Terahertz dielectric spectroscopy of human brain gliomas and intact tissues ex vivo : double-Debye and double-overdamped-oscillator models of dielectric response

: Terahertz (THz) technology offers novel opportunities in the intraoperative neurodiagnosis. Recently, the significant progress was achieved in the study of brain gliomas and intact tissues, highlighting a potential for THz technology in the intraoperative delineation of tumor margins. However, a lack of physical models describing the THz dielectric permittivity of healthy and pathological brain tissues restrains the further progress in this field. In the present work, the ex vivo THz dielectric response of human brain tissues was analyzed using relaxation models of complex dielectric permittivity. Dielectric response of tissues was parametrized by a pair of the Debye relaxators and a pair of the overdamped-oscillators – namely, the double-Debye (DD) and double-overdamped-oscillator (DO) models. Both models accurately reproduce the experimental curves for the intact tissues and the WHO Grades I–IV gliomas. While the DD model is more common for THz biophotonics, the DO model is more physically rigorous, since it satisfies the sum rule. In this way, the DO model and the sum rule were, then, applied to estimate the content of water in intact tissues and gliomas ex vivo . The observed results agreed well with the earlier-reported data, justifying water as a main endogenous label of brain tumors in the THz range. The developed models can be used to describe completely the THz-wave – human brain tissues interactions in the frameworks of classical electrodynamics, being quite important for further research and developments in THz neurodiagnosis of tumors. dielectric permittivity using


Introduction
Human brain gliomas are classified by the World Health Organization (WHO) into Grades I to IV [1]. They are among the most common and deadly pathologies of the brain, constituting ∼ 26% of For the freshly-excised tissues, they were attributed to the increased water content in a tumor due to the abnormal microvascularity, edema, and body fluids around necrotic debris. In turn, for the paraffin-embedded tissues, differences occurred owing to changes of cell density in a tumor. Furthermore, a contrast between the gray matter and the white matter was clearly observed, reportedly originating from higher content of myelin in the white matter [30]. In Ref. [31], paraffin-embedded intact tissues and glioma model (GL261 cell line) from mice were studied ex vivo using THz pulsed spectroscopy. The observed results justified a contrast between intact and tumorous tissues in paraffin blocks and revealed some optimal spectral bands and features for the differentiation between healthy and pathological tissues. In Ref. [32], intact tissues and glioma model (C6 cell line) from rats were studied using THz pulsed spectroscopy. By analyzing complex refractive index of tissues, the authors estimated the increased water content in a tumor and claimed it as the main factor underlying a contrast between intact and tumorous tissues. In Refs. [33][34][35], a potential of THz reflectometry and imaging in the intraoperative neurodiagnosis was highlighted involving ex vivo and in vivo glioma models (eGFP+GSC-11, C6, and U87-MG cell lines) from mice and rats, as well as few ex vivo high-grade gliomas of the human brain.
Despite the aforementioned favorable results of the ex vivo and in vivo THz measurements of glioma models, the latter only partially mimic biophysical properties of human brain tumors, while their THz dielectric response can differ in a large extent. Therefore, in our previous works [36,37], we applied THz pulsed spectroscopy to study ex vivo the optical properties of gelatin-embedded intact tissues and WHO Grades I-IV gliomas of the human brain, where gelatin embedding allowed us to preserve tissues from hydration/dehydration and, thus, to sustain their THz response unaltered as compared to the freshly-excised ones [38]. From Fig. 1, we noticed statistical differences between the THz response of intact tissues and gliomas of all WHO grades, while the response of edematous tissue was close to that of a tumor. Evidently, edema (along with traumatic brain injuries [39,40]) can be confused with a tumor, thus, complicating delineation of its margins. In spite of such a rapid progress in THz diagnosis of gliomas, there is still a lack of physical models describing dielectric permittivity of healthy and pathological human brain tissues in the THz range. In order to mitigate this problem, in this work, we analyze the ex vivo THz complex dielectric permittivity of intact tissues and WHO Grade I-IV gliomas using two distinct models -namely, the double-Debye (DD) model, based on a pair of the Debye relaxators, and the double-overdamped-oscillator (DO) model, based on a pair of the Lorentz oscillators with a high damping constant. We demonstrate a simple mathematical relation between these models. Both models are shown to accurately reproduce experimental data for the intact tissues and gliomas. While the DD model is more usual in THz biophotonics, the DO model is more physically rigorous, satisfying the sum rule. Therefore, in this work, the DO model and the sum rule are applied to estimate the content of water in human brain tissues. The observed results agree with the earlier-reported data, justifying water as a main endogenous label of brain tumors in the THz range. Thus developed models of THz dielectric response of human brain tissues could bring significant contribution to further research and engineering efforts in the THz intraoperative neurodiagnosis.

