strument.

Until recently, exploring the relationship betweencell structure and light scattering was difficult becausethere was not a mathematical model for predictingscattering for complex biological cells. Because of thesize of scatterers in cells relative to the wavelengthsused in optical imaging, electromagnetic methods arerequired to describe scattering. Mie theory has beenused extensively to approximate scattering but gener-ally requires modeling a cell as a homogeneous sphere.Schmitt and Kumar have presented a useful expan-sion of this idea by applying Mie theory to a volume ofspheres with various sizes distributions.5 Anotherapproach more general than Mie theory involves cal-culating the light scattered from a coated sphere. Inthe coated sphere model, the sphere and the coatingeach have a distinct index of refraction, which makes itpossible to model a spherical cell containing a homo-geneous spherical nucleus.6 More recently, anoma-lous diffraction approximations,7 multipole solutions,8and T-matrix computations9 have been proposed andimplemented. Each of these techniques offers signif-icant advantages over conventional Mie theory; how-ever, all require limiting geometric and refractive-index assumptions.

1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS 3651

m

aTaltsutmpai

3

A more flexible approach is provided by a three-dimensional finite-difference time-domain ~FDTD!

odel of cellular scattering.10 Although computa-tionally intensive, this model allows the computationof scattering patterns from inhomogeneous cells ofarbitrary shape. The aim of the research describedin this paper is to develop an increased understand-ing of how light interacts with tissue on a cellularlevel using the FDTD model to predict cellular scat-tering patterns. The research reported in this paperexpands upon our previous research in the field. Weimplement cell dielectric structure in a more realisticmanner, for example, by incorporating fluctuations inrefractive index within the nucleus rather than mod-eling the nucleus as a homogeneous structure. Weinvestigate how scattering is influenced by changesin nuclear size and refractive index, by changes inorganelle size, refractive index, and volume fraction,and by changes in the refractive index of the mediumsurrounding a cell. We simulate scattering from acollagen fiber. We investigate the wavelength de-pendence of scattering patterns and scattering crosssection. Finally, to provide a comparison for theo-retical FDTD predictions, an automated goniometeris used to experimentally measure the phase func-tions and asymmetry parameters of cell suspensions.

B. Cell Structure

Because scattering arises from mismatches in refrac-tive index, when considering a cell from the perspec-tive of how it will interact with light, the cell is viewedmore appropriately as a continuum of refractive-indexfluctuations than as a single object containing a num-ber of discrete particles. The magnitude and spatialextent of the index of refraction fluctuations arise fromthe physical composition and size of the componentsthat make up the cell. Organelles and subcompo-nents of organelles having indices different from theirsurroundings are expected to be the primary sources ofcellular scattering. The cell itself may be a significantsource of small-angle scatter in applications such asflow cytometry in which cells are measured individu-ally; however, for in vivo scattering-based diagnostics,the cell as an entity is not as important because cellswill be surrounded by other cells or tissue structures ofsimilar index.

Certain organelles in cells are important potentialsources of scattering. The nucleus is significant be-cause it is often the largest organelle in the cell andits size increases relative to the rest of the cellthroughout neoplastic progression. Other potentialscatterers include organelles whose size relative tothe wavelength of light suggests that they may beimportant backscatterers. These include mitochon-dria ~0.5–1.5 mm!, lysosomes ~0.5 mm!, and peroxi-somes ~0.5 mm!. Mitochondria may be particularlyinfluential in those cells that contain significant mi-tochondrial volume fractions because of this or-ganelle’s unique folded membrane structure. Forexample, Beauvoit et al. have found that mitochon-dria contribute 73% of the scattering from hepato-cytes.11 In addition, melanin, traditionally thought

652 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999

of as an absorber, must be considered an importantscatterer because of its size and high refractive index.Finally, structures consisting of membrane layerssuch as the endoplasmic reticulum or Golgi appara-tus may prove significant because they contain indexfluctuations of high frequency and amplitude. Al-though the research presented here primarily con-cerns cells, in understanding scattering from tissue,fibrous components such as collagen and elastin mustbe considered in addition to cellular matter. Therelative importance of fibrous and cellular compo-nents depends on tissue type.

2. Methods

A. Yee’s Method

Yee’s method can be used to solve Maxwell’s curlequations using the FDTD technique.12 The algo-rithm takes Maxwell’s curl equations and discretizesthem in time and space, yielding six coupled finite-difference equations. The six electric and magneticfield components ~Ex, Ey, Ez, Hx, Hy, Hz,! are spatiallynd temporally offset on a three-dimensional grid.he grid spacing must be less than ly10 to yieldccurate results. Except when otherwise noted, ay20 grid was used for the simulations presented inhis paper. As the six finite-difference equations aretepped in time, the electric and magnetic fields arepdated for each grid point. To simulate propaga-ion in an unbounded medium, boundary conditionsust be applied to the tangential electric field com-

onents along the edges of the computational bound-ry at each time step. The Liao boundary conditions used in this paper.13 The incident wave is a sinu-

soidal plane-wave source.The FDTD method computes the fields in a region

around the cell that lies in the near field, which isthen transformed to the far field. Parameters suchas asymmetry parameter and scattering cross sectioncan be computed from the scattering pattern. Thedetails of the FDTD model used in this research andthe relevant equations have been published previ-ously and are not repeated here.10

B. Simulation Parameters

The cell is constructed by assigning permittivity val-ues to each cell component. If desired, a range ofpermittivity values may be assigned to one compo-nent if that component, for example, the nucleus, isinhomogeneous. The equation for complex permit-tivity contains a frequency-dependent conductivityterm.14 However, for a purely real refractive index,the dielectric constant at a particular wavelength issimply the square of the frequency-dependent refrac-tive index.

