2.1.

Basic Formulas of NIRS

Figure 1 shows a schematic diagram of an NIRS measurement on a human body. Here, $I_{\text{in}}(\lambda ,t)$ and $I_{\text{out}}(\lambda ,t)$ are the incident and detected light intensities for the wavelength $\lambda$ at time $t$, respectively. According to the modified Beer–Lambert law (MBLL), by using two different probe wavelengths and solving for the matrix calculation, the oxy- and deoxyhemoglobin NIRS signals $N_{{\mathrm{HbO}}_2}(t)$ and $N_{\mathrm{Hb}}(t)$ can be expressed as follows:^21–25

Fig. 1

Schematic of NIRS measurement.

2.2.

Derivation of Metabolic Index

First, consider decomposing each component of Eqs. (1) and (2) into AC and low-frequency (LF) components. Figure 2 shows a conceptual diagram of NIRS signal decomposition into AC and LF signals. Here, it is assumed that heart rate variation is negligible, and the AC component is completely periodic as long as the time window is short enough. In this way, $N_{{\mathrm{HbO}}_2}(t)$ and $N_{\mathrm{Hb}}(t)$ can be rewritten as follows:

Fig. 2

Conceptual diagram of NIRS signal decomposition into AC and LF components.

Here, the subscripts AC and LF indicate the AC and LF components of the corresponding physical quantity, respectively.

Similarly, $c_{{\mathrm{HbO}}_2}(t)$, $c_{\mathrm{Hb}}(t)$, and $L(t)$ can be rewritten as combinations of AC and LF components as follows:

By substituting Eqs. (5)–(7), Eqs. (1) and (2) are expressed as follows:

Furthermore, by expanding Eqs. (8) and (9) and by extracting the AC components

Next, to simplify Eqs. (10) and (11), the following additional constraints are introduced:

Here, Eq. (12) means that the molar concentration of the total hemoglobin in the blood is conserved within a limited measurement period, and Eq. (13) means that the increase in $c_{{\mathrm{HbO}}_2,\mathrm{AC}}(t)$ is equal to the decrease in $c_{\mathrm{Hb},\mathrm{AC}}(t)$ and vice versa, and Eq. (14) means that the oscillation amplitude of the optical path length is much smaller than the LF component of the optical path length.

In many prior studies, the optical path length $L(t)$ has been considered constant, leading to the interpretation of oxy- and deoxyhemoglobin NIRS signals as volume concentrations in the light-probed region. However, this study assumes that the optical path length $L(t)$ pulsates and changes slowly in correlation with the blood flow and heartbeat and that the oxy- and deoxyhemoglobin concentration $c_{{\mathrm{HbO}}_2}(t)$ and $c_{\mathrm{Hb}}(t)$ are molar concentrations in the blood. To explain the validity of these assumptions, Fig. 3 illustrates the schematic of the change in the optical path length.

Fig. 3

Schematic of the change in the optical path length in correlation with the blood volume.

Since hemoglobin is the most dominant light absorber in the NIR region, light absorption by the dermis can be ignored and the dermis can be treated as a scattering-dominant component. Therefore, the typical absorption path length can be expressed as the total length of capillary blood probed by the light. Based on this idea, the optical path length varies along with changes in both blood flow and heartbeat due to the variability in the capillary diameter. However, the assumption that the scattering change due to blood flow change and heartbeat is negligible forms the basis of this premise. The dermis meets this premise by taking into account the capillary density.²⁶

By using Eqs. (12) and (13) in Eqs. (10) and (11), the following relationship can be derived:

Then, by substituting Eq. (15) into the first term on the right side of Eq. (11) and by simplifying with respect to $c_{\mathrm{Hb},\mathrm{AC}}(t)$, the following can be derived:

Since $c_{\mathrm{Hb},\mathrm{AC}}(t)$ corresponds to the oxygen consumption during each heartbeat cycle generated by cell respiration, it is hypothesized that the metabolic levels in the human body can be estimated from the amplitude of $c_{\mathrm{Hb},\mathrm{AC}}(t)$.²⁸

