In impedance cytometry, various design and post processing techniques have been developed to improve measurement accuracy. One of the parameters that most afflicts cytometry measurements is the dependence of the signal on the position of the cell within the channel so that identical cells, passing in different positions, give rise to different signals.
The following report analyzes a planar electrode configuration and the dependence of the signals on the position within the microfluidic channel. A compensation technique is then analyzed and applied on a dataset deriving from measurements on families with three different electrical diameters.
A mapping according to the straight line interpolating the shape parameter and the normalized diameter allows you to remove the position contribution and separate the signals of families with different diameters, then go back to the single distributions eliminating the position contribution.
The following report is an extract from a project carried out for the course of Modeling and Simulation of Physiological Systems, held for Medical Engineering at the University of Rome Tor Vergata.
Introduction
Systems for microfluidic analysis implement technological solutions to count, trap, focus, separate and identify the properties of the cell. The properties of individual cells can be identified based on differences in size and dielectric properties, using non-invasive techniques and without requiring the aid of cellular labeling.
Among the different techniques, the impedance analysis in microfluidics stands out. Multiple pairs of electrodes, placed around the microfluidic channel, are used to apply a voltage at discrete frequencies and measure fluctuations in electric current.
These fluctuations are due to the passage of the cell inside the channel and therefore within the non-uniform electric field generated by the electrodes. These systems can be more or less accurate depending on the configuration used.
This report presents the analysis of a compensation technique to eliminate the dependence on the position in the channel that affects the measured signals and does not allow a correct estimate of the cell diameter.
This compensation technique exploits the dependence of the electrical diameter on the shape parameter, linked to the properties of the measured signal. Although the different families analyzed (different diameters) appear as indistinguishable signals, the mapping using this compensation procedure allows to produce signals where the different families are clearly distinguishable.
Background
The impedance cytometer is a microfluidic device used to measure the perturbations of the electric field within a microchannel through which a cell passes. The cell passes through a specific electrode pattern on which an alternating voltage is applied and this leads to a variation of the measured current.
In other words, we consider a microchannel filled with a conductive buffer inside which electric currents pass. In the device in question, a potential is applied to the central electrode and a differential current is measured between the two side electrodes.
Low frequencies are used to determine cell size properties as the signal is typically proportional to cell volume. High frequencies, on the other hand, are used to obtain information on the conductivity of the cell membrane.
The following report considers a coplanar electrode configuration for cytometry measurements represented in fig. 1.
Through the measurement of differential current it is possible to estimate some properties of the cell passing through the channel. In particular, as the cell passes, a signal with a bipolar Gaussian waveform is measured.
Through the peak amplitude it is possible to estimate the electrical diameter. The signal is proportional to the volume of the cell so the diameter will be related to the amplitude of the signal (๐) as:
\begin{equation}
D=G a^{1 / 3}
\end{equation}Where ๐บ is a gain that is affected by the specific electrical properties of the device used.
Bipolar Gaussian
The bipolar Gaussian is a waveform characteristic of a cytometric signal composed of two Gaussians identified by the generic equation:
\begin{equation}
g(t)=a\left[e^{g_{+}(t)}-e^{g_{-}(t)}\right]
\end{equation}That is, considered a reference amplitude ๐ (i.e. the maximum peak value), the overall waveform is given by the sum of two Gaussians over time, the second of which overturned. The peak-to-peak distance is equal to ๐ฟ and a centering parameter ๐ก๐ is introduced. The two Gaussians share the same standard deviation ๐ and are identified by eqs. (3) and eq. (4).
\begin{equation}
g_{+}(t)=\frac{-\left(t-\left(t_{c}+(\delta / 2)\right)\right)^{2}}{2 \sigma^{2}}
\end{equation}\begin{equation}
g_{-}(t)=\frac{-\left(t-\left(t_{c}-(\delta / 2)\right)\right)^{2}}{2 \sigma^{2}}
\end{equation}Fitting procedure
Starting from the experimental data it is necessary to introduce a numerical fitting procedure to identify the Gaussian, and therefore its four descriptive parameters, such as to represent the analyzed signal [1].
