Fast measurements of concentration profiles inside deformable … · digital holography: the Fourier method. Digital holography allows one to numerically investigate a vo- ... three

Fast measurements of concentration profiles insidedeformable objects in microflows with reduced

spatial coherence digital holography

Christophe Minetti,1,* Natacha Callens,1 Gwennou Coupier,2

Thomas Podgorski,2 and Frank Dubois1

1Microgravity Research Center, Université Libre de Bruxelles, 50 Avenue F. Roosevelt,CP 165/62, B-1050 Brussels, Belgium

2Laboratoire de Spectrométrie Physique, Université Joseph Fourier—Grenoble I, 140 Rue de la Physique,F-38402 St. Martin d'Hères Cedex, France

*Corresponding author: [email protected]

Received 9 June 2008; revised 13 August 2008; accepted 3 September 2008;posted 8 September 2008 (Doc. ID 97163); published 7 October 2008

We investigate the use of a digital holographic microscope working with partially coherent spatial illu-mination to study concentration profiles inside confined deformable bodies flowing in microchannels. Thestudied phenomenon is rapidly changing in time and requires the recording of the complete holographicinformation for every frame. For this purpose, we implemented one of the classical methods of off-axisdigital holography: the Fourier method. Digital holography allows one to numerically investigate a vo-lume by refocusing the different planes of depth, allowing one to locate the objects under investigation inthree dimensions. Furthermore, the phase is directly related to the refractive index, thus to the concen-tration inside the body. Based on simple symmetry assumptions, we present an original method for de-termining the concentration profiles inside deformable objects in microconfined flows. Details of theoptical and numerical implementation, as well as exemplative experimental results are presented.© 2008 Optical Society of AmericaOCIS codes: 090.0090, 100.0100, 180.0180, 070.0070, 090.1760.

1. Introduction

Optical microscopy is limited by the small depths offocus due to high numerical apertures of the micro-scope lenses and the high magnification ratios. Theextension of the depth of focus is thus an importantgoal in optical microscopy. In this way, digital holo-graphy microscopy (DHM) yields the reconstructionin depth [1] and is very promising in optical micro-scopy because it considerably expands the field ofinvestigation.In DHM, holograms are recorded with a CCD cam-

era and provide both phase and intensity maps. Thecomplex amplitude can be computed to refocus the

depth images of a thick sample slice by slice (by im-plementing the Kirchhoff–Fresnel equation). In ad-dition, the phase is the significant informationused to quantitatively measure the optical pathlength of a sample, which is not available from mea-surements by classical optical methods [2]. DHM hasbeen applied in numerous applications of interest,such as observation of biological samples [3–6], livingcells culture analysis [7–9], and accurate measure-ments of refractive indices inside cells and even3D tomography [10,11]. Optical scanning holography,an unconventional form of digital holography, hasbeen used to explore 3D microscopy [12].

DHM is very flexible for implementing powerfulholographic information processing techniques [13].For example, methods to correct phase aberration[14–16], to perform 3D pattern recognition [17–19],

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to process border artifacts [20], to emulate classicalphase contrast imaging [6,9], to perform phase ima-ging [21], to implement autofocus algorithms [22,23],and to perform object segmentation [24] have beenproposed.In parallel, several optical methods have been im-

plemented to extract the phase and amplitude infor-mation from the recorded interference pattern[1,25,26]. For dynamic phenomena, a crucial pointis the acquisition time. Indeed, phase stepping meth-ods need the acquisition of several images, which re-stricts the applications to objects slowly varying intime. Conversely, the Fourier method [27–29] is verysuitable for fast phenomena because it requires onlyone recorded hologram in order to compute the com-plex amplitude.In this paper, quantitative phase map analysis is