Here, within a limited time window, the AC-NIRS signal $N_{{\mathrm{HbO}}_2,\mathrm{AC}}(t)$ and $N_{\mathrm{Hb},\mathrm{AC}}(t)$ can be approximated by sinusoids with slowly varying amplitudes by neglecting gradual changes in heart rate as follows:

Here, by properly selecting a measurement location on the body where the arteries connect to the cellular respiration site, and by selecting the distance between the detector and the light source of the NIRS probe to allow for a shallow penetration depth of the light path, the following equation can be applied:

In this study, e.g., the capillary-rich fingertip is identified as one of the suitable sites, and a relatively short detector-light source distance should also be utilized to prevent reaching bone depth. Then, by substituting Eqs. (19)–(22), Eq. (17) becomes

Here, Eq. (23) means that pulsation in deoxyhemoglobin (or oxyhemoglobin) concentration does not exist and is counterintuitive. To resolve this contradiction, consider modifying Eq. (20) as

Then, by substituting Eqs. (19) and (24), Eq. (23) becomes as follows:

Then, assuming that $A_{{\mathrm{HbO}}_2}(t)$, $A_{\mathrm{Hb}}(t)$, $L(t)$, and $\mathrm{\Delta }\theta (t)$ change on a much slower time scale than $2\pi /\omega$, the oscillation amplitude of Eq. (28). $C_{\mathrm{Hb},\mathrm{AC}}(t)$ is obtained as follows:

In addition, from Eqs. (15), (19), and (24), the following relation is derived:

Finally, by substituting Eq. (30) into Eq. (29)

Looking at Eq. (31), the pulsation amplitude of deoxyhemoglobin $C_{\mathrm{Hb},\mathrm{AC}}(t)$ increases as the total hemoglobin concentration $c_0$, the relative AC optical path length amplitude $L_{\mathrm{AC},\text{amplitude}}(t)/L(t)$, and the absolute phase delay $|\mathrm{\Delta }\theta (t)|$ increase. Here, $L_{\mathrm{AC},\text{amplitude}}(t)/L(t)$ is supposed to be a value similar to the perfusion index (PI), which is widely used in the field of pulse oximetry to evaluate signal quality.³⁰ On the other hand, $C_{\mathrm{Hb},\mathrm{AC}}(t)$ decreases with increasing arterial oxygen saturation ${\mathrm{SaO}}_2(t)$. In particular, $C_{\mathrm{Hb},\mathrm{AC}}(t)$ decreases to zero as ${\mathrm{SaO}}_2(t)$ reaches 100%. This change indicates that aerobic respiration does not occur in areas with 100% oxygen saturation, such as in arteries. This change also highlights the importance of carefully selecting the measurement location on the body.

Since the $c_0$ for each specific subject does not change significantly unless there is drastic dehydration or anemia, and since $L_{\mathrm{AC},\text{amplitude}}(t)/L(t)$ should also be nearly constant without exercise and temperature change of the measurement portion, $\mathrm{MI}(t)$ is the dominant factor in oxygen consumption which should strongly correlate with the body metabolism in a resting state.

Furthermore, Eq. (32) reveals that every term on the right-hand side is derived solely from the AC component of the oxy- and deoxyhemoglobin NIRS signals $N_{{\mathrm{HbO}}_2,\mathrm{AC}}(t)$ and $N_{\mathrm{Hb},\mathrm{AC}}(t)$, without incorporating the LF component. This fact implies that the drift in the light source intensity does not significantly impact $\mathrm{MI}(t)$ since the light source drift has a negligible effect on the AC component of the calculated NIRS signal in the MBLL.

In the following sections, an attempt to investigate the correlation between the metabolic index MI and blood glucose is explained through a series of experiments.

3.1.

Evaluation Using a Smartwatch-Based Prototype

A conceptual validation of the proposed MI method through a smartwatch-based prototype is presented in this section.

In the initial evaluation, the Samsung Galaxy Watch 4 44 mm (SM-R870) was used as the experimental unit. Figure 4(a) shows the rear view of the SM-R870. Red and IR LEDs are located in the center, and their center wavelengths are $\sim 650$ and 930 nm, respectively. Eight photodetectors are arranged radially around the LEDs. The sampling frequency of the red and IR LED signals is 100 Hz. Here, the experimental device has no hardware modification from its commercially available state. Figure 4(b) shows a schematic of the measurement. Originally, the LEDs and photodetectors in the smartwatch were designed to be used while wearing it on the wrist. However, due to its capillary density, the human wrist is not the best position for capturing a strong pulse wave signal. Therefore, instead of wearing it on the wrist, the smartwatch was wrapped on a finger pad.

