2.1.

Sample Preparation

Blood samples were taken from healthy donors by venopuncture with EDTA as an anticoagulant. Preparation of lymphocyte samples was carried out by a density-gradient separation procedure.²⁶ This procedure consists of diluting $2\mathrm{ml}$ of whole anticoagulated blood in $6\mathrm{ml}$ of phosphate buffered saline (PBS) and centrifuging at $\approx 450\mathrm{g}$ for $20\min$ through a Ficoll®-sodium metrizoate solution with a density of $1.077\mathrm{g}∕\mathrm{ml}$ . Then a thin layer of mononuclear cells from the plasma/Ficoll interface was aspirated, washed with PBS in a centrifuge, and stained with immunofluorescence markers CD3-FITC and CD19-PE (ImmunoTools, Friesoythe, Germany) for T- and B-lymphocytes, respectively. After $20\min$ of staining, the sample was washed again and diluted to a concentration of about ${10}^6\text{cells}∕\mathrm{ml}$ . The prepared samples were analyzed with a scanning flow cytometer (SFC) measuring the LSPs and fluorescence signals from each cell. Clusters containing all lymphocytes were gated by light-scattering signals. In conventional flow cytometry, identification of lymphocytes can be performed using forward and side scattering, with intensities plotted on the well-known “side scattering versus forward scattering” $(\mathrm{SSC}\times \mathrm{FSC})$ map. Analogously, in SFC we used absolute values of the integrated whole light-scattering trace instead of the side scattering [Fig. 1a ]. Dot plots of these two integral parameters were verified to be similar to and display the same population percentages as ones obtained by the standard flow cytometer Coulter Epics (Beckman Coulter, Fullerton, California) (data not shown). Then T- and B-lymphocytes subsets from this cluster were distinguished by specific fluorescence using the $\mathrm{CD}3\times \mathrm{CD}19$ map [Fig. 1b].

Fig. 1

2-D cytograms used in identification of T- and B-cells.

2.2.

Scanning Flow Cytometer and Confocal Microscopy

The schematic layout of the SFC’s optical system is presented in Fig. 2 . The SFC was assembled on the bench of a FACStar Plus BD flow cytometer (BD, Franklin, Lakes, New Jersey). An argon-ion laser (laser, $488\mathrm{nm}$ , $60\mathrm{mW}$ ) was used for generation of the scattering pattern, for excitation of fluorescence, and for triggering the electronic unit. The laser beam split by mirror 1 is directed coaxially with the stream by mirrors 2 and 3, and a lens (lens 1, $f=45\mathrm{mm}$ ) through a hole in the mirror (mirror 4). The hydrofocusing head (not shown) produces two concentric fluid streams: a sheath stream without particles and a probe stream that carries the particles. The fluidics system directs a probe stream with a $12\mu \mathrm{m}$ diam into the optical cell (Fig. 2). The operational function of the optical cell was previously described in detail.^{13, 27} The light scattered by a single cell is reflected by mirror 4 to the photomultiplier tube (PMT 2). The other portion of the laser beam is focused by objective 2 $(\mathrm{NA}=0.2)$ into the capillary of the optical cell. The light scattered in the forward direction is collected by objective 1 $(\mathrm{NA}=0.2)$ and detected by PMT 1. The beam stop prevents illumination of PMT 1 by the incoming laser beam. The fluorescence of specific dye molecules linked at the single cell is collected by objective 3 $(\mathrm{NA}=0.4)$ and detected by PMT 3. The bandpass optical filter provides measurement of the specific fluorescence with an appropriate signal-to-noise ratio.

Fig. 2

Schematic layout of the optical system of the scanning flow cytometer.

