This paper presents our study regarding diffracted intensity distribution in Fresnel and Fraunhofer approximation from different cell types. Starting from experimental information obtained through digital holographic microscopy, we modeled the cell shapes as oblate spheroids and built their phase-only transmission functions. In Fresnel approximation, the experimental and numerical diffraction patterns from mature and immature red blood cells have complementary central intensity values at different distances. The Fraunhofer diffraction patterns of deformed red blood cells were processed in the reciprocal space where, the isoamplitude curves were formed independently for each degree of cell deformation present within every sample; the values on each separate isoamplitude curve are proportional with the percentage of the respective cell type within the sample.
© 2012 OSA
For many natural or man-made micronic structures, we can numerically or experimentally find their diffraction pattern (DP), which carries information about the structural and morphological properties of the objects. The reversed problem, that of extracting information about the object from its diffraction pattern, is rather more complicated. There are many studies about the DPs on different cell types (muscle fibers, tumor cells or blood cells) in given conditions and much useful information is obtained [1–6]. Also, the diffraction halo produced by a tear indicates the presence of different modifications of the structural characteristics of the cells .
Both Fraunhofer and Fresnel approximations are used in different applications. The far field DP is used in the technique known as ektacytometry  and also in order to complete and bring additional information to other already commercially available instruments . In diffraction phase microscopy  and digital holographic microscopy , the 3D image of a single cell is reconstructed from its Fresnel diffraction pattern superposed onto the reference beam . Numerical models for cell shapes have been introduced in order to calculate the diffracted intensity distribution  so that now computing the diffraction patterns rigorously or in scalar approximation is enabled by many commercial software programs (RSoft, VirtualLab, BPM, etc).
In this paper we present our simulations and experimental results regarding the DPs obtained from different types of red blood cells (RBCs) in different conditions, in Fraunhofer and Fresnel approximations. We design several models for the shapes of mature and immature RBCs, whose dimensions were found experimentally using digital holographic microscopy (DHM).
From phase-only transmission functions modeled for mature and immature RBCs, we calculate the DPs in Fresnel approximation. We hence show that the DPs of these two cell types exhibit complementary behavior with regards to the evolution of their respective central intensity values at different distances. The same behavior was also observed experimentally.
In the Fraunhofer approximation, we calculate the DPs from a phase-only transmission function modeled for mature red blood cells which are supposed to be deformed in one direction (as is the case in ektacytometric studies). A problem appears in practice when not all cells are deformed by the same amount and the studied sample has in fact proven to contain many kinds of deformed cells. Recently, the case when the sample contains two types - normally and poorly deformable RBCs - was considered . We study the case when we have two or three cell types with similar degrees of deformation. To find the percentage of each type within the total number, we develop a method based on the study of the isoamplitude curves in the reciprocal space.
2. Mature and immature RBC identification from Fresnel diffraction patterns
To prepare the samples, certified phlebotomists collected blood from patients in a hospital, respecting all the appropriate hygienic and legal regulations. The blood was then stored in coated containers at 2-3°C, following the standard procedures of in-vitro preservation. A blood drop was smeared on a glass slide and a cover slip was placed on top. This sample was introduced in the standard experimental setup for DHM based on the Mach-Zehnder interferometer with two additional identical objectives placed in both arms to have the same curvature of the wavefront on the CCD sensor. Holograms were recorded within only a few minutes to avoid sample alteration and drying.
Figure 1(a) shows a hologram and Fig. 1(b) shows the 3D object image reconstructed from phase information. Profiles for different blood cell (BC) types are shown in Fig. 1(c) - mature RBC, Fig. 1(d) - immature RBC and Fig. 1(e) - white blood cell. For a healthy person, immature RBCs represent 1-2% from the total RBCs and this percentage is important in different diseases. Our procedure for reconstruction steps is described elsewhere . From these images, we calculate the dimensions in a plane perpendicular to the propagation axis (diameters for mature RBCs in the range of 5.6-7.9μm and for immature RBCs in the range of 7.6-9.6μm). The cell thickness (dimension along propagation axis) was computed from phase shift information, using refractive index values extracted from literature [16–21].
