Terahertz Women Reproductive Hormones Sensor Using Photonic Crystal Fiber With Behavior Prediction Using Machine Learning

In this article, we propose a rectangular hollow core Photonic Crystal Fiber (PCF) sensor for high detection of women’s reproductive hormones (progesterone and estradiol) in the blood sample at THz regime (0.8 THz to 1.7 THz). The numerical sensing performances are evaluated by the full vector finite element method (FVFEM). We have achieved improved relative sensitivity with minimal loss for sensing progesterone concentrations and estradiol concentrations by optimizing structural parameters. The obtained maximum relative sensitivity is 99.87% for 10 n mol/L (progesterone) and 99.88% for 3 n mol/L (Estradiol) at 1.35 THz. Further, we have obtained low effective material loss (EML) and a large effective mode area. The primary takeaway from this research is that it’s critical to monitor estradiol and progesterone levels in order to ensure that a woman’s reproductive system is functioning in a balanced manner and in general health systems. Also, this biosensor can be fabricated by current fabrication technologies. Moreover, the prediction carried out with the help of Locally Weighted Linear Regression, and hyperparameter tuning, we can conclude that for weighting hyperparameter value of 0.02. We have achieved the maximum prediction accuracy with the unity R2 score and this model can be employed for the prediction of relative sensitivity for various parameters.


I. INTRODUCTION
A wide range of sensor technologies have been developed throughout the year for sensing biomolecules [1], [2], [3]. Among these, optical biosensors have made a significant impact on the research for the detection of biomolecules.
The associate editor coordinating the review of this manuscript and approving it for publication was Norbert Herencsar .
In the biomedical field, hormones play an important role in the human body. Imbalanced levels of hormones lead to dangerous diseases like cancer, osteoporosis, hyperplasia of the cardiovascular system, etc. [4], [5]. Particularly, progesterone and estradiol hormones connect in each and every cell and tissue of female bodies. Both hormones cooperate and are in charge of the changes that occur during pregnancy. For a woman's reproductive system to function properly and to keep track of her general health systems, it is crucial to detect the levels of these hormones [6], [7], [8]. So, the researcher has developed a sensor for sensing the concentration level of hormones in the human body. So far, immunometric assaybased approaches have been employed in the biomedical industry to detect various hormone quantities [9]. However, it has certain drawbacks, including a poor level of sensitivity and accuracy. An optical biosensor is used to create highly sensitive photonic biosensors in order to solve these problems.
The development of novel types of photonic biosensors has received significant research attention to dates such as tumor detection [10], DNA (Deoxyribonucleic acid) sensor [11], support to neural activity [12], and implantable photonic components for medical treatment, etc. Optical fiber-based biosensors also play an important role in the identification of bio components. Bacterial detection [13], glucose in serum [14], cholesterol detection [15], and plasmonicaided fiber optics sensor [16] are only a few examples. In fiber sensing technology, last few years several authors worked in the Photonic crystal fiber (PCF) based biosensor. Because PCF sensors have numerous advantages for sensing biomedical components such as high sensitivity, low detection limit, compact size, high resolution, and high design flexibility. Further, PCF-based sensors can withstand high voltages, high temperatures, and dangerous chemicals. Due to its many potential uses in non-invasive medical imaging and biological sensing, terahertz frequency is currently the subject of extensive research. Researchers have suggested a variety of guided media for THz propagation during the past few decades, including bare metal wires, metal-coated dielectric tubes, metallic wires, slot waveguides, etc. Yet, the high bending loss, reduced coupling efficiency and maximum material absorption loss of the aforementioned guided mediums force their disregard. The photonic crystal fiber entered the light as a result of innovative light guidance qualities, such as low dispersion, minimum material loss, maximum core power fraction, etc. Recently, many researchers are working on PCF-based biosensors in Terahertz (THz) regime with different structures of the core such as hexagonal, octagonal, elliptical, hollow, and solid cores. Hollow core PCF is more suitable for excellent sensing which is offered strong bonding between light and lower refractive index bio-sample in the core region. Furthermore, PCF offers a precise and efficient detecting capacity in the THz frequency band.
