Quantitative image analysis of birefringent biological material

Quantitative image analysis of birefringent biologicalmaterial

S. ROSS,* R. NEWTON,* Y.-M. ZHOU,* J. HAFFEGEE,* M.-W. HO,* J. P. BOLTON† & D. KNIGHT‡*Bioelectrodynamics Laboratory, Biology Department and †Physics Department, Open University,Walton Hall, Milton Keynes MK7 6AA, U.K.‡Bioscience Group, King Alfred’s College, Sparkford Road, Winchester SO22 4NR, U.K.

Key words. Interference colours, liquid crystals, noninvasive quantitative imaging.

Summary

A previously described optical technique for enhancinginterference colour contrast in polarized light microscopy isadapted for quantitative image analysis. This is based on alinear relationship for colour intensity vs. molecular bire-fringence and degree of phase alignment, derived usingnematic liquid crystal approximations, and verified experi-mentally. The image analysis is described and illustrated ona section of the marginal rib of the dogfish egg case.

Introduction

Interference colours are generated in birefringent specimens intransmitted white light between two polarizers whose vibra-tional directions are at 90 8. A plane-polarized light ray, onpassing through the birefringent specimen with its optic axisperpendicular to the direction of light propagation, is splitinto two mutually perpendicularly vibrating rays thatpropagate through the specimen at different velocities. Theretardation of the slow ray relative to the fast ray – relativeretardation measured in nanometres – generates a phasedifference, d, between the rays as they emerge from thespecimen,

d ¼ 2pdðne ¹ noÞ=l ð1Þ

where d is the thickness of the crystal, ne and no are therefractive indices for the extraordinary ray vibrating in aplane parallel to the optic axis, and for the ordinary rayvibrating perpendicular to the optic axis of the birefringentspecimen, and l is the wavelength of the light. The slowand fast rays are recombined into a single ray as they passthrough the second polarizer, so they interfere either

destructively or constructively depending on the phasedifference introduced by the birefringent specimen. Forwhite light with a full spectrum of wavelengths in thevisible range, from 390 to 780 nm, the phase differencevaries across the spectrum, so that a precise hue ofinterference colour will be generated depending on therelative retardation (of the slow vs. the fast ray).

When the relative retardation of the specimen is below200 nm, brilliant interference colours will not be generatedunless a compensator crystal plate of sufficient relativeretardation is added in series with the sample to bring thenet retardation to within wavelengths of light in the visiblerange. For weakly birefringent materials with relativeretardations less than 50 nm, however, little or no colourcontrast is obtained when the compensator plate is placedwith its vibrational directions at the conventional angle of45 8 from the crossed polarizers. Instead, colour and colourcontrast are greatly enhanced when that angle is small (Ho& Lawrence, 1993; Ho & Saunders, 1994; Newton et al.,1995). For best results over the range of weak birefrin-gences frequently found in biological materials, the anglewas ascertained to be 4.5–7.5 8, and the relative retarda-tion of the compensator <560 nm – the wavelength ofgreen/yellow light. The same procedure was first discoveredby Wright (1911) in petrology, and described briefly againby others more recently (see Newton et al., 1995), but itdoes not appear to be widely used or developed in biologicalresearch.

We have described the applications of the technique forobtaining high-contrast colour images of live organismswhich reveal their anatomical details relatively noninvasively(Ho & Lawrence, 1993; Ho & Saunders, 1994; Newton et al.,1995). In this paper, we derive a linear relationship forinterference colour intensity vs. effective relative retarda-tion, based on which a quantitative imaging procedure isdeveloped. The latter enables us to determine and plot thedistribution of retardations and angles of maximumretardation in biological specimens. In those respects, it is

Journal of Microscopy, Vol. 187, Pt 1, July 1997, pp. 62–67.Received 17 July 1996; accepted 17 February 1997

62q 1997 The Royal Microscopical Society

Correspondence to: Mae-Wan Ho. Fax: 01908 654167; E-mail: [email protected]

Polarized light microscopy U.K. patent GB2270774A, US patent, 5559630, devel-

oped for quantitative imaging by S. R., J. H., R. N., Y. M. Z. and M. W. H. in collabora-

tion with Prior Scientific and Data Cell under an EPSRC-LINK scheme; the software is

an extension of the Media Cybernetics Image Pro image processing program.

complementary to the method described by Oldenbourg &Mei (1995). We illustrate how our method is used in asection of the marginal rib of the dogfish egg case.