Fig. 4

(a) Rear view of the smartwatch experimental unit. (b) Schematic diagram of the measurement using a smartwatch worn on a fingertip.

Functions to retrieve raw photodetector data from Galaxy Watch devices require the Samsung Privileged Health SDK, which is not publicly available. Therefore, the Samsung Partner App Program was used to gain access to the smartwatch functions needed for this experiment.

3.1.1.

Experimental protocol

Figure 5(a) shows the schematic of the clinical test setup. A protocol was established while referencing factors that can affect reading values in the case of a pulse oximeter.³¹ Healthy and non-diabetic subjects were asked to sit still during the experiment with the smartwatch resting on their finger pad to reduce fluctuations in peripheral blood flow caused by minor changes in posture and local blood pressure. Subjects were also asked to rest their elbow on the smartwatch-wearing side on an armrest. Bubble wrap was placed on the armrest to relieve pressure and prevent hypoperfusion. In addition, before starting the measurement, subjects held a hand warmer for a few minutes to ensure adequate peripheral blood flow.

Fig. 5

(a) Schematic diagram of the clinical trial setup. (b) Clinical trial procedure.

Figure 5(b) shows the typical course of the clinical test. A CGM device (Abbot, FreeStyle Libre^®) was used as a reference for the BGLs. Since it is reported that CGM sensors do not exhibit their best performance during the first few days after installation,³² CGM sensors were installed at least 2 days before the experiments for aging purposes. In addition, to correct for the systematic offset and delay specific to each CGM sensor, SMBG strips (Abbot, FreeStyle Precision^®) were used as needed before and after data recording by the smartwatch. To prevent possible distortion of the PPG signal due to body movements resulting from SMBG usage, no SMBG strips were used during the recording period. Data recording then started after at least 2 h of fasting, oral challenges were given about 30 min after the start of recording, and data recording continued about 60 min after the oral challenges. As a result, $\sim 90\min$ of PPG data were collected in each data recording.

Sugar-containing carbonated beverages (The Coca-Cola Company, Coca-Cola^® Original 350 mL) and glucose-containing jelly beverages (Morinaga, in Jelly^® Energy) were used for oral challenges. Typically, a special, highly concentrated glucose solution is used for this type of oral challenge test. However, since this is a concept test, more human-friendly oral challenges have been adapted this time. In addition, sugar-free carbonated beverages (The Coca-Cola Company, Coca-Cola^® Zero Sugar 350 mL) and non-carbonated bottled water were used for sugar-free oral challenges in the control experiments. Care was taken not to make all oral challenges excessively cold since this would reduce peripheral blood flow due to lowering of the body temperature. Finally, the main nutritional values of the oral challenges are listed in Table 1.

Table 1

Key nutritional facts of oral challenges per serving.

Item	Coca-Cola® Original	In Jelly® Energy	Coca-Cola® Zero Sugar	Bottled water
Serving size	350 mL	180 g	350 mL	500 mL
Calories (kCal)	140	180	0	0
Fat (g)	0	0	0	0
Carbohydrate (g)	39	45	0	0
Protein (g)	0	0	0	0

All clinical trials described in this paper were conducted in accordance with the Clinical Trials Act of the Ministry of Health, Labor and Welfare of Japan, published on the basis of the Declaration of Helsinki, and were approved by the Ethical Committee of Hamamatsu Photonics K. K. Informed consent was obtained from all subjects before measurements were performed. All clinical tests were performed under the supervision of the co-author with a medical license.

3.1.2.

Data processing

Figure 6 shows the process flowchart for the metabolic index MI measurement used in this study and visual explanations of several operational steps. First, raw PPG data from red and IR LEDs were retrieved and accumulated to a certain data length.

Fig. 6

Process flowchart of the metabolic index MI measurement, and visual explanations of several calculation steps.