In 2019, Hossain et al. designed a blood sensor using a hollow core for detecting water, plasma, red and white blood cells, and hemoglobin. The obtained maximum relative sensitivity is 93.6% at 2 THz [17]. Further, Kawsar Ahmed et al. have also designed THz based refractive index sensor (RIS) for the detection of blood components using PCF with a sensitivity of 80.56% [18]. Habib [20]. The proposed study has utilized hollowcore PCF in asymmetrical core construction for biosensor application to detect female reproductive hormones by adhering to the same features. Graphene-coated FET (field-effect transistor), is used to monitor parathyroid hormone receptor (PTHR) and G protein-coupled receptor (GPCR), while investigating women's health assessment [21]. In the fiber optic principle, numerous authors reported hormones sensor using tapered optical fiber for dopamine sensor [22], laser detection for thyroid in blood sensor [23], optical fiberbased plasmon nanoparticle endocrine system for controlling body metabolism [24], 1D (one dimensional) Photonic crystal waveguide as women reproductive hormones [25] and plasmon fiber grating [26]. Further, machine learning-based biosensor has been used for many applications that include forecasting, recommendations, recognition, etc. Furthermore, Machine learning algorithms have been vastly employed in photonic devices for the purpose of resource reduction and saving simulation time through the prediction and forecasting of intermediate values [31], [32], [33], [34], [35].
In this article, the asymmetric structure of hollow core PCF is designed for sensing progesterone and estradiol hormones in the THz regime. These hormones work together in every cell and tissue of a woman's body and throughout her life. Further, it is very crucial for reproductive development and sexual in humans. Therefore, a hollow core PCF-based RIS is designed for sensing women's reproductive hormones in order to achieve the healthiness and effective control of the female ovaries based on previously published studies. Further, we have predicted relative sensitivity with help of the Locally Weighted Linear Regression algorithm. The upcoming chapters are showed the proposed sensor design; analyze the sensing performances; and machine learning algorithm for predicting sensor behavior.

II. STRUCTURAL DESIGN
In order to design high relative sensitivity in the THz regime, we have designed rectangular-shaped air holes hollow-core PCF by using the finite element method as depicted in Fig.1. The proposed sensor was designed using the commercially available full vector finite element method (FVFEM) software COMSOL multiphysics v5.3. In order to block stray energy from the fiber axis and absorb radiant light, the perfectly matched layer (PML) thickness is 780 µm. Moreover, the scattering boundary condition is utilized in conjunction with PML to cut down on reflected energy. The cross-section of the sensor is accommodated by a 2D mesh grid made using the finite element technique. Maxwell's equations can be written as a matrix eigenvalue problem to get the imaginary and real effective indices of core modes. We kept the element size extremely fine. The complete mesh consists of 44106 triangular elements, 2180 boundary elements, and 98 vertex elements, and the total mesh area is 506.8 µm 2 . The diameter of the optical fiber is 4.85 mm. The rectangular air VOLUME 11, 2023 holes are considered to surround the core for effective light confinement in the analyte-filling area.