Relationship of interference colour intensity vs. relativeretardation and degree of coherent phase alignment

Our optical system is best analysed in terms of the intensityof light emerging from two superposed crystal plates – thebiological specimen and the compensator full-wave plate –between crossed polarizers (see Hartshorne & Stuart,1970):

I=Io ¼ Io ¹ sin 2ðW2 ¹ W1Þ sin 2W1 cos 2W2 sin2 d1=2

þ sin 2ðW2 ¹ W1Þ cos 2W1 sin 2W2 sin2 d2=2

þ cos2ðW2 ¹ W1Þ sin 2W1 sin 2W2 sin2ðd1 þ d2Þ=2

¹ sin2ðW2 ¹ W1Þ sin 2W1 sin 2W2 sin2ðd1 ¹ d2Þ=2 ð2Þ

where d1 and d2 are the phase differences, respectively, ofthe wave-plate and the biological ‘crystal’, and W1 and W2

are the angles that the vibrational directions of their slow(or fast) wave makes with the polarizer.

Using Eq. (2), Newton et al. (1995) plotted the intensity ofmonochromatic red, green and blue light (700, 560 and450 nm) as a function of compensator plate alignment, andshowed that the effect of the small W1 angle is to reduce thecontributions of red and blue relative to the green, with theresult that the sample appears more colourful to the eye. Forsample retardations (R-values) smaller than 50 nm, thesmall angle gives much more colour contrast.

As the wave-plate with fixed relative retardation (d2 ¼

2pR=L) is kept at a constant angle with respect to thepolarizers, Eq. (2) can be simplified for small birefringences,using the approximation, sinW < W for small W, to give thefollowing relationships between colour intensity in theþ45 8 orientation of blue, Ib, and red, Ir , as a function ofrelative sample retardations, Rb and Rr .

Ib ¼ Io;b½0:032 þ 1:8 × 10¹3Rb þ 4:54 × 10¹5R2bÿ

Ir ¼ Io;r½0:023 þ 1:1 × 10¹3Rr þ 1:84 × 10¹5R2r ÿ ð3Þ

The difference (DI) between the peak intensity (þ45 8)and the minimum intensity (¹45 8), gives the followinglinear relationship:

DIb ¼ Io;b½3:6 × 10¹3Rbÿ

DIr ¼ Io;r½2:2 × 10¹3Rrÿ: ð4Þ

Ignoring absorption, depolarizing scattering and disper-sion effects, these relationships are exact for mineralcrystals, or crystals with perfect molecular order. For liquidcrystals, however, only the effective relative retardation isbeing measured, for the light intensity depends not just onintrinsic molecular (and form) birefringence but also on the

degree of coherent alignment of the individual molecules, asexpressed by the order parameter, S (see below).

For uniaxial nematic liquid crystals, which include nearlyall biological liquid crystals to first approximation, the orderparameter, S, measuring the state of alignment of themolecules along the direction of the nematic axis, is given by

S ¼ hð3 cos2 v ¹ 1Þi=2 ð5Þ

where h i denotes the average and v is the polar anglebetween the direction of the nematic axis (the averagedirection of alignment of the molecules) and the long axis ofthe individual molecule. S varies from 0 in isotropicmaterial to 1 when full alignment of all the molecules isattained. Isotropic material is dark at all angles of rotationwhereas fully aligned material is maximally bright at 458

between crossed polarizers. In Appendix 1, an expression isderived to show that, for small birefringences, the intensityof light transmitted does indeed vary linearly with the orderparameter as well as the intrinsic birefringence. Insituations where one parameter is constant, the other canbe estimated.

Experimental configuration

The measurements are made using a Prior polarizingmicroscope (Prior Scientific Instruments, Cambridge),which has been modified by Prior to include a motorizedrotating stage. The motor is driven by a Prior stagecontroller; this enables the angular position of the stage tobe computer controlled, and also provides automaticfocusing.

