# Monitoring Theophylline Concentrations in Saline Using Terahertz ATR Spectroscopy
## 3 Results and Discussion  
The amplitude of THz wave reflectance of the samples was measured by comparing the sample spectrum and reference spectrum. Reference spectra were obtained using a mirror. Here, we discuss the evanescent field penetration depth, \(d_{p}\), which is the distance between the interface and a point where the intensity of light becomes 1/e. Thus, \(d_{p}\) is described as  
\[d_{p}=\lambda_{2}/2\pi\sqrt{({n_{1}}^{2}/{n_{2}}^{2}){\rm sin}^{2}\theta_{1}-1} \tag{1}\]  
where \(\lambda\) is wavelength of the light, \(n_{1}\) is the refractive index of ATR prism, \(n_{2}\) is the refractive index of the sample, and \(\theta_{1}\) is the incident angle of the light. The frequency dependence of the penetration depth \(d_{p}\) in this study is shown in Figure 2. As the wavelength increases, the penetration depth, \(d\), increases. This indicates that the evanescent wave at low frequency propagates inside objects, rather than a high frequency wave in the ATR method. Thus, we can expect that a terahertz wave at a lower frequency will have a larger interaction between light and a solution. Thus, we used data for a lower frequency THz wave for the analysis in this study. In terms of NaCl solution, physiological saline is 0.9% NaCl solution. In the process of making saline solutions, we observed ATR spectra across different concentrations of NaCl solution using the ATR method. Reflection spectra were estimated by a division of the sample spectra by the reference spectra. As the concentration of NaCl increased, the amplitude of reflection spectra decreased (Figure 3). It is assumed that the decrease was caused by a change in the distance of the penetration depth due to the change of refractive index with the increase in NaCl concentration. The penetration depth is calculated with Equation (1). A change in refractive index causes a change in the value of \(d_{p}\). Because water and solutions have strong absorptions for THz waves, a large change in ATR spectra occurs with a small change in the distance. Thus, it is possible to observe the concentration of solutions using THz ATR measurements. Next, we measured the concentration dependence of theophylline in saline. The amplitude of the reflected spectra of the theophylline solution also changed with the change in theophylline concentration. The reflectance at 0.164 THz changed from 0.071 to 0.073 from the change of the concentration from 2.7 mg/mL to 5 mg/mL (Figure 4). The data shown in Figure 4 are from triplicate experiments. The concentration change was smaller than that of the NaCl solution. In the theophylline solution, the refractive index did not change with the concentration change, but amplitude changes in the ATR spectra were observed. These changes seemed to have occurred due to a change in the state of the solution. For water, strong absorptions exist below 1 THz. One of the absorptions is due to the Debye relaxation mode (\(\tau_{1}\)), around 0.01 THz, and a second is due to the fast relaxation mode (\(\tau_{2}\)) around 0.8 THz [3]. Theophylline in solution exists in molecular form and a hydrated state, because its dissociation constant is low. Thus, some water molecules make hydrogen bonds with theophylline, and they will change motions. The \(\tau_{1}\) mode is from the relaxation of the rotation of a water molecule, which connects with theophylline via a hydrogen bond, and the \(\tau_{2}\) mode is from the rotation of a free  
Figure 3: NaCl concentration dependence of amplitude of reflection spectra obtained at 0.86 THz. Figure 2: Frequency dependence of the penetration depth \(d_{p}\). water molecule. Thus, the \(\tau_{2}\) mode does not shift, but part of the \(\tau_{1}\) mode shifts to a lower frequency region when more theophylline is present. The shift in the \(\tau_{1}\) mode causes decreased absorption of THz waves by water, so the amplitude of the reflected THz spectra becomes larger. This is consistent with the experimental results.