Estimation of the concentration of particles in suspension based on envelope statistics of ultrasound backscattering
Introduction
Due to its innocuous, noninvasive, fast and low-cost nature, ultrasound techniques are an attractive tool for the analysis of the internal structure of materials, tissues and liquids, especially for those applications that require no modification of the properties of the propagation medium, such as non-destructive testing in industry or in vivo clinical applications. Under certain conditions, ultrasound can be used to obtain quantitative information of the medium under study, like the sound speed [1], the attenuation coefficient [2], [3], the size and/or concentration of particles in suspensions [4], [5], [6] or the microstructural parameters of inhomogeneous media [7], among others. Nevertheless, the medium complexity and the presence of unknown variables make the determination of quantitative parameters a great challenge.
This work proposes the development of a methodology for evaluating the concentration of scatterers like particles or cells in suspension using ultrasound based on combined ultrasound imaging and quantitative analysis. Much work has been developed in this area and many strategies have been used so far to achieve this concentration estimation. The interest of assessing the concentration of a given number of scatterers goes beyond the medical applications, as it is the case of the automatic and non-invasive evaluation of biomass in marine ecology or in fish farming industry [8], [9], or the characterization of suspensions made of nano-particles (nanofluids) [10], and micro-particles.
Ultrasound applications in the clinical setting are mainly based on the qualitative analysis of the contrast image formed as a result of the impedance changes between tissues. In the last decades and in the line pointed by the Doppler technique, which is able to extract quantitative information of blood velocity and flow [11], a remarkable research effort was directed towards the determination of quantitative metrics from tissues to inform about the health condition of patients. Some examples include the analysis of the mineral status of bones as a biomarker of osteoporosis [12], the analysis of soft tissues to detect tumor proliferation [13], fibrosis [14], etc.
In this clinical context, there is a growing interest in the development of in vivo noninvasive methods to detect and evaluate the cellularity in biological tissues or fluids. Anomalies in cellular concentration can be related to multiple infectious diseases, cancerous processes and other health problems [15]. In fact, the cell concentration measurement in body fluids is often a diagnostic marker of the health status of a patient. At present, cell counting is performed ex vivo, requiring invasive methods for collecting the sample, such as blood extraction, lumbar puncture and biopsy. This is a complex and time-consuming procedure that requires trained personnel and involves a potential risk of injuries and infections [16], [17]. At this juncture, ultrasounds may play an important role for the development of in vivo cell counting techniques.
This work is focused on the concentration assessment of cells, in particular, for the characterization of serous body fluids like the cerebrospinal fluid, the pericardial fluid, the lacrimal fluid, etc. where a wide range of cell concentrations can be found depending, among other factors, on the health status. Mustonen et al. (1998) [18] assessed the cell populations in healthy human corneas using a confocal microscopy, obtaining density ranges above 800 cells/µl for all cell types, reaching 5000 cells/µl in the case of basal epithelial cell. Normal concentration of white blood cells in the cerebrospinal fluid (CSF), a typically pluricellular space, varies depending of the age and whether an infection or other pathological process has occurred. For a healthy child, Kestenbaum et al. (2010) [19] proposed mean reference values of cerebrospinal fluid white blood cells of 3 cells/µl and 2 cells/µl for infants 0–28 and 29–56 days of age, respectively. Martín-Ancel et al. [20] determined a maximum value of 5 white blood cell/µl for neonates without infections of the central nervous system. This values are within an accepted normal range of up to 20 cells/µl in neonates [21]. Detecting concentrations above these values is generally associated to the occurrence of inflammatory and infectious diseases, which typically increase the cellularity, and high concentrations can be reached in advanced stages; as in the case of untreated bacterial meningitis which is characterized by a cellularity of 1000–5000 cells/µl or more [22]. On this basis, a concentration range for analysis between 5 and 3000 particles/µl was established for the simulations and experiments presented in this work, which corresponded to 0,003–1,8 cells/resolution cell for the ultrasound system used, being the resolution cell defined as the particle echo size at 3 dB below the maximum amplitude.
Techniques developed for counting the number of scatterers can encompass different principles. One of the most common methods relies on the estimation of the backscattering coefficient or, in general, on the energy level of signal backscattered by the given suspension. The method has proven effective in measuring the suspended sediment concentration in the ocean [23], evaluating the concentration of yeasts in suspension [24] or quantifying leukocytes [25] as well as erythrocytes [26], [27] in suspension in vitro. To get a concentration estimation, this technique requires the knowledge of the energy reflected by a single scatterer, either through theoretical considerations or by calibration, and the energy reaching the suspension.
There are other kinds of techniques used to obtain an absolute scatterer concentration when the energy reaching the suspension is unknown, which are based on counting the number of echoes coming from individual particles in a given volume [28], [29]. This strategy, which was directed towards clinical applications, has proved to be efficient for low scatterer concentrations. However, as the concentration increases and the echoes coming from the scatterers overlap, the method becomes no longer valid. Mentioned references [28], [29] reported application limits around 100 particles/µl for pulse central frequencies in the range 20–75 MHz. Specially for these low concentrations of cells, the central frequency of the ultrasound pulse is critical for in-vivo applications and might be carefully chosen according to the tissues involved: if the frequency is too high, the pulse would be attenuated by tissues placed between the ultrasound probe and the region of interest. On the contrary, if the frequency is too low, the pulse backscattered by cells would be too weak, as the backscattering coefficient increases with the forth power of frequency.