In the experimental setup, we blocked the reference arm, and in this case, the CCD camera sensor only records the Fresnel diffraction patterns separately for every cell. Figure 2 shows some intensity distributions recorded experimentally at different distances along the propagation axis. The differences between the DPs from mature and immature RBCs are clearly distinguished at all distances (the DP of immature RBC is marked). The principal characteristic that can be easily monitored is the behavior of the central intensity in the DP of each cell type. These experimental observations were first introduced in our previous work . There we labeled RBCs as mature or immature based on reconstructed images from experimentally recorded holograms, using gradient values as distinguishers. Conversely, the present study refers only to the diffracted image pattern recorded with reference arm blocked.
In order to compute the DPs in the Fresnel approximation, we model each cell as an oblate ellipsoid  with dimensions deduced from experimental information. Based on this, the phase-only transmission functions of the cells were designed in MATLAB. We simulate the propagation through these phase objects using the Fresnel operator implemented in its convolution form . The equations associated with this computation are described elsewhere . The diffraction patterns for different distances are shown in Fig. 3 (we employed our MATLAB code and also the commercial software VirtualLAB).
In a plane perpendicular to the direction of the incident light, the isointensity curves are circles. We monitor the values of the central intensity for mature and immature RBCs. Its behavior is exemplified in Fig. 4 for an interval where one can observe the interchange between the central intensity values of each cell DP. When the external radius for mature and immature RBC is slightly modified (keeping the proportion between them as established experimentally), the same behavior is observed but the maximum values are shifted (see Figs. 4(a) and 4(b)). When only the concavity radius for mature RBC is modified, the behavior is maintained, albeit for different values (see Figs. 4(c) and 4(d)).
3. Far field diffraction pattern used for the differentiation of deformed cells
In normal conditions, when all BCs are healthy, they will be deformed in the same way when they are subjected to a tension. When certain diseases are present, not all cells are deformed in the same way . Some of them maintain a circular projection and some take on a shape with the elliptical projection oriented in the same direction, having the major axis perpendicular to the tension direction (the projection is considered on a plane perpendicular to the propagation axis when the sample is illuminated with a laser beam). Using the diffracted pattern and an ellipse-fitting program the elongation index of deformed RBC can be determined .
In , the authors experimentally and numerically studied the DP from the simple case where it is considered that all cells are reduced by the shear stress to the same dimensions. In  the authors examined the case where only two kinds of cells exist in the mature RBC population: normally and poorly deformable cells. There, the authors can tell the percentage of these two types if they know the intensity values in the diffraction patterns when only one cell type or the other is present in the blood sample. In the first case , when all cells have the same deformation, the isontensity curves of the Fraunhofer DPs have simple elliptical shapes and in the second case , when only a certain percentage are deformed, the isointensity curves have a thick cross-like shape, where the information about different cell types are mixed. In the regular case, when there are no deformed cells, the Fraunhofer DPs have circular isointensity curves.
We simulate the case when a laser beam traverses a suspension of randomly distributed RBCs which are deformed in a perpendicular direction. Throughout this study, we consider two arbitrarily chosen axis ratios for such deformed RBCs: 1.3 (type1 or wide RBCs) and 1.6 (type2 or narrow RBCs), respectively. The diffracted intensity is then monitored and its angular distribution is considered. We obtain the same behavior in our simulations as those from the literature (Fig. 5 top). In all cases, the cross-like, circular and elliptical curves are rather thick and it is hard for someone to establish their dimensions, trajectory or intensity values within a reasonable approximation.
To overcome this impediment, the resultant DPs (an intensity image, square amplitude) are Fourier transformed. The images obtained in this way in the reciprocal space exhibit isoamplitude curves which are thin (see Fig. 5 bottom). Another advantage is the fact that the isoamplitude curves for different cell types are formed independently in the reciprocal space. We simulate the case when in the RBC population, two subpopulations exist with different percentages: deformed and undeformed (with elliptical and circular projection). The evolution of the isoamplitude curves in the reciprocal space when the percentage of deformed cells varies between 0% and 100% and for undeformed cells between 100% and 0%, can be seen in the movie from Fig. 6(a) (Media 1).