Next, the raw PPG data were converted to the NIRS signals $N_{{\mathrm{HbO}}_2}(t)$ and $N_{\mathrm{Hb}}(t)$ according to the steps shown in Sec. 2.1. Here, the transmitted light intensity is assumed to be proportional to the PPG signal, and the extinction coefficients of oxyhemoglobin and deoxyhemoglobin compiled by Scott Prahl³³ were used for the calculation.

A second-order Butterworth band-pass filter (BPF) was then applied to each NIRS signal to remove unnecessary high-frequency components and LF components due to respiratory cycles. Here, the passband, the maximum loss in the passband, and the minimum attenuation in the stopband were 0.8 to 10 Hz, 3 dB, and 10 dB, respectively. After applying the BPF, the excess end portions in each waveform were trimmed so that the first and last points of the data corresponded to the beginning and end of the pulse wave.

A fast Fourier transform (FFT) was then applied to the trimmed NIRS signals. Each trimmed signal was resampled to make the data length the smallest power of two greater than or equal to the original length, and the Hamming window was selected for a window function.

Then, ${\mathrm{SaO}}_2(t)$ and $\mathrm{\Delta }\theta (t)$ were calculated from the previously mentioned FFT spectra. First, by comparing the magnitude of each main peak whose frequency corresponds to the heart rate, ${\mathrm{SaO}}_2(t)$ can be calculated according to Eq. (21). Second, $\mathrm{\Delta }\theta (t)$ can be calculated by comparing the FFT phase at the main peak.

Finally, the metabolic index $\mathrm{MI}(t)$ was calculated according to Eq. (32), and the calculated $\mathrm{MI}(t)$ was successively appended to a data buffer.

After the recording was completed, a series of recorded MI data were further processed as follows. Figure 7 shows a visual explanation of the post-processing of the calculated MI values. First, the MI values were divided into 1-min chunks. Then, the mean within each chunk was calculated after removing obvious outliers. Finally, the Savitzky-Golay filter was applied to the one-minute averages to smooth out fluctuations due to body motion, measurement errors, etc. Here, the polynomial order and window length for the Savitzky-Golay filter were 1 and 29, respectively. Although Fig. 7 contains outliers due to the above reasons, under ideal conditions, three to five data sets each consisting of about four heartbeat signals, are sufficient to derive a stable and instantaneous average MI value. In this ideal case, the total measurement time required is $\sim 7.2$ to 20 s assuming that the normal heart rate of adults is 60 to 100 bpm.

Fig. 7

Visual explanation of the post-processing of the calculated MI values.

3.1.3.

Results from the smartwatch-based prototype

Figures 8(a)–8(d) show the evaluation results of the smartwatch-based prototype according to different types of oral challenges for the same male subject who has no underlying health conditions. Here, the vertical dashed line in each plot indicates the time when the oral challenge was given to the subject, and the 10% vertical error bar range was applied to the reference CGM values considering the typical accuracy of the sensor.³⁴

Fig. 8

(a)–(d) Evaluation results of the smartwatch-based prototype for the different oral challenge types.

In Fig. 8(a), the metabolic index MI agreed well with the reference glucose concentration transition induced by the sugary drink ingestion. On the other hand, in Figs. 8(c) and 8(d), both MI and reference values were maintained nearly flat during each oral challenge test. Obviously, the sugar-free drink does not affect the subject’s BGL that much. However, the finding that MI values did not change as much during the sugar-free oral challenge tests suggests that MI is not simply a reflection of the body’s hydration status. At the same time, judging from the results that both carbonated and non-carbonated sugar-free drinks showed similar flat MI trends, the result in Fig. 8(a) is unlikely to be due to the carbonation status of the oral challenge. Regarding Fig. 8(b), an offset of about 15 min that can be seen between MI and CGM is likely due to the lag between blood glucose and interstitial fluid (ISF) glucose. In general, 15 min of CGM sensor delay is a bit large compared to the typical delay reported by the manufacturer. However, a previous study³⁵ shows that CGM delay can reach 15 min. Then, by compensating for 15 min of offset in Fig. 8(b), the plot can be modified as in Fig. 9(a). In this case, the MI also follows the up and down trend of the CGM well, in the same way as seen in Fig. 8(a). The proposed MI method response time can be estimated as 5 to 15 min from Figs. 8(a)–8(d) considering that the CGM device time delay is 5 to 15 min and the MI exhibited a similar trend.