The width and height of A 1 , A 2 , A 5 , and A 6 are equal to 1125 µm and 500 µm, respectively. Similarly, the width and height of A 3 and A 4 are also identical to 1200 µm and 500 µm, respectively. Further, the width and height of A 7 are 250 µm and 1050 µm, respectively. The gap between two rectangular air holes (d) is equal to 10 µm. The background material of the proposed fiber is Zeonex (Refractive Index (RI) = 1.53). It has numerous advantages such as minimum absorption loss, high bio-compatibility, and hightemperature insensitivity compared with other THz materials [27], [28]. Notably, Zeonex and Topas have comparable optical properties, but Zeonex has superior biocompatibility, high glass transition temperature, fabrication flexibility, and maximum chemical resistance than Topas. Although the RI of the backdrop material is closely related to the effective material loss (EML), it also helps to minimize EML in specific grades of Zeonex, such as 330R, 480R, and 480. In this proposed work, we have infiltrated blood samples in the rectangular core region for sensing progesterone and estradiol hormones. The width and height of the rectangular core (A 8 ) sizes are 450 µm and 690 µm. The simulation makes use of the RI of both hormone concentration levels that were discovered by experimentation. Using a pocket refractometer (Atago PAL-RI), the RI is calculated with respect to different concentrations of progesterone and estradiol in a blood sample as shown in Fig. 2 [25]. The typical range of progesterone and estrogen RI values for women during and after their reproductive years is between 0 and 10 nmol/L and 0 and 3 nmol/L, respectively. As a result, we examined the sensor for reproductive hormones in the range. Further, the proposed asymmetric rectangular PCF can be fabricated with help of 3D printing and extrusion methods which also support the fabrication of complex structures of rectangular air holes [29], [30].   Fig. 3 indicates electric field distribution for X-polarization, and Y-polarization at 1.35 THz.

III. RESULTS AND ANALYSIS
The important parameter of the designed sensor is relative sensitivity which indicates the concentration of sensing analytes. It can be expressed by [27], where n a represents the refractive index of an analyte, n eff means effective refractive index, and PF means power fraction in the core region which is expressed by [29], In order to obtain maximum relative sensitivity, we adjust the structural dimensions. Initially, we considered the distance between two rectangular air holes (d) to be 10 µm and 20 µm. From this optimization, we obtained maximum relative sensitivity at 10 µm due to effective light confinement in the sample region. Fig.4 is shown relative sensitivity for various women's reproductive progesterone hormones with respect to different frequency range at d = 20 µm. From the progesterone hormones, we have obtained maximum relative sensitivity as 97.24% at 1.3 THz for X polarization and 98.43% at 1.35 THz for Y polarization at 10 nmol/L due to maximum refractive index and high power in the core as shown in Fig.4. For estradiol, the maximum sensitivity is 97.26% for X polarization at 1.35 THz and 98.6% at 1.3 THz for Y polarization at 3 n mol/L as shown in Fig.5. A detailed examination of the figures reveals that for X polarization, sensitivity declines after 1.3 THz and for Y polarization, after 1.35 THz. Even though the core power fraction is expected to increase with frequency, it hits its peak at a certain frequency   and as frequency increases, some useful power begins to leak into the cladding region, causing the core power to weaken.
Further, we have plotted relative sensitivity for d = 10 µm with respect to frequency as shown in Fig. 6 and Fig.7. We have found maximum relative sensitivity as 97.25% at 1.3 THz for X polarization and 98.6% at 1.35 THz for Y polarization at 10 n mol/L as shown in Fig.6. For estradiol, the maximum sensitivity is 97.3% for X polarization at 1.35 THz and 98.63% at 1.3 THz for Y polarization at 3 n mol/L as shown in Fig.7. The obtained relative sensitivity of progesterone and estrogen for different d is shown in Table 1. In the proposed structure, we found the highest relative sensitivity in Y polarization mode, and the reason is the shape of the hollow core maximum in the y direction. And also, it is produced a greater power fraction in the core region. So, we have considered the Y polarization mode for further optimizations.
Then, we increased the hollow core size from the optimum value to +2.5% and decreased it to -2.5%. Fig.8 (a-d) exhibits relative sensitivity for different progesterone concentrations with respect to frequency. The obtained relative sensitivity is  99.87% for increasing core size and 97.34% for decreasing core size at 1.35 THz. These data are taken at 10 n mol/L as shown in Fig. 8(d). Fig. 8 (a-c) reveals relative sensitivity of 99.81%, 99.83%, and 98.85% at 1.35 THz (increment of 2.5%) for 0 n mol/L, 1 n mol/L, and 3 n mol/L, respectively. Among the four concentrations, the sensitivity is high for 10 n mol/L due to it having a maximum refractive index that causes a more effective light interaction with the analyte.
