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Stable sensing platform for diagnosing electrolyte disturbance using laser-induced breakdown spectroscopy

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Abstract

Electrolyte disturbance is very common and harmful, increasing the mortality of critical patients. Hence, rapid and accurate detection of electrolyte levels is vital in clinical practice. Laser-induced breakdown spectroscopy (LIBS) has the advantage of rapid and simultaneous detection of multiple elements, which meets the needs of clinical electrolyte detection. However, the cracking caused by serum drying and the effect of the coffee-ring led to the unstable spectral signal of LIBS and inaccurate detection results. Herein, we propose the ordered microarray silicon substrates (OMSS) obtained by laser microprocessing, to solve the disturbance caused by cracking and the coffee-ring effect in LIBS detection. Moreover, the area of OMSS is optimized to obtain the optimal LIBS detection effect; only a 10 uL serum sample is required. Compared with the silicon wafer substrates, the relative standard deviation (RSD) of the serum LIBS spectral reduces from above 80.00% to below 15.00% by the optimized OMSS, improving the spectral stability. Furthermore, the OMSS is combined with LIBS to form a sensing platform for electrolyte disturbance detection. A set of electrolyte disturbance simulation samples (80% of the ingredients are human serum) was prepared for this platform evaluation. Finally, the platform can achieve an accurate quantitative detection of Na and K elements (Na: RSD < 6.00%, R2 = 0.991; K: RSD < 4.00%, R2 = 0.981), and the detection time is within 5 min. The LIBS sensing platform has a good prospect in clinical electrolyte detection and other blood-related clinical diagnoses.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Electrolyte disturbance is one of the most common and critical clinical conditions [1]. Especially disorders of serum potassium (K) or sodium (Na) are usually triggered by other diseases, including cirrhosis, diabetes, cancer, diarrhea, extensive burns, and other common diseases, also can be triggered by external factors such as surgery and medication use [26]. The electrolyte disturbance can affect the normal functions of the heart, nerves, and muscles. When it occurs to emergency and critically ill patients, the electrolyte disturbance can accelerate the deterioration of other diseases and lead to a higher risk of death [3,7,8]. Doctors often diagnose diseases through the types and contents of elements, proteins, and blood cells in blood [1,9,10]. Among them, serum electrolyte detection is the only diagnostic method for electrolyte disturbance. Element detection can quickly determine the severity of the electrolyte disturbance through simple element content change information. Therefore, rapid, economical, and high-precision detection of blood electrolyte elements is particularly necessary.

At present, electrochemical methods [11] and colorimetry/complexometry methods [12] are the major methods used in clinical electrolyte detection due to the advantages of easy integration and low cost. However, the detection accuracy and stability of these two methods need to be improved. Mass spectrometry such as the inductively coupled plasma mass spectrometry (ICP-MS) is identified as one of the most accurate methods for electrolyte detection, which can accurately detect more than 60 different elements at once [13]. The spectroscopic methods are also used in electrolyte detection, including atomic absorption spectrometry and flame emission spectrometry [14]. However, for the disadvantages of complex sample pre-preparation, rigorous environmental requirements, or higher cost, these methods are difficult to be widely used in clinical testing. There is an urgent need for a new element detection method, which can guarantee the requirements of rapidity, accuracy, simplicity, and economy at the same time.

As a nascent elemental analytical technique, laser-induced breakdown spectroscopy (LIBS) is commonly used in the qualitative and quantitative analysis of elements by analyzing the wavelength and intensities of the collected plasma spectra. LIBS can rapidly provide elemental composition information for samples regardless of the sample phase. With its advantages of fast detection speed, less sample demand, simultaneous analysis of multiple elements, and in situ detection, LIBS is widely used in the field of environmental monitoring, space exploration, archaeology, and industrial applications [15,16], and these advantages can also meet the needs of clinical detection.