This fitting procedure is implemented according to an optimization algorithm. That is, we try to reduce the difference between the measured data [๐] ๐ and the fitting template ๐ (๐ก๐) at the same instant in time. Defined the error function as that difference:
\begin{equation}
\underline{e}=[d]_{i}-g_{i}\left(t_{i}, a, t_{c}, \delta, \sigma\right)
\end{equation}We try to minimize the objective function defined just like the error norm:
\begin{equation}
\mathrm{E}\left(a, t_{c}, \delta, \sigma\right)=\frac{1}{2} \sum_{i}\left\|d_{i}-g\left(t_{i}, a, t_{c}, \delta, \sigma\right)\right\|^{2}
\end{equation}Accuracy in cytometry
Microfluidic devices can be more or less accurate. This is related to the electrode configuration, geometry, suspension medium and flow velocity [2].
In particular, for the reference case, a dependence of the signal on the position within the channel is observed. Although the width of the Gaussian peak is correlated with the cell diameter, a dependence on the position is also observed.
That is, with the same diameter, a cell passing near the electrodes will give a peak of greater amplitude to the same cell passing at a greater distance [3]. Although in the configuration used it is not possible to have a direct measurement of the position within the channel, it is possible to obtain an estimate by making a signal compensation taking into account the correlation between the position and the shape parameters.
In particular, the relationship between the shape parameter ๐๐๐๐๐ and the normalized diameter that follows a linear law is valid:
\begin{equation}
\frac{D}{d}=c_{1}+c_{2}\left(\frac{\sigma}{\delta}\right)
\end{equation}Where:
\begin{equation}
\frac{\sigma}{\delta}=\text { shape }
\end{equation}You can then use these straight line coefficients, calculated for each signal, to correct the electrical diameter by removing the position effect [4].
Dataset
The reference dataset is a set of raw data obtained from impedance cytometry measurements on a sample of test beads.
There are cytometric signals from three families of test beads with nominal diameters of 5.2, 6 and 7๐๐. For the device in question the gain is equal to 10.5 ๐m / A 1/3 and the inter-electrode distance is equal to ๐ฟ = 40 ๐๐. The signals are sampled with a frequency ๐๐ = 115 kHz with a total of over 50,000 signals.
Results
Initially the procedure is applied on a restricted dataset considering only 400 cytometry signals. The index signals from 400 to 800 are then selected. Subsequently, the procedure will also be applied to a more extended reference.
Numerical data fit
Starting from the single signal within the entire dataset, it is possible to fit the bipolar Gaussian by estimating the four descriptive coefficients.
Considering the template in eq. (2) You can use the Matlab fit() command providing the template itself, the data and the initial values โโfor the least squares estimation. Furthermore, since the signal is very small โ 10โ6, it is useful to normalize the amplitude signal and scale the time axis bringing it back in seconds in the range [0; 1 / ๐๐ ] with an equivalent number of samples, where ๐๐ is the sampling frequency whose inverse indicates the last time sample.
This must be followed by a rescaling in the original domain in order to compare the signals and carry out the compensation procedure correctly. An example of fitting is shown in fig. 3 and it is evident that the bipolar Gaussian is descriptive of the measured signal.
| ID | Standard deviation | Mean | Coefficient of variation | Nomina diameter |
|---|---|---|---|---|
| # 1 | 0.187 | 5.196 | 0.035 | 5.2 um |
| # 2 | 0.184 | 6.016 | 0.031 | 6 um |
| # 3 | 0.205 | 7.005 | 0.029 | 7 um |
Compensation
For the compensation procedure it is necessary to estimate the coefficients of the straight line resulting from the linear fitting. This fitting must be done in the space [๐ฟ / ๐; ๐ท / d] and it is therefore necessary to normalize the electrical diameters with respect to the individual nominal diameters of the three families.