Fig. 9

(a) Delay-compensated plot of Fig. 8(b). (b) Scatter plot of MI values versus reference CGM values generated from the data from Figs. 8(a), 8(c), and 8(d) and Fig. 9(a). (c) Transition of $\mathrm{\Delta }\theta (t)$ in Fig. 8(a).

By compiling the plot data from Figs. 8(a), 8(c), 8(d), and 9(a), a scatter plot Fig. 9(b) can be obtained. Here, each numerical data of MI corresponding to each CGM data point has been calculated by interpolation. In Fig. 9(b), the correlation coefficient $r$ calculated by the linear least squares (LLS) fitting reached 0.78, which indicates that the proposed metabolic index MI has a strong correlation with BGL.

In addition, the arterial oxygen saturation ${\mathrm{SaO}}_2(t)$ remained between $90\pm 2\%$ during a series of experiments, which means that ${\mathrm{SaO}}_2(t)·[1-{\mathrm{SaO}}_2(t)]$ remained between 0.07 and 0.11 in Eq. (32), and the maximum to minimum ratio of ${\mathrm{SaO}}_2(t)·[1-{\mathrm{SaO}}_2(t)]$ was at most $\sim 1.4$. Here, the decrease in ${\mathrm{SaO}}_2(t)$ from its normal range is attributed to the measurement section where ${\mathrm{SaO}}_2(t)$ approaches ${\mathrm{StO}}_2(t)$.³⁶

Therefore, the increase in the metabolic index MI was largely due to the change in $\mathrm{\Delta }\theta (t)$: the phase delay between oxy- and deoxyhemoglobin. For reference, the $\mathrm{\Delta }\theta (t)$ transition in the experiment of Fig. 8(a) is shown in Fig. 9(c). In this study, the results consistently showed that $\mathrm{\Delta }\theta (t)$ was positive during each oral challenge test, indicating that the oxyhemoglobin NIRS signal $N_{{\mathrm{HbO}}_2}(t)$ consistently precedes the deoxyhemoglobin NIRS signal $N_{\mathrm{Hb}}(t)$. The MI trend is therefore almost identical to the non-absolute $\mathrm{\Delta }\theta (t)$ trend. For this reason, only the MI trends are plotted in this paper to evaluate the proposed method, rather than separately plotting $\mathrm{\Delta }\theta (t)$ and ${\mathrm{SaO}}_2(t)$.

Thus, the validity of the assumptions in Eq. (24) is verified.

3.2.

Evaluation Using a Smartphone Camera-Based Prototype

The conceptual validation of the proposed MI method has been confirmed in Sec. 3.1. This section examines the reproducibility of the proposed method using a smartphone-based prototype.

Although the number of smartwatches is currently increasing worldwide, the penetration rate of smartwatches still has not reached that of smartphones. The utilization of smartphones as blood glucose monitoring devices has the potential to drastically expand the accessibility of the proposed MI method, given the global penetration of smartphones. Moreover, the active area and sensitivity of photodiodes in smartwatches are the minimum necessary due to spatial constraints, the main processors of smartwatches are less powerful, and the flexibility in designing smartwatch applications is limited compared to those of smartphones. In addition, several previous studies on smartphone camera-based non-invasive blood glucose monitoring have recently been reported.^37,38

For the above reasons, the smartphone camera-based prototype is newly introduced for the repeatability test instead of the smartwatch-based prototype.

Figure 10(a) shows an overview of a smartphone-based prototype developed for this research. The prototype consists of a smartphone and a light source unit mounted over the main camera module of the smartphone. Figure 10(b) shows the internal structure of the light source unit. Here, Samsung Galaxy S21 Ultra 5G (SC-52B) was selected as the smartphone and its wide-angle camera has a pixel count of 108 megapixels.

Fig. 10

(a) Overview of the smartphone camera-based prototype. (b) Internal structure diagram of the light source unit.