To accurately determine scattering patterns, it isnecessary to model the cells in as physically realistica manner as possible. Information about the size,quantity, and dielectric structure of cellular or-ganelles was obtained from the literature and by ex-amining phase contrast images and electronmicrographs of cell and organelles of interest. Re-

Fct

tdanaptutaal

Table 1. Index of Refraction Values Obtained from the Literature

fractive index is a function of the concentration ofmacromolecules in a particular organelle. However,an organelle’s composition can vary significantlyamong different types of cells. For example, it hasbeen reported that the refractive index of the nucleusis higher than that of the cytoplasm in Chinese ham-ster ovary cells15 and breast epithelial cells,16

whereas the cytoplasm was found to have a higherindex than the nucleoid in E. coli.17 Thus accuratemodeling of dielectric structure requires specificknowledge of organelle composition for the cell type ofinterest. Because this information is not readilyavailable, refractive-index values from the literaturewere assembled as a starting point for the simula-tions. The nucleus was always modeled with someinhomogeneities in refractive index. The distribu-tion of index variations employed in the simulationswas based on Fourier analysis of 1003 phase contrastimages of normal and cancerous human breast epi-thelial cells. The general ranges of refractive indexvalues used in the FDTD simulations are shown inTable 1.15,18–20

C. Goniometer

A goniometer can be used to experimentally measurethe phase function of an optically thin tissue sectionor dilute cell suspension. These measurements canthen be compared with FDTD predictions or to otherphase function approximations such as Mie theory orHenyey–Greenstein formulations. The goniometersetup is depicted in Fig. 1. An unpolarized 5-mWHe–Ne laser ~l 5 612 nm! is incident upon a cylin-drical sample cell ~1 cm!. The cell is contained in a31-cm tank filled with water for index-matching pur-poses. An incoherent fiber-optic bundle ~1.5875 mmin diameter! is actuated about 160° beginning fromthe laser point of entry. A stepper motor under thecontrol of a computer moves the fiber. The fiber is

Fig. 1. Schematic diagram of a goniometer built to measure lightscattering as a function of angle.

Organelle Refractive Index Reference

Extracellular fluid 1.35–1.36 18Cytoplasm 1.36–1.375 15Nucleus 1.38–1.41 15Mitochondria and organelles 1.38–1.41 19Melanin 1.6–1.7 20

attached to the arm of the goniometer, which extendsfrom the motor shaft and is designed to have an ad-justable length ~5–14 cm!, permitting trade-offs be-tween detected power and angular resolution. Aprism with an aluminized hypotenuse for near 100%reflection is used to direct light into the fiber. Anaperture on the prism face limits the effective N.A. ofthe fiber. The fiber delivers the light to a photodiode~Newport Model 835!. Power is recorded to the com-puter using a general-purpose interface bus. A typ-ical measurement with 2° angular resolution takesapproximately 2 min.

3. Results: Simulations

A. Verification

The simulation program was verified by computingthe scattering patterns of homogeneous spheresranging in diameter from 0.5 to 10 mm and comparingthe results with Mie theory. For small spheres, thetwo curves agree closely for all angles at horizontaland vertical polarizations. For larger spheres, theFDTD pattern agrees well for most angles but issomewhat greater than the Mie pattern for angleshigher than 160°. The artificial increase is due toimperfect boundary conditions, resulting in artificialreflections at the edges of the computational domain.The scattering pattern of a 5-mm sphere is shown in

ig. 2. The asymmetry parameter g and scatteringross section differ from theoretical values by lesshan 0.2%, despite the artificial reflections.

In this paper we present the output of our model inwo ways. We show scattering patterns, which weefine to be a plot of the intensity of scattered light asfunction of angle. When the scattered power is

ormalized to the value at 0°, we refer to the curve asnormalized scattering pattern. Alternatively, we

resent the scattering phase function, which presentshe scattered power versus angle, where the areander the curve is normalized to one. In the scat-ering pattern, it is visually easier to assess the rel-tive magnitude of scattered power at particularngles when comparing results from multiple simu-ations. However, when comparing FDTD data with

Fig. 2. Validation of the FDTD code. Comparison of FDTD re-sults with Mie theory predictions for a 5-mm sphere ~m 5 1.04; l5 900 nm!.

1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS 3653

bmtacc

a

aavitwnc

ddirselibt

5btf

3

Mie theory or Henyey–Greenstein phase functions,data must first be normalized to an area of one.Methods for obtaining the phase function, asymme-try parameter, and scattering cross sections from rawdata have been described previously.10

B. Influence of the Nucleus

It is important to understand the effect of nuclearmorphology on scattering properties because nuclearmorphology is dramatically altered in cancerouscells. In general, cancerous cells have an increasednuclear-to-cytoplasmic ~NyC! ratio.21 In addition,ecause cancerous cells divide more rapidly than nor-al cells and often have extra chromosomes, the pro-

ein concentration of the nucleus may be higher,ltering nuclear refractive index. Also, there arehanges in nuclear shape and texture as cells becomeancerous.21

We used the FDTD model to study the relationshipbetween the physical structure of the nucleus and itsscattering properties, examining separately the ef-fects of nuclear size and nuclear refractive index onthe scattering pattern. Unless otherwise specified,all simulations presented in this paper involved a15-mm-diameter spherical cell containing cytoplasm,a nucleus, and organelles of multiple sizes andshapes. Wavelength of incident light was 900 nm.To isolate changes that are due to the nucleus, thevolume fraction of other organelles was kept low~0.02%!, and a small mismatch between cytoplasmnd extracellular fluid ~m 5 1.014! was employed.