The 50-W lamp in the microscope is connected to a Priorcomputer-controlled power supply, which enables the lightlevel to be set accurately, and any variations in lampbrightness to be compensated for. The compensator platehas been modified by Prior to allow manual adjustment ofthe angle while it is still inserted in the microscope.

A JVC colour CCD camera is attached to the microscopeand provides the red, green, blue (rgb) signals to the framegrabberboard in the computer (Imaging Technology CFG).The framegrabber is controlled by an image processingprogram (Image Pro, Media Cybernetics), running underthe Microsoft Windows operating system. Special softwarehas been developed to meet the specific requirements of ourfunctions to be combined with the standard Image Profunctionality to allow a wide range of image manipulationoptions.

A set of mica calibration slides are used with retardationsin the range 5–100 nm.

Software processing

The software provides general functions to enable the userto control the position of the stage, focusing and the light

q 1997 The Royal Microscopical Society, Journal of Microscopy, 187, 62–67

QUANTITATIVE IMAGE ANALYSIS OF BIR EFRING ENT B IOLOGICAL MATERIAL 63

64 S. ROSS ET AL.


level. The calibration routines record the light variation as acalibrations slide is rotated through 1808. This informationis stored in the computer so that it can be used later toderive accurate values of retardations. To make quantitativemeasurements from a biological sample, several steps arerequired. First the sample slide is rotated on the stage from 0to 1808, in 2–158 steps. At each step, the image from thecamera is sent to the disk. Next, the centre of the image isdetermined either manually, or by an automatic scanningprocess. A software rotation algorithm is then used toremove the effect of the stage’s rotation on the image, sothat they are all correctly aligned. From the sequence ofimages, a set of five composite images are then produced, inwhich the pixel values represent:1 maximum brightness during rotation;2 minimum brightness during rotation;3 variation in brightness during rotation (maximum minusminimum, Max–Min);4 angle at which maximum brightness occurs;5 angle at which minimum brightness occurs.

The composite frames can be used to make manymeasurements of interest. For example, the retardationvalues can be derived from the Max–Min image, and theangle of orientation from the angle at which maximumbrightness occurs. The measured angle at which theminimum occurs is much more subject to noise than thatof the maximum, and is mainly used as a check, as it shouldbe 908 from the angle at which the maximum occurs.

The computer software records the image in colour usingthe three image planes, rgb (red, green and blue).Measurements of retardation and orientation are made ona specific plane, normally red or blue, as these show thegreatest variation in intensity.

Quantitative image analysis

Quantitative image analysis is developed based on the linearrelationship described in Eq. (4). Colour intensities aremeasured with a CCD colour camera in red, green and blue.The experimental values will deviate from the theoretical asthe camera detects a bandwidth for each colour, and thespectrum of the microscope light will deviate from the‘white’ ideal. This can be corrected for in the theoreticalcurve, but is unnecessary if the measurement system iscalibrated. Thus, provided the response of the camera (andvideo board) is linear, colour intensity values can be linearlytransformed into retardances. (In cases where the responsedeparts from nonlinearity, the values are curve-fitted to

include higher order terms and reasonable estimates ofretardances can still be obtained.)

The output of the camera is digitized by a PC videoacquisition card and stored on the computer’s disk.Calibration is achieved with the set of standard mica platesof known retardations to correct for variations associatedwith different microscope objectives and levels of lightingnecessary to produce good images. The retardations of thestandards were independently checked by measurementsusing a Senarmont compensator on a Zeiss Universalmicroscope fitted with a mercury vapour light source andinterference filter to isolate the green line at 546 nm. Foreach standard, the þ458 blue and red colour intensitieswere determined by automated stepwise rotation at 58

intervals for 1808. The calibration curve of DI vs.retardation should cover the range of retardations underinvestigation. Calibration is repeated for each set of recor-ding conditions so that different recordings can be com-pared where required. The orientation of the sample isdetermined with respect to some reference axis, such as thevibrational direction of the polarizer or the analyser. Thesample is then rotated as for the standards and the colourintensities recorded.