proposed to study concentration profiles inside phos-pholipid vesicles in a microfluidic flow. Vesicles arelipid membranes enclosing an inner solution andsuspended into an outer one. Vesicles can be consid-ered as model systems to reproduce the mechanicalproperties of living cells, especially simple ones likered blood cells. For that reason, the study of their dy-namics under flow is the object of increasing atten-tion. When flowing in a straight channel whosewidth and thickness are of the same order as theirdiameter, vesicles adopt a stationary centered bullet-like or a parachutelike shape, resulting from thebalance between the hydrodynamic stresses andthe membrane elasticity [a typical example is shownin Fig. 2(a) below] [30,31]. These shapes are also ob-served for capsules, droplets, or blood cells. Beyondthe description of this stationary shape of the mem-brane, and given the fact that membranes are perme-able to water, one can wonder how this affects theflow field and solute concentration profiles insideand outside vesicles. Since the solutions are gener-ally transparent aqueous solutions, phase contrastmicroscopy is a advantageous tool to investigatethese properties.The use of DHM provides the phase information

and offers the refocusing capability useful for quan-titatively studying, for every hologram, a completevolume, which is not possible with a classical phasecontrast microscope [2,5,19]. From the phase map,one can easily derive the optical path length of theobject and the refractive index difference linked tothe concentrations inside and outside the body. How-ever, owing to the integration along the optical axis,deriving concentration profiles from a single holo-graphic image is not straightforward and requiresthe implementation of specific image processingtechniques. We propose an original method for deter-mining the concentration profiles by making simplesymmetry assumptions on the studied bodies. Bycoupling an automatic best focus [22] determinationwith a phase map compensation and a segmentationprocess, we will present a way to derive concentra-tion profiles inside a flowing vesicle confined in amicrochannel. Integrated refractive index measure-

ment has been proposed previously for living cell ap-plications [32–34].

Usually, DHM is implemented with a coherent la-ser beam. However, highly coherent sources are verysensitive to any defect in the optical path in such away that the results can be badly corrupted by thecoherent artifact noise. To reduce the effect of thisnoise and avoid multiple reflection interferences,we developed a DHM with a partial coherent illumi-nation in a Mach–Zehnder configuration [5,35]. Be-cause we are working with dynamic phenomena, itis mandatory to get the complete digital holographicinformation for every recorded image with a short ex-posure time. The illumination is performed by a laserdiode, and the reduced spatial coherence is obtainedby focusing the laser beam close to a rotating groundglass. It has been demonstrated that partial spatialillumination significantly increases the quality of theholographic images [35], in particular when the ex-perimental cell, as it is the case for this paper, cannotbe optimized with respect to the optical quality.

In Section 2, the optical setup is briefly described,and the technical specifications of the microscope areprovided. Section 3 is devoted to the image proces-sing techniques used to perform the determinationof the concentration profile. These techniques are im-plemented on the exemplative case of flowing vesi-cles, according to an experimental setup that isdescribed in Section 4. Results and discussion arepresented in Section 5.

2. Optical Setup Description

This section describes the DHM setup used for ourexperimental investigation. The optical setup is de-scribed in Fig. 1. It is a Mach–Zehnder interferom-eter in a microscope configuration working with a

Fig. 1. Optical setup of the digital holographic microscope. L1,focusing lens; RGG, rotating ground glass for spatial coherence re-duction; L2, collimating lens; L3, L4, identical microscope lensesð×20Þ; L5, refocusing lens; CCD, charge-coupled device camera;M1–M3, mirrors; BS1, BS2, beam splitters.

5306 APPLIED OPTICS / Vol. 47, No. 29 / 10 October 2008

partial spatial coherent source. The spatial partialcoherence is achieved by focusing a coherent beam(a monomode laser diode, λ ¼ 635nm) with a lens(L1) close to a rotating ground glass (RGG). Thespatial partial coherence of the source is adjustedby changing the position of the focused spot onthe RGG plane [27]. The use of partial coherencereduces the coherent noise and provides a temporal-coherence-like effect that eliminates the coherentmultiple reflection effect [35]. In our experiment,the setup has been adjusted to achieve a coherencelength of about 150 μm.The field of view of the DHM is 524 μm × 420 μm.