This optical setup of the SFC (PMT 2 in Fig. 2) measures the following combination of scattering matrix elements:¹³

In general, experimental data have a different signal-to-noise ratio over regions measured that assume a multiplication of theoretical data with a weighting function to reduce an effect of the noise on the fitting results. In the current study we propose

The operation angular range of the SFC is limited by geometrical parameters of the optical cell and alignment of the laser beam. We determined the operation range from analysis of polymer microspheres. The angular range, where experimental weighted LSP $w\left(\theta \right)I\left(\theta \right)$ is perfectly fitted by weighted LSP calculated from Mie theory, constitutes the operational range of the SFC for measurement of the light-scattering properties of blood cells. The result of a representative result is shown in Fig. 3, where the inset shows the measured light-scattering trace of a single microsphere. Note that for microspheres in the size range of $0.5\text{to}10\mu \mathrm{m}$ and refractive index range of 1.56 to 1.59, such excellent results are obtained routinely with SFC.¹³ The comparison of experimental and theoretical weighted LSPs allows us to set the operation region of the current SFC setup to the range from $12\text{to}50\mathrm{deg}$ .

Fig. 3

Experimental and Mie theory best-fit weighted differential cross sections of an individual polymer microsphere. The Mie result corresponds to a microsphere with a size of $1.84\mu \mathrm{m}$ and refractive index of 1.606. The measured light-scattering trace of the same microsphere is shown in the inset.

A Carl Zeiss LSM 510 META confocal microscope was used to develop our optical model of lymphocytes. The typical parameters of the microscopy used in this work were as follows: objective—Plan-Apochromat $100\times 1.4$ Oil DIC; wavelength— $543\mathrm{nm}$ and $405\mathrm{nm}$ ; filters—BP $560–615\mathrm{nm}$ and LP $420\mathrm{nm}$ ; scaling— $30\mathrm{nm}$ $(\Delta x,\Delta y,\Delta z)$ .

2.3.

Discrete Dipole Approximation

The discrete dipole approximation (DDA) is a general method to compute scattering and absorption of electromagnetic waves by particles of arbitrary geometry and composition.²⁸ The ADDA computer code is an efficient DDA implementation on a cluster of computers, parallelizing a single DDA computation;²⁹ ADDA’s source code and documentation is freely available.³⁰ Simulations of light scattering by the nonsymmetric optical model of a lymphocyte (see Sec. 3.2) were performed with ADDA v.0.78.2. For each particle, we computed the LSP [see Eq. 1] for $\theta$ from $12\text{to}50\mathrm{deg}$ with steps of $0.5\mathrm{deg}$ , using steps of $5.6\mathrm{deg}$ for integration over $\phi$ . All simulations were run on the Dutch compute cluster Lisa.³¹

2.4.

Global Optimization

The solution of the inverse problem is performed by a least-squares method, i.e., by minimizing the weighted sum of squares:

We use the following cellular characteristics: cell diameter $D_c$ , ratio of nucleus and cell diameters $D_n∕D_c$ , refractive indices of cytoplasm $m_c$ , and nucleus $m_n$ . The region $\mathbf{B}$ is defined by their boundary values: $D_c∊[4.5,10]\mu \mathrm{m}$ , $D_n∕D_c∊[0.7,0.95]$ , $m_c∊[1.34,1.41]$ , and $m_n∊[1.41,1.58]$ , excessively covering the range of lymphocyte morphological characteristics as reported in recent literature. ^{7, 32, 33, 34}

2.5.

Errors of Parameter Estimates

Many textbooks on nonlinear regression deal with errors of parameter estimates (see e.g., Refs. 35, 36). However, most of the theory is worked out for the case of normally distributed and independent experimental errors. Unfortunately, there is significant dependence between residuals $z_i$ in our case, which can be characterized by a sample autocorrelation function ${\rho }_k$ .³⁵ A typical example of this function for a lymphocyte is shown in Fig. 4 . The correlation is caused by: 1. finite angular resolution of the SFC, causing dependence between nearby ${\theta }_i$ , and 2. model errors, i.e., systematic difference between lymphocyte shape and a coated sphere (see Sec. 3.2). The correlation due to angular resolution (about $0.5\mathrm{deg}$ ¹³) may extent only up to approximately $k=3$ ; therefore, a major cause for correlation is model errors. These errors are definitely not purely random, but they are determined by a number of parameters, such as, e.g., an offset of nucleus from the cell center or its inhomogeneity, which have large biological variability in the sample and hence can be considered random for any particular lymphocyte. There are also certain errors associated with imperfect alignment of SFC and noncentral positions of the particle in the flow.¹³ However, we consider these errors as part of model errors, since the exact structure and cause of the latter is not important for our characterization method, nor can they be distinguished from real model errors.

