Finite element analysis and detection of melanoma cells under dual-cell photoacoustic interference conditions

Circulating tumor cells (CTCs) are cancer cells that are shed from primary tumors within the body and circulate in the bloodstream. These cells have the potential to implant in other body regions, forming secondary tumors. The detection of CTCs plays a crucial role in the early diagnosis of cancer, prognosis assessment, and the formulation of personalized treatment plans. Methods for detecting CTCs include flow cytometry, PCR technology, optical imaging systems, size/density differences, and photoacoustic detection.¹ Intrinsic melanin in melanoma cells makes circulating melanoma cells (CMCs) particularly detectable through photoacoustic detection. Hemoglobin in red blood cells and melanin in melanoma cells serve as ideal endogenous contrast agents, enabling the rapid and accurate detection of cell shape and size by photoacoustic microscopy without the necessity for staining. Photoacoustic microscopy, an emerging optical imaging technology, boasts advantages such as high resolution, non-invasiveness, real-time capability, and multimodal imaging, and it has seen rapid development in recent years.² The principle of photoacoustic detection is based on the photoacoustic effect: when a pulsed laser is absorbed by the target object, it generates acoustic waves. Subsequently, these waves are detected and converted into electrical signals. Analyzing these signals yields the physical parameters of the target object.³ 

There is a close relationship between cell shape and function, and thus, measurements of cell and organelle morphology provide crucial information for understanding the cell function and disease diagnosis. In 2013, Eric M. and colleagues proposed a method that utilizes photoacoustic microscopy to enable quantitative analysis of morphological changes in individual red blood cells. The method is based on the observation of the photoacoustic signal characteristics of cells above 100 MHz. When red blood cells are different in size, shape, orientation, and composition, they show different periodic ups and downs in the photoacoustic spectra, and there are differences in the amplitude of the curves, their shapes, and the positioning of extremal points.⁴ Analyzing the frequency-domain sound pressure level curves gives detailed information about the size and morphology of red blood cells. In 2014, researchers developed and refined a photoacoustic finite element model of melanoma cells, enhancing the congruence between the model’s signal and that of the actual measured melanoma cells.⁵ In 2015, Eric M. Strohm's group combined simultaneous high-frequency ultrasound and photoacoustic measurements to identify and differentiate types of single cells in blood samples containing melanoma cells.⁶ In 2019, a team headed by Eric M. Strohm and associates devised a novel three-dimensional hydrodynamic flow focusing technique that combines ultrasonography, photoacoustic detection, and microfluidics.⁷ By 2022, the system was further advanced through the integration of various signal post-processing methods, facilitating high-throughput, label-free counting, and measurement at the single-cell level.^8–11 

However, the current method for determining the cell type through cell photoacoustic detection still involves detecting individual cells one by one. Achieving simultaneous detection of multiple cells would greatly improve the efficiency.^9–12 To address this problem, this paper investigates a method for detecting cell parameters under the interference of optical acoustic waves from two cells. According to the principle of acoustic wave interference, the distance of the sound source and the position of the detection point are adjusted in order to realize the simultaneous detection of multiple cells. Furthermore, the previous signal processing method was only applicable for analyzing single-cell signals without interference and significant curve differences. It failed to quantify the differences between similar curves and was not suitable for signal comparison and analysis under interference conditions.⁶ In order to realize the comparison and analysis of signals under interference, the previous eigenvalue method is improved to the similarity function comparison method.¹³ This paper aims to improve the detection and signal post-processing methods to determine the presence of melanoma cells in two cells under the mutual interference of photoacoustic waves.

In order to determine whether melanoma cells exist in a two-cell system under the interference of photoacoustic signals, in this paper, a single-cell model and a two-cell model are established, and a finite element analysis of the effects of two-cell photoacoustic spectral parameters is performed.

First, the dominant frequency of a single red blood cell with a diameter of 7.33 µm (mean diameter) is determined by analyzing its photoacoustic spectrum, and the unit wavelength is calculated based on this frequency. Then, in the two-cell model, the distance between two red blood cells with a diameter of 7.33 µm is set according to multiples of the unit wavelength. By comparing the similarity of signals with those from a single red blood cell, the optimal distance between the cells and the position of the detection points in the interference enhancement zone are determined. Next, under the condition that the two cells are at an optimal distance and the detection points are in the interference enhancement zone, the acoustic fields are analyzed for two scenarios: when both cells are normal red blood cells and when a single melanoma cell appears on either the upper or lower side. Finally, using the Frechet distance as a metric, the sound pressure level curves of both scenarios are compared with those of a single red blood cell. The Frechet value is employed to distinguish between the two scenarios by measuring the differences between the curves.