The maximum relative sensitivity for estradiol hormone is 99.8%, 99.82%, 99.85%, and 99.88% at 1.35 THz (increment of 2.5%) for 0 n mol/L, 0.1 n mol/L, 1 n mol/L, and 3 n mol/L, respectively as shown in Fig. 9 (a-d). The obtained relative sensitivity of progesterone and estradiol for optimization of the analyte filling area is shown in Table 2. EML is an important parameter for hollow core PCF-based THz sensors which limits the sensing performance. It can by calculated by [27], Where µ 0 and ϵ 0 represent permeability and permittivity in a vacuum, respectively. α max is material absorption loss and n mat is refractive index. E and S z indicate electric field distribution and pointing vector in the z-direction.
The obtained EML values are plotted with respect to the frequency at the optimized structure as shown in Fig. 10. At minimum frequency, EML value is high for both progesterone and estradiol hormones. When increasing frequency, the interaction of light is very close in the core material which leads to minimum EML. Also, the minimum EML is obtained at a higher refractive index analyte of hormones. Additionally, Fig. 11 depicts the behavior of the proposed core structure's effective mode area in terms of frequency. It can be calculated by [29], where I(r) indicates the transverse electric field intensity. We found that the effective mode area decreases with an increase in frequency. In addition, the effective area is high at a lower refractive index of hormones due to better light confinement in the core region. The value of effective mode area is noticed as 2.12 × 10 5 µm 2 (10 n mol/L) for progesterone and 2.34 × 10 5 µm 2 (0 n mol/L) for estradiol at 1.35 THz frequency. Further, the detection limit ( n a ) of the proposed THz sensor is 0.0005 and 0.0026 for progesterone and estrogen, respectively.

IV. PREDICTION WITH THE LOCALLY WEIGHTED LINEAR REGRESSION
As it is required to have an optimal structure that can yield high performance and for that simulation is the best way to identify the best-performing structures with optimal parameters. It is more feasible to simulate the structure first and identify the best-performing parameters and for some designs, there would be many parameters to optimize and it would take time for these simulations. Here comes the ML in the picture. ML algorithms can be employed to predict the output for middle wavelengths/frequencies to save simulation time and the simulation can be run for higher step sizes and middle values can be predicted using ML algorithms. This, in turn, will give outputs in lesser time and the optimal structure can also be achieved similarly. Nowadays, Machine learning-based algorithms are vastly applied for the prediction in various photonics devices. Here, a Non-Parametric Learning Method called Locally Weighted Linear Regression (LWLR) will be used. [36], [37]. We'll explore the weighting Function, the predict Function, and finally, visualize the predictions with Python's NumPy and Matplotlib and seaborn libraries. During data training, the ideal value for a parametric algorithm's parameters, such as theta, is sought.
The loss function (J(θ)) for LR is given as: The updated J(θ) for the LWLR is: where, w (i) is the weight and can be expressed as: The prediction will be made for x. To designate the i th training exercise is symbolled as x (i) . This function's value is bounded by zero and one. Therefore, if we examine the function, we see that when |x (i) − x| is small, the weight is close to 1. With a big value of |x (i) − x|, weight is near to 0. Weight is close to 0 for x (i) s that are far from x and near to 1 for x (i) s close to x. As a result, error terms of J(θ) are multiplied by virtually zero for x (i) s very far from x and by almost one for x (i) s close to x. The error terms are only added up for the x (i) values somewhat close to x. In order to determine the size of the circle, we incorporate a hyperparameter tau (τ ) into the weighting function. Circles' diameters can be made wider or narrower by adjusting the hyperparameter (τ ). w (i) now can be written as: 75428 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.    Theta can also be calculated as: Once we have theta, the following expression can be used for predictions: R 2 score, determined in Eq. (11) is used to assess the accuracy of the model.