But in this dataset it is not possible to identify, with an automatic procedure, the family to which the measurement belongs, it is possible to exploit the scatter plot of the electrical diameter vs shape parameters to visually distinguish three families (fig. 6a). Then, using the inpolygon() function, it is possible to manually select the values โโcorresponding to the three families, separate them and then normalize them for the respective nominal diameter.
It is therefore possible to calculate the line that best approximates the shape parameter trend as a function of the normalized electrical diameter (fig. 4). Therefore it is possible to extract the coefficients:
- ๐๐ท = 2.59
- ๐๐ธ = โ6.20
Within the Matlab environment it is possible to use the fitlm() command so as to directly obtain the two coefficients of the line and also an estimate of the determination coefficient ๐2 = 0.96. Through these coefficients it is possible to correct the values through the eq. (7). The separated families are then obtained both in the scatter plot of the shape parameter and of the velocity, in fig. 6 and fig. 7.
Distributions
Clearly this procedure, by separating the families, allows the calculation of their distribution. By analyzing the distribution of signals, before compensation, it is evident that no peak can be identified in the histogram graph in fig. 5a. This graph is fully representative of how the signals are actually mixed due to the position contribution.
After the compensation, however, it is clearly possible to distinguish the different families as is evident from the three clearly distinguishable peaks in fig. 5. The distributions of the between families are then reported, each with a Gaussian trend in which parameters are present in Table 1. It is evident that the majority of the marbles fall into the third family, with a larger diameter, but this value also reflects a greater dispersivity . The two families with the smallest back, on the other hand, present a comparable dispersion.
Extension of the domain of interest
The same procedure can also be applied to a considerably greater quantity of cytometric signals. Therefore, in the same reference dataset, the cytometric index signals at 200 to 35200 are considered.
The procedure can be applied in the same way but the scatter plots are replaced by density plots that allow to better observe the distribution of the values (fig. 8). Again we obtain the two fitting parameters ๐1 = 2.63, ๐2 = โ6.28 through which to apply the compensation procedure.
In a similar way, the histogram follows where the three families are separated (fig. 8d). In this case, the majority of the marbles still fall into the family with a larger diameter, but the family with a smaller diameter has a greater dispersion.
Conclusions
The compensation procedure allows to separate the different families in the cytometry signal with respect to the electrical diameter and is reliable both on a few signals and on larger samples.
This procedure requires you to know in advance the diameters of the spheres used in cytometry analysis. In the event that the individual cytometry signals cannot be associated with the relative nominal diameter with certainty, a manual identification procedure is required which, by its nature, will induce a certain error in the classification and linear fitting, therefore on the compensation procedure.
This error is mainly reflected in a dispersion in the distributions of the marbles. The standard deviation of the family distributions remains below 0.23 and macroscopically the three families and their distributions remain highly distinguishable.
Clearly, due to the numerical error present and the large amount of outliers in the cytometric signal, it is not as easy to classify the single signal as to classify the macroscopic distribution.
Code availability
The code is available at the project’s GitHub repository.
References
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- [2] Tao Sun and Hywel Morgan. โSingle-cell microfluidic impedance cytometry: a reviewโ. en. In: Microfluidics and Nanofluidics 8.4 (Apr. 2010), pp. 423โ443. ISSN: 1613-4982, 1613-4990. DOI: 10 . 1007 / s10404 – 010 0580 – 9.
- [3] Daniel Spencer et al. โHigh accuracy particle analysis using sheathless microfluidic impedance cytometryโ. en. In: Lab on a Chip 16.13 (2016), pp. 2467โ2473. ISSN: 1473-0197, 1473-0189. DOI: 10 . 1039 / C6LC00339G.
- [4] Vito Errico et al. โMitigating positional dependence in coplanar electrode Coulter-type microfluidic devicesโ. en. In: Sensors and Actuators B: Chemical 247 (Aug. 2017), pp. 580โ586. ISSN: 09254005. DOI: 10 . 1016 / j . snb . 2017 . 03 . 035.