In addition, a high-brightness green LED OptoSupply OSG59L5B61Y was selected as the light source. The reason why green LED was chosen instead of white LED is described in Figs. 11(a)–11(c) and 11(d1)–11(d3). Figure 11(a) shows the typical spectrum of the green LED and its transmission spectrum after the subject’s index finger. Note that the $y$-axis is a logarithmic scale, and the spectra have been filled in with a color corresponding to each wavelength for visual support. Because LEDs have a broad spectrum compared to laser diodes (LDs), the green LED has a slight amount of reddish components in the tails of its spectrum.

Fig. 11

(a) Typical spectrum of a green LED and its transmitted spectrum after passing through a fingertip in log Y scale. (b) Typical efficiency curves of an RGB CMOS camera. (c) The normalized transmitted spectrum of the green LED after passing through a fingertip (d1)–(d3) Light intensity of the transmitted light perceived by each RGB sensor of the CMOS camera.

Next, Fig. 11(c) shows the transmitted spectrum of the green LED on a linear scale. Due to the strong absorption by oxy- and deoxyhemoglobin around 550 and 400 nm, the initial Gaussian-like spectrum of the green LED is transformed into a two-hump shape. Then, Fig. 11(b) shows the typical quantum efficiency of an RGB color CMOS camera. Since the quantum efficiency data of the Samsung Galaxy S21 Ultra 5G smartphone camera was not available, data from a compact scientific CMOS color digital camera from Thorlabs³⁹ was used as a typical value instead. Finally, by multiplying the quantum efficiency by the transmitted light, the light intensity spectrum perceived by the blue, green, and red CMOS sensors can be obtained as shown in Figs. 11(d1)–11(d3), respectively. Looking at Figs. 11(d1) and 11(d3), it can be seen that each spectrum is composed of almost a single prominent peak. Therefore, by combining the light intensity data from blue and red CMOS sensors, a kind of visible spectroscopy for the human fingertip can be configured.

The experimental protocol for the smartphone prototype largely followed that described in Sec. 3.1.1. Figure 12 illustrates the test setup for the smartphone prototype. The index finger of a healthy, non-diabetic subject with no underlying health conditions was placed between the light source unit and the smartphone camera, and a bubble wrap sheet was placed against the hypothenar to relieve pressure and prevent hypoperfusion. Since the smartphone prototype utilizes a transmissive optical configuration, the distance between the light source and detector was optimized by adjusting the fingertip insertion depth. The autofocus, auto exposure, and auto white balance functions of the smartphone camera were disabled. In addition, the camera aperture and frame rate were kept constant throughout the clinical trial. The camera sensitivity was kept to approximately the same value.

Fig. 12

Test setup for the smartphone camera prototype.

With regard to oral challenges, Coca-Cola^® Original or in Jelly^® Energy was administered to the subject and no sugar-free challenge was used, and oral challenges were administered between 15 and 30 min after the start of each test. Furthermore, stricter expiration date controls were also applied to the CGM sensors. In addition to the first two days for aging, the CGM sensors were not used for clinical testing during the last two days of their specified expiration date to improve performance.³²

The data analysis process was the same as explained in Sec. 3.1.2 and Figs. 6 and 7 except for the method for acquiring raw PPG data. Instead of red and IR PPG data, the average pixel values of blue and red CMOS data were used. Note that camera images captured by the video stream function of Android OS are formatted in YUV420, so each YUV420 image must be converted to RGB format. Furthermore, since the extinction coefficient dataset of oxy- and deoxyhemoglobin used in Sec. 3.1.2 did not work properly in the case of pseudo-visible spectroscopy by the smartphone camera prototype, the extinction coefficient matrix was manually optimized to obtain a smartphone prototype result comparable to that of the smartwatch prototype. Finally, for the BPF and Savitzky–Golay filters, the same parameters were used as in Sec. 3.1.2.

3.2.1.