In recent years, the application of LIBS in the biomedical field has attracted more and more attention. For example, Y. Gimenez et al. manage to 3D image some elements in the kidneys of rodents injected with nanoparticles, and the image resolution reaches 10 µm, demonstrating the feasibility of LIBS profiling to tissue elements [17]. Zhang et al. use ultrasound-assisted alkali dissolution combined with laser-induced breakdown spectroscopy method to achieve accurate quantification of trace elements in hair. The detection limits of zinc and copper reach to 0.3517 and 0.0146 mg/kg, respectively [18]. The works above prove that LIBS has great application prospects in the biomedical field. Moreover, blood detection is one of the important applications of LIBS in the biomedical field. By analyzing the LIBS spectrum of blood, the qualitative and quantitative analysis of elements in blood and the classification and identification of diseases are realized. Chu et al. successfully realize the diagnosis of nasopharyngeal carcinoma and a variety of blood cancers through the extreme learning machine and random subspace method, the detection accuracy rate is more than 95% [19,20]. Yue et al. develop a method for distinguishing normal individuals, ovarian cyst patients, and ovarian cancer patients, and the sensitivity and specificity of LIBS serum diagnosis are 71.4% and 86.5% [21]. By fixing serum on gel and adding rubidium and barium as internal standard elements, Bardarov et al. quantify the electrolyte elements Na, K, Mg, and Ca in serum, and the limit of detection (LoD) for Na, K, Ca, and Mg is 6, 0.6, 1.9 and 1.7 mg/kg, respectively [22].

However, most of the current LIBS researches, related to blood detection, focus on the treatment of samples or data analysis, ignoring the treatment of substrates. Researchers typically use common substrates such as glass [23], filter paper [24], graphite [21], silicon wafers [22,25], and boric acid tablets [19] to convert serum from liquid to solid because they are easy to obtain and manipulate. These substrates are affected by the uneven solute distribution and unstable sample structure caused by the coffee-ring effect [26,27], which lead to instability of LIBS spectra and thus affect the results of qualitative and quantitative analysis [28,29]. So, how to solve the influence of the coffee-ring effect is the key to further improving the performance of blood detection by LIBS.

In this work, we developed a stable sensing platform, based on silicon wafers with laser-microprocessed arrays, for electrolyte disturbance diagnosis using LIBS. To suppress the coffee-ring effect and improve the uniformity and stability of the coagulated serum film, a superhydrophilic substrate was obtained by etching an ordered array of microstructures on the silicon wafer surface. The structure of the new substrate was characterized, and the optimal substrate area was optimized based on the LIBS spectral quality of Na and K in serum. The results of Na and K detection also demonstrated that the sensing platform has stable detection performance. Furthermore, a group of spiked serum samples containing different degrees of high potassium, high sodium, low potassium, low sodium, and normal control was detected by the sensing platform, and standard curves were drawn to evaluate the diagnostic performance of the sensing platform for electrolyte disorders.

2. Methods

2.1 Materials and sample pretreatment

The human serum sample is acquired from healthy volunteers in the Institute of Hematology and Blood Diseases Hospital. The healthy volunteer is ruled out for electrolyte abnormalities by clinical electrolyte testing. Informed consent has been obtained for experiments on human subjects, and the study is approved by the ethics committee at the Institute of Hematology and Blood Diseases Hospital (XKT2020007-EC-1).

The silicon wafer (SW, thickness: 0.1 mm, size: 25 mm × 75 mm) was applied for laser processing. Different concentrations of Na - K solutions were prepared by mixing NaCl powder (NaCl, 99%, Shanghai Macklin Biochemical, China) and KCl powder (KCl, 99%, Shanghai Macklin Biochemical, China) into deionized (DI) water.

We prepared a set of spiked serum samples to simulate electrolyte disturbance with different degrees. The spiking process is shown in Fig. 1. Each sample was prepared by mixing 80% serum with the solution and deionized water (the total ratio was 20%) to ensure the consistency of the sample matrix. The healthy serum has been clinically tested to rule out electrolyte disturbance, in which the Na and K levels are determined by ICP-MS to 2900 and 220 mg/L, respectively. In the clinic, each increase or decrease of serum sodium level by 10 mmol/L (approximately 230 mg/L) out of the normal range is regarded as one level of increase in severity, while each fluctuation of serum potassium level by 0.5 mmol/L (approximately 20 mg/L). Therefore, we set the change gradients of sodium and K for each sample to 230 mg/L and 20 mg/L, respectively. Table 1 shows the elements’ content of the samples. Sample 1 and sample 2 represent the different degrees of hypokalemia and hyponatremia. Sample 4 and sample 5 represent varying degrees of hyperkalemia and hypernatremia. The level of Na and K in sample 3 is the same as in healthy serum.

 figure: Fig. 1.