THz pulsed spectroscopy of human brain tissues ex vivo
In this work, we consider the same set of 26 human brain glioma samples ex vivo, which was used in our previous paper [36]. It comprised of 2, 9, 4, and 11 samples of glioma tissue featuring WHO Grades I to IV, respectively. After resection in the Burdenko Neurosurgery Institute (Moscow, Russia), the tissue specimens were fixed by gelatin slabs for further characterization using THz pulsed spectroscopy in the frequency range of 0.2-1.5 THz. Gelatin embedding of tissues allowed to sustain hydration and, thus, THz response of tissues ex vivo unaltered (as compared to those of freshly-excised tissues ex vivo) during transportation and THz measurements [38].
For tissue characterization, a homemade THz pulsed spectrometer was used, the detailed description of which is given in Refs. [37,41]. It is equipped with a housing for the THz beam path purging with nitrogen gas, which yields suppression of a water vapors' impact on the measured data. At the same time, the measured sample is handled atop of the flat crystalline quartz window at an ambient room atmosphere, while the window serves simultaneously as a sample holder and as a spacer between the nitrogen gas and an ambient atmosphere. The applied spectrometer operates in reflection mode and uses a pair of gold-coated off-axis parabolic mirrors in order to focus the THz beam on the "crystalline-quartz -tissue" interface and to collimate the reflected part of the THz beam. Detailed description of this experimental setup and outline of the mathematical procedures applied for reconstruction of the tissue THz optical properties can be found in Ref. [36].
After the spectroscopic measurements, the tissue samples were fixed in formalin and transferred to the Burdenko Neurosurgery Institute for further routine histopathological examination, which involves, in particular, H&E-stained histology; see Fig. 1. It is worth noting that, along with the WHO Grade I-IV gliomas, we analyzed perifocal tissues, which often surround a tumor and are comprised of intact (healthy) and edematous tissues with the distinct THz response, as evident from Fig. 1.
Thus reconstructed optical properties of intact tissues, edema, and WHO Grade I-IV gliomas of the human brain are shown in Fig. 1. These data form a basis for our analysis of picosecond relaxation dynamics in tissues involving methods of dielectric spectroscopy. For this aim, it is convenient to represent the THz spectroscopy data in form of a complex dielectric permittivity˜︁ ε defined as˜︁ where ε ′ and ε ′′ are its real and imaginary parts, respectively. It is related to the complex refractive index˜︁ n, or the refractive index n and absorption coefficient α (by field) as︁ where, n ′ ≡ n and n ′′ stand for its real and imaginary parts, correspondingly, c 0 ≃ 3 × 10 8 m/s is the speed of light in free space, ν is an electromagnetic-wave frequency.

Relaxation models of tissue dielectric response in the THz range
In our analysis, two distinct models of complex dielectric permittivity, which account for the picosecond relaxation dynamics of tissue water, are considered. First one is the DD model, which is comprised of the two Debye relaxation terms and widely exploited in THz biophotonics [24,25,42,43] where ω = 2πν is an angular frequency, ∆ε 1 , ∆ε 2 are magnitudes that regulate a contribution of "slow" and "fast" Debye relaxations with the time constants τ 1 , τ 2 to the tissue dielectric response, while ε ∞ is a constant dielectric permittivity at high-frequencies (compared to the analyzed spectral range, ν ≫ (︁ 2πτ 1,2 )︁ −1 . This model is quite convenient and yields accurate parametrization of the THz spectroscopic data for water, water solutions, biological liquids and tissues; only 5 independent coefficients are required for this aim -ε ∞ , ∆ε 1 , ∆ε 2 , τ 1 , and τ 2 . In Fig. 2, the DD model is shown for the dielectric response of liquid water with the parameters ε ∞ = 4.1, ∆ε 1 = 72.2, ∆ε 2 = 2.5, τ 1 = 10.6 ps, and τ 2 = 0.18 ps. reported in Ref. [42]. This DD model describes two broad absorption bands in ε ′′ -curve, which are attributed to the "slow" and "fast" relaxations of water and centered at the inverse relaxation times of (2πτ 1 ) −1 , (2πτ 2 ) −1 . In turn, ε ′ -curve tends to decrease with increasing frequency featuring the high-frequency magnitude ε ∞ and the low-frequency magnitude ε ∞ + ∆ε 1 + ∆ε 2 .  4)) models of complex dielectric permittivity˜︁ ε for a liquid water: (a),(b) real ε ′ and imaginary ε ′′ parts of˜︁ ε, respectively. The DD model is plotted based on the parameters from Ref. [42], while the DO model is calculated via Eq. (5).