Fig. 4

Typical autocorrelation function of residuals, resulting from fitting of a lymphocyte LSP by LSPs calculated from the Mie theory for coated spheres.

Rigorous methods to deal with correlated residuals exist,³⁵ but they are quite cumbersome. Here we propose to use an approximate but much simpler approach. It is known³⁷ that the sum of squares of correlated residuals can be approximately described by a ${\chi }^2$ distribution, if correlations can be described by a stationary Gaussian process, i.e., if it is fully characterized by ${\rho }_k$ . The number of effective degrees of freedom of the resulting ${\chi }^2$ distribution is

Fig. 5

Distribution of effective number of degrees of freedom over a sample of T-lymphocytes from donor 6.

Next, we apply a Bayesian inference method with standard noninformative (or homogeneous) prior $P(\sigma ,\mathbf{\beta })\sim {\sigma }^{-1}$ .³⁵ The posterior probability density of $\mathbf{\beta }$ given a specific experimental LSP is then $P\left(\mathbf{\beta }\right|I_{\exp })=\kappa {\left[S\left(\mathbf{\beta }\right)\right]}^{-n∕2}$ , where $\kappa$ is a normalization constant obtained as

Since we know the complete probability distribution of model parameters for any measured particle, we can infer any statistical characteristics of this distribution. In this work, we calculate the expectation value $\mathbf{\mu }=⟨\mathbf{\beta }⟩$ and the covariance matrix $\mathbf{C}=⟨(\mathbf{\beta }-\mathbf{\mu }){(\mathbf{\beta }-\mathbf{\mu })}^T⟩$ using Eq. 7, since in most cases the probability distribution is well described by a multidimensional normal distribution (data not shown). Other quantities of interest are the highest posterior density (HPD) confidence regions, defined as ${\mathbf{B}}_{\mathrm{HPD}}\left(P_0\right)=\left\{\mathbf{\beta }\right|P\left(\mathbf{\beta }\right|I_{\exp })>P_0\}$ .³⁵ Its confidence level is

We use these confidence regions to assess the influence of parameter space boundaries on the results. Although $\mathbf{B}$ is quite large, its boundaries cut off tails of the distribution, causing bias in determination of both $\mathbf{\mu }$ and $\mathbf{C}$ . The bias increase with relative weight of the cut tail, so we define a confidence level $\alpha$ of the particle (more precisely, of associated probability distribution over $\mathbf{\beta }$ ) as the confidence level of the HPD region touching the boundary of $\mathbf{B}$ . It is important to note that in 4-D parameter space, $1-\alpha$ is much larger then the weight of the tail cut by the boundary. Therefore, choosing, e.g., $\alpha =0.8$ implies a small bias of $\mathbf{\mu }$ and $\mathbf{C}$ .

The main purpose of $\alpha$ is to assess biases of $\mathbf{\mu }$ and $\mathbf{C}$ . It can also be used as a measure of goodness of fit but only approximately, because it depends on the position of $\mathbf{\mu }$ relative to the boundary of $\mathbf{B}$ . Relative goodness of fit is better assessed by the matrix $\mathbf{C}$ .

2.6.

Sample Characterization

Having determined characteristics of each lymphocyte with certain errors (i.e., ${\mathbf{\mu }}_i$ and ${\mathbf{C}}_i$ , $i=1,\dots ,N_s$ , where $N_s$ is the number of cells analyzed), we propose the following method to estimate population mean and variances. General normal distributions are assumed for both the measurement results of individual particles and the true distribution of characteristics over a sample, the latter being described by a mean value $\mathbf{h}$ and covariance matrix $\mathbf{A}$ . We use an estimator proposed by Fuller,³⁸ whose first step is based on means of ${\mathbf{\mu }}_i$ and ${\mathbf{C}}_i$ and a sample covariance matrix based on ${\mathbf{\mu }}_i$ :

The second step is the following:

The advantage of this procedure is the automatic decrease of importance of measurements with large errors. Therefore, we can seamlessly include all measurements into the estimation of cell characteristics, even those that seem unreliable due to large sums of squares and/or standard deviations of the characteristics. However, doing so is not the best option because of the bias due to the region boundaries, which are especially important for particles with small confidence level $\alpha$ . We also found that the iterative procedure fails to converge when certain cells with large estimate errors are included in the processed sample (data not shown). In the current study, we excluded such cells—they all had $\alpha <0.35$ and their total amount was less than 0.7% of all cells in each sample.