Results from the smartphone camera-based prototype

The repeatability test of the smartphone-based prototype was repeated a total of 19 times on a single male subject with no underlying health conditions. Figure 13 shows four typical evaluation results of the smartphone-based prototype from the 19 tests. Here, the test results are plotted in the same way as in Fig. 8, and the CGM delays were adjusted by using the sensor-specific constant value of each sensor, which was determined through comparison with SMBG readings during the preparation period. Figures 13(a) and 13(b) are examples of good correlation with the reference CGM, while Figs. 13(c) and 13(d) are examples of moderate correlation. In each of Figs. 13(a)–13(d), it can be seen that the metabolic index MI responded to the oral challenges and followed the CGM trends. Here, it should be noted that the trend of MI in each of Figs. 13(a)–13(c) appears to increase even under fasting conditions, which can be attributed to the smoothing effect of the Savitzky–Golay filter.

Fig. 13

(a) and (b) Typical evaluation results of the smartphone-based prototype with a good correlation to the reference. (c) and (d) Typical results with a moderate correlation to the reference.

Figure 14(a) shows a scatter plot of the metabolic index MI versus the reference CGM made from the repeatability test results. Here, each MI numerical data corresponding to each CGM data point was calculated by interpolation in the same way as in Fig. 9(b), and the correlation coefficient obtained by LLS fitting was 0.66. Although this correlation coefficient value $r$ is not as high as that obtained in Fig. 9(b), it is still high enough to infer a strong positive association.

Fig. 14

(a) Scatter plot of metabolic index MI values versus reference CGM values generated from repeatability test results. (b) Perkes error grid for type 1 diabetes generated from the repeatability test results.

Figure 14(b) also shows a Perkes error grid for type 1 diabetes⁴⁰ generated from the repeatability test results and the conversion coefficients obtained by the LLS fitting. Here, the color scale in the plot indicates the spatial density of nearby points. Although blood glucose values obtained by venous blood glucose testing or capillary blood glucose testing are generally used as the reference values for error grid analysis, CGM values were used as the reference values in this research. Because of this substitution, note that Fig. 14(b) has combined errors from the MI method and the CGM. Nevertheless, most of the data points in Fig. 14(b) are distributed between zones A and B. Specifically, $\sim 69\%$ and 31% of the data are located in zones A and B, respectively, and there were no data points in other zones. This result compares favorably with some other previous non-invasive optical blood glucose monitoring research.^41,42

3.3.

Attempt to Improve the Accuracy of the Proposed MI Method

In this section, an attempt is made to improve the accuracy of the proposed MI method by combining the $L_{\mathrm{AC},\text{amplitude}}(t)$ correction.

In this study, the metabolic index MI shown in Eq. (32) has been used so far for a non-invasive BGL index. However, as shown in Eq. (31), the oscillation amplitude of deoxyhemoglobin $C_{\mathrm{Hb},\mathrm{AC}}(t)$ is also affected by $c_0$, $L(t)$, and $L_{\mathrm{AC},\text{amplitude}}(t)$. Although the total hemoglobin concentration $c_0$ should be less susceptible to variation without drastic changes in body hydration level or acute anemia, the oscillation amplitude of the optical path length $L_{\mathrm{AC},\text{amplitude}}(t)$ and the optical path length $L(t)$ may change depending on perfusion status, pressure, and other factors. In other words, BGL estimation based on a metabolic index MI may still be capable of better accuracy by taking $L(t)$ and $L_{\mathrm{AC},\text{amplitude}}$ correction into account. However, in principle, it is impossible to measure the effective optical path length $L(t)$ based on the MBLL. Therefore, the following assumptions and approximations are introduced.

Since it can be assumed that the AC amplitude of the optical path length $L_{\mathrm{AC},\text{amplitude}}(t)$ should increase as its counterpart $L_{\mathrm{LF}}(t)$ increases and that the divergence speed of $L_{\mathrm{AC},\text{amplitude}}(t)$ should not be greater than that of $L_{\mathrm{LF}}(t)$, the relationship between $L_{\mathrm{AC},\text{amplitude}}(t)$ and $L(t)$ can be approximated as follows:

Here, to make the rightmost term of Eq. (34) dimensionless, a dimensionless correction coefficient $\alpha$ is introduced as follows:

Finally, multiplying $\alpha (t)$ by $\mathrm{MI}(t)$ gives the following amplitude-corrected metabolic index ${\mathrm{MI}}^{\prime }(t)$