Second, we consider the well-known in condensed matter physics DO model [44,45], which is comprised of two Lorentz terms with a high damping constant︁ where ∆ε 1 , ∆ε 2 , ω 01 , ω 02 , and γ 1 , γ 2 stand for magnitudes, quasi-resonant frequencies (given by the restoring forces) and damping constants, respectively, of the "slow" and "fast" relaxation processes, that are equivalent to the "slow" and "fast" Debye relaxations in Eq. (3). It is worth noting that a constant ε ∞ and magnitudes of each i-th Debye relaxation term ∆ε i are equal to that of a corresponding overdamped oscillator. Other parameters of the i-th overdamped oscillator can be easily and univocally calculated based on parameters of the corresponding Debye relaxation term by resolving the following equations: Given that the inequality C ≫ 1 is satisfied, particular value of constant C is not so important. For example, in this work, we assumed this parameter to be as large as C = 10 2 . In the overdamped oscillator, parameters ω 0,i , γ i loose their separate physical meaning, and only their combination ω 0,i /γ i = τ −1 i defines maximum in the loss spectrum -i.e. in the ε ′′ -curve [44]. In Fig. 2, the DO model is calculated for water using Eq. (5) based on the abovementioned parameters of the DD model from Ref. [42]. It should be noticed that the overdamped oscillators reproduce the shape of dielectric permittivity curves ε ′ , ε ′′ .
From Fig. 2, it is clear that both the Debye kernel and the overdamped oscillator are practically equivalent at low-frequencies up to ω ∼ (2πτ i ) −1 <ω 0,i ≪ γ i ; but for ω> (2πτ i ) −1 , the Debye model predicts higher losses that those of the overdamped oscillator [44]. Unlike the overdamped oscillator, the Debye kernel does not fulfill the sum rule for the oscillator strengths and, thus, predicts infinite number of charge carriers / dipoles N underlying the dielectric response of a medium [44,46,47] N ∝ In turn, integration of the conductivity given by the i-th overdamped-oscillator term results in a finite value ∆ε i ω 2 is the number of dipoles contributing to the i th relaxation process, while q i and m i are effective charge and mass of each dipole. Therefore, for the high-frequency range and for the estimation of a number of charges/dipoles in a medium, the overdamped oscillator model is preferable, being more physically rigorous than the Debye or related models, such as the Cole-Cole [48,49], Davidson-Cole [50], or Havriliak-Negami [51]. Such an advantage of the overdamped oscillator over the Debye relaxator makes it more appropriate for THz biophotonics.

Modeling complex dielectric permittivity of human brain tissues ex vivo
The described DD and DO models of dielectric permittivity were applied for fitting the results of the ex vivo THz pulsed spectroscopy of intact tissues and WHO Grade I-IV gliomas of the human brain, which are reprinted in Fig. 1.
For this aim, the THz responses of all tissues were recalculated in form of the real ε ′ and imaginary ε ′′ parts of the complex dielectric permittivity˜︁ ε (Eqs. (1), 2). Then, parameters of the DD model were calculated by fitting of the experimental data with Eq. (3), for which we exploited the weighted least-square method in MATLAB software. For all considered tissue types, we set the time constants of "slow" and "fast" relaxation processes τ 1 , τ 2 to be equal to that of water from Ref. [42], which reduced the problem dimensionality and simplified the fitting procedure, providing appropriate model parameters. It should be noticed that the experimental results of each THz measurement for each considered sample were fitted by the DD model independently, which, then, allowed us to calculate both mean values and standard deviations of the calculated model parameters. Finally, parameters of the DO model (Eq. (4)) were calculated based on those of the DD one via Eq. (5).