Fig. 1. The spiking process of electrolyte disturbance simulation samples

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Tables Icon

Table 1. The elements (Na, K) content of electrolyte disturbance simulation samples

Each 10 µL serum sample or spiked serum sample is dropped on the processed area. To avoid liquid splash, leading to spectral instability, the liquid serum is heated to solid (60 °C, 1 min).

2.2 Equipment

Figure 2 shows the LIBS experimental setup: a laser beam from a Q-switched Nd: YAG laser (wavelength: 532 nm; pulse energy: 40 mJ; repetition rate: 10 Hz; pulse width: 8 ns; Nimma 400, Beamtech, China) passes the reflector and focusing mirror (focal length: 150 mm) onto the sample surface. The Motorized positioning systems (Y110TA150, Jiangyun, China) can move samples on the horizontal plane, to make each laser pulse onto a fresh surface. The plasma emission is gathered by a light collector, then transmits by fiber to the spectrometer. The LIBS signal is obtained by a spectrometer (AvaSpec-ULS4096CL-EVO, Avantes B.V, Netherlands, spectral range: 200-950 nm) coupled with complementary metal-oxide-semiconductor (CMOS), the gate delay and gate width of COMS is set to be 2 and 9 µs, respectively. The laser scanning confocal microscope (VK4, Keyence) and the optical microscope (Axioscope 5, Zeiss) are applied to characterize the morphology and structure of the substrate. A digital camera (80D, Canon, Japan) is used to take physical pictures and detect the contact angle.

 figure: Fig. 2.

Fig. 2. The Schematic diagram of the LIBS device

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As is shown in Fig. 3, the laser micro-processing platform (Customization, Shenzhen Tsing Rui Technology Co., LTD, China) is used to process the SW. The laser beam from the fiber laser (wavelength: 1064 nm; repetition rate: 55 kHz; pulse width: 100 ns; laser power: 20 W; YDFLP-E-20&30-M7-S-R, Shenzhen JPT Opto-electronics Co., Ltd., China) focuses onto the SW surface by the galvanometer (scan speed: 1000 mm/s). Under the galvanometer control, the laser beam moves laterally and vertically to construct the ordered microarrays silicon substrate (OMSS), and each processing takes only 5 sec.

 figure: Fig. 3.

Fig. 3. The Schematic diagram of the laser micro-processing platform

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2.3 Data analysis

Figure 4(a) The schematic diagram of LIBS mapping. b) The direction of laser scanning. The line with black indicates the scanning path and the point with red indicates the laser ablation pit. c) Visualization of ROI extraction. A pixel corresponds to a pair of spectra, each pixel is a mixture of blue and yellow, with more blue representing a weaker spectrum of carbon and vice versa. The effective spectrum can be extracted by the positions contained in the red circle. d) Output the effective spectrum.

 figure: Fig. 4.

Fig. 4. a) The schematic diagram of LIBS mapping. b) The direction of laser scanning. The line with black indicates the scanning path and the point with red indicates the laser ablation pit. c) Visualization of ROI extraction. A pixel corresponds to a pair of spectra, each pixel is a mixture of blue and yellow, with more blue representing a weaker spectrum of carbon and vice versa. The effective spectrum can be extracted by the positions contained in the red circle. d) Output the effective spectrum.

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Figure 4 shows the LIBS mapping flow. As Fig. 4(b) shows, the laser beam scan line by line to form a square scan area that covers the serum-filled processing area. But what we are interested in is the serum spectra, Fig. 4(c) shows the region of interest (ROI) extraction. Using the carbon spectrum (247.883 nm) to get the LIBS element mapping image which shows the distribution of the serum, because the carbon only can be found in the serum, rather than in the SW or OMSS. Then we use image processing to get the location of the serum spectra and output the spectra (Fig. 4(d)). The ROI extraction and spectra output are completed by MATLAB (R2019b, MathWorks, America).