The target of the study is to extend the use of ultrasound techniques to evaluate the concentration of cells when the energy reaching them is unknown and the methods based on individual echo counting fail due to echo overlapping. The former is especially relevant because the variability of patients (specific anatomy, age, health status, etc.), the different gain setting used and the coupling conditions between the transducer and the skin may result in the fact that the ultrasound energy which reaches a region of interest is unknown.
To this end, the analysis of the statistical distribution of the envelope signal [30] was considered. This statistical distribution might provide information about the characteristics of the signal and, by extension, its propagation medium. Based on the fact that the echo signal is formed by a superposition of waves resulting from multiple reflections generated from the scatterers in the medium, the envelope of this signal can be modeled as the result of a statistical distribution of the scattering points. Several models have been proposed for the distribution of the amplitudes of the echo envelope such as Rayleigh, Rice, K distribution and Homodyned-K distribution, among others. This kind of analysis has proved highly valuable in the study of physical properties of tissues in a clinical setting, being used in the valuation of the pathological state of different cancers such as breast cancer [31], [32] and thyroid cancer [33], detection of metastases lesions [34], diagnosis of liver fibrosis [35], [36] or tissue characterization in echocardiographic imaging. From a more general point of view, statistical analysis showed to be useful for modeling and interpreting the signal in remote sensing technologies, as radar or sonar [37], [38], [39], [40], [41]. The characterization of the amplitude distribution of echoes was effective to discriminate target and non-target echoes, detect obstructions or heterogeneities in the propagation medium and identify and remove noise from the signal. Furthermore, this method already proved its potential to evaluate the density of echoes beyond sonography in applications intended to quantify marine animals using sonar [42], [43].
In this work, a scattering point size of the order of 10 µm (size order of leukocytes) is assumed for the experimental verification. In addition, signal frequency, concentrations and particle material used in the experimental testing and other settings concerning materials and methods were chosen for their relevance in clinical applications. However, the algorithms and estimation methodologies presented are general, and may be applied to other situations where a scatterer concentration assessment may be needed.
The present paper is structured as follows. First, in the “Theory” section, the aforementioned statistical distribution models are presented in order to review the theoretical context underlying the technique proposed for estimating concentration. The “Materials and method” section describes the simulation and experimental procedures used to test the methodology. In the “Results” section, the sensitivity of different distributions and parameters to the scatterer concentration is analyzed with the help of ultrasound wave scattering simulations. In addition, these simulations are also used to study the influence of different variables like the presence of noise or the standard deviation of the scatterer sizes. Finally, the results from experimental tests are presented to analyze the performance of the algorithm proposed under a more realistic situation using polystyrene (PS) 7 and 12 µm particle suspensions in water.
Section snippets
Theory. Distributions to characterize scattering media
The echo signal returned from a medium containing a dispersion of cell or particles is a sum of reflections from numerous scatterers. According to the theory, there is a mathematical relation between the statistical distribution of the amplitude values constituting the envelope of this signal and the general physical properties of the suspension (statistical distribution of the scatterers, concentration, material characteristics, etc.) [44]. Therefore, finding a statistical distribution that
Simulations
The theoretical acoustic field radiated by the focused transducer used for the experiments was modeled [47] to evaluate the performance under controlled conditions of the envelope techniques previously described. Echo signals reflected by suspensions of particles dispersed in an aqueous saline solution (0,9% NaCl) were simulated using the Faran model [48]. To this end, random coordinates in all three dimensions were assigned to each particle for their spatial distribution, and the number of
Parameter assessment for different distribution models
The three parameters (, s and µ) characterizing the distributions described in Section 2 were estimated from the simulated images. The main objective of these tests was to assess how these parameters relate to the scatterer concentration. As explained, the calculation was performed using the first three even moments equation (6) presented in Section 2, fixing the parameters to the corresponding value of each distribution according to the Table 1.
Fig. 4 shows the results achieved for σ, s and µ
Conclusions
The performance of statistical envelope methods to characterize the concentration of particles in suspension from ultrasound images was studied in this work. First, statistical distributions often used to model backscattered ultrasound signals and the corresponding parameters involved were reviewed. Then, the calculation of these parameters was carried out using simulations of the field emitted/received by a transducer and the waves scattered by particle suspensions in order to better
Funding
This work was supported by the Instituto de Salud Carlos III, projects PI16/00738 and PI16/01822, and co-financed by FEDER resources, by the project PID2019-111392RB-I00 (Spanish Ministry of Science and Innovation) and by the contract GARJUV_CAM_18_00217 from the Youth Employment Program (Madrid Department of Education and Research), co-financed by the European Social Fund (ESF). ISGlobal also receives support from the Spanish Ministry of Science and Innovation through the “Centro de Excelencia
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Nanoparticle sizing by focused-beam dynamic ultrasound scattering method
2022, UltrasonicsCitation Excerpt :As a matter of fact, ultrasonic scattering techniques have been used on a wide range of solid, liquid, and gaseous materials, including particle suspensions [3,4], emulsions [5–7], and nanobubbles [8]. Recently, novel ultrasonic techniques have been proposed, such as an attempt to analyze dynamic viscoelasticity using cross-correlation analysis of diffuse shear fields [9] and a more robust method to evaluate concentration in suspension that can be used in clinical settings [10]. Although megahertz dynamic ultrasound technology has a longer wavelength than visible light, it has recently been used to analyze particles size as small as a few tens of nanometers.