The detection method in this paper is based on the acoustic wave interference principle. The sound waves of two identical red blood cells satisfy the principle of superposition when they propagate. When sound waves with a fixed phase difference of the same frequency are superimposed, a stable strengthening and weakening zone will be formed in the space, and the signals in the interference enhancement zone are not only attenuated but also strengthened. Based on the principle of acoustic wave interference, two identical red blood cells (mean diameter) are used as the sound source, and the distance between the cells is adjusted so that the detection points are distributed in the interference enhancement zone of the two-cell sound field by analyzing the sound field of the two cells at different distances. However, when melanoma cells are present in the two-cell system, the sound sources are no longer identical. Consequently, after the superposition of sound waves, the detection points that were previously located in the enhancement zone no longer fall within it. As a result, the obtained sound pressure level curve differs from that of a single cell. The determination of the presence of melanoma cells in a two-cell system is realized by comparing the difference between the two states when the same cell and different cells are present.

For signal post-processing, we selected the discrete Frechet distance method, based on the similarity function definition, to evaluate curve differences. Previous studies have primarily used the eigenvalue method for the quantitative analysis of cellular photoacoustic spectra. This method measures curve differences by comparing their mean, variance, slope, and extreme value, facilitating cell classification. However, this method is only applicable when there are large differences in the curves. The similarity function definition method quantifies curve differences by comparing their distances and is sensitive to minor differences. In this study, the amplitude of the sound pressure level curves of the cells has a small difference, and the difference is mainly reflected in the slope, shape, and position of the extreme point of the curve. Therefore, the Frechet distance method, which evaluates curve similarity based on shape, is more appropriate. In addition, the smaller the Frechet value, the smaller the difference between the curves.¹⁴ 

When there are two or more columns of waves in the same sound field, a sound field of sound wave synthesis is formed. When sound waves with the same frequency and a fixed phase difference are superimposed, the phenomenon of interference of sound waves occurs.¹⁵ 

Therefore, it can be obtained that p = (p₁ + p₂), that is, the sound pressure of the sound field synthesized by the two columns of sound waves is equal to the sum of the sound pressure of each column of sound waves, which is the principle of superposition of sound waves.

Assume that the phase difference φ = φ₁ − φ₂ between the two columns of sound waves arriving at the position does not change with time, that is, the two columns of sound waves always arrive at a certain place with a certain phase difference.

Therefore, the synthesized sound pressure is still a fixed frequency acoustic vibration. However, the amplitude of the synthesized sound pressure is not equal to the sum of the sound pressure amplitudes of the two columns, but it is equal to the phase difference of the sound wave in the two columns. In two columns of sound waves with the same frequency and a fixed phase difference after the superposition of the synthesized sound field, in some positions, the sound waves are strengthened, and the amplitude of the synthesized sound pressure for the two columns is the sum of the amplitude of the sound wave. In other positions, the sound waves cancel each other, and the amplitude of the synthesized sound pressure is the difference between the sound wave amplitude of the two columns, which is the interference phenomenon of sound waves.

This study utilizes COMSOL Multiphysics software for finite element analysis, aiming to model and simulate various types of single cells as well as double cells at different distances. The theoretical model is obtained by extending the model developed by Evans and Fung.¹⁶ The model assumes that cells are spherical and homogeneous, which holds for suspended spherical cells, and the assumption has been confirmed experimentally and theoretically.^17,18 The study developed a two-dimensional axisymmetric model that utilizes the combination and segmentation of basic shapes to obtain cell geometries, adds pressure acoustic transient physical fields, establishes a global Cartesian coordinate system, and divides the cell model geometries into a free-profile triangular mesh.

The shape of normal human red blood cells is a biconcave round cake, with two concave sides in the center and thicker edges, demonstrating a mean diameter of about 7.33 µm and a uniform size within the range of 7.0–7.6 µm. The single-cell model established in this paper is a double-concave red blood cell placed flat in the center of a spherical water environment with a radius of 45 µm. Taking the center of the cell as an origin, a polar coordinate system was established, with a radius of 10 µm and a range of −90° to 90°, and one detection point was arranged every 5°, with a total of 37 detection points. Figure 1(a) presents the overall schematic diagram of the red blood cell, the water environment, and the detection points, and Fig. 1(b) shows the enlarged view of the cell and the detection points in Fig. 1(a).