The camera is a JAI CV-M4 camera with a CCD ar-ray of 1280 × 1024 pixels (cropped in a 1024 × 1024pixel window to match the fast Fourier transformcomputation). The exposure time is set to 200 μs, al-lowing us to record flowing objects without blurringeffect. The angle between the reference and the ob-ject beam is adjusted in such a way that a gratinglikethin interference pattern is recorded on the sensor.This results in a high-fringe-density hologram (onefringe corresponds to 6 pixels) that is used to imple-ment the Fourier method to compute the complexamplitude of the object beam [28,29].

3. Description of Algorithms

In the general case, it is impossible to determine ex-actly the 3D distribution of the refractive index fromthe phase obtained with DHM without making addi-tional assumptions [36]. To get a 3D map, one shoulduse tomographic techniques, which require severalimages and are slow. Refractive index tomographyhas been successfully implemented in various appli-cations [37–41]. However, in our case, by makingsymmetry assumptions for the objects under investi-gation, it is possible to determine the 2D refractiveindex map integrated along the optical axis and de-rive the mean value of the refractive index inside theobjects. Note that with symmetry assumptions onecould recreate the different views obtained by atomograph and retrieve the 3D repartition of the re-fractive index inside the object with only one imageobtained by digital holography. Nevertheless, be-cause the phase is defined modulo 2π and becausethe precise measurement of the refractive index dif-ference inside a vesicle from the phase map is notstraightforward, specific image processing techni-ques are required. This section describes the differ-ent algorithms developed to extract the opticalpath length in each point of the object from the re-corded hologram. We start this section with a globaloverview of the processing sequence and follow witha more detailed description of the algorithms devel-oped for the processing.

A. Overview Description

The developed sequence used can be described asfollows: we consider one object (vesicle) flowing ina microchannel. Starting from the recorded holo-gram, we extract the amplitude and the phase and

compute the complex amplitude Uðx; yÞ, using theFourier method. We crop a region of interest (ROI)around the approximate position ðx0; y0Þ of the vesi-cle and seek the best focus position z0 of the object byreconstructing the cropped complex amplitudeU 0

dðx; yÞ slice by slice, where U 0dðx; yÞ is the cropped

complex amplitude around ðx0; y0Þ reconstructed at adistance d from the focus plane of the microscope.From this position, z0, we compute the amplitude[and the intensity, Fig. 2(a)] and the phasemap [Fig. 2(b)]. As the phase presents discontinuities (definedmodulo 2π), we compute a compensated phase by re-moving the background phase. Then we perform asegmentation of the object to detect the boundariesof the body and derive the optical path length in eachpoint of the body.

B. Best Focus Determination

Digital holography allows one to numerically inves-tigate a volume by refocusing the different slices butdoes not provide any information about the focus sta-tus. We have recently proposed a focus plane detec-tion criterion by defining a new focus metric. Thismetric is the integral of the complex amplitude mod-ulus. It has been shown [22] that this integral is max-imum for pure phase objects when the reconstructeddistance reaches the best focus plane. However, thiscriterion is based on an integral and gives a valuethat is influenced by all the objects distributed inthe volume. In our application, the studied objectsare confined in a microchannel whose walls interferewith the object when the best focus distance of theobject is computed. For this reason, we perform localanalysis by cropping the complex amplitude in a ROIaround the object. We choose a ROI size very close tothe object size (typically 140 × 140 pixels) to avoidany unwanted contribution of the background orthe walls of the channel, the ROI size always beingsmaller that the channel width. To avoid diffractiondue to the borders of the hologram and that appearsin digital holographic reconstruction, we extend theROI size (typically to 256 × 256 pixels) with valuesthat minimize the borders diffraction [20]. By cou-pling those two methods (best focus and border ex-tension), we automatically determine the bestfocus distance of the object. Because we are inter-ested in the phase measurement, we first have topropagate the complex amplitude up to the best focusplane of the object before computing the phase map.Indeed, the digital propagation creates phasechanges that distort the optical path length determi-nation, leading to additional measurements errors inout-of-focus planes.