Therefore, we consider only particles with $\alpha ⩾{\alpha }_0$ for characterization of the samples. To choose the threshold ${\alpha }_0$ , we plot the resulting parameter estimates versus ${\alpha }_0$ . Such a plot for mean cell diameter over sample including errors of mean is shown in Fig. 6, together with the fraction of processed cells (those above the threshold). One can see that there are a number of cells, for which $\alpha$ equals exactly zero. That is because the maximum of the probability distribution is reached at the boundary of the parameter space. With decreasing ${\alpha }_0$ , the error of mean first decreases due to the increasing number of processed cells (better statistics). Then it stabilizes, however, since new accepted measurements have relatively large errors and hence contribute very little new information. The mean value, on the contrary, shows a systematic trend with decreasing ${\alpha }_0$ (at least after a certain value), which is due to the “boundary” biases discussed before. As a compromise, we chose a round value ${\alpha }_0=0.8$ for this work; however, we stress that moderate variation of ${\alpha }_0$ does not significantly change the results. Plots for other parameters and samples support this choice (data not shown).

Fig. 6

Dependence of estimated mean cell diameter and fraction of processed cells on confidence level threshold for a sample of T-lymphocytes of donor 6. Confidence interval for $⟨D_c⟩$ is shown (expected $\text{value}\pm \text{standard}$ error). The discontinuity at ${\alpha }_0=0$ is caused by a significant fraction of cells having $\alpha =0$ .

For 4-D normal distributions without correlations, a confidence level of 0.8 corresponds to a distance between mean and boundary equal to roughly 2.5 standard deviations. Using a confidence level threshold does introduce a certain bias, because measurements close to the boundary are less likely to pass the threshold, but we estimate it to be within errors of mean for samples studied in this work. However, it makes sense to broaden the parameter space in future studies to remove such biases altogether, especially when studying samples from nonhealthy donors.

It is important to note that we estimate a full covariance matrix of the sample of individual lymphocytes as well, including correlations between different characteristics. However, we have shown and analyzed only the diagonal elements of the matrices $\mathbf{A}$ and $\mathbf{Z}$ , i.e., variances of the characteristics over the sample and errors of mean values. Correlations are significant, e.g., the correlation between $D_c$ and both $m_c$ and $m_n$ are generally between $-0.6$ and $-0.8$ , but we leave this interesting and potentially biologically significant observation for future study.

3.1.

Differential Light-Scattering Cross Section of Lymphocytes

The essential feature of the SFC is the ability to measure the absolute differential light-scattering cross section of single particles of any shape and structure. This feature is realized by measurement of a mixture of unknown particles and polymer microspheres.³⁹ The LSP of the polymer microsphere, measured with the SFC, gives a high accuracy agreement with the Mie theory.⁴⁰ This allows calibration in absolute light-scattering units. The differential cross section was calculated from the following equation:

To determine the differential cross section for lymphocytes, we simultaneously analyzed blood leukocytes and polystyrene microspheres with a size of $1.8\mu \mathrm{m}$ . 20 LSPs of individual T-lymphocytes and one microsphere are presented in Fig. 7 . The experimental LSP of the microsphere, which provides the scale for the $y$ axis of the figure, is the same as shown in Fig. 3. The LSPs of lymphocytes demonstrate substantial variations in their intensity and structure over the sample. Following our analysis of confocal 3-D images of the cells, we suggest that variations in the LSP structure are caused by inhomogeneity and irregular shape of the nucleus. The variations in the LSP intensity result from differences in effective refractive index of the nucleus and cellular volumes.