The indicators, including the standard deviation (SD), the relative standard deviation (RSD), and the coefficient of determination (R2) are used for stability and quantitative effect evaluation. The equations of the indicators are listed below:

$$SD\textrm{ }\textrm{ = }\textrm{ }\sqrt {\sum\nolimits_{i\textrm{ } = \textrm{ }1}^t {\frac{{{{({{x_i} - \overline x } )}^2}}}{{t - 1}}} } \textrm{ },$$
$$RSD\textrm{ }\textrm{ = }\textrm{ }\frac{{\sqrt {\sum\nolimits_{i\textrm{ } = \textrm{ }1}^t {\frac{{{{({{x_i} - \overline x } )}^2}}}{{t - 1}}} } }}{{\overline x }}\textrm{ } \times \textrm{ 100\%},$$
$${R^2}\textrm{ }\textrm{ = }\textrm{ }\frac{{{{\left[ {\sum\nolimits_{i\textrm{ } = \textrm{ }1}^n {({{x_i} - \overline x } )({{y_i} - \overline y } )} } \right]}^2}}}{{{{\sum\nolimits_{i\textrm{ } = \textrm{ }1}^n {({{x_i} - \overline x } )} }^2}\textrm{ }\cdot \textrm{ }\sum\nolimits_{i\textrm{ } = \textrm{ }1}^n {{{({{y_i} - \overline y } )}^2}} }}\textrm{ },$$
in which t is the spectra numbers of per target sample, n represents the sample amount. xi stands for the corresponding intensities, yi represents the added analytes concentration. $\bar{x}{\; }$and $\bar{y}$ stand for the average intensities of xi and yi.

3 Result and discussion

3.1 Characterization of the OMSS

Figure 5(a) exhibits the physical pictures of the serum (10 µL) on the OMSS and the SW. The contact angle of the serum on the OMSS is close to 0°, and the contact angle of the serum on the SW is 45°, which indicates that the ordered microarrays silicon substrate is superhydrophilic. Figure 5(b) shows the physical pictures of the dried serum on the OMSS and SW. The dried serum on the OMSS only has slight cracks. In contrast, the dried serum on the SW has obvious cracks. After laser ablation, the dried serum on the SW has been shattered and peeled off by the shock wave but the difference is that the OMSS has better adhesion for serum (Fig. 5(c)). Figure 5(d) and 5(e) are the microscope images before and after laser ablation respectively. The average depth of the OMSS is 3 µm, the roughness of the OMSS is Ra = 1.68 µm, while the roughness of the SW is Ra = 0.02 µm. Based on Fig. 5(d) and 5(e), the local details are further obtained by laser scanning confocal microscope. Figure 5(f) shows only intermittent fine cracks in the dried serum on the OMSS. Figure 5 g and 5 h show the ablation pit of the dried serum on the OMSS after laser ablation, the serum around the ablative pit is well attached to the OMSS.

 figure: Fig. 5.

Fig. 5. a) The side view of the 10 µL serum droplet on the OMSS and SW. b) The dried serum (on the OMSS) with a diameter of 10 mm and the dried serum (on the SW) with a diameter of around 4 mm. c) The dried serum after laser ablation. d) and e) The microscope images before and after laser ablation. f) The laser confocal microscope images for the details of dried serum on the OMSS. g) and h) The microscope image for the details of dried serum on the OMSS after laser ablation.

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Compared with the SW, when the serum is dropped on OMSS and dried, its crack phenomenon is significantly reduced. Especially after laser ablation, the dried serum can still adhere well to OMSS, but not to SW. This indicated that the serum on OMSS is stable when the laser interacted with the serum, which is one of the keys to ensuring the stability of the LIBS spectral signal.

3.2 Optimization of ordered microarrays for LIBS analysis

To complete the LIBS analysis, we optimized the processing area of the OMSS. The dried serum on the SW forms a circular region with a diameter of 4 mm, so we prepared a range of OMSS with different diameters from 4 mm to 12 mm. Figure 6 shows the diffusion of 10 µL serum on the SW and OMSS with different size areas. The serum could diffuse completely on the OMSS with a diameter less than or equal to 10 mm but could not diffuse completely in the 12 mm diameter OMSS, and the size of serum diffusion on 12 mm diameter OMSS approximates a 10 mm diameter circle. Therefore, it can be considered that the 10 µL serum can diffuse completely on the 10 mm diameter OMSS.

 figure: Fig. 6.