To extend our method, we also consider the case when three types of mature RBCs, with three different, but close values for the deformed radius exist in the sample. The evolution of the isoamplitude curves in the reciprocal space when the percentage of two types of deformed cells varies between 0% and 40%, 60% respectively, and for undeformed cells between 100% and 0%, can be seen in the movie from Fig. 6(b) (Media 2). Furthermore, the isoamplitude curves maintain the same axis ratios as the corresponding ellipses found in the sample images, within an expected error margin: 0.96, 1.23 and 1.56 for the isoamplitude curves corresponding to the ellipse ratios of 1, 1.27 and 1.63 respectively. The static to the outside of these movies is mostly a consequence of each sample image being generated by randomly positioning RBC projection model shapes using the Matlab uniformly distributed pseudorandom integer generating function, randi.
In this case as well, the Fraunhofer DP has isointensity curves with cross-like shapes, but these curves are thick (see Fig. 7(a) ) and the information about each type are mixed. In the reciprocal space, the situation is clearer, because the isoamplitude curves are thin and separated for each cell type (see Fig. 7(b)).
In order to obtain quantitative information from isoamplitude curves in reciprocal space, we automatically monitor the values in some pixels on them (marked in reciprocal space images from figures below). The values are stored in our built-in MATLAB function and no approximation is needed. The shape and the dimensions of the isoamplitude curves in reciprocal space are independent of the number of cells sharing the same degree of deformation, but rather depend on the geometry of the different cell types; the respective cell percentages only directly influence the amplitude values of the resultant transformed image in the reciprocal space. The shape of the isoamplitude curve which corresponds to one cell type is independent of the presence of other cell types in the sample. This behavior is due to the fact that in the reciprocal space we have represented the spatial frequencies from the direct space shapes
In Fig. 8 , the pixels from which the appropriate amplitude values were extracted in order to build the plots on the bottom are highlighted in the corresponding reciprocal space images on the top row. Figure 8(a) bottom shows the evolution of the amplitude values in the marked points, when we consider that two subpopulations exist in the sample population: type1 and type2 ellipses. When we consider that the sample population contains undeformed cells as well as cells which were deformed to become, in turn, type1 or type2 ellipses in projection, the evolution of amplitude values becomes as shown in Figs. 8(b) and 8(c) (bottom), respectively.
In the case when we have three subpopulations: circular projection, as well as type 1 and type 2 ellipses, the behavior of the values monitored in three points (highlighted in Fig. 9(a) ) of the isoamplitude curves is shown in Fig. 9(b). The sequence of sample images used to build this plot simulates the succession of phases that RBCs within a volume of real blood might undergo when submitted to compressive stress, each cell gradually increasing its degree of deformation, which is quantified by its axial ratio.
In Fig. 8 (bottom) the values from a set have been normalized through division by the greatest of all the values from the same set. This maximum intensity value of the monitored pixel on the isoamplitude curve corresponds with the case when in the sample population there is only one cell type; accordingly, the minimum value corresponds with the case when in the sample population there is no cell from the respective type (this value is different from zero because there is background noise). Between these extreme values, the evolution is linear. As such, one can find the unknown percentage of a certain cell type within a population of cells, drawing from values in known conditions, i.e. when a single cell type exists in the sample.
Figure 10 presents the linear fitting of the relative normalized intensity to cell type percentage plot, for which the standard error is calculated. This gives an indication of what error to expect when confronting a random sample with an unknown proportion of cells with a certain degree of deformation with the previously compiled baseline linear plot in order to identify the unknown percentage. The first plot (Fig. 10(a)) produces a root-mean-square error (RMSE) of 0.039 for disks and 0.024 for type1 ellipses, whereas the second plot (Fig. 10(b)) presents an RMSE of 0.041 for disks and 0.019 for type2 ellipses.