C. Phase Map Compensation

The phase map gives the optical path length inte-grated along the propagating axis of the illuminatingwave, which is the basic information needed for thisstudy (see Subsection 5.B). The phase map presentsa nonuniform background phase owing to twosources of optical defects. First, the optical windows


of the experimental cell are not completely flat andare not exactly parallel to each other. Second, there isalways a small misalignment of the interferometer.Therefore, it is necessary to implement a phasemap correction. Because of the symmetry betweenthe object and reference channels of the interferom-eter we are using, we implemented a fast methodbased on the phase map derivative. The complexamplitude Uðx; yÞ is expressed by

Uðx; yÞ ¼ Aðx; yÞ exp½iφðx; yÞ�: ð1Þ

The measured phase φðx; yÞ is the sum of the phaseinduced by the object and the background phase thatcorresponds to the phase obtained when no object ispresent in the optical system:

exp½iφðs; yÞ� ¼ exp½iβðx; yÞ� exp½iϕðx; yÞ�; ð2Þ

where βðx; yÞ is the background phase and ϕðx; yÞ isthe object phase that we want to measure. This phaseaddition results in global phase differences (over thefield of view) larger than 2π, introducing jumps in thephase map that complicate the refractive index var-iations from the measured phase. To solve this pro-blem, one usually unwraps the phase beforeconverting it to refractive index data. Several meth-ods have been proposed to solve the complex problemof phase unwrapping [42]. However, those methodsrequire heavy computations that can be limiting toprocess large amount of data. Furthermore, phaseunwrapping does not compensate for aberrationsand is therefore not a good way to measure the phasedelay introduced by the object. In our application, themeasured refractive index difference between the ob-ject and the surrounding solution is sufficiently smallto avoid phase jumps bigger than 2π when the phasemap is adequately corrected. Therefore, we use acompensation method to remove the backgroundphase βðx; yÞ. Because of the symmetry of the inter-ferometer, we assume that the background phaseβðx; yÞ is modeled by a quadratic phase expressed by

βðx; yÞ ¼ σxðx − x0Þ2 þ σyðy − y0Þ2; ð3Þ

where ðx0; y0Þ are the coordinates of the center of theparabola and σx and σx are the curvatures along the xand y axes. Note that a similar modeling of the phasemap by a polynomial function has been successfullyproposed and implemented [43] for phase compensa-tion aberration. As outlined above, the phase changeinduced by the objects is relatively small. One canthus assume that the measured phase φðx; yÞ is rela-tively close to the background phase βðx; yÞ and canbe determined by fitting φðx; yÞ by Eq. (3). However,this fitting cannot be computed directly from φðx; yÞowing to phase jumps. To remove those discontinu-ities, the fitting is performed on the derivative ofthe phase map [44]. The parameters x0, y0, σx, andσx of the quadratic phase factor result from a least

Fig. 2. (a) Intensity image of a vesicle in a microchannel of 80 μmwidth. The vesicle is at a distance of 19 μm from the DHM focusplan. (b) Phase image of a vesicle in a microchannel. The phasevalues are remapped to 256 gray levels. (c) Compensated phasemap of Fig. 2(b). The phase values are remapped to 256 gray levels.


mean squares method by minimizing the followingexpression:

ψðσ; x0; y0Þ ¼XNx¼1

XNy¼1

�� δδx

�expfiφðx; yÞg

× expfi½σxðx − x0Þ2 þ σyðy − y0Þ2�g��

2

þ�� δδy

�expfiφðx; yÞg expfi½σxðx − x0Þ2

þ σyðy − y0Þ2�g��

2; ð4Þ

which gives the values of σx, σx, x0, and y0 anddetermines βðx; yÞ.The difference in the refractive index of the fluid

and the microchannel walls introduces strong phasevariations between the zone of interest and the restof the field of view. For this reason, the compensationdefined by Eq. (3) is computed only on the phase va-lues inside the microchannel. The compensatedphase is then subtracted from the measured phaseas follows:

ϕðx; yÞ ¼ φðx; yÞ − βðx; yÞ ifφðx; yÞ > βðx; yÞ;ϕðx; yÞ ¼ 2π − φðx; yÞ þ βðx; yÞ ifφðx; yÞ < βðx; yÞ; ð5Þ

which results in a phase map ϕðx; yÞ with a homoge-nous background inside the channel and a smoothbright shape representing the optical path lengthof the studied object. A typical example of phasemap compensation is shown in Fig. 2(c). The opticalpath length of the body is then obtained by compar-ing the value of the phase in each point inside thebody with those outside the microchannel. This istrue under the assumption that the compensatedphase is perfect. In practice, the reference phasemay not be exactly quadratic and is obtained througha fitting procedure. Nevertheless, around the objectunder study (in the ROI), we can consider the re-maining compensated phase to be linear, and itcan be approximated by a plane with a specific orien-tation. To avoid a tilting effect, we remove this back-ground phase by computing the plane fitting theborder of the ROI around the object. This providesa ROI with the object surrounded by a constant back-ground. To get the optical path length of the object,we now have to extract the object from its surround-ing background.

D. Segmentation

The segmentation process aims at delimiting theboundaries of the object under study and gets ridof the noisy small values of the optical path lengthjust around the edges of the vesicle (inherent tothe Fourier method). The simplest way is to applya threshold computed on the basis of the mean valueof the background. This is convenient for objects withsharp edges but not sufficiently precise for vesicles

whose boundaries are very smooth. We will see inSubsection 5.B that our symmetry assumptions re-quire a fine determination of the body shape.Furthermore, the Fourier method used to extractthe phase has a blurring effect on the object due tothe low-pass filtering. Active contour techniquesare iterative processes that attempt to minimize adefined function and are much more suitable for ap-plication with smooth boundaries. Starting from aninitial contour centered on the object, the successiveiterations deform the contour until the function isminimized and the deformed contour fits the bound-aries of the object. We used an active contour algo-rithm based on the Mumford–Shah functionalminimization [45]. Our motivation for choosing theactive contour technique is guided by the following:we know the position ðx0; y0Þ of the object and its ap-proximate shape, around which we have cropped theROI. Thus, as an initial contour we can reasonablychoose a circle with a ray equal to 2=3 of the sizeof the ROI. From the contour, we extract the matrixΦðx; yÞ corresponding to the phase in each point ofthe object and having null values outside the object.The error on the segmentation of the borders of theobjects is estimated to be about 2% of the objectwidth. This error is estimated by comparing thesegmented shape obtained on ten different vesicles(of different sizes) to their width and height obtainedmanually.

The optical path length eðx; yÞ of the body is thendefined by

eðx; yÞ ¼ Φðx; yÞλ2π ; ð6Þ

where λ is the wavelength of the illuminationsource (635nm).

4. Objectives and Experimental Description

The techniques described in the previous sections areapplied on vesicles under flow. The core of the experi-mental device consists in a straight channel made bysoft lithography in a polydimethylsiloxane (PDMS)piece glued to a glass slide. Except for a few dust par-ticles in the PDMS, the whole system is homoge-neous and transparent. Vesicles are preparedfollowing the electroformation method [46]. Theyare made of a dioleoylphosphatidylcholine (DOPC) li-pid bilayer enclosing an internal solution of sucrosein water. Samples are diluted in a slightly hyperos-motic external solution of glucose. In the following,we consider a 80 μm × 80 μm channel of square crosssection in which the Poiseuille flow is controlled byapplying a pressure difference between the inletand outlet. The flow rate varies from around3 × 10−4 to 10−2 μl=s. Vesicles flowing in this micro-channel have a length between 20 and 40 μm anda width between 10 and 30 μm. To measure how theyare deflated relative to a sphere, they are character-ized by their reduced volume ν ≤ 1 defined byν ¼ V=V0, where V is the volume of the vesicle