The study uses spherical mouse melanoma cells (B16–F1) containing nuclei, with an average diameter of 17 µm and a range of 13–21 µm. The nucleus was 81.8% of the overall cell volume, a reasonable assumption considering the cell size. Melanin is mainly distributed in the cytoplasm of the cell, and there is almost no melanin in the nucleus.⁵ Therefore, the cytoplasm is the main absorber of incident light as well as the main source of radiation after absorbing energy. The initial pressure of the cytoplasm, nucleus, and water is 0 Pa. After laser beam irradiation, the cytoplasm’s pressure increases to 1 Pa, while that of the nucleus and water remains at 0 Pa. The melanoma cells exhibit a mass density of 1000 kg/m³ and a sound speed of 1560 m/s.

Red blood cells contain hemoglobin (an ideal endogenous contrast agent) and are surrounded by Duchenne’s Modified Essential Medium (DMEM). This medium is a watery liquid at 37.8 °C. Since DMEM does not absorb the energy of incident light, the red blood cells become the primary absorber of incident light and subsequently act as the primary source of radiation. The initial pressure value of the cell is 0 Pa, and the initial pressure value of the water is 0 Pa. After irradiation by the laser beam, the pressure value of the cell is 1 Pa and the pressure value of the water is 0 Pa. The mass density and speed of sound in water are 1000 kg/m³ and 1520 m/s, while those within the red blood cell are 1100 kg/m³ and 1650 m/s, respectively. The detection range is 0–10 ns, with a step size of 0.125 ns in the time domain and 0–600 MHz in the frequency domain.

Figure 2(a) depicts a two-dimensional schematic diagram illustrating the red blood cell and water environment, while Fig. 2(b) presents a two-dimensional schematic diagram of the melanoma cell and water environment. Figures 2(c) and 2(d) illustrate the three-dimensional schematic diagrams of the red blood cell and melanoma cell within the water environment, respectively. The model parameters for both red blood cells and melanoma cells are provided in Table I.

The radius of the spherical water environment where the two cells are located is 45 µm, and the distance between the cells is the distance between the cell centers. In the side view, taking the center of the circular water environment as the origin, the direction of the long diameter of the cells is the X-axis, the direction of the short diameter of the cells is the Y-axis, and the centers of the cells are located on the Y-axis. The distance from the center of the cell to the origin is the same for the upper and lower sides, and the two cells are symmetrical about the X-axis. For example, when the distance between the cells is 22.19 µm, the coordinates of the center of the cells on the upper side are (0, 11.095), and the coordinates of the center of the cells on the lower side are (0, −11.095).

The detection points were distributed in the lower side cell, the polar coordinate system was established with the center of the lower side cell as the origin, the radius was 11.095 µm, the range was −90° to 90°, and one detection point was set every 5°, with a total of 37 detection points. Figure 3(a) illustrates the 2D model of two red blood cells with a diameter of 7.33 µm at a distance of 22.19 µm. Figure 3(b) provides an enlarged view of the portion of cells and detection points from Fig. 3(a). Figure 3(c) displays the sound pressure level distribution at 137.5 MHz for two red cells with a diameter of 7.33 µm at a distance of 22.19 µm. Observing Fig. 3(c), the red straight lines and multiple hyperbolas in the figure are interference enhancement zones. Figure 3(d) depicts a 2D schematic of the melanoma cell on the upper side and the red blood cell on the lower side. Figure 3(e) offers an enlarged view of the portion of cells and detection points from Fig. 3(d). Figure 3(f) illustrates the sound pressure level distribution at 137.5 MHz for two cells separated by 22.19 µm, where the melanoma cell is positioned above and the red blood cell is positioned below. Time-domain sound pressure signals at different cell angles are collected at various detection points, and frequency domain sound pressure level curves are obtained via fast Fourier transform. The sound pressure level, expressed in decibels (dB), is defined as the logarithmic ratio of the measured photoacoustic signal’s sound pressure to the reference sound pressure, multiplied by 20.