More evidence to the reliability of this method is shown by the following finding. By plotting the relative pixel intensity as a function of the corresponding RBC type percentage for a chosen cell type – disks, type1 or type2 ellipses – we obtain three linear plots, one for each of the examined sample configurations – samples containing the chosen cell type along with one of the remaining two cell types, or both. Generally, by representing these three linear plots simultaneously, each normalized by division by the greatest value of the respective set, it is easily observed that these plots overlap nearly perfectly (Figs. 11(a) and 11(b)), which highlights the fact they share the same slope. We can thus affirm that the behavior of the values on the isoamplitude curve of a specific RBC type is independent of other cell types which might occur within the sample.
A possible error source can be when the points in which the isoamplitude curves are monitored are not chosen properly. This occurs when the monitored pixel on the contour of the respective isoamplitude curve lies within the inside of one of the other such curves in the image, which itself is varying in intensity from one sample to another (Figs. 11(c) and 11(d)). This addition masks the true nature of the evolution of any individual isoamplitude curve. As Fig. 11(b) and Fig. 11(c) point out, for the same isoamplitude curve, the choice of pixel on which to base the plots influences whether or not this useful information is obtained, thus some consideration must be taken in this regard.
4. Discussions and conclusions
The DPs from different cell types were extensively studied in Fresnel and in Fraunhofer approximations and some useful information was extracted. Experimental and simulation results reveal that in the Fresnel approximation the diffraction pattern is formed separately for every cell and the identification of individual cell is possible; in the Fraunhofer approximation, the figure formed with the diffracted intensity contains a superposition of information from all cells, and only statistical information, namely the percentages of the different cell types contained in a given sample, can be extracted. For numerical considerations, we built the phase-only transmission functions based on the experimental results obtained in DHM reconstruction step for cells shapes and dimensions.
In the same setup, with the reference wave blocked, the Fresnel DP of mature and immature RBCs was recorded experimentally on the CCD camera and the central intensity values were shown to be complementary at various distances along the propagation axis. In this way, an immature RBC is clearly observed between the more numerous mature RBCs at a simple inspection. The same behavior was numerically obtained in commercial software (VirtualLAB) and with our code in MATLAB using simulations in Fresnel approximation on samples which contain both mature and immature RBCs. We change different geometrical parameters (external radius for both cell models or the internal radius for mature RBC concavity) and numerically monitor the central intensity values of the DP on an interval along the propagation axis. The behavior is the same, although values are changed and the position is shifted. These conclusions complement our previously work, bringing valuable quantitative evidence, achieved through simulation, in accordance with our experimental observations.
The DPs in Fraunhofer approximation are computed in the case when the existence in the sample of several cell types - regarding their deformation stage - is simulated. The isointensity curves of these DPs contain thick curves with a cross-like shape. In order to find the proportion of normal or compressed mature RBCs within the cell population, we process the Fraunhofer DP in the reciprocal space. The shape and dimensions of the isoamplitude curves in the reciprocal space are independent of the number of cells sharing the same degree of deformation and of the other subpopulatios in the sample, but rather depend on their geometry. The pixel values in properly chosen points of the isoamplitude curve depend on the percentages of the respective cell types. In the axial ratios of the isoamplitude curves we can find information about the axial ratios of the cell types present in the cell population. By monitoring the amplitude values in some properly chosen points, we can extract information about the percentage of normal or compressed RBCs, provided that the values in the same points are known for the case when one cell type is present in the sample. This method enables different types of deformed RBCs to be identified and their contribution to the cell population in the sample to be calculated.
RBC deformability is an important parameter in blood flow through small capillaries of the microcirculation. An abnormal behavior of RBCs in this process is a sign of different diseases. Our method offers a start point with perspectives of future applications in which image processing in reciprocal space would be used as a quick procedure without any approximation to find the percentage of different deformed RBC types. It has the capability of differentiating cells with close axial deformation ratio values.
The research presented in this paper is supported by the Sectorial Operational Program for Human Resource Development financed by the European Social Fund and by the Romanian Government under the contract no. POSDRU/89/1.5/S/63700. The equipment used in these experiments were acquired using funds from the Romanian contract 4/CP/I/2007-2009. We want to thank to Mihaela Scarlat for sample preparation and useful advice.
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