Note that we define m to be a relative index of refrac-tion ~in this case, between cytoplasm and extracellu-lar fluid! and n to be an absolute index of refraction.Nuclear index variations were uniformly distributedbetween Dn 5 60.03 about the mean nuclear index,n 5 1.39, at spatial frequencies ranging from 2 to 20mm21.

Scattering patterns were calculated for cells withnuclei of diameter 0, 5, 7.5, and 10 mm. Increases innuclear size create noticeable changes in the scatter-ing pattern at small angles. Figure 3 shows thesmall-angle scattering patterns of a 15-mm cell with

Fig. 3. Influence of nuclear structure on scattering pattern.Low-angle scatter is shown on a linear scale. There were novisible differences in scattering patterns at angles over 20°.

654 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999

no nucleus, a small nucleus ~5 mm in diameter!, andlarge nucleus ~10 mm in diameter!. The higher-

ngle ~over 20°! scattering of cells containing nuclei ofarious sizes is highly dependent on factors such asncident wavelength and refractive-index structure ofhe nuclei. For example, when cells are modeledith homogeneous nuclei, high-angle scattering isot significantly altered as nuclear size is in-reased.22 However, when the nucleus is modeled as

a heterogeneous structure, high-angle scattering in-creases with increasing nuclear size.22

Scattering cross section significantly increases asthe NyC ratio rises @Fig. 4~a!#. In this paper we

efine the NyC ratio as the diameter of the nucleusivided by the diameter of the cytoplasm. Scatter-ng cross section also increases as the relative NyCefractive index is increased @Fig. 4~b!#. For nuclearizes and index of refraction values within the rangexpected for biological cells, there appears to be ainear relationship between relative NyC refractivendex and scattering cross section; the relationshipetween the NyC ratio and the scattering cross sec-ion is of higher order.

C. Influence of Organelles

To investigate the effect of small cytoplasmic or-ganelles, three simulations were conducted. Foreach of the three cases, a 15-mm spherical cell with a-mm nucleus ~mean m 5 1.02; spatial fluctuationsetween 2 and 20 mm21! was created. In addition tohe nucleus, the first cell contained an 8.5% volumeraction of melanin ~n 5 1.65; 1-mm sphere!. Mela-

Fig. 4. ~a! Influence of NyC ratio on scattering cross section. ~b!Influence of relative NyC refractive index on scattering cross sec-tion.

1

imtmn

nin was modeled with and without absorption.There was little difference in obtained results whenabsorption was incorporated as an imaginary compo-nent of the refractive index. Despite its high absorp-tion coefficient, melanin causes little attenuationwithin a cell because path lengths are small. Tocompare the effects of melanin with an organelle oflower refractive index, a second cell was modeledcontaining an 8.5% volume fraction of organelles sim-ilar in size, shape, and refractive index to mitochon-dria ~ellipsoids ;0.5 mm 3 0.5 mm 3 1.5 mm; n 5.41!. A third cell ~amelanotic! was constructed sim-

ilarly to the second cell but contained only a 3% or-ganelle volume fraction. The scattering patternsfrom these simulations are shown in Fig. 5.

Figure 5 demonstrates that internal structure canhave a strong influence on scattering pattern, partic-ularly for angles over 40°. In the cells containingmelanin and mitochondria, the total volume of or-ganelles in the cell is identical. The difference in thetwo curves is due to the higher index of melaninrelative to mitochondria. The melanin results in ascattering pattern that covers a smaller magnituderange than the other two curves. We define themagnitude range of a scattering pattern as the scat-tering pattern’s maximum intensity value divided bythe minimum intensity ~ImaxyImin! value. Scatter-ng cross section was also larger for a cell containing

elanin rather than lower-index organelles. Scat-ering cross sections of cells containing melanin anditochondria ~8.5% volume fraction! and the amela-otic cell ~3% volume fraction! were 861, 729, and 639

mm2, respectively. In general, increasing organellevolume fraction increases scattering cross sectionand makes the scattering pattern more isotropic.

D. Influence of Morphology

We investigated the influence of morphology on scat-tering patterns. For example, does it matterwhether a cell is constructed by placing a certain

Fig. 5. Influence of internal structure on scattering pattern.Three simulations are shown: a cell containing organelles of re-fractive index and size consistent with melanin ~8.5% volume frac-tion!, a cell containing organelles with index and size ofmitochondria ~8.5% volume fraction!, and an amelanotic cell con-taining organelles with index and size of mitochondria but asmaller volume fraction ~3%!.

number of organelles of specific size and refractive

index at explicit locations rather than randomly gen-erating a dielectric structure, using a range of refrac-tive indices and a chosen frequency of indexfluctuations? To examine the difference betweenthese two cases, the scattering pattern of a cell gen-erated with specific morphology was compared withthe scattering pattern of a cell constructed using ran-dom refractive-index assignments. To generate thecell without morphological structure, all grid pointswithin the sphere were assigned refractive-index val-ues, based on the desired frequency of spatial fluctu-ations in refractive index ~2–20 mm21! and a chosenrange of refractive-index values, uniformly distrib-uted about a mean index ~n 5 1.40 6 0.05!. The cellwith known morphology is similar to the melanoticcell described in Subsection 3.C. Results are shownin Fig. 6. The overall shape and range of the scat-tering pattern is similar for a cell with random di-electric structure and a cell with specific morphology.

Figure 7 demonstrates the effect of changing thefrequency of spatial variations while keeping themean refractive-index value constant for a cell with

Fig. 6. Comparison of scattering pattern from a cell with a spec-ified internal structure ~cytoplasm, nucleus, and organelles! with acell with randomly assigned dielectric structure. Mean index ofboth cells is identical, n 5 1.4.