Figures 15(a)–15(d) show the evaluation results of the $\alpha$-corrected ${\mathrm{MI}}^{\prime }(t)$ obtained by the smartphone prototype, which correspond to and are plotted in the same manner as Figs. 13(a)–13(d), respectively. Note that $n=0.5$ of the power exponent was temporarily adapted in Eq. (36), which means that the AC amplitude of the optical path length is proportional to the square root of $L_{\mathrm{LF}}(t)$, and typical values of $A_{{\mathrm{HbO}}_2}(t)$ and $A_{\mathrm{Hb}}(t)$ in the resting state were chosen for $A_{{\mathrm{HbO}}_2,0}$ and $A_{\mathrm{Hb},0}$. By comparing each plot in Fig. 15 with that of Fig. 13, it can be seen that the rise and fall in the metabolic index become clear with $\alpha$ correction. Among these, Figs. 13(c) and 13(d), which showed a moderate correlation with the CGM values, showed a better correlation in Figs. 15(c) and 15(d) by applying the correction.

Fig. 15

(a)–(d) Correction-applied evaluation results of the smartphone-based prototype corresponding to Figs. 13(a)–13(d), respectively.

In addition, a scatter plot of the $\alpha$-corrected metabolic index ${\mathrm{MI}}^{\prime }$ versus CGM values and a Perkes error grid for type 1 diabetes with $\alpha$ correction, generated based on the same experimental data set of Fig. 14, are shown in Figs. 16(a) and 16(b), respectively. Here, the same correction parameters were used as in Fig. 15. As can be seen in Fig. 16(a), the correlation coefficient $r$ computed by the LLS fitting improved slightly to 0.73 with the $\alpha$ correction. In addition, in Fig. 16(b), $\sim 79\%$ and 21% of the data points are distributed in zones A and B, respectively, and the zone A ratio has increased by 9% compared to Fig. 14(b).

Fig. 16

(a) Scatter plot of $\alpha$-corrected metabolic index MI’ values versus reference CGM values generated from the repeatability test results. (b) Perkes error grid for type 1 diabetes generated from the $\alpha$-corrected repeatability test results.

According to these results of $\alpha$ correction, although $n=0.5$ of the power exponent is a tentative value and there is no biomedical basis to support this number yet, it can be seen that $\alpha$ correction can improve the MI-based non-invasive BGL estimation to some extent.

As a reference, Fig. 17 shows a transition of the correlation coefficient $r$ computed by LLS fitting after applying $\alpha$ correction for the same data set of the smartphone-based prototype according to different power exponents $n$. Here, the $n=1$ spot corresponds to the case without $\alpha$ correction. As can be seen from Fig. 17, $r$ showed a gradual trend around $n=0.4$ to 0.7, which implies that the AC amplitude of the optical path length should be roughly proportional to $L_{\mathrm{LF}}(t)$ to the power of $\frac{1}{2}$ to $\frac{2}{3}$.

Fig. 17

Transition of the correlation coefficient $r$ computed by the LLS fitting according to different power exponents $n$.

Finally, as a summary, Table 2 shows a comparison of the key performance indicators of BGL estimation with and without $\alpha$ correction. Here, the term MARD stands for mean absolute relative difference, which is the average of the error between the estimated value and the reference value divided by the reference value, and RMSE stands for root-mean-square error, which is the square root of the average of the squared errors between the estimated values and the reference values. MARD and RMSE are typical metrics used to evaluate the performance of glucose monitoring systems.^9,32 Note that the absence of data other than in zones A and B is due more to an insufficient number of data points than to the good performance of the proposed MI method, and that MARD and RMSE here have combined errors from the MI method and the reference CGM.

Table 2

Comparison of key performance indicators of BGL estimation with and without α-correction.

According to the manufacturer, all FreeStyle Libre^® CGM sensors in the United States have a MARD of <10%³⁴ and a third-party evaluation has shown a roughly similar result.⁴³ Although the proposed MI method, even with $\alpha$ correction, has an even larger MARD compared to the $<10\%$ MARD of the commercially available CGM, implementing hardware improvements and algorithm optimization may further reduce the MARD of the proposed method to below 10%.

For reference, Table S1 in Sec. S2 of the Supplementary Material shows the performance comparison between the proposed MI method and other non-invasive blood glucose monitoring methods.