Fig. 7

Differential light-scattering cross section of T-lymphocytes and a polymer microsphere at a wavelength of $488\mathrm{nm}$ (in logarithmic scale). The polymer microsphere is the same as shown in Fig. 3.

3.2.

Lymphocyte Optical Model

A lymphocyte has a nearly spherical shape and a single nucleus.³² 3-D images of a few cells were obtained from confocal microscopy, and one slice of them is presented in the left part of Fig. 8 . One can see that the coated sphere is an adequate first-order model, and this model was used for the solution of the inverse light-scattering problem. However, it is also clear that the nucleus is certainly not perfectly homogeneous. To study the effect of such nuclear inhomogeneity on the light scattering, we enhanced the optical model by introducing an oblate spheroid into the nucleus (Fig. 8, right). This model was used to mimic the effect that nuclear inhomogeneity may have on the LSPs and to verify the results of global optimization algorithms in the presence of model errors. We stress once more that our global optimization is based on the simpler coated sphere model, while the more complicated model was only used for a few simulations.

Fig. 8

Slice from the confocal image (left) and (improved) optical model (right) of a lymphocyte.

3.3.

Characterization of Mononuclear Cells from Light Scattering

We measured leucocytes of seven donors, identifying T- and B-cells using the procedures described in Sec. 2.1. The global optimization algorithm described in Secs. 2.4, 2.5 was applied to each cell. Results of this procedure for four typical cells from a sample of T-cells of one donor are shown in Fig. 9, together with the confidence level $\alpha$ (see Sec. 2.5). Characterization of the sample is performed according to Sec. 2.6. The ratio of LSPs that passed the confidence level threshold varied from 10 to 20% from sample to sample. The results of characterization are presented in Table 1 (shown sample size is after applying the threshold). We characterized samples of B-cells only for two donors because of the small amount of B-lymphocytes above the threshold for the other donors. Results for these two donors comparing B- and T-lymphocytes are shown in Table 2 .

Fig. 9

Results of global optimization for four experimental LSPs of T-lymphocytes from donor 6, showing experimental and best-fit weighted LSPs. The graphs correspond to a different reliability level of the fit $\alpha$ , decreasing from (a) to (d). Estimates of the cell characteristics (expectation $\text{value}\pm \text{standard}$ deviation) are also shown.

Table 1

Results of characterization of T-lymphocytes of seven donors. ⟨.⟩, σ(.) , and σ(⟨.⟩) denote estimated mean, standard deviation, and standard error of mean, respectively.

Table 2

Comparison of model characteristics of T- and B-lymphocytes of two donors. ⟨.⟩, σ(.) , and σ(⟨.⟩) denote estimated mean, standard deviation, and standard error of mean, respectively. Results for T-lymphocytes duplicate corresponding columns of Table 1.

We tested the effect of applying the confidence level threshold by processing all data without it. For illustration, sample parameters related to cell diameter are shown in Table 3 . Comparing it with Tables 1, 2, one can see that for all samples the standard deviation is about 1.5 times larger and the standard error of mean is about two times smaller. This can be explained by the acceptance of values close to the parameter space boundary and larger sample size, respectively. Moreover, the mean value is significantly shifted (especially for T-lymphocytes of donors 5 and 7), showing larger variability between different donors. However, as discussed in Sec. 2.6, these results are systematically biased. Therefore, we consider them less reliable and stick to results obtained with confidence level thresholds in further discussions.

Table 3

Results of characterization of all lymphocyte samples using no confidence level threshold. Only parameters related to cell diameter are shown. These results are expected to be significantly biased by parameter space boundaries.

