Tri-exponential syllectogram representation

The conventional mathematical representation of the syllectogram contains two time-constants and describes the curve from the peak onward. (References 6 Bauersachs R.M., Wenby R.B., Meiselman H.J., Determination of specific red blood cell aggregation indices via an automated system, Clin. Hemorheol., vol. 9, pp. 1-25, 1989., 23 Firsov N.N., Bjelle A., Korotaeva T.V., Priezzhev A.V., Ryaboshapka O.M., Clinical application of the measurement of spontaneous erythrocyte aggregation and disaggregation. A pilot study, Clin. Hemorheol. Microcirc., vol. 18, pp. 87-97, 1998., 31 Hardeman M.R., Goedhart P.T., Dobbe J.G.G., Lettinga K.P., Laser-assisted Optical Rotational Cell Analyser (LORCA); A new instrument for measurement of various structural hemorheological parameters, Clin. Hemorheol., vol. 14:(4), pp. 605-618, 1994.) The tri-exponential representation (References 15 Dobbe J.G.G., Engineering developments in hemorheology, PhD Thesis, University of Amsterdam, Sept. 2002., 16 Dobbe J.G.G., Streekstra G.J., Strackee J., Rutten M.C.M., Stijnen J.M.A., Grimbergen C.A., Syllectometry: Effect of aggregometer geometry in the assessment of red blood-cell shape recovery and aggregation, IEEE-Transactions on Biomedical Engineering, vol. 50:(1), pp. 97-106, 2003.) introduces a third time-constant to include the upstroke caused by RBC shape recovery. (Reference 5 Baskurt O.K., Meiselman H.J., Determination of red blood cell shape recovery time constant in a couette system by the analysis of light reflectance and ektacytometry, Biorheology, vol. 33:(6), pp. 489-503, 1996.) Thus, the intensity curve I(t) was fitted using a tri-exponential function containing three time-constants associated with RBC-shape recovery (T_r), rouleaux formation (T_f) and 3D aggregate formation (T_s):

(Equation 4)

where I_r, I_f, and Is denote the contribution of shape recovery, (fast) rouleaux formation and (slow) 3D aggregate formation, respectively.

The curve fit is performed using the Levenberg-Marquardt algorithm (Reference 46 Press W.H., Teukolsky A.A., Vetterling W.T., Flannery B.P., Levenberg-Marquardt method, in: Numercal recipes in C, The art of scientific computing, Cambridge University Press, 2nd ed., pp. 683-688, 1992 (ISBN: 0-521-43108-5).) for fitting non-linear functions. Since the syllectogram is sampled uniformly most data points stem from the tail of the curve. To prevent the fitting algorithm from focusing on this region, the curves were re-sampled by selecting 300 points uniformly distributed on a logarithmic time scale.

Several aggregation parameters are derived from the syllectogram as indicated in Figure 6. To be able to distinguish “fitted” intensity parameters from “conventional” parameters (not resulting from the curve fit), the latter are give the symbol Isc.

The amplitude (Amp=Isc_top – Isc₀) is used to describe the extent of aggregation. (References 6 Bauersachs R.M., Wenby R.B., Meiselman H.J., Determination of specific red blood cell aggregation indices via an automated system, Clin. Hemorheol., vol. 9, pp. 1-25, 1989. ,31 Hardeman M.R., Goedhart P.T., Dobbe J.G.G., Lettinga K.P., Laser-assisted Optical Rotational Cell Analyser (LORCA); A new instrument for measurement of various structural hemorheological parameters, Clin. Hemorheol., vol. 14:(4), pp. 605-618, 1994.) Aggregation kinetics are described by the time-constants T_f and T_s but also by t½. The latter is the time that elapses until the peak intensity is reduced by half the amplitude (to Isc½). The overall aggregation behaviour of the suspension in described by a single parameter: the aggregation index. This aggregation index (AI) is a value between 0 and 100% and depends on both the extent and kinetics of aggregation. (References 6 Bauersachs R.M., Wenby R.B., Meiselman H.J., Determination of specific red blood cell aggregation indices via an automated system, Clin. Hemorheol., vol. 9, pp. 1-25, 1989., 22 Donner M., Siadat M., Stoltz J.F., Erythrocyte aggregation: Approach by light scattering determination, Biorheology, vol. 25:(1/2), pp. 367-375, 1988., 31 Hardeman M.R., Goedhart P.T., Dobbe J.G.G., Lettinga K.P., Laser-assisted Optical Rotational Cell Analyser (LORCA); A new instrument for measurement of various structural hemorheological parameters, Clin. Hemorheol., vol. 14:(4), pp. 605-618, 1994., 57 Stoltz J.F., Donner M., Erythrocyte aggregation: Experimental approaches and clinical implications, Int. Angiol., vol. 6, pp. 193-201, 1987.) It is determined from the areas A and B bounded by t = t_top and t = t_{top + 10} as AI = 100% * A/(A+B) (see Figure 6).

The time elapsed until the occurrence of the peak is sometimes used as an indication of the RBC-shape recovery time. The peak occurs when the first derivative of equation 4 equals zero. The first derivative depends on all parameters of equation 4 including the aggregation parameters. The fact that t_top is influenced by the aggregation process makes it an unsuitable candidate for representing the RBC-shape recovery time.

The aggregation software allows the user to display the results of a fixed syllectogram model or to display the best performing one (auto-mode):