Fig. 7. Scattering pattern of two cells with randomly assigneddielectric structure. The spatial frequency of index fluctuations ishigher in the top curve ~labeled high frequency! than in the lowercurve ~labeled low frequency!. Mean refractive index is the samefor both cells.

1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS 3655

n

ti1ac1

lodTeiTb

6g

ats

ia

3

randomly generated dielectric structure. The nor-malized scattering pattern for two cells are plotted.Both cells have a mean index of n 5 1.4, with uni-formly distributed variations between n 5 1.35 and

5 1.45. In one cell the spatial frequency of theindex variations ranges from 5 to 20 mm21; in theother cell the spatial frequency of the variationsranges from 2 to 10 mm21. The curves demonstratethat, as the frequency of refractive-index fluctuationsis increased, the scattered intensity becomes higherat large angles.

E. Influence of Extracellular Medium

Typically, in published simulations of cell scattering orgoniometric measurements, the cell is surrounded by alow-index medium such as extracellular fluid ~n 51.35!, blood ~n 5 1.345!, growth medium ~n 5 1.34!, orphosphate-buffered saline ~n 5 1.33!.8,9,15,16,23 How-ever, in tissue, cells are likely surrounded by othercells or tissue structures of higher index of refraction.When an object is bordered by other objects of similarindex, rather than a medium of significantly lowerindex, scattering will be reduced because of indexmatching. To demonstrate this concept, we repeat anexperiment first performed by Barer in the 1950’s.24

Figure 8~a! shows two images of cuvettes filled with anequal concentration of breast cancer cells ~106yml!.In the cuvette on the left, the cells are immersed insaline solution ~n 5 1.33!. Because of scattering fromthe cells, it is not possible to read the text behind thecuvette. In the cuvette on the right, the cells are im-mersed in an albumin solution ~n 5 1.37!. Scatteringis reduced and the text is readable.

The FDTD model predicts the index-matching ef-fect. Figure 8~b! shows the scattering pattern of anovarian cancer cell immersed in solutions of differentindex of refraction: n 5 1.35 and n 5 1.37. Whenhe index of the medium surrounding the cytoplasms increased to more closely match the cytoplasm ~n 5.37!, scattering at the lowest angles is reduced bylmost an order of magnitude relative to the n 5 1.35ase. Scattering cross section is reduced from 729 to92 mm2 simply by decreasing the index mismatch by

Dn 5 0.02.

F. Scattering from a Collagen Fiber

To investigate the scattering pattern of collagen, acollagen fiber was modeled using a cylindrical geom-etry ~3 mm in diameter, 20 mm in length!. The col-agen fiber was composed of a number of fibrilsriented along the long axis of the cylinder. Theiameter of the fibrils ranged from 60 to 240 nm.he fibrils contained cross striations approximatelyvery 60 nm in length. These cross striations weremplemented through refractive-index fluctuations.he refractive index of collagen has been measured toe between n 5 1.46 and 1.55.25 The high end of this

range was measured using dried collagen and is notrealistic for in vivo tissues. The simulation reportedhere used a mean value for collagen of n 5 1.46 ~Dn 5

0.04!. The extracellular matrix surrounding colla-en can vary from watery to a gellike consistency,

656 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999

ltering its refractive index. For this experiment,he index of refraction of the extracellular matrix waset at n 5 1.36.The calculated scattering pattern of a collagen fiber

s shown in Fig. 9. The scattering patterns shownre the patterns for three different orientations of the

Fig. 8. ~a! As a visible demonstration of reduction in scatteringthat is achievable by immersing cells in a solution of like index, twocuvettes are shown with equal concentrations of cells. In thecuvette on the left, cells are immersed in saline ~n 5 1.33!; in thecuvette on the right, cells are immersed in a higher-index albuminsolution ~n 5 1.37!. Index matching reduces the scattering, andthe text behind the cuvette is readable. ~b! Influence of externalmedium on scattering pattern. The scattering patterns of thesame cell immersed in media of varying index ~n 5 1.35, n 5 1.37!are compared.

Fig. 9. Scattering pattern of a collagen fiber. Curves shown arefor three orientations of the collagen fiber with respect to an in-coming plane wave along the 1z axis. The collagen fibers wereoriented along the x, y, and z axes, respectively.

tH

w

dim

isfddp

s1hif

collagen fiber with respect to an incoming plane wavein the 1z direction. The collagen fiber was placedalong the x, y, and z axes for the three trials. As isthe case for cells, the scattering pattern is highlypeaked in the forward direction. The number of or-ders of magnitude covered by the scattering pattern~ImaxyImin! is lower for collagen than a cell by 1 to 2orders of magnitude.

G. Wavelength Dependence of Scattering Pattern

To investigate the wavelength dependence of cellularscattering, scattering from the same cell was simu-lated for wavelengths of 514 nm, 780 nm, and 2 mm.The results are shown in Fig. 10. As evidenced byFig. 10, changes in wavelength, even in the visibleportion of the spectrum, produce significant changesin the scattering. As the wavelength increases withrespect to the size of the scatterers, the number ofpeaks in the curves decrease. This can be predictedqualitatively from a decrease in the average Mie sizeparameter. It should be noted that little experimen-tal data are available concerning the dispersion of cellconstituents other than water. The simulationsshown here illustrate only the influence of changes inincident wavelength relative to the size of the scat-tering particles, not the influence of dispersion.However, over the range of wavelengths simulatedhere, the relative indices of refraction do not changesignificantly.