The prediction plot derived from this algorithm is presented in Fig. 12 -14. Fig. 12 demonstrates the actual vs. predicted relative sensitivity values for Progesterone and Estradiol hormones for various concentrations when the distance between consecutive rectangle holes is kept at 20 µm for the weighting hyperparameter value of 0.02. Fig. 12 (a) shows the actual vs. predicted value for Progesterone hormone for the 0 n mol/L concentration and we can see that, after the relative sensitivity of 97.00 it identically fits the actual values. Fig. 12 (b) demonstrates the actual vs. predicted relative sensitivity of Progesterone hormone for the concentration of 10 n mol/L and we can observe that when relative sensitivity reaches 97.05 the data equivalently fits the actual value. Fig. 12 (c) shows the actual vs. predicted value for Estradiol hormone for the 0 n mol/L concentration and we can see that, after the relative sensitivity of 96.95 it identically fits the actual values. Fig. 12 (d) demonstrates the actual vs. predicted relative sensitivity of Estradiol hormone for the concentration of 3 n mol/L and we can observe that the data equivalently fits the actual value from the beginning.
For the optimization of LWLR, there are several methods and one of the simple ways is to tune the hyperparameters. For the proposed study, τ is the hyperparameter and we have tuned it to optimize the LWLR. And the tuning of hyperparameter, τ has been reported in the manuscript to showcase this. Later, we checked the tuning of the weighting hyper-parameter, τ . For this, we varied the τ in 20, 2, 0.2, 0.02 for both the hormones, and the results are presented in Fig. 13 and 14 separately for the progesterone and estradiol hormones for the concentration of 0 nmol/L. Fig. 13 (a-c) demonstrates the actual vs. predicted values for the progesterone hormone for the τ = 0.2, 2, and 20, respectively. It is quite clear that as we increase the weighting hyperparameter τ , the prediction accuracy decreases drastically. This behavior is also evidentially clear from the Fig. 13 (d) which demonstrates the R 2 score for progesterone hormone for various concentrations of 0 nmol/L, 1 nmol/L, 3 nmol/L, and 10 nmol/L and weighting hyper-parameter, τ n 20, 2, 0.2, 0.02. It is clear that for τ = 0.02, we achieve the highest R 2 score of 1.0 for all concentrations, and for τ = 0.2, the R 2  score in the range of 0.99 is obtained and later the prediction accuracy drastically decreases for τ = 2 and τ = 20. Fig. 14 (a-c) illustrates the actual vs. predicted values for the estradiol hormone for τ = 0.2, 2, and 20, respectively. It is evident from the results that as the weighting hyper-parameter is increased, the prediction accuracy decreases substantially. This is also evident from Fig. 14 (d), which depicts the R2 score for the progesterone hormone at different concentrations of 0 nmol/L, 0.1 nmol/L, 1 nmol/L, and 3 nmol/L and weighting hyper-parameter, τ in 20, 2, 0.2, 0.02. It is clear that for τ , we obtain the highest R 2 score of 1.0 for all concentrations; for τ = 0.2, we receive an R 2 score in the vicinity of 0.99; and the prediction accuracy declines severely for τ = 2 and τ = 20.

V. FABRICATION FEASIBILITY OF THE PROPOSED FIBER
The designed hollow-core PCF-based sensor has a few rectangular-shaped air holes, as can be seen in Fig. 1. Today, capillary stacking, stack and draw, drilling, extrusion, sol-gel, and 3D printing are the most frequently used methods for fabricating PCF [29], [41], [42], [43], [44]. Stack and draw, capillary stacking, and sol-gel techniques are supported to fabricate circular-shaped air hole PCF. Any form of asymmetrical PCF structure can be fabricated by using extrusion and 3D printing technology. The extrusion and drawing by a 3D printer fabrication procedure have been introduced by the National Oceanography Centre, UK, and Optoelectronics Research Centre, UK in order to fabricate suspended structured PCF. The Max Plank Institute, The University Of Adelaide, Australia fabricated several complicated PCF structures using extrusion technology, including rectangular shape air holes and spider-web-shaped PCF [45]. The proposed PCF can be fabricated using the current PCF fabrication technology. Therefore, it is not currently a difficult task to fabricate the proposed rectangular hollow-core PCF.