The mean values of lymphocyte diameter reported in the literature for normal donors as determined by the Coulter principle were $6.7\mu \mathrm{m}$ ⁴¹ and $7.3\mu \mathrm{m}$ ,^{42, 43} by electron microscopy were $6.5\mu \mathrm{m}$ ⁴⁴ and $5.4\mu \mathrm{m}$ ,⁴⁵ and by optical microscopy a range from $7.68\text{to}7.88\mu \mathrm{m}$ was obtained (for six donors, samples contained 4 to 8% monocytes),⁴⁶ from $6\text{to}8.3\mu \mathrm{m}$ (for 12 donors),³² and from $6.6\text{to}8.2\mu \mathrm{m}$ (for six donors).³³ Standard deviations of $D_c$ are $0.6\mu \mathrm{m}$ ,⁴¹ from $0.4\text{to}1.2\mu \mathrm{m}$ (12 donors),³² and from $0.4\text{to}0.8\mu \mathrm{m}$ (six donors),³³ showing no significant difference between different methods. Mean diameter ratios $D_n∕D_c$ as determined by electron microscopy were 0.78⁴⁴ and 0.56,⁴⁵ and as determined by optical microscopy varied from 0.82 to 0.90 (three donors).³² Data specific to T-lymphocytes are also available for two donors.³² $D_c=7.8\pm 0.7$ and $8.3\pm 0.8\mu \mathrm{m}$ ( $\text{mean}\pm \text{standard}$ deviation), and $⟨D_n∕D_c⟩=0.80$ and 0.85.

To summarize, there is disagreement between different methods, and our results for cell and nucleus diameter fall within the broad range of literature data. However, our data there are unusually small variations in mean size of T-cells and their nucleus from donor to donor (even accounting for error of mean). This fact disagrees with recent microscopic measurements,^{32, 33} and currently we cannot explain this. There seems to be no reason for our processing algorithm to decrease the natural variability of mean T-cell sizes between the donors, e.g., by somehow concentrating it to $6.4\mu \mathrm{m}$ . This is also supported by the fact that mean sizes of B-cells are significantly different. Therefore, we can only hypothesize that either donors studied in this work are more uniform (in age, health condition, etc.) than in previous studies, or the microscopic method together with corresponding sample preparation is susceptible to some measurement errors that increases the visible variability between the donors. For instance, the authors of the optical microscopy study of several donors^{32, 33} discussed the nonsphericity of the cells, but did not specify how exactly the diameter was determined. We plan to perform a detailed microscopy study in combination with our method in the future to resolve these issues.

As for the refractive index, we are aware of only one study where the cytoplasmic refractive index of human lymphocytes was measured.⁷ The index-matching method was used leading to a mean value of $m_c$ equal to $1.3572\pm 0.0002$ (standard error of mean) for several normal donors, which is significantly lower than values obtained in the current study. This is because our method determines the integral (effective) refractive index of the whole cytoplasm, including all granules and other inclusions, while index matching yields the value for the ground substance of the cytoplasm.

The fact that we use only a small part of LSPs measured with the SFC for sample characterization is due to the small confidence levels of the majority of the experimental LSPs, or, (approximately) equivalently, large standard deviations of model parameter estimates. As we discussed in Sec. 2.5, this is mostly due to model errors, i.e., the fact that real lymphocytes are not perfect coated spheres. To better understand these errors, we performed a limited set of DDA simulations. First, we tried to shift the nucleus inside the coated sphere model, but that showed no considerable effect on the structure of the LSP, except for scattering angles larger than $40\mathrm{deg}$ . Such LSPs are successfully processed, with the global optimization resulting in a confidence level above 0.95 and accurate estimation of the model parameters (data not shown). This is partly due to the large nucleus diameter used $(D_n∕D_c=0.9)$ , which is why it may seem to contradict earlier simulations using smaller nucleus diameters, e.g., Ref. 47.

Second, we tested the enhanced model including nucleus inhomogeneity (see Sec. 3.2 and right part of Fig. 8). The estimated mean parameters of the T-cells of donor 6 were used in simulation of light scattering for the cell model (see corresponding column of Table 1): $D_c=6.38\mu \mathrm{m}$ , $D_n∕D_c=0.903$ , $m_c=1.3765$ , and average $m_n=1.4476$ . Since now the nucleus consists of two domains, we chose for the inner spheroid the same refractive index as the cytoplasm (to maximize the contrast) and a larger refractive index (1.457) for the rest of the nucleus. Then the average (effective medium) refractive index of the nucleus equals the one specified before. The spheroid has axis ratios 1:2:2, its volume is 0.108 of the whole nuclear volume, its center coincides with the center of the cell, and the orientation of the symmetry axis with a respect to the light incidence direction ${\theta }_0$ was varied from $0\text{to}90\mathrm{deg}$ with a step of $30\mathrm{deg}$ . Results of processing the simulated LSPs with the global optimization are shown in Fig. 10 .