We also investigated the wavelength dependence ofscattering cross section. Recently, Perelman et al.26

presented a technique for extracting nuclear size dis-tributions from measurements of tissue backscatter-ing. A fundamental assumption of the technique isthat the optical scattering cross section, ss~l, l !, ofhe nucleus of a cell can be predicted using the van deulst ~anomalous diffraction! approximation27:

ss~l, l ! 512

pl2H1 2sin~2dyl!

dyl1 Fsin~dyl!

dyl G2J , (1)

here d 5 pncl~m 2 1!. In Eq. ~1!, nc is the refrac-tive index of cytoplasm, m is the refractive index ofthe nucleus relative to that of cytoplasm, and l is the

Fig. 10. Wavelength dependence of scattering pattern. The nor-malized scattering pattern of one cell is shown for three wave-lengths ~514 nm, 780 nm, 2 mm!.

iameter of the nucleus. The van de Hulst approx-mation assumes that the nucleus of a cell can be

odeled as a homogeneous sphere.We compared FDTD predictions for optical scatter-

ng cross section for inhomogeneous nuclei with re-ults obtained using the van de Hulst approximationor wavelengths between 500 and 900 nm. The vane Hulst approximation was computed for a 5-mm-iameter nucleus, n 5 1.04, surrounded by cyto-lasm, nc 5 1.36. Two sets of FDTD computations

were considered. First, the nucleus was modeled asa 5-mm sphere, m 5 1.04, surrounded by cytoplasm,nc 5 1.36. This produces the same results thatwould be predicted by Mie theory. The comparisonof FDTD homogeneous sphere results with resultsfrom the van de Hulst approximation ~Fig. 11! dem-onstrates that the van de Hulst approximation isvalid in this wavelength and particle diameter re-gion. For the second set of FDTD simulations, thenucleus was modeled as an inhomogeneous spheresurrounded by cytoplasm, with nuclear index uni-formly distributed between m 5 1.01 and m 5 1.07~mean m 5 1.04!. The frequency of spatial varia-tions ranged from 2 to 30 mm21. A 33-nm grid spac-ing was used for all wavelengths so that the samespatial frequency fluctuations could be simulated forall cases. As shown in Figure 11, FDTD results foran inhomogeneous cell predict a similar cross sectionversus wavelength trend to that predicted by the vande Hulst approximations. However, the cross-sectional values predicted for an inhomogeneous nu-cleus are a factor of 2–3 higher.

4. Results: Experiments

A. Validation

To compare theoretical predictions of light scatteringto physical measurements, a goniometer was con-

Fig. 11. Scattering cross section versus wavelength for nucleussurrounded by cytoplasm. Three cases are shown. First, thevan de Hulst approximation ~anomalous diffraction approxima-tion! is used to calculate scattering for a 5-mm nucleus ~m 5 1.04;ncytoplasm 5 1.36!. Second, the FDTD model is used to calculatecattering for a homogeneous 5-mm nucleus ~m 5 1.04; ncytoplasm 5.36!. Third, the FDTD model is used to calculate scattering for aeterogeneous 5-mm nucleus, with mean relative refractive index

dentical to the homogeneous case and spatial fluctuations rangingrom 2 to 30 mm21.

1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS 3657

~dcdT1twsrrttw

sllcboOMt

wTf2

tmwu

f

3

structed. To demonstrate that the goniometer wasfunctioning properly, measurements were made ondilute suspensions of 0.2-mm polystyrene spheresBangs Laboratories!. A linear polarizer was intro-uced to allow measurement of horizontal and verti-al polarizations. The two polarizations result inifferent scattering patterns for 0.2-mm spheres.he measured scattering patterns are plotted in Fig.2 along with the corresponding curves from Mieheory. Commercially available software ~MIETAB!as used for the Mie theory calculation. The mea-

ured scattering patterns agree well with the theo-etical curves. The increased measured valueselative to Mie theory predictions near 90° are due tohe finite angular resolution of the goniometer ratherhan a limitation in dynamic range of the detectorhich is 5–6 orders of magnitude.

B. Cell Measurements

The scattering pattern of a suspension of culturedovarian cancer cells ~OVCA-420 cells! was measuredusing the goniometer. The cells were measured sus-pended in isotonic phosphate-buffered saline ~n 51.33! at concentrations of approximately 105 cellsyml.This concentration was determined by repeated dilu-tion of a concentrated solution of 106 cellsyml. Aftereach dilution, the scattering pattern of the cells wasmeasured in the goniometer. Measurements wererepeated with progressively more-dilute solutions un-til further dilution did not influence the results in themanner described by Mourant et al.23 The totaltransmission of the final dilution was ;0.94 at l 5612 nm. The diameter of the cells was approxi-mately 10 mm with a 6-mm nucleus.

The measured phase function for a dilute suspen-ion of OVCA cells is plotted in Fig. 13. Values be-ow 6° are obtained by linear extrapolation ofogarithmic intensity data. Figure 13 also shows aomparison of measured results with those predictedy the FDTD model ~averaged over three orientationsf the cell with respect to the incident light! for anVCA cell measured and simulated at l 5 612 nm.odeled cell parameters were based on phase con-

rast observations of OVCA cells. The modeled cell

Fig. 12. Validation of goniometer performance. Comparison ofmeasured data and Mie theory predictions for 0.2-mm polystyrenespheres.

658 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999

as a 10-mm cell with a 6-mm-diameter nucleus.he nucleus contained spatial variations in the re-

ractive index at a spatial frequency of approximatelymm21 with Dn 5 60.03. Data concerning the vol-

ume fraction of organelles in OVCA cells were notavailable; a volume fraction of 10% was used in thesimulation. The results show that, with someknowledge about a cell’s structure and by averagingFDTD results for the same cell at a few orientationsrelative to incident light, it is possible to reasonablymodel the physically measurable scattering pattern.The accuracy could be improved by increased knowl-edge of cellular contents and dielectric structure.

