2.1.

Sample Preparation

Blood was taken from a healthy donor by venipuncture and collected in a vacuum tube containing sodium citrate as anticoagulant ($9:1$ blood to sodium citrate). Within 1 h of the collection, platelet plasma was obtained by centrifugation at 900 g for 15 min at room temperature. The sample was 100-fold diluted in 0.9% saline, and 1 and 2 μm polystyrene microspheres (Molecular Probes, USA) were added for a scaling of the SFC optics. Part of this sample was immediately analyzed with the SFC measuring the LSPs of all particles in the sample, including platelets and other constituents of blood plasma. Another part was stimulated by the addition of 10 μM ADP, delayed for 1 min, and then analyzed with the SFC. It is important to note that the dilution greatly reduced platelet aggregation; therefore, stimulation by ADP resulted only in activation of platelets.

We plot all measured particles on a map of LSP integrated from 10 deg to 60 deg (see Sec. 2.2) versus LSP integrated from 10 deg to 11 deg (Fig. 1). This is analogous to the SSCxFSC map used in ordinary flow cytometers. The clusters corresponding to polystyrene 1 μm microspheres and theirs dimers are clearly visible; they define gates G1, G2, and G3. A characteristic area of blood platelets is also visible and define gate G4. After the addition of ADP, the area of platelets narrows in agreement with ordinary flow-cytometric experiments.¹⁵ However, we used the same gate G4 for identification of platelets after activation.

Fig. 1

The scatter plot of the integrated LSP in the angular range [10 deg, 60 deg] versus that in the range [10 deg, 11 deg] in log-log scale for (a) the native sample and (b) the sample treated by 10 μM ADP. Polystyrene microspheres of 1 and 2 μm are gated by G1 and G3, respectively, while dimers of 1 μm polystyrene microspheres are gated by G2. Events in G4 (selected for further processing) are mainly formed by blood platelets.

2.2.

Scanning Flow Cytometer

Technical features of the SFC and the operational function of the optical cell were previously described in detail elsewhere.²⁵ The actual SFC was fabricated by the Cytonova Ltd. company (Novosibirsk, Russia, http://cyto.kinetics.nsc.ru/). It is equipped with a 40 mW laser of 660 nm (LM-660-20-S) for generation of LSPs of individual particles. Another 15 mW laser of 488 nm (Uniphase 2214-12SLAB) is used for generation of the trigger signal. The measured LSP is as follows:

Fig. 2

Optical model of a platelet (oblate spheroid). Incident radiation propagates along the $z$-axis, $z^{\prime }$ is the symmetry axis of spheroid, $\mathrm{\Psi }$ is the angle between axes $z$ and $z^{\prime }$, and $\theta$ is the polar scattering angle.

2.3.

Light-Scattering Simulations

We use an oblate spheroid as an optical model of a blood platelet. With regard to LSP calculation this model has four parameters, which we chose to be an equi-volume sphere radius $r$, an aspect ratio $\epsilon$, a refractive index $n$, and an orientation angle $\mathrm{\Psi }$ (Fig. 2).

Theoretical LSPs for oblate spheroids were calculated using the discrete dipole approximation (DDA).²⁷ In particular, we used open-source code ADDA v.1.0.²⁸ The LSPs of platelets were simulated for $\theta$ ranging from 10 deg to 70 deg with a step of 1 deg. We used 12 dipoles per a wavelength of incident radiation or 12 dipoles per a thickness of the spheroid when it was smaller than the wavelength in order to more precisely describe the shape of the platelet. The integration over $\phi$ was performed from 0 deg to 360 deg with 32 steps. To estimate the accuracy of the DDA simulations we increased the number of dipoles per wavelength to 40 for six typical LSPs. Based on the proven convergence of the DDA results to the exact value with refining discretization,²⁹ we used these simulations as a reference against which to test the standard DDA simulations. The ratio of the corresponding error norm to the norm of the whole LSP, both defined by Eq. (2), was smaller than 0.7% for these six cases.