In Fig. 3, we illustrate the experimental ε ′ , ε ′′ -curves, which are overlapped with the DD and DO models of complex dielectric permittivity, as well as compared with the DD model of liquid water from Ref. [42]. It is clear from these results that these two models appear to be equal in the considered spectral range, accurately reproducing the experimental data and accommodating the experimental error bars. As expected, both ε ′ -and ε ′′ -curves of tissues response are below those of a liquid water, but feature quite similar frequency-dependent character. Fig. 3. Ex vivo THz complex dielectric permittivity˜︁ ε of the intact tissues, edematous tissues and gliomas of the human brain: (a),(b) measured real ε ′ and imaginary ε ′′ dielectric permittivity parts for the intact brain tissues, as compared with the DD (Eq. (3)) and DO (Eq. (4)) models; (c)-(l) the same data for the edematous tissues and gliomas; (m) the DD model parameters for water and tissues, from which parameters of the DO model can be calculated using Eq. (5). In (a)-(l), dielectric properties of tissues are compared with the DD model of liquid water from Ref. [42]. Here, GI-GIV stands for the WHO Grade I-IV gliomas.
Here, one can notice that slightly larger error bars are observed for the intact tissues as compared to those of the edema and gliomas, which can be attributed to the unaccounted difference between the THz optical properties of white matter and gray matter. Indeed, due to small amount of intact tissues studied in Ref. [36], we did not differentiate these two types of intact tissues during measurements of perifocal regions of a tumor. At the same time, white matter and gray matter possess distinct response at THz frequencies due to (predominantly) the different content of myelin [29,30].
The dielectric parameters ε ∞ , ∆ε 1 , ∆ε 2 , which are equal for the DD and DO models, are presented in Fig. 4 for all considered groups of human brain tissues. From panel (a), it is evident that ε ∞ is higher for the edematous tissues and gliomas of all grades, as compared to that of the intact tissues. This difference can be attributed to the increase water content and the increased cell density in edematous tissue and a tumor [29,52,53]. The increased magnitudes of both "slow" and "fast" relaxations ∆ε 1 , ∆ε 2 for all tumor grades, observed from Fig. 4, might be attributed to the higher content of polar H 2 O molecules in such tissues [26]. At the same time, dielectric parameters ε ∞ , ∆ε 1 , ∆ε 2 are smaller for tissues than for water, as expected, since water content is limited by f <100%. Fig. 4. Dielectric parameters ε ∞ , ∆ε 1 , ∆ε 2 for the intact tissues, edema and gliomas, which are equal for the DD and DO models: (a) dielectric constant at high frequencies ε ∞ ; (b) magnitudes of the "fast" and "slow" relaxation processes ∆ε 1 , ∆ε 2 . The error bars in (a) and the colored areas in (b) represent ±3σ confidential intervals. The data is compared with the DD model of liquid water from Ref. [42]. Here, GI-GIV stand for the WHO Grades I-IV gliomas. In (b), we show a blue line, at which a ratio between the two relaxation terms ∆ε 1 /∆ε 2 is constant and equal to that of a liquid water. Different magnitudes of the two relaxation processes ∆ε 1 , ∆ε 2 , observed in Fig. 4 for the liquid water and tissues can be analyzed from the viewpoint of water bonding and segregation. As demonstrated in Ref. [54], dielectric constants (ε ∞ , ∆ε 1 , ∆ε 2 ) are much smaller for bound water than for free. Furthermore, "slow" relaxation term of bound water (∆ε 1 ) appears to be suppressed much stronger than that of the "fast" one (∆ε 2 ), which reducing the ratio ∆ε 1 /∆ε 2 , as compared to a liquid water. In turn, segregation can change dynamics of water molecules and, thus, a ratio ∆ε 1 /∆ε 2 in a complex and less-predictable manner [55]. When liquid water interacts with a medium without any pronounced changes in its dynamics, we can expect it to appear on a blue line in Fig. 4(b), which define a constant ratio ∆ε 1 /∆ε 2 equal to that of a liquid water. For the WHO Grade IV gliomas, this ratio is similar to a liquid water, which might be attributed to a large amount of free water in such tissue due to necrotic debris. This ratio is smaller for the intact tissues and WHO Grade I, III gliomas, which might point out water bonding. Finally, an increased ratio is inherent to the WHO Grade II gliomas and edema, which is difficult to quantify at this time, and more investigations are in order. One should notice that THz spectroscopy seems to be sub-optimal for the analysis of tissues water bonding and segregation owing to very broadband absorption peaks given by the "fast" and "slow" relaxations; see Fig. 1. More physically-reasonable analysis of water dynamics in healthy and pathological tissues, aided by different instruments (for example, infrared Fourier spectroscopy [45], backward-wave oscillator spectroscopy [56], and Raman spectroscopy [57]), seems to be prospective for addressing this challenge.