A biconcave red blood cell was positioned flat at the center of a spherical water environment with a radius of 45 µm. In the side view, the center of the cell was taken as the center of the circle, detection points were set up with a radius of 10 µm, and an interval of 5° was taken from −90° to 90°, with a total of 37 detection points. Various detection angles generated distinct periodic undulation curves, exhibiting different amplitudes, shapes, and extreme point positions. Figure 4 displays frequency domain sound pressure level curves at 0°, 20°, 40°, 60°, −20°, −40°, and −60°. Using MATLAB, calculations revealed that the maximum or near-maximum vertical coordinates corresponded to a horizontal coordinate of 137.5. The amplitude of all frequency-domain sound pressure level curves for an individual cell peaked around 137.5 MHz. Consequently, the predominant frequency of the photoacoustic signal for a biconcave red blood cell with a diameter of 7.33 µm is 137.5 MHz.

When two identical point sources interfere with each other, an interference enhancement zone occurs if the distance between the two sources is a multiple of the wavelength. In this study, two identical red blood cells serve as the two identical point sources. By setting the distance between the identical cells as a multiple of the wavelength and fixing the position of the detection point, the optimal distance is found among multiple distances, ensuring the detection point is located within the interference enhancement zone. The dominant frequency of the red blood cell at the mean diameter is 137.5 MHz, and the wavelength of the 137.5 MHz sound wave in water is 11.095 µm. Using 11.095 µm as the unit wavelength, six different sets of cell spacing distances were established, starting from 5/4 times the wavelength and increasing by increments of 1/4 times the wavelength. The six sets of spacing distances are 13.869, 16.643, 19.416, 22.190, 24.964, and 27.738 µm. The radius of the detection point of the lower side cell was also set to 11.095 µm. Detection points are arranged around the cell with a radius of the unit wavelength. By comparing the similarity with the signal of a single 7.33 µm red blood cell, the points overlapping with the interference enhancement zone in these detection points are identified.

The curves obtained from the detection points at each angle at the six distances were compared with the curves obtained from the detection point at the same angle of a single 7.33 µm red blood cell, and the average Frechet value for all the detection points at each distance was calculated. Figure 5(a) presents a histogram showing the average Frechet value of the sound pressure level curves of the detection points at each distance, compared with the curves of the detection points at the same angle of a single red blood cell at each of the six distances. From the observation of Fig. 5(a), it can be seen that the Frechet values at different distances are similar. This is because there are too many detection points from the non-interference enhancement zone included in the calculation. Therefore, it is necessary to exclude the detection points from the non-interference enhancement zone (with larger Frechet values).

For each of the six distances, three detection points with smaller Frechet value and a fixed angular separation were selected. These detection points, with a smaller Frechet value, are likely to fall within the interference enhancement zone, while the angular separation ensures that the detection points are situated in different enhancement zones. Figure 5(b) shows histograms displaying the average Frechet value of the curves obtained from the three selected detection points at each distance. Upon examining the histograms, it can be observed that the average Frechet value is minimized at a distance of 22.190 µm. Therefore, 22.190 µm is chosen as the optimal distance. Figure 5(c) illustrates a histogram of the Frechet value for the 37 detection points within the range of −90° to 90°, at a distance of 22.190 µm between the two cells. The Frechet distance was used as a metric tool to find the point among the 37 detection points that was most similar to the signal of a single cell and to determine the position of the detection point within the interference enhancement zone. From the signals obtained by the 37 detection points, three detection points with small Frechet value and specific angular spacing angles were selected, at positions of 35°, −30°, and −70°. These positions lie within the range of 0°–90°, which is between the two point sources and encompasses more interference information. Therefore, the 35° position was chosen as the detection point in the interference enhancement zone for subsequent calculations.

Taking the center of the spherical water environment as the origin, in the side view, with the horizontal direction as the X-axis and the longitudinal direction as the Y-axis, the distance between the two cells was set to 22.19 µm, the center coordinates of the upper side cell were (0, 11.095), and the center coordinates of the lower side cell were (0, −11.095). Using the center of the lower cell as the origin, the polar coordinates are established, and the detection point is set at the 35° position with a radius of 11.095 µm.