and V0 is the volume of the sphere of same surface asthe considered vesicle. The typical reduced volumesare in the range 0:9 ≤ ν ≤ 1. According to the flowrate, the vesicles flow at a velocity varying from100 to 4000 μm=s.The results presented in the next section have

been widely observed for vesicles whose size, reducedvolume, and velocity lie in the ranges presentedabove. As demonstrative examples, we focus ontwo typical vesicles, whose characteristics aresummarized in Table 1.

5. Results

A. Specific Processing

Images are recorded at a frame rate of 24 frames=swith an exposure time of 200 μs. On every recordedhologram, the complex amplitude (intensity andphase) is computed and the processing sequence de-scribed in Section 3 is applied (best focus determina-tion, phase compensation, and segmentation). TheROI, centered on the vesicle, is given by hand;further processing steps are performed automati-cally to obtain, for each occurrence of the object,the matrix hðx; yÞ corresponding to the thickness ofthe object at each point and null values outsidethe object. We observed that, only a few micrometersaway from the vesicle, the phase in the external so-lution is not influenced by the passage of the vesicle;therefore we can consider the refractive index n1 ofthe external solution to be constant, and the result-ing background phase can be approximated by aquadratic phase map.

B. Experimental Demonstration

For every frame, we can consider the optical pathlength e of the vesicle given by Eq. (6), which is alsodefined by

eðx; yÞ ¼ 2hðx; yÞΔnðx; yÞ: ð7Þ2hðx; yÞ is its real thickness, while Δnðx; yÞ ¼n2ðx; yÞ − n1 is the refractive index difference be-tween the internal and the external solutions andn2ðx; yÞ is the mean value along the z axis of the localindex n2ðx; y; zÞ.In the most general case, one has to deal with two

unknown functions of x and y, namely, hðx; yÞ andΔnðx; yÞ, of which only the product is known. There-fore, to determine them, one needs additional as-sumptions or information to extract relevant datafrom Eq. (7).An important simplification of the problem is ob-

tained when one considers a straight channel witha square cross section. In this case, when the vesicle

is centered in the channel, its transverse cross sec-tions can be assumed to be as thick as wide for sym-metry reasons.

For each cross section defined by its xi along thelongitudinal axis, one can thus consider the profileeðxi; yÞ=2. Two such profiles for vesicle 1 are shownin Fig. 3. For every value of xi, the path length profileis well fitted by half an ellipse: eðxi; yÞ=2 ¼αðxiÞ½RðxiÞ2 − y2�1=2:

In particular, the square shape of the channel crosssection has no footprint on the vesicle shape. Thiswas expected, since the flow in the central part isquasi axisymmetric [47] and the elasticity of themembrane also tends to favor an axisymmetricshape. Then, we can consider that the profilehðxi; yÞ is a half-circle, hðxi; yÞ ¼ ½RðxiÞ2 − y2�1=2, andtherefore that the scaling factor αðxiÞ is equal toΔnðxi; yÞ, which is then y independent.

The variations of ΔnðxÞ as a function of x areshown in Fig. 4 for vesicle 1 for three different velo-cities (189, 916, and 4006 μm=s). Each curve is ob-tained by averaging over ten different positions inthe channel to decrease the noise, whose importancewill be discussed later.

One can clearly observe a gradient from back tofront inside the vesicle, which increases with the flowrate. To our knowledge, this is the first experimentalobservation of this phenomenon, which was not pre-dicted theoretically. The uncertainties inherent tothe method should therefore be discussed. We firstfocus on those specific to the nonoptical part (thatis, the channel) in order to evaluate those associatedwith the optical setup.