Figure 14 shows the measured data compared withHenyey–Greenstein and Mie theory predictions.The Henyey–Greenstein phase function used anasymmetry value of g 5 0.97, computed from themeasured scattering pattern. The Mie phase func-tion was calculated using unpolarized light ~l 5 612nm! for a 10-mm sphere, n 5 1.38, surrounded byphosphate-buffered saline, n 5 1.33. For this cell,he Henyey–Greenstein phase function underesti-ated high-angle scatter. The Mie phase functionas a poor fit, underestimating high-angle scatter byp to 2 orders of magnitude.

Fig. 13. Comparison of measured and FDTD data for OVCA-420cells. The FDTD data are the average over two orientations of thecell with respect to incident light.

Fig. 14. Comparison of measured scattering pattern with Mietheory ~10-mm sphere, m 5 1.04, l 5 612 nm! and Henyey–Greenstein predictions. Asymmetry value of g 5 0.97, calculatedrom measured data, was used in the Henyey–Greenstein model.

h

C. Influence of Acetic Acid

Acetic acid is used during colposcopy to enhance dif-ferences in the diffuse reflectance of normal and dis-eased regions of the cervical epithelium.28 Transientwhitening of tissue after the application of acetic acidserves as a simple and inexpensive method for identi-fication of areas that may eventually develop into cer-vical cancer. To explore the effect of acetic acid on cellscattering, cervical cancer cells ~He–La cell line! wereexposed to 6% acetic acid. The scattering pattern ofthe cells was measured immediately before and afterexposure to acetic acid. The measured scattering ofHe–La cells before and after exposure to acetic acid isshown in Fig. 15. Increased scattering is apparent atmost angles.

As another test of the influence of acetic acid, FDTDsimulations were conducted on a cell before and afterthe addition of acetic acid. Changes to the simulatedcell were based on phase contrast and confocal micro-scope images of cells before and after acetic acid.29

These images indicate that the predominant effect ofacetic acid is an alteration of the refractive-index struc-ture of the nucleus. In the FDTD simulation of a cellbefore and after addition of acetic acid, nuclear indexfluctuations were of increased frequency ~before addi-tion, 2–10 mm21; after addition, 2–20 mm21! and

igher magnitude ~before addition, Dn 5 60.03; afteraddition, Dn 5 60.06! after the introduction of aceticacid. A cross section for the cell after addition of ace-tic acid increased from 519 to 843 mm2.

5. Discussion

An obvious question about a model as computation-ally intensive as the FDTD method is whether suchan approach is really necessary. Are simple analyt-ical descriptions such as those provided through Mietheory or Henyey–Greenstein approximations suffi-cient? It is our belief that simple analytical descrip-tions of cellular scattering are only adequate for somepurposes. For example, if a Monte Carlo model isused to calculate total tissue reflectance or transmit-tance, approximations such as Henyey–Greenstein orMie theory are likely to produce acceptable results.However, if the same Monte Carlo model is used to

Fig. 15. Measurements of a cervical cancer cell line before andafter addition of 6% acetic acid ~AA!.

simulate a typical endoscopic situation, in which thelight delivery and collection fibers are near to eachother, the choice of phase function will significantlyaffect the number of photons collected. The influ-ence of phase function on light transport for smallsource detector separations has been demonstratedpreviously by Mourant et al.30 The results of Mou-rant’s simulations demonstrate the necessity ofchoosing a phase function that accurately reflects theprobability of high-angle scatter when small sourcedetector separations are used.

We believe a more flexible and realistic approachsuch as that provided by the FDTD solution is theonly way to estimate the accuracy of current Mietheory and Henyey–Greenstein approximations. Inaddition, we believe that there is valuable informa-tion about the interaction of light with a single cellthat can be obtained from our more complicated ap-proach. For example, we find that the overall shapeof the nucleus influences small-angle scatteringwhereas the effects of small intracellular organelles,or high-frequency index of refraction fluctuations, aremore evident at higher angles. We believe that un-derstanding these types of trends not only developsfundamental knowledge of the interaction of lightwith a single cell, but also serves a practical purposein facilitating the design of more effective optical di-agnostic systems by allowing prediction of changes inscattering as a function of cell morphology.

Currently, limitations in the knowledge of cellularcomposition and dielectric structure make it difficultto construct a specific type of cell, simulate it, andconclude that the obtained scattering pattern is ex-actly what would be obtained through experimentalmeasurements of the scattering from that kind of cell.However, the simulations presented in this paperclearly establish that scattering patterns change sig-nificantly based on the cell’s internal structure, thesurrounding environment, and the wavelength of il-luminating light. Thus using one phase functionsuch as the Henyey–Greenstein phase function or aMie theory approximation to represent a typical cellis probably unrealistic. Henyey–Greenstein phasefunctions have been shown to poorly describe high-angle scattering. This was documented recently byMourant et al.23 and is further supported by the ex-perimental measurements presented in this re-search. Mie theory approximations are severelylimited because they do not account for internalstructures, which are expected to be the primarysource of cellular scattering in in vivo measurements.

The FDTD simulations document the influence oforganelle volume fraction and refractive index on cel-lular scattering. Both volume fraction and refrac-tive index of cell components will be highly variablein biological cells and will depend on cell type. Al-though it is possible to qualitatively predict trends inphase function or scattering cross section as the re-fractive index and volume fraction are varied, theFDTD method provides more quantitative informa-tion. The simulations suggest that it is possible toconsider increasing the size and spatial variations of

1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS 3659

mhiccmaiicio

s

o 23

3

the nucleus as analogous to changing organelle vol-ume fraction. The similarity of cells simulated withspecific morphology to those with randomly gener-ated dielectric structure suggests that it is the overallfrequency and magnitude of index of refraction fluc-tuations, rather than a particular spatial arrange-ment of cell components, which has the more crucialeffect on a cell’s scattering pattern. The effect ofhigh-frequency fluctuations, which can be viewed assmall scatterers, is particularly apparent at high an-gles. In a cell, high-frequency fluctuations in refrac-tive index might be created by components such asmicrotubules, membranes, chromatin, or other or-ganelle constituents.