Fig. 6. The extent of serum diffusion on different areas of OMSS

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Further, the optimal area of OMSS for LIBS detection is explored. Figure 7(a) exhibits the spectra of the OMSS area with (serum) or without (blank) serum. The spectrum of the OMSS (without serum) indicated that the OMSS does not contain Na and K, demonstrating that using the OMSS as the substrate does not interfere with the detection of Na and K in serum. Figure 7(b) and 7(c) show Na (Fig. 7(b)) and K (Fig. 7(c)) spectra of serum on the untreated SW and the OMSS with different diameters. The spectral intensities of serum on the untreated SW are the lowest. The spectral intensities on the OMSS rise with the increase of the area. Furthermore, Fig. 7(d) and 7(e) show the quantitative comparison of the spectral intensities (Na (Fig. 7(d)) and K (Fig. 7(e))) and the relative spectral stability (RSD) of the serum on the SW and the OMSS. Both Na and K spectral intensities increase with the diffusion area of serum, and the spectral stability also is improved. The spectral intensity of serum on the 4 mm diameter OMSS is similar to serum on the SW, but serum on the 4 mm diameter OMSS has higher spectral stability (Fig. 7(d) and 7(e)). This is due to the OMSS having better adhesion for serum so that each laser pulse hits on the serum and produces an amount of consistent ablation to keep the spectral stability. The superhydrophilic of the OMSS also suppressed the coffee-ring effect, which made the solute after drying more evenly distributed on the substrate, so the spectrum is more stable. Compared with the spectrum obtained by using SW as the substrate, the intensity of the spectrum obtained by using the optimized OMSS as the substrate increases by two to three times, and the RSD of the spectrum is reduced from 80.00% to less than 15.00%. The serum on the 10 mm diameter OMSS has the strongest spectral intensity and the lowest RSD, so we finally select the 10 mm diameter OMSS as the LIBS serum detection substrate.

 figure: Fig. 7.

Fig. 7. a) Comparison of Na and K spectrum of the OMSS and the dried serum on the OMSS. b) Na Spectra of 10 µl serum on the SW and different areas of OMSS. c) K spectra of 10 µl serum on the SW and different areas of OMSS. d) Comparison of the intensity and RSD of Na spectra on the SW and different areas of OMSS. e) Comparison of the intensity and RSD of K spectra on the SW and different areas of OMSS.

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3.3 Diagnosis of electrolyte disturbance

To verify the diagnostic and stable performance of the platform, a set of spiked serum samples with different Na and K contents were analyzed. As mentioned in Section 2.1, in this set, sample 1 (Na: 2320 mg/L, K: 176 mg/L) corresponds to moderate hyponatremia and moderate hypokalemia, sample 2 (Na: 2610 mg/L, K: 198 mg/L) corresponds to mild hyponatremia and hypokalemia, sample 3 (Na: 2900 mg/L, K: 220 mg/L) corresponds to normal sodium and potassium content, sample 4 (Na: 3110 mg/L, K: 238 mg/L) corresponds to mild hypernatremia and mild hypokalemia, sample 5 (Na: 3320 mg/L, K: 256 mg/L) corresponds to moderate hypernatremia and moderate hyperkalemia. Each sample is composed of 80% volunteer serum and 20% standard solution or deionized water to ensure that the sample matrix is close to the actual serum matrix. 10 ul of each sample was added to the OMSS. After drying, 20 LIBS spectra were collected in the middle region and averaged, and the collection was repeated five times.

Figure 8(a) and 8(b) indicate the spectra of each sample, the spectral intensity increases with the increase of Na (Fig. 8(a)) and K (Fig. 8(b)) concentration. Because serum Na level is higher than serum K, Na can fluctuate over a wider range. So the difference of Na content in each sample is 230 mg/L, while the difference of K content is only 20 mg/L. This leads to more obvious differences in the spectra of Na from different samples. To evaluate the diagnostic performance of this platform for electrolyte disturbance, the calibration curves are built for Na and K. In Fig. 8(c), the relationship between the Na intensities and the Na content of the serum (moderate hyponatremia, mild hyponatremia, normal level, mild hypernatremia, and moderate hypernatremia) are built with a calibration curve, the R2 of the curve is 0.991. Similarly, Fig. 8(d) shows the calibration curve of the relationship between the K intensities and the K content of the serum, the R2 of the curve is 0.981. Table 2 shows the RSD of each sample, the spectral fluctuation of Na is greater than that of K, which is also due to the difference of one order of magnitude between the two element contents, but the RSD of each sample is still less than 6.00%. Also, the total time from sample processing to calibration curve drawing is about 20 min. When a calibration curve is available, it only takes 2 to 3 min to complete the detection of a new sample. The results show that the sensing platform has a stable and fast diagnostic capability.