Samples to be compared are recorded under the sameconditions, as are the calibration curves. For time-lapsestudies carried out over long periods, the stability of thecolour response is monitored by the background values,which in our case are found to remain constant to within65%. The main causes of fluctuation are the light level andthe focus (especially where long time-lapse sequences areinvolved). These are minimized by feedback voltage controlto the microscope lamp, and autofocus interfacing betweenthe computer and the microscope, both of which are builtinto the software.

Molecular orientations in collagen ribbons of dogfish eggcase

Interference colour imaging was used to determine mole-cular orientations in ribbons of collagen in the marginal ribof the dogfish egg case extruded from the extrusion dies inthe nidamental gland. The image of the section, the plot ofretardation and angle of orientation along a transect, a two-dimensional orientation graph indicated by vectors, and agrey-tone image giving relative retardations over the wholearea of interest are shown in Fig. 1. The results of thisinvestigation, reported in detail elsewhere (Knight et al.,1996), suggest that rheological interaction with nematic


Fig. 1. (a) Interference colour image of section of dogfish egg capsule marginal rib cut at 158 to the long axis, where the plane of the lamellaecurves. The line marking a transect is 400 mm in length. (b) Plots of relative retardation and angle of orientation along the transect indicatedin (a). (c) Two-dimensional plot of orientation of collagen fibres overlaid on the image. The length of individual vectors is proportional to therelative retardation. (d) Grey-tone plot of relative retardation over the area of interest in the image. Note the loss of resolution between adja-cent orange and blue ribbons.


liquid crystallization is responsible for producing the bentnematic arrangement seen in this material. This interpreta-tion is consistent with observations from low-angle X-raydiffraction, bright-field microscopy, electron microscopy andconfocal microscopy performed on the same material(Knight et al., 1996).

The quantitative imaging technique described in thispaper is unique in that it gives detailed anatomicalinformation as well as physical information concerningbirefringence, orientation and degree of coherent phaseordering of the molecules making up the tissues. Forrelatively ‘inert’ materials and sections, such as the dogfishegg case and human cornea, it complements measure-ments obtained with other techniques such as X-raydiffraction and electron microscopy. It is especially suitablefor characterizing liquid crystalline mesophases of poly-mers and other biomimetic materials. Because of itsnoninvasive/nondestructive nature, it can yield informa-tion on rapid changes in liquid crystalline mesophases inliving organisms and cells that cannot be obtained by anyother means (see Ho et al., 1996), and is equally suitablefor following fast dynamics of liquid crystalline phasetransitions in vitro.

We have shown that, for small birefringences, theintensity of light transmitted varies linearly with the degreeof alignment as well as intrinsic birefringence. In situationswhere one parameter is constant, the other can beestimated. In the development of Drosophila larva bodywall musculature, for example, it is known that themolecular constituents of muscle are randomly arrangedwhen first formed, to become organized into regularmyofilament bundles typical of muscle some time afterwards(Abmayr et al., 1995). We have observed a rapid increase incolour intensity in time-lapse images of the maturingDrosophila larva which may be correlated with the increasein coherent alignment of the molecular constituents duringthe condensation of the body wall musculature (Ho et al.,1996).

Acknowledgments

This research was supported by an EPSRC-LINK grant toM.W.H. and funding from King Alfred College (D.P.K.). Wethank Lyndon Davies for his interest and support incoordinating the LINK grant, Prior Scientific for supplyingthe Prior polarizing microscope and accessories and DataCell for the imaging software for which our image analysisis an extension. We thank Dr D. J. Goldstein of SheffieldUniversity for advice on measuring birefringence, and forlending us the quarter-wave plate for 546 nm. Thanks arealso due to Michael Lawrence for help with microscopy andsome of the video recordings, to Mike Dodd for advice onstatistics, to Phil Bland and Mike Stewart for the loan ofmicroscopes, and Ian Wood of Hitachi for the loan of a CCD

colour camera. Helpful comments were provided by twoanonymous referees.

References

Abmayr, S.M., Erickson, M.S. & Bour, B.A. (1995) Embryonicdevelopment of the larval body wall musculature of Drosophilamelanogaster. Trends in Genetics, 11, 153–159.

