Classification of different erythrocyte cells by using bioimpedance surface acoustic wave and their sedimentation rates

The effective permittivity for the cells within the plasma can be calculated from the following eqution [10]:

Ak is

the axial polarization factor of the cell as function of its shape [10]. ε*s${{\varepsilon }^{*}}_{s}$ and ε*p${{\varepsilon }^{*}}_{p}$ are the equivalent permittivity of the plasma and the cell, respectively.

Surface acoustic wave (SAW) is an acoustic wave generated and received between two transducers called interdigital transducers (IDTs). The first IDT generates the wave to the receiving one on the other side passing through the medium between them.

Depending on the electrical characteristics of this medium between them, a corresponding variation is associated with this generated wave and received by the second IDT. This variation is called the wave's perturbation, which is classified as frequency (velocity) perturbation and amplitude perturbation [11]. The related equations for the time-frequency perturbation is written as [9]:

Where ΔvvandΔff$\frac{\Delta v}{v}\,\text{and}\frac{\Delta f}{f}$ are the velocity and frequency shifts, respectively, as compared to the input or the transmitted signal. ε_r is the reference solution (the plasma-RBCs). ω is the angular frequency, K² is the mechanical coupling. ε_p is the dielectric constant of the substrate and ε₀ is the dielectric constant of free space.

From this equation we can deduce that the response of this SAW sensor is a signal with shifted frequency, Δf, but this shift varies with time, since the concentration of the RBCs varies during the sedimentation.

To clarify the SAW response, during the sedimentation process two solutions will be formed in the top of the sedimentation tube, which are RBCs-plasma with subscript 1 and pure plasma solution with subscript 2 as shown in Fig.1.

Fig.1

As shown in Fig.1a, during the sedimentation process, RBCs will separate from the plasma and move down. So three layers will be formed, which are the plasma (top) and the RBCs-plasma (medium) with normal concentration and RBCs-plasma with higher concentration (bottom), respectively.

With the progress of time, the heights of these two liquids will vary. The time rates of changing for these heights are related to the sedimentation velocity, where higher velocity means high rates of these lengths or heights. The sedimentation velocity of the RBCs as function of their radii and the volume fractions is given by [13].

ρ_RBC and ρ_plas are the densities of the RBCs and plasma, respectively. d is the cell radius, and μ is the viscosity of the free plasma. g is the acceleration of gravity.

If we consider dividing the tube blood sample into three divisions as shown in Fig.1b, the conductivity σ(t) of the solution under the IDT can be calculated by [9]:

Also, the equation of the total dielectric constant of the solution under the IDT, ε(t) is given by [9]:

As applying equations 6 and 7 in equation 4, the frequency response will be deduced as function of the mechanical and the electrical characteristics of the RBCs.

So the mechanism of the SAW-bioimpedance is summarized in the following steps:

Step 1: Calculate v and l.

Step 2: Calculate σ(t) and ε(t) (Eqs. 6 and 7).

Step 3: Calculate Δvv$\frac{\Delta v}{v}$ as a function of the frequency and the calculated parameters in step 3.

Step 4: l will increase with increasing time.

Step 5: Repeat the steps again from step 1.

Fig.2 shows the response of the SAW with different volume fractions. The reference or the normal value of the volume fraction is between 40% and 50% and it will be the reference case to compare the cases of the RBCs.

Fig.2

Changing the volume has an influence on the sedimentation rate as was shown in Eqn.5, and consequently on the value of the variation length under the sensor and so the conductivity and the permittivity of the solution as in Eqs.6 and 7.

The variation of the volume affects directly the ESR of the RBCs with an inverse relation. This relation is shown clearly in Fig.3 with the same start value because the same volume fraction but with different radii.

Fig.3

Increasing the volumes of the RBCs accelerate the rate of the sedimentation that corresponds to the transfer length, l. Increasing the volume fraction changes the rate of the length, l from one time to another (see Fig.1).

This case is equivalent to the influence some diseases have on the RBCs.

From the modelling equations, different volume fractions means different time responses of the SAW sensor as shown in Fig.4.

Fig.4

Fig.4 shows the responses of the RBCs with four different volume fractions, which are 35%, 40%, 45% and 50%. All responses are shown as parallel lines, where the same volumes of the RBCs give different values of the frequency shifts, because of different volume fractions.

Microcytes and macrocytes are practical cases corresponding to the variation of the volumes of the RBCs that correspond to some cases like e.g. influence of alcohol.

Fig.5 shows the responses of the SAW for the microcytes and macrocytes as compared to the normal RBCs.

Fig.5

Oblate cells are red blood cells inherent to some pathological cases. Detecting this kind of cells depends on a different feature called polarization factor as function of the dimensions and the form of the cell [10, 12].

The simplest cell's form is the spherical cells so it is considered as the reference case for calculating the other electrical parameters as referred to the parameters of the spherical cells [9].

The estimated electrical values of the oblate cells (from [9]) will be applied to this model to study the response of these cells. Fig.6 shows the electrical response of the SAW corresponding to the electrical parameters of the oblate cells for three different volume fractions (0.3, 0.4, and 0.5) referring to the spherical cases with the same volume fractions.

Fig.6