Tables Icon

Table 2. The RSD of Na and K spectra in Simulation samples

 figure: Fig. 8.

Fig. 8. Samples test using OMSS. a) The Na spectra of the simulation samples. b) The K spectra of the simulation samples. c) The calibration curves for Na. d) The calibration curves for K.

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4. Conclusion

In summary, we develop a stable and fast LIBS sensing platform for electrolyte disturbance diagnostic. The OMSS, novel substrates suitable for the conversion of serum liquid to solid in LIBS detection, are obtained by laser microprocessing. The structure is characterized, and the inhibition effect on the coffee-ring and the dehiscence of dried serum are investigated. Then, according to the LIBS detection result of Na and K elements in serum, the optimal area of OMSS is selected, and the optimal area suitable for 10 µL serum is a circle with a diameter of 10 mm. Compared with the RSD of more than 80.00% when using the silicon wafer substrates, the RSD of the obtained LIBS spectrum is less than 15.00% using the optimized OMSS as the substrates, illustrating the significant improvement in stability. Finally, to verify the diagnostic ability of the platform for different levels of electrolyte turbulence, we test the simulated hypokalemia, hyponatremia, hypernatremia, hyperkalemia, and normal control samples. The RSD of the spectrum is less than 6.00%, the R2 of the calibration curve of Na and K are 0.991 and 0.981, respectively. The proposed LIBS sensing platform shows stable, fast, and accurate electrolyte disturbance detection capability, and also has great potential in clinical diagnostic.

Funding

National Natural Science Foundation of China (62075069).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (8)

Fig. 1.
Fig. 1. The spiking process of electrolyte disturbance simulation samples
Fig. 2.
Fig. 2. The Schematic diagram of the LIBS device
Fig. 3.
Fig. 3. The Schematic diagram of the laser micro-processing platform
Fig. 4.
Fig. 4. a) The schematic diagram of LIBS mapping. b) The direction of laser scanning. The line with black indicates the scanning path and the point with red indicates the laser ablation pit. c) Visualization of ROI extraction. A pixel corresponds to a pair of spectra, each pixel is a mixture of blue and yellow, with more blue representing a weaker spectrum of carbon and vice versa. The effective spectrum can be extracted by the positions contained in the red circle. d) Output the effective spectrum.
Fig. 5.
Fig. 5. a) The side view of the 10 µL serum droplet on the OMSS and SW. b) The dried serum (on the OMSS) with a diameter of 10 mm and the dried serum (on the SW) with a diameter of around 4 mm. c) The dried serum after laser ablation. d) and e) The microscope images before and after laser ablation. f) The laser confocal microscope images for the details of dried serum on the OMSS. g) and h) The microscope image for the details of dried serum on the OMSS after laser ablation.
Fig. 6.
Fig. 6. The extent of serum diffusion on different areas of OMSS
Fig. 7.
Fig. 7. a) Comparison of Na and K spectrum of the OMSS and the dried serum on the OMSS. b) Na Spectra of 10 µl serum on the SW and different areas of OMSS. c) K spectra of 10 µl serum on the SW and different areas of OMSS. d) Comparison of the intensity and RSD of Na spectra on the SW and different areas of OMSS. e) Comparison of the intensity and RSD of K spectra on the SW and different areas of OMSS.
Fig. 8.
Fig. 8. Samples test using OMSS. a) The Na spectra of the simulation samples. b) The K spectra of the simulation samples. c) The calibration curves for Na. d) The calibration curves for K.

Tables (2)

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Table 1. The elements (Na, K) content of electrolyte disturbance simulation samples

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Table 2. The RSD of Na and K spectra in Simulation samples

Equations (3)

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S D    =    i   =   1 t ( x i x ¯ ) 2 t 1   ,
R S D    =    i   =   1 t ( x i x ¯ ) 2 t 1 x ¯   ×  100\% ,
R 2    =    [ i   =   1 n ( x i x ¯ ) ( y i y ¯ ) ] 2 i   =   1 n ( x i x ¯ ) 2     i   =   1 n ( y i y ¯ ) 2   ,
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