The T-matrix approach is faster but does not converge for large aspect ratio.³⁰ However, we also tested the T-matrix method with discrete sources,³¹ which, unlike the conventional T-matrix method, can handle particles with highly nonspherical geometries.³² In particular, we used null-field method with discrete sources (NFM-DS) package v.1.1 by Schmidt et al.³³ The agreement between the two methods was fine for some values of the model parameters, especially for relatively small platelets [Fig. 3(a)]. However, the accuracy of the NFM-DS deteriorated with increasing platelet volume. In the region of platelets with typical volume and large aspect ratio it showed unsatisfactory results [Fig. 3(b)], even with the number of multipoles (Nrank) two times larger than the value recommended in the supporting documentation supplied by the program. We contribute the difference between the two curves in Fig. 3(b) to the NFM-DS, because the estimated DDA errors are much smaller (see above). Moreover, the DDA was well tested against independent methods in this range of size and refractive index^34,35 and, in particular, for red blood cells,^35,36 while the T-matrix and similar methods are expected to suffer increasing numerical instability with increasing particle size.³⁷

Fig. 3

Weighted LSPs [Eqs (1) and (3)] for two platelet models calculated from the discrete-dipoles approximation and the null-field method with discrete sources for (a) small and (b) typically sized platelets with relatively large aspect ratio $\epsilon =6$.

3.1.

Database Construction

We calculated the database consisting of $N=500,000$ theoretical LSPs with parameters randomly distributed in the intervals shown in Table 1.

Table 1

Boundaries of the LSP database for oblate spheroids utilized in the solution of the inverse light-scattering problem.

Random sampling of the database elements from a space of model parameters has several features compared to a regular grid. Firstly, it automatically accounts for different sensitivity of LSP to model parameters. For example, consider an artificial example, when one model parameter is irrelevant (LSP is completely independent of it), but this is unknown a priori. Then a regular grid will lead to redundant multiple calculations of the same LSPs, while random sampling will be equivalent to manual exclusion of irrelevant parameter and random sampling from the space of remaining parameters. Secondly, size of the randomly sampled database can be easily increased by an arbitrary number of elements, which is especially convenient for adaptive construction of the database.³⁸ Finally, formulae for statistical analysis of characterization results (Sec. 3.3) are rather simple.

All simulations were run on the compute cluster of the supercomputing center of the Novosibirsk State University. Typical simulation time of single LSP for a platelet is 1 min on a single core of Intel X5355 processor (2.66 GHz).

3.2.

Global Optimization

In this and in the next section we develop a general method to solve the inverse light-scattering problem for nonspherical particles described by a few parameters. This method utilizes the ideas originally proposed in Ref. 21, which we adapt to use in discrete global optimization with a database of solutions of the direct light-scattering problem. Additionally, we analyze all related uncertainties.

The problem is transformed into the global minimization of the weighted sum of squares:

Equation (2) also defines the norm (distance) in the $k$-dimensional space of LSPs, which is further elaborated in Appendix.

We consider a general definition of a precalculated database, as a set $\{{\mathbf{\beta }}_i,{\mathbf{p}}_i,I({\mathbf{\beta }}_i)\}$, $i=1,\dots ,N$. Values $p_i$ are relative probabilities (or weights) of database elements, which are defined normalized by $\sum _{i=1}^N{p}_i=1$, and are determined by a structure of the database. For a particular case of random sampling (Sec. 3.1), $p_i=1/N$. Another important case follows from deterministic division of the initial region of particle parameters $\mathbf{B}$ (with the volume $V$) into nonoverlapping subregions $V_i$ with ${\mathbf{\beta }}_i\in V_i$, then $p_i=V_i/V$. For example, this case applies to the global-optimization method DiRect, thus this paper can be considered a generalization of the approach proposed in Ref. 21.

Once the database is calculated, processing of any experimental LSP is performed by comparison with all LSPs from the database, i.e., the inverse problem is solved by nearest-neighbor interpolation on the database, using distance defined by Eq. (2). In addition to approximately finding a global minimum of $S(\mathbf{\beta })$ with the best estimate ${\mathbf{\beta }}_0$, it provides an approximate description of the whole surface of $S(\mathbf{\beta })$ by a set of values $[S({\mathbf{\beta }}_i)]$.

3.3.

Errors of Parameters Estimates

Bayesian approach is used to calculate a probability density function $P(\mathbf{\beta })$ over parameter space for a given experimental LSP:

The function $P(\mathbf{\beta })$ provides a complete description of the information deduced from an experimental LSP. In particular, one can calculate a mathematical expectation of any quantity $f(\mathbf{\beta })$:

Since $S(\mathbf{\beta })$ is known in a set of points ${\mathbf{\beta }}_i$ with weights $p_i$, the integral can be approximated by a discrete sum, as a ratio of two sample means:

According to Eq. (6) we calculate the mathematical expectation $\mathbf{\mu }=\langle \mathbf{\beta }\rangle$ (generally different from ${\mathbf{\beta }}_0$) and the covariance matrix $\mathbf{C}=\langle (\mathbf{\beta }-\mathbf{\mu }){(\mathbf{\beta }-\mathbf{\mu })}^T\rangle$. One can also obtain highest-posterior density (HPD) confidence regions, defined as $\mathbf{R}(P_0)=\{\mathbf{\beta }|P(\mathbf{\beta })>P_0\}$. $P_0$ is a variable threshold, which is implicitly determined by the confidence level $\alpha$:

It is important to note that the accuracy of determination of $\mathbf{C}$, and hence of HPD confidence regions, largely depends on accuracy of $S({\mathbf{\beta }}_0)$. The latter can be thought of as an expected change of $S({\mathbf{\beta }}_0)$ if the whole database is randomly resampled. When database is large enough $S({\mathbf{\beta }}_0)$ is close to the result of exact fit $S({\mathbf{\beta }}_{\perp })$, which is determined solely by experimental noise for a particular processed LSP (see Appendix). In that case the accuracy of $S({\mathbf{\beta }}_0)$ is fine (see Sec. 5.1). However, when theoretical testing LSP is processed (see Sec. 4), $S({\mathbf{\beta }}_{\perp })=0$ and variance of $S({\mathbf{\beta }}_0)$ is large for any size of the database. Thus, the proposed method to estimate characterization uncertainties is inadequate for and cannot be tested with theoretical LSPs without noise.

5.1.

Verification of the Assumption about the Refractive Index

In order to estimate an adequacy of the database size $N$, we conducted the following virtual experiment. First, we performed global optimization with the complete database containing $N=5\times {10}^5$ LSPs, and obtained a set of $N$ values $[S({\mathbf{\beta }}_i)]$. Then, we calculated the quantity $s(Q)=\underset{i\le Q}{\min }S({\mathbf{\beta }}_i)$, which is equal to values of $S({\mathbf{\beta }}_0)$ obtained by global optimization with first $Q$ LSPs from the database. These quantities were then averaged over all experimental LSPs, and are shown in Fig. 6 varying $Q$ from ${10}^5$ to $5\times {10}^5$ with a step of ${10}^5$. Theoretical dependence of $\langle s(Q)\rangle$ on the database size $Q$ is given by (A7). This analysis allows us to fit the points in Fig. 6 by a theoretical curve and determine the asymptotic value $A=\underset{Q\to \infty }{\lim }\langle s(Q)\rangle =6.5\times 10^{-2}$, which is a measure of the experimental noise, i.e., deviation of experimental LSPs from theoretical model of oblate spheroid. The remaining deviation due to sparsity of the database [$s(Q)-A$] is only $0.8\times {10}^{-2}$ for the largest $Q$, which is much smaller than the level of the experimental noise. Hence, further increase of the database size should not significantly change the results. The same analysis applies to any part of the database with proportional scaling of $Q$ and $N$, because of the random sampling used for its construction.

Fig. 6

Dependence of the average over sample best-fit value $S({\mathbf{\beta }}_0)$ on the database size for experimental LSPs. The coefficients $A$ and $B$ are obtained by fitting with the theoretical curve (see Appendix) are also shown as $\text{values}\pm \text{standard errors}$.

We also examined the errors of Monte Carlo integration [Eq. (8)] that also depends on $N$. Using the complete database, the statistical errors for evaluation of $\mathbf{\mu }$ and $\mathbf{C}$ were not larger than 0.5%.

5.2.

Verification of the Assumption about the Refractive Index

In order to verify the validity of restricting the refractive index, we performed several tests, carrying out global optimization with the complete and truncated (i.e., with the refractive index restricted to the range [1.37, 1.39]) databases.

First, for each experimental LSP we determined whether the HPD 95%-confidence region (using the complete database, see Sec. 3.3), or more specifically its projection on the axis of refractive index, intercepts the range [1.37, 1.39]. This condition implies that the hypothesis of restricted refractive index is acceptable in the probabilistic sense with 95% probability. We found that the interception take place for the vast majority (93%) of platelets LSPs. For the remaining 7% of platelets the restriction of the refractive index also produces adequate results, so we did not exclude them from considerations.

Second, we examined a best-fit weighted sum of squares $S({\mathbf{\beta }}_0)$. Distributions of platelets LSPs over $S({\mathbf{\beta }}_0)$ are shown in Fig. 7 for both complete and truncated databases. The distributions are well-described by log-normal curves, and corresponding fit parameters are shown in each plot. One can see that the restriction of the refractive index only slightly deteriorates the agreement between the theoretical and experimental LSPs.

Fig. 7

Distributions over best-fit weighted sum of squares $S({\mathbf{\beta }}_0)$ for (a) the complete database and for (b) the database with refractive index restricted to the range [1.37, 1.39]. Results of the lognormal fit (mode, median, and mean values for log-normal distributions) are shown.