The FDTD method can also provide informationabout the relative importance of scattering from cel-lular matter and extracellular fibrous proteins in tis-sue. Comparing the scattering pattern of collagenwith that of a cell, it is evident that the likelihood ofhigh-angle scattering events is markedly increasedfor fibrous tissue components such as collagen. Foroptical techniques sensitive to high-angle scatter, it ispossible that collagen could dominate scattering fromcells when the volume of interrogation includes alayer of cells lying on top of a layer of collagen. Po-tentially, scattering from collagen could prove prob-lematic in the detection of dysplasia in epithelialtissues. Dysplastic changes originate near the basallayer and propagate upward. Scattering from thereticular fibers in the basal layer and underlyingstroma might override more subtle scatteringchanges caused by nuclear abnormalities and mitoticactivity in mildly dysplastic cells. This suggeststhat weakly confocal probes that restrict the volumeof interrogation to the epithelium might be useful toavoid measuring scattering from collagen when ex-amining epithelial tissues.

The simulations and measurements presentedhere illustrate the profound differences that the in-dex of the surrounding medium can have on the cellscattering pattern. Thus it is important to recognizethe difference between measuring and simulatingcells in suspensions compared with those in real tis-sue. A large portion of the measured or predictedscattering for cells suspended in saline occurs be-cause of the mismatch in refractive index between thecells and its surrounding, a mismatch that may notbe realistic for in vivo tissue. Goniometric measure-

ents on cells should be performed in a medium ofigher index to measure scattering that is due to

nternal structure, rather than scattering from theell and phosphate-buffered saline border. Practi-ally, this is not easily accomplished because the im-ersion medium must be a substance that does not

lter or harm the cell. Albumin, a useful protein forndex matching in phase contrast microscopy exper-ments, scatters too significantly to be used to in-rease the refractive index of the immersion mediumn goniometry. Proteins smaller than albumin mayffer a potential alternative.The measurements of scattering from cell suspen-

ions reported here are similar to the measurements

660 APPLIED OPTICS y Vol. 38, No. 16 y 1 June 1999

f fibroblast suspensions by Mourant et al. In bothcases, the data presented do not show the interfer-ence peaks found in Mie theory predictions. A bio-logical cell contains scatterers with a wide range ofsizes. This wide range of scatterer sizes may elimi-nate the peaks that are present in Mie theory predic-tions that assume scatterers of only one size.Schmitt and Kumar have shown previously thatwhen a distribution of scatterer sizes is assumedrather than a single size, the phase functions becomesmoother curves with fewer peaks, which is muchmore similar to what we measure.5 Another factorthat may contribute to the smoothness of measuredphase functions is the random orientation of cells insuspension.

Finally, the simulations and measurements pre-sented here illustrate the potential importance of ace-tic acid as a contrast agent in optical diagnostics.Whether considering FDTD predictions, goniometricmeasurements, or the images from both phase con-trast and confocal microscopy recently presented bySmithpeter et al.,29 acetic acid has been shown toincrease scattering from a cell. It has been sug-gested in the literature that the acetowhitening effectseen in cervical tissue is due to coagulation of nuclearproteins.28 Based on our own phase contrast micros-copy observations of the effects of acetic acid,29 wesuspect that acetic acid changes the size of scatterersin the nucleus, resulting in an increase in the mag-nitude and frequency of fluctuations in refractive in-dex. Although the importance of acetic acid inaccurate colposcopic diagnosis has been recognizedfor many years, we believe that acetic acid may alsoprove extremely significant in quantitative opticaldiagnosis of precancerous conditions because of aceticacid’s ability to selectively enhance nuclear scatter.

6. Conclusions

We have presented the application of the FDTD tech-nique for examination of light-scattering propertiesof biological cells and tissue structures. Althoughcomputationally intensive, the method places no lim-itations on the geometry or index structure of theobjects under question. The model can be used tostudy the effect of changes of wavelength and of cel-lular biochemical and morphological structure. Thedata obtained from the model can be used to calculatephase functions, scattering cross section, and asym-metry parameter. As the data demonstrate, rela-tively small changes in internal structure, externalmedium, and wavelength can significantly changeboth phase function and scattering cross section.This suggests that phase functions that are moreflexible than Mie theory or Henyey–Greenstein maybe useful to improve the accuracy of optical modelingtechniques. Currently, the utility of the FDTD tech-nique is limited by lack of adequate knowledge con-cerning cellular dielectric structure. As thisknowledge is developed, the FDTD method will be-come increasingly valuable.

15. A. Brunsting and P. Mullaney, “Differential light scattering

The authors thank the High Performance Comput-ing Facility at the University of Texas at Austin for acomputer time grant on a Cray J90 supercomputer.The authors also express their appreciation to Kim-berly Kline, Marla Simmons-Mechaca, Robert Bast,and Dafna Lotan for assistance in obtaining and pre-paring cell samples.

References1. M. Rajadhyaksha, M. Grossman, D. Esterwitz, R. Webb, and R.

Anderson, “In-vivo confocal scanning laser microscopy of hu-man skin: melanin provides strong contrast,” J. Invest. Der-matol. 104, 946–952 ~1995!.

2. J. Izaat, M. Hee, D. Huang, A. Swanson, C. Lin, J. Schuman,C. Puliafito, and J. Fujimoto, “Micron-resolution biomedicalengineering with optical coherence tomography,” Opt. Photon.News 4, 14–19 ~1993!.