De Jeu, W.H. (1978) The dielectric permettivity of liquid crystals.Liquid Crystals (ed. by L. Liebert), pp. 109–145. Academic Press,New York.

Hartshorne, N.H. & Stuart, A. (1970) Crystals and the PolarizingMicroscope, p. 302. Edward Arnold, London.

Ho, M.W., Haffegee, J., Newton, R., Ross, S., Zhou, Y.M. & Bolton,J.S. (1996) Organisms as polyphasic liquid crystals. Bioelectro-chem. Bioenergetics, 41, 81–91.

Ho, M.W. & Lawrence, M. (1993) Interference colour vital imaging:a novel noninvasive microscopic technique. Microscopy andAnalysis, September, 26.

Ho, M.W. & Saunders, P.T. (1994) Liquid crystalline mesophases inliving organisms. Bioelectrodynamics and Biocommunication (ed.by M.W. Ho, F.A. Popp and U. Warnke). World Scientific,Singapore.

Knight, D.P., Hu, S.W., Gathercole, L.J., Rusaouen-Innocent, M.,Ho, M.W. & Newton, R. (1996) Molecular orientations in anextruded collagenous composite, the marginal rib of the eggcapsule of the dogfish Scyliorhinus canicula; a novel lyotropicliquid crystalline arrangement and how it is defined in thespinneret. Philos. Trans. R. Soc. London, 351, 1205–1222.

Newton, R., Haffegee, J. & Ho, M.W. (1995) Colour-contrast inpolarized light microscopy of weakly birefringent biologicalspecimens. J. Microsc. 180, 127.

Oldenbourg, R. & Mei, G. (1995) New polarized light microscopewith precision universal compensator. J. Microsc. 180, 140–147.

Wright, F.E. (1911) The Methods of Petrographic-MicroscopicResearch, Their Relative Accuracy and Range of Application.Carnegie Institute, Washington.

Appendix 1. The relationship between the light intensityand the order parameter

The light intensity differences between þ458 and ¹458

orientation of the sample are given by Eq. (4),

DIb ¼ Io;b½3:6 × 10¹3Rbÿ

DIr ¼ Io;r½2:2 × 10¹3Rrÿ ðA1Þ

where Rb,r = d(√

e1 –√

e2); Rb,r are the retardations of thesample for blue and red; e1 and e2 are small dielectricconstants parallel and perpendicular to the optic axis of thebiological sample, and d is the thickness of the sample. Inthe mean field approximation (De Jeu, 1978),

e1 ¼ 1 þ 4pNrhF=Mða þ 2Þ=3fS

þ ðFm2=3KBTÞð1 ¹ ð1 ¹ 3 cos2 vÞSÞ

e1 ¼ 1 þ 4pNrhF=Mða þ 2Þ=3fS

þ ðFm2=3KBTÞð1 þ 1=2ð1 ¹ 3 cos2 vÞSÞ ðA2Þ

66 S. ROSS ET AL.


where r is density, h ¼ 3e=ð2e þ 1Þ, e is the mean dielectricconstant, F ¼ 1=ð1 ¹ af Þ, f ¼ 4pNrð2e ¹ 2Þ=3Mð2e þ 1Þ, a

is the mean polarization, m is the dipole moment, M is themolecular weight, S is the order parameter and f is thepolarizability anisotropy. Equation (A2) can be simplified:

e1 ¼ 1 þ Aða þ b þ BSÞ

e2 ¼ 1 þ Aða þ b þ CSÞ ðA3Þ

where A, B, C and b are constants unrelated to S.Equation (A1) can be simplified in the case of small

retardations, where e1 and e2 are small, that means bothA(x + BS) and A(x + CS) are much smaller than 1.Substituting the expressions of dielectric constants (A3) intothe expression for retardation (A1) and expanding it gives

R ¼ d0:75ABS

ð1 þ AxÞ1=2 ¹0:09375A2B2S2

ð1 þ AxÞ3=2 þ O½Sÿ3� �

: ðA4Þ

Effective retardation, R, and hence the intensity of lighttransmitted, is approximately linearly related to S, becausethe second-order term, S2, is much smaller than the first-order term, and higher order terms, O[S]3, are even smaller.



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Quantitative image analysis of birefringent biological material