5.3.

Typical Results of Global Optimization

Projections of HPD confidence region obtained by global optimization with the complete and truncated databases for the typical experimental LSP are shown on Fig. 8. Regions on the upper panel follow the same pattern as the clouds of 50 nearest parameters in Sec. 4 (cf., right column of Fig. 4). The LSP itself is shown in Fig. 9 together with the best-fit LSPs from the complete and truncated databases. $S({\mathbf{\beta }}_0)$ is slightly larger for the truncated database, as expected. However, the corresponding best-fit LSP visually seems more appropriate, i.e., better describes the second maxima. This illustrates one of the promising future directions of research: replacing weighted sum of squares by a more physically based distance measure, e.g., based on explicit consideration of the positions of LSP extrema. Anyway, the main effect of restricting the refractive index, as shown in Fig. 9, is the decrease of the uncertainties of parameter estimates by a factor from 1.3 to 2 (while for some other LSPs uncertainties decreased 14-fold for aspect ratio and 5-fold for equi-volume sphere radius), which was our design goal. This effect is also demonstrated through constriction of confidence regions in Fig. 8.

Fig. 8

Projections of 95%-confidence region for a typical experimental LSP for (a) to (c) the complete and (d) to (f) truncated databases.

Fig. 9

Typical experimental LSP (dots) and best-fit LSPs (gray lines) obtained with (a) the complete database and (b) the database with the refractive index restricted to the range [1.37, 1.39]. Results of global optimization, including parameters of the nearest LSP from the database (NN), mathematical expectation (ME), and standard deviation (SD) of parameters are also shown.

The following results were obtained using the truncated database. In particular, Fig. 10 presents characterization results for four typical blood platelets that were chosen based on their $S({\mathbf{\beta }}_0)$, cf., Fig. 7(b). We emphasize the precision of determined $r$ (median uncertainty is 125 nm, while for 35% of platelets) less than 100 nm, which is very good for optical methods. This corresponds to a median precision of 0.9 fl for the platelet volume in the wide range of platelet aspect ratios. The precision of the aspect ratio is 0.62 (median) and $<0.3$ for 23% of platelets. Distributions of the platelet sample over uncertainties of parameters are approximately log-normal (data not shown). Large uncertainties between platelet variations of the parameter can be caused by several factors, including varying signal-to-noise ratio of the measured LSP and varying deviations of the real platelet shape from the oblate spheroid model.

Fig. 10

Typical experimental and best-fit theoretical LSPs as obtained by global optimization using the database with restricted refractive index. Parameters of the nearest LSP from the database (NN), mathematical expectation (ME), and standard deviation (SD) of parameters are shown in inset tables.

5.4.

Results of Characterization of Blood Platelet Population

In Fig. 11 we present obtained results of characterization (best-fit values) for resting platelets (left column) and platelets activated by 10 μM ADP (right column). In particular, maps of platelet volume versus aspect ratio are shown in Fig. 11(a) and 11(b). The area of platelets evidently moves to the region of the smaller aspect ratio. The corresponding distributions over the platelet volume [Fig. 11(c) and 11(d)] are in a qualitative agreement with the conventional curves measured by Coulter counters. Mean platelet volume (MPV) values of 10.0 and 9.4 fl for resting and activated platelets, respectively, fall within the reference range of 8 to 12 fl. MPV for activated platelets is slightly less, which can be supported by the phase-contrast microscopic measurements¹⁷ and centrifugal volumetric experiments,⁴⁴ although the statement that the platelet volume increases during activation is more generally accepted.² Platelet distribution width, defined as width at the level of 20% of maximal peak, is also within the reference range of 9 to 14 fl for both cases.

Fig. 11

Maps of blood platelet volume versus aspect ratio (best-fit values) for (a) native sample (1883 cells) and (b) after the addition of 10 μM ADP (2050 cells). Distributions over a volume and an aspect ratio for [(c) and (e)] resting and [(d) and (f)] activated platelets are shown, respectively.

Distributions over the aspect ratio (best-fit values) for both resting and activated platelets are shown in [Fig. 11(e) and 11(f)]. The first distribution shows a peak with a relatively large amount of platelets with aspect ratio near 2, although, according to the literature, such platelets are a minor part in the normal state.⁵ It can be explained by the fact that platelets are partly activated, which always occurs during blood collecting and preparation of a platelet-rich plasma. This is supported by the fact that after addition of 10 μM ADP almost all platelets transfer to that peak, leaving a little trace in the region of aspect ratio $>2$, corresponding to unactivated platelets or platelets which shape is still changing.