3. J. Mourant, I. Bigio, J. Boyer, R. Conn, T. Johnson, and T.Shimada, “Spectroscopic diagnosis of bladder cancer with elas-tic light scattering,” Lasers Surg. Med. 17, 350–357 ~1995!.

4. J. Fishkin, O. Coquoz, E. Anderson, M. Brenner, and B. Trom-berg. “Frequency-domain photon migration measurements ofnormal and malignant tissue optical properties in a humansubject,” Appl. Opt. 36, 10–20 ~1997!.

5. J. Schmitt and G. Kumar, “Optical scattering properties of softtissue: a discrete particle model,” Appl. Opt. 37, 2788–2797~1998!.

6. A. Brunsting and P. Mullaney, “Light scattering from coatedspheres: model for biological cells,” Appl. Opt. 11, 675–680~1972!.

7. G. Streekstra, A. Hoekstra, E. Nijhof, and R. Heethaar, “Lightscattering by red blood cells in ektacytometry: Fraunhoferversus anomalous diffraction,” Appl. Opt. 32, 2266–2272~1993!.

8. G. Videen and D. Ngo, “Light scattering multiple solution for acell,” J. Biomed. Opt. 3, 212–220 ~1998!.

9. A. Nilsson, P. Alsholm, A. Karlsson, and S. Andresson-Engels,“T-matrix computations of light scattering by red blood cells,”Appl. Opt. 37, 2735–2748 ~1998!.

10. A. Dunn and R. Richards-Kortum, “Three-dimensional compu-tation of light scattering from cells,” IEEE J. Sel. Top. Quan-tum Electron. 2, 898–905 ~1996!.

11. B. Beauvoit, T. Kitai, and B. Chance, “Time-resolved spectros-copy of mitochondria, cells, and rat tissue under normal andpathological conditions,” in Photon Transport in Highly Scat-tering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V.Priezzhev, and V. V. Tuchin, eds., Proc. SPIE 2326, 127–136~1995!.

12. L. Junquerira, J. Carneiro, and O. Kelly, Basic Histology~Appleton and Lange, New York, 1992!.

13. Z. Liao, H. Wong, B. Yang, and Y. Yuan, “A transmittingboundary for transient wave analysis,” Sci. Sin. Ser A 27,1063–1076 ~1984!.

14. E. Hecht, Optics ~Addison-Wesley, Reading, Mass., 1987!.

from spherical mammalian cells,” Biophys. J. 14, 439–453~1974!.

16. R. Drezek, “Finite difference time domain simulations andgoniometric measurements of light scattering from cells,” M.S.thesis ~University of Texas at Austin, Austin, Texas, 1998!.

17. J. Valkenburg and C. Woldringh, “Phase separation betweennucleoid and cytoplasm in E coli as defined by immersiverefractometry,” J. Bacteriol. 160, 1151–1157 ~1984!.

18. J. Maier, S. Walker, S. Fantini, S. Franceschini, and E. Grat-ton, “Possible correlation between blood glucose concentrationand the reduced scattering coefficient of tissues in the nearinfrared,” Opt. Lett. 19, 2062–2064 ~1994!.

19. H. Liu, B. Beauvoit, M. Kimura, and B. Chance, “Dependenceof tissue optical properties on solute-induced changes in re-fractive index and osmolarity,” J. Biomed. Opt. 1, 200–211~1996!.

20. I. Vitkin, J. Woolsey, B. Wilson, and R. Anderson, “Optical andthermal characterization of natural ~sepia oficinalis! melanin,”Photochem. Photobio. 59, 455–462 ~1994!.

21. C. Boone, G. Kelloff, and V. Steele, “The natural history ofintraepithelial neoplasia: relevance to the search for inter-mediate endpoint biomarkers,” J. Cell. Biochem. 16, 23–26~1992!.

22. A. Dunn, “Light scattering properties of cells,” Ph.D. disserta-tion ~University of Texas at Austin, Austin, Texas, 1997!.

23. J. Mourant, J. Freyer, A. Hielscher, A. Eick, D. Shen, and T.Johnson, “Mechanisms of light scattering from biological cellsrelevant to noninvasive optical-tissue diagnostics,” Appl. Opt.37, 3586–3593 ~1998!.

24. R. Barer, “Refractometry and interferometry of living cells,” J.Opt. Soc. Am. 47, 545–556 ~1957!.

25. Y. Yarker, R. Aspden, and D. Hukins, “Birefringence of artic-ular cartilage and the distribution of collagen fibril orienta-tions,” Connect. Tissue Res. 11, 207–213 ~1983!.

26. L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoha-ran, A. Nustrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I.Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation ofperiodic fine structure in reflectance from biological tissue: anew technique for measuring nuclear size distribution,” Phys.Rev. Lett. 80, 627–630 ~1998!.

27. H. C. van de Hulst, Light Scattering by Small Particles ~Dover,New York, 1957!.

28. M. Anderson, J. Jordon, A. Morse, and F. Sharp, A Text andAtlas of Integrated Colposcopy ~Mosby, St. Louis, Mo., 1993!.

29. C. Smithpeter, A. Dunn, R. Drezek, T. Collier, and R. Richards-Kortum, “Near real time confocal microscopy of culturedamelanotic cells: sources of signal, contrast agents, and lim-its of contrast,” J. Biomed. Opt. 3, 429–436 ~1998!.

30. J. Mourant, J. Boyer, A. Hielscher, and I. Bigio, “Influence ofthe scattering phase function on light transport measurementsin turbid media performed with small source-detector separa-tions,” Opt. Lett. 21, 546–548 ~1996!.

1 June 1999 y Vol. 38, No. 16 y APPLIED OPTICS 3661