A first source of error is associated with the uncer-tainties in the determination of the width and thick-ness of the channel. The latter was determined bymeasuring the thickness of the channel mold witha profilometer, of precision �100nm. Within this er-ror, the thickness was found to be constant all alongthe channel.

As one can see in Fig. 2, the edges of the channelare not totally straight because of the quality of thelithography mask. This leads to uncertainties in thewidth of �3 μm. This could introduce a variation ofaround 5% of the scaling factor of the cross section.Under the (unrealistic) assumption that the vesiclecompletely reproduces this deviation from the sym-metry, this leads to a maximum error of 5% in thedetermination of Δn.

Second, the measurement can be altered by someinhomogeneities in the field of view, such as dusttrapped in the PDMS or floating around the vesicle.This alteration can be estimated by comparingmeasurements made at different positions in the

Table 1. Characteristics of the Vesicles Presented for Demonstration

Inner Solution Outer Solution Size Reduced Volume

Vesicle 1 300mM sucrose 350mM glucose 24 μm× 30 μm 0.96Vesicle 2 300mM sucrose 370mM glucose 32 μm× 40 μm 0.99


channel. The related error (standard deviation inthis set of measures) has been estimated to less than10%. This error drops to a minimum value of 1%when averaging is performed over a few dozenframes. This value, which can be estimated by con-sidering two equivalent passages of the vesicle,can be considered the one associated with the opticaldevice. Note that this is also the error associatedwith the pixelization of the signal in the Oxy plane;no other source of error has been identified.

Consequently, the variations of Δn with x and theflow rate can be trusted. This gradient is directlylinked with a gradient in the sugar concentrationC2. The corresponding variations are of a few milli-molar concentrations [48], to be compared with themean value hC2i, which is around 300mM. Note thatit would be practically impossible to measure such

Fig. 3. Half optical width eðx0; yÞ=2 along two cross sections lo-cated at two different positions x0 along the longitudinal axis ofvesicle 1. Inset, location of these cross sections on the compensatedphase image of the vesicle. Solid curves show the fits with a half-ellipse.

Fig. 4. Evolution of the index ΔnðxÞ along the longitudinal axis of the vesicle for three different vesicle velocities (average over tenmeasurements for each velocity, to get rid of the local inhomogeneities of the channel: for clarity, only one typical error bar is shownon each curve). x ¼ 0 corresponds to the rear of the vesicle, which is moving from left to right. The right-hand part of the figure showsthe corresponding shapes given by the compensated phase image.

Fig. 5. Index gradient inside vesicle 2 doing a U-turn. The timestep between two curves is 1=24 s. During the 1=12 s correspondingto the three curves shown, the vesicle has just reversed its fore-aftshape but has only moved a few micrometers. Nevertheless, thegradient inversion is clear. Note that each curve corresponds toa single frame, and is thus noisier than the curves of Fig. 4.


tiny variations of concentrations by techniques suchas light absorption or fluorescence intensity.The existence of such gradients around a vesicle

while the concentration in the carrier fluid is homo-geneous has never been reported in the literature.Since the vesicle’s membrane is permeable to waterbut not to big molecules such as sugar, the flowthrough this membrane is induced by both osmoticand hydrodynamic pressure. The hydrodynamicpressure, which could be a motor for gradient crea-tion through sugar advection, is generally much low-er than the osmotic pressure difference induced by aslittle as a 1mM concentration difference across themembrane. It is thus generally considered that osmo-tic pressure is the leading contribution and thustends to homogenize the concentration profiles. How-ever, these considerations are generally based on aquasi-static view of the problem, in which case thenotion of osmotic pressure is well defined. In this si-tuation, membrane permeability measurements canbe easily carried out, for instance, by sucking a partof a vesicle into a micropipette. The permeabilitiesassociated with the hydrodynamic and osmotic pres-sure were shown to be roughly of the same order[49,50]. However, Nardi et al., who studied the osmo-tic-pressure-induced movement of a vesicle placed inan externally imposed concentration gradient,showed that a model based on such a quasi-staticview yields a drift velocity more than 3 orders of mag-nitude lower than the measured one [51]. To ourknowledge, no theoretical model is available in theliterature to explain this. As noticed by Nardiet al., in their experiments—as in ours—the vesicleis placed in a dynamic situation for which the notionof osmotic pressure is not well defined, or at leastcannot be as well decoupled from the hydrodynamicpressure as in the quasi-static case. Moreover, themembrane permeabilities can be modified by theflow stress and may vary locally, since they dependon the membrane tension [49], which is not uniformalong the flowing vesicle [52].The understanding of how the solute species are