Fig. 10

Results of global optimization for four simulated LSPs corresponding to the enhanced optical model of a lymphocyte. Each graph shows simulated and best-fit weighted LSPs, and corresponds to different orientation of the oblate spheroid inside the nucleus. Estimates of the cell characteristics (expectation $\text{value}\pm \text{standard}$ deviation) are also shown.

One can see that the nuclear inhomogeneity (with volume fraction of only 10%) significantly distorts the LSPs, resulting in large errors of parameter estimates. Therefore, the proposed nuclear inhomogeneity may be a significant part of optical model errors for lymphocytes. But, of course, more simulations are required to make any definite conclusions. On the other hand, the global optimization performs reliably on these distorted LSPs, i.e., the real values of the initial model are well inside the confidence intervals (although the latter may be large). From the obtained confidence levels, we can suggest that using the threshold ${\alpha }_0=0.8$ leaves only cells with a relatively homogeneous nucleus (but possibly shifted from the center of the cell). We may suggest that these cells are in G1-phase of a cell cycle.⁴⁸

3.4.

Comparison of Cellular Characteristics for T- and B-Cells

Comparing the results of the global optimization for characterization of T- and B-lymphocytes (Table 2), one can see that the main difference in T- and B-cell morphology is the mean cell size $⟨D_c⟩$ . Although the difference in $⟨D_c⟩$ of about 5% is statistically significant, it is smaller than the estimated biological variability $\sigma \left(D_c\right)$ inside each sample. Hence, at present we cannot reliably classify a single lymphocyte as being T- or B-, using only its morphological characteristics determined from LSP. However, one may try to estimate the relative count of T- to B-lymphocytes in a sample, processing the distribution of all lymphocytes over morphological characteristics as a bimodal distribution (e.g., a sum of two multidimensional normal distributions). A similar method was proposed by Terstappen ⁴⁹ using the distribution of lymphocytes over the side scattering signal. However, further research is required to assess the accuracy of such relative count estimates.

Another way of looking at differences between T- and B-lymphocytes is studying average LSPs. First, we averaged 100 experimental LSPs from the samples of T- and B-lymphocytes for donor 6, resulting in Fig. 11a . There is a difference in intensity for scattering angles from $15\text{to}35\mathrm{deg}$ . However, this difference is much smaller than the natural variability of LSPs inside each sample (see Fig. 7). Second, we simulated LSPs of lymphocytes specified by estimates of the mean values of model characteristics, resulting in Fig. 11b. The structure of these LSPs in the angular range from $20\text{to}27\mathrm{deg}$ is different from the rest of the interval, because it contains almost no oscillations. Moreover, in this range, the LSP of the average T-lymphocyte has smaller intensity than that of the average B-lymphocyte, which agrees with Fig. 11a. This is explainable, since oscillations smooth out during averaging over the sample, while differences in monotonous parts of the LSP are more likely to persist. It will be interesting to identify which morphological characteristics are responsible for this angular interval in the LSP. But currently we can only conclude that the difference in mean diameter at least partly explains the difference in averaged LSPs.

Fig. 11

The LSPs of T- and B-lymphocytes for donor 6 (in logarithmic scale): (a) experimental averaged over a sample; and (b) simulated using the estimates of the mean values of model characteristics over sample.

We also note that B-cells are generally better fitted by the coated sphere model than T-cells. An example is presented in Fig. 12, showing that standard deviations of estimated $D_c$ for donor 6 are on average smaller for B-lymphocytes. Results for other characteristics and donor 1 are similar (data not shown). We hypothesize that this fact can be caused by smaller deviation of the shape of B-lymphocytes from the coated sphere, in particular a relatively smaller inhomogeneity of the nucleus.

Fig. 12

Distribution of standard deviations of estimated cell diameter for samples of T- and B-lymphocytes of donor 6.