The mean diameter of normal red blood cells is 7.33 µm, ranging from 7 to 7.6 µm. The red blood cell models with diameters of 7, 7.2, 7.4, and 7.6 µm were established, which were obtained by equal scaling of 7.33 µm red blood cells. The four diameters of red blood cells were combined two by two and placed on the upper and lower sides of the spherical water environment to obtain a total of 16 states. The frequency-domain sound pressure level curves at the 35° position of the detection point in each state were calculated, and the Frechet value in each state was obtained after comparing the similarity with the curves at the 35° position of a single 7.33 µm red blood cell. Figure 6 shows the histogram of the Frechet value for the curves at the 35° position for the 16 states when both the upper and lower sides are red blood cells. It can be observed that when both cells are normal red blood cells, the Frechet value is less than 48.

Melanoma cells were mouse melanoma cells (B16–F1) with a mean diameter of 17 µm, ranging from 13 to 21 µm. Geometric modeling of 13, 14, 15, 16, 17, 18, 19, 20, and 21 µm melanoma cells at 1 µm intervals was established. Melanoma cells of 9 diameters were combined with red blood cells of 4 diameters two by two. When the lower side of the spherical water environment was red blood cells and the upper side was melanoma cells, there were 36 states. There were also 36 states when the spherical water environment had red blood cells on the upper side and melanoma cells on the lower side. The frequency domain sound pressure level curves of the detection point at 35° position in each state were calculated and compared with the curves at 35° position of a single 7.33 µm red blood cell for similarity to get the Frechet value in each state. Figures 7(a) and 7(b) shows the histograms of the Frechet value at the 35° position for 36 states when the upper side is the red blood cell. Among them, Fig. 7(a) shows the 18 states when the upper side is 7 µm as well as 7.2 µm for red blood cells; Fig. 7(b) shows the 18 states when the upper side is 7.4 µm as well as 7.6 µm for red blood cells. Figures 7(c) and 7(d) show histograms of the Frechet value at the 35° position for 36 states when the lower side was red blood cells. Among them, Fig. 7(c) shows the 18 states when the lower side red blood cell is 7 µm as well as 7.2 µm; Fig. 7(d) shows the 18 states when the lower side red blood cell is 7.4 µm as well as 7.6 µm. By observing Fig. 7, it can be concluded that in the presence of melanoma cells, the Frechet values for all states are greater than 52.

Based on the principle of acoustic interference, by adjusting the distance between the cells, arranging the detection points in the interference enhancement zone, and measuring the difference in the curves using the Frechet distance as a metric, it is possible to determine the presence of melanoma cells in two cells under conditions in which the cellular acoustic waves interfere with each other.

In this study, we analyzed the acoustic field under the condition of cellular signal interference. By adjusting the cell distances, positioning the detection points, and employing a similarity function definition for signal processing, the information of both cells can be obtained using a single detection point, allowing for the determination of the presence of melanoma cells in the two cells. Compared to previous studies, this study improves the detection method and signal processing, leading to enhanced detection efficiency. This study successfully detects cell parameters under cellular signal interference, with the analysis method and signal processing approach remaining applicable for simultaneous multi-cell detection. Based on cell manipulation technology, if multiple cells can be detected at the same time, the detection efficiency will be greatly improved. However, the conclusions of this study need to be experimentally verified as there may be other factors impacting the experimental results in practical operations. Therefore, all factors need to be taken into account to ensure that the conclusions are more aligned with the actual situation.

First, the dominant frequency of a single mean diameter red blood cell was calculated to obtain a unit wavelength of 137.5 MHz. Then, the distances between identical red blood cells were adjusted to be multiples of the unit wavelength to determine the optimal interference distance and the position of the detection points. It was found that the optimal interference distance was twice the wavelength, with the detection points located at an angle of 35°. Next, at a distance of twice the wavelength, with the detection points positioned at the 35° angle, the Frechet values were calculated for two scenarios: one with both cells being normal red blood cells and the other with the presence of a melanoma cell among the two cells. A mean diameter red blood cell was used as the reference standard. Finally, upon comparing the two scenarios, it was observed that the Frechet values for normal red blood cells were all below 48, while the Frechet values for the presence of melanoma cells were all above 52. These data show that by adjusting the intercellular distance and detection point position and utilizing the Frechet distance as a curvilinear metric tool, it is feasible to determine the presence or absence of melanoma cells in the context of mutual interference between two cells.

This project was sponsored in part by the Project Supported by Shanxi Scholarship Council of China (No. 2023-127) and the Natural Science Foundation of Shanxi Province (No. 202103021224201).

The authors have no conflicts to disclose.

Rongrong Zhao: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Jianning Han: Funding acquisition (equal).

The data that support the findings of this study are available within the article.