distributed around a moving vesicle is thus still anopen problem. Qualitatively, the existence of a gradi-ent can be understood through general considera-tions. The key feature in the fact that the vesiclespeed is lower than the flow speed in the middle ofthe channel [30]: the vesicle is only partly pushedby the flow, while a water stream still goes throughit. Therefore, inside the vesicle, sugar must be ad-vected to the front, leading to a concentration gradi-ent inside the vesicle. Simultaneously, due to waterpermeation through the membrane, glucose concen-tration inhomogeneities build up in a boundary layeroutside the vesicle. This leads to another gradientnear the membrane. The final result is given bythe sum of both gradients, whose relative weight iscontrolled by the permeation laws across the mem-brane, which to date are not known in such a dy-namic case. Note that our analysis takes intoaccount the possible existence of a boundary layer

outside the vesicle. This layer must be thin, sinceno modification of the phase is observed a few micro-meters away from the vesicle. Thus the axisymmetryassumptions that were made for a vesicle remain va-lid for the vesicle and its surrounding layer.

It is also interesting to take advantage of the fastacquisition time of the DHM to explore transientstates. An example is given in Fig. 5, where we showthe gradients inside vesicle 2 doing a U-turn becauseof a flow reversal. We can see that the gradientinversion is very fast and is well measured by ourtechnique.

6. Concluding Remarks

This paper describes a technique to measure concen-tration profiles inside deformable transparent bodies(vesicles) in microconfined flows at high frame rate.We used a digital holographic microscope workingwith a spatial partially coherent source created byfocusing a laser diode beam close to a rotating groundglass. The use of partial coherence strongly reducesthe inherent coherent noise generated by the opticaldefects of the experimental cell (microchannel) thatis not of high optical quality. By coupling the phaseinformation provided by interferometry and the refo-cusing capability of digital holography, we can derivethe optical path length of vesicles everywhere in thechannel depth at a high frame rate (24 frames=s). Tocope with the vesicle velocities in the channel, we im-plemented the Fourier method.

Several algorithmic methods have been coupled toextract, from the recorded hologram, the thicknessmatrix of the vesicle. We first seek the best focusplane by using a specific criterion. The compensationof the phase map allows us to avoid discontinuitiesdue to the periodicity of the phase map. Finally,the segmentation extracts the vesicle from its back-ground. By making simple physical symmetry as-sumptions, we simplify the problem of preciselymeasuring the refractive index difference betweenthe internal and the external solution. It has beenshown that our method makes it possible to measurein a very reproducible way concentration profiles in-side the flowing vesicle with a very good resolution.Note that these profiles are measured in situ with nospecific preparation of the solutions, and the methodcould be applied to bodies such as red blood cellswithout taking the risk to modify their properties.This technique also opens new perspectives for themeasurement of time-varying concentrations atmicroscopic scales in objects moving in an unsteadyflow.

The authors acknowledge financial support fromthe European Space Agency (ESA) and CentreNational d’Etudes Spatiales (CNES). This workwas supported by the SSTC/ESA-PRODEX (ServicesScientifiques Techniques et Culturels/EuropeanSpace Agency-Programmes de Développementd’expériences) contract 90171.


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