VI. CONCLUSION
An asymmetric rectangular hollow-core PCF sensor is designed for high detection of progesterone and estradiol hormones in the blood sample at the THz regime (0.8 THz to 1.7 THz). Zeonex is used in the background material for attaining extraordinary sensing properties in the THz region. Using FEM, the numerical sensing performances are evaluated for different concentrations of progesterone and estradiol hormones. The structural parameters such as the strut and core size of the proposed PCF biosensor are optimized to attain maximum relative sensitivity with low loss. The obtained maximum relative sensitivity is 99.87% for 10 n mol/L (progesterone) and 99.88% for 3 n mol/L (Estradiol) at 1.35 THz regime. The Effective material loss (EML) is found around 0.00221 cm −1 and 0.00242 cm −1 for 10 n mol/L (progesterone) and 3 n mol/L (Estradiol), respectively at 1.35 THz regime. Further, the proposed biosensor offered a large effective area such as 2.12 × 10 5 um 2 and 2.34 × 10 5 um 2 for 10 n mol/L (progesterone) and 3 n mol/L (Estradiol), respectively at 1.35 THz regime. Also, this biosensor can be fabricated by current fabrication technologies. The prediction carried out with the help of Locally Weighted Linear Regression, and hyperparameter tuning, we can conclude that for τ = 0.02, the maximum prediction accuracy with the unity R 2 score is observed and this model can be employed for the prediction of relative sensitivity for various parameters. APPENDIX See Table 4.  Table. SHOBHIT K. PATEL (Senior Member, IEEE) received the Ph.D. degree in electronics and communication engineering from the Charotar University of Science and Technology, Changa, India. He is currently working in the areas of photonics, metamaterial, antenna, optics, and artificial intelligence. He has published several research articles in high-impact SCI journals. He has also filed seven Indian patents on different novel research done by him. He received DST International Travel Grant, in 2014, to present a paper at the IEEE APS-URSI Symposium in Memphis, USA. He also received the DST International Travel Grant, in 2017, to present a paper at the PIERS Symposium, NTU, Singapore. He was named in the list of ''top 2% scientists worldwide identified by Stanford University,'' in 2021. He is currently working on many graphene-based projects and has received funding from SERB and DST, for his research. He has been honored with awards for his achievements in the area of research field. S. N. DEEPA is currently an Associate Professor in electrical engineering with the National Institute of Technology Arunachal Pradesh, Jote, Arunachal Pradesh. She possesses 18 years of teaching and research experience and has published ten books with national/international publishers, 106 international journal articles, 12 national journal articles, and 53 papers at national/international conferences. Her research interests include linear and non-linear control system design and analysis, soft computing techniques, evolutionary strategies, adaptive and robust control systems, and medical image processing techniques. She is recognized in the Top 2% of the world's scientist list ranked by Stanford University, USA. With respect to her research attributes, she possesses 9174 citations, Google Scholar H-index of 22 and an i10-index of 58, her Scopus H-index is 16, and the Web of Science H-index is 12 as on date. Her research gateway score is 28.43. FAHAD AHMED AL-ZAHRANI received the B.Sc. degree in electrical and computer engineering from Umm Al-Qura University, Makkah, Saudi Arabia, in 1996, the M.S. degree in computer engineering from the Florida Institute of Technology, in 2000, and the Ph.D. degree in computer engineering from Colorado State University, in 2005. He is currently a Professor with the Computer Engineering Department, Umm Al-Qura University. He has taught several computer network courses and supervised related research projects. From 2011 to 2016, he was the IT Dean of Umm Al-Qura University and has had several other responsibilities. His research interests include high-speed network protocols, sensor networks, optical networks, performance evaluation, the IoT, blockchain architecture, and performance analysis. He is a member of the International Society for Optical Engineering and the Optical Society of America.