It is worth noting that dielectric parameters ε ∞ , ∆ε 1 , ∆ε 2 can serve as physically-reasonable principle components for the differentiation between healthy and pathological human brain tissues, when using THz pulsed spectroscopy as an intraoperative diagnosis tool, as suggested in Ref. [58] for the THz diagnosis of basal cell carcinoma of the skin. However, we postponed a detailed analysis of such an opportunity to our future studies.

Estimation of water content in tissues
Water content in tissues f can be approximately estimated involving the sum rule where integration of imaginary part ε ′′ of the complex dielectric permittivity within the considered spectral range of ω min to ω max is involved for tissues, in the numerator, and for a liquid water, in the denominator. In our opinion, such an estimation relies on the first principles and, thus, forms a favorable alternative to the simplified linear spectral decomposition technique [25], the method proposed by Yamaguchi et al. [32], or others [59]. As we would show below, it yields quite physically reasonable data, agreeing well with other considerations. In Fig. 5(a), the calculated water content f is shown for the intact tissues, edema, and WHO Grade I-IV gliomas ex vivo. We observed the water content of 71.1 ± 6.8% in intact tissues, while that in edema and all glioma Grades is ∼ 5-10% higher. WHO Grade II-IV gliomas show the highest content of water, while for the WHO Grade IV gliomas it can be as large as 80.6 ± 8.6%. In Fig. 5(a), we also notice quite large error bars, which are given by fluctuations of the dielectric permittivity model parameters over the considered set of tissue specimens; they are attributed to the native variability of endogenous optical properties of tissues [60,61]. Other possible reason of such variability originates from uncertainties of the experimental data fitting by the two broad absorption bands given by Eqs. (3) and (4), which are centered slightly above (or at the edges) of the spectral operation range of THz pulsed spectrometer (see Figs. 1, 2) [24].  5. Water content in healthy and pathological tissues of the brain measured using different experimental techniques: (a) intact tissues, edema, and WHO Grade I-IV gliomas of the human brain ex vivo measured by the THz pulsed spectroscopy in this work; (b) intact tissues and C6 glioma model from rat brain ex vivo measured by the THz pulsed spectroscopy in Ref. [32]; (c) healthy human brain tissues and tumoral edema in vivo measured by MRI in Refs. [62][63][64], where WM and GM stand for white matter and gray matter, respectively; (d) healthy rat brain tissues ex vivo measured by a pycnometer in Ref. [65]. Here, error bars represent fluctuations of water content within the considered set of tissue specimens.
Thus estimated water content in human brain tissues can be compared with few well-known studies, which were based on different experimental techniques. In Fig. 5(b), we show water content in freshly-excised intact tissues and C6 glioma model from the rat brain ex vivo, estimated from the THz pulsed spectroscopy data by Yamaguchi and co-authors [32]. Despite they revealed slightly higher average water content in both intact tissues and glioma model, the observed difference between them is quite similar to our estimations, thus, demonstrating ∼ 5-10% higher water content in tumors. Such a variance between the results of the two compared studies can be due to both different methods applied for water content estimation based on processing of the THz pulsed spectroscopy signals and (possibly) distinct water content, which changes the THz response of human and rat brain tissues (both healthy and pathological).
Next, in Fig. 5(c), we consider water content in healthy brain tissues (including white matter and gray matter) and tumoral edema of the human brain studied by MRI in Refs. [62,64]. These MRI-based estimations demonstrate that our data is quite reasonable. The MRI data on water content in tumoral edema from Ref. [63] appeared to be very close to that in intact tissues from Ref. [62], which appears to be surprising and even illogical; thus, we attribute this effect to possible differences of methods and signal processing routine applied in these two MRI-based studies. Estimations from Ref. [64] demonstrate that water content in white and gray matters statistically varies, which should lead to distinct response of such tissues in the THz ranges; thus, this effect should be accounted in our future studies.
Finally, by means of the results from Ref. [65], we consider estimation of water content in healthy rat brain tissues using pycnometry. This technique relies on a completely different physical principle of water content estimation, but shows quite similar results as compared to our data; thus, additionally justifying its reasonability.

Discussion
As mentioned above, the developed DD and DO models of complex dielectric permittivity (see Eqs. (3)-5 along with Fig. 3), allow us to describe the THz-wave interactions with both intact tissues and WHO Grade I-IV gliomas of the human brain ex vivo. In particular, this provides an ability to: • model the THz-wave interaction with layered media using a plane-wave approximation along with the Fresnel equations (for describing the THz-wave transmission and reflection at interfaces between media), and the modified Bouguer-Lambert-Beer law (for describing the THz-wave phase shifts and absorption in a bulk medium); such modeling is of great importance in THz spectroscopy of tissues [24,25]; • simulate numerically the THz-wave interaction with biological objects of a complex shape using methods of computational electrodynamics or statistical Monte-Carlo approaches; such modeling is of high importance for the correct understanding of THz radiation transport in biological tissues [24,25]; • apply parameters of the DD or DO models as physically-reasonable principal components for the discrimination between different tissue types [58].
Therefore, the developed models can be quite useful for further research and engineering efforts in THz biophotonics, in general, and THz neurodiagnosis of tumors, in particular. It is worth noting that the two analyzed relaxation bands, that underlie the dielectric response of water and tissues in the GHz-THz range (see Fig. 2), are centered beyond or at the edges of the considered spectral range of our THz pulsed spectrometer. This can lead to some inaccuracies of the experimental data parametrization by the dielectric permittivity models. Evidently, method of broadband dielectric measurements, which combines backward-wave oscillator spectroscopy, THz pulsed spectroscopy, Fourier-transform infrared spectroscopy and other techniques [45,56], can help to obtain much accurate data about the picosecond relaxation dynamics of liquid water and tissues. Despite attractiveness of such broadband measurements, they still remain quite a daunting task due to technical difficulties and time constraints of tissue measurements using the above-mentioned instruments. Therefore, in our study, we used only the THz pulsed spectroscopy with its quite broad spectral operation range and technical flexibility, while we postponed broadband measurements to our future studies.
Real biological systems might possess more complex dielectric response in the THz range. Particularly, its dielectric permittivity model can be comprised of three or even more relaxation terms, each of which represents a particular fraction of free/bound or segregated water, with its distinct electrodynamic characteristics [54,55]. In general, dielectric response of such media can obey more complex regularities, involving integration of the relaxation process kernel over a continuous spectrum of the relaxation states; for example, for the Debye relaxation kernel, such integral response has the form [44,66,67] where g (τ) is a continuous relaxation-time spectrum. Overall, we still have quite limited understanding of relations between the complex molecular dynamics of liquid and tissue water [68] and its broadband electrodynamic response. Nevertheless, the detailed analysis of such complex dielectric permittivity models and of underlying physical mechanisms is a topic of further comprehensive research work. Such models can be overabundant for the considered broadband relaxation electrodipole excitations of matter, while the considered DD and DO models provide quite accurate description of biological tissue response in most cases [24,25]. All data presented in this paper were obtained and analyzed relying on an assumption of homogeneous tissue character at the scale posed by the THz wavelengths, which is typical for the THz biophotonics [24,36]. This assumption allows for using the effective medium theory and parametrizing the experimental data by different dielectric permittivity models. However, an increasing number of research papers demonstrates scattering properties of tissues in the THz range. For example, in Ref. [69], the authors shown that the cross-polarization THz imaging provides some contrast for the tumor margins delineation, while such polarization changes cannot be predicted in the framework of the effective medium theory. Furthermore, our latest results on the THz solid immersion microscopy [27,70,71] reveal structural inhomogeneities of different biological tissues ex vivo, that can become the origin of the Mie scattering effects. For the considered intact tissues and gliomas of the human brain, structural heterogeneity can occur thanks to the different optical properties of healthy white and gray matters [30], the neurovascular structural elements of the brain, the presence of necrotic debris in the WHO Grade IV gliomas [18], or other factors. Nevertheless, such a heterogeneous character of human brain tissues still have not been studied in the THz range, and it deserves further comprehensive research work.
The described methods for estimation of water content in biological tissues relying on the THz pulsed spectroscopy signal processing and the sum rule holds strong potential in THz biophotonics, where water plays a role of endogenous label of pathological processes in most cases [24,25]. At the same time, estimation of the water content in tissues based on THz data might be carried using alternative approaches. Among them is the simplified linear spectral decomposition technique used in earlier studies. Otherwise, number of water dipoles in system can be calculated by analytical integration of the sample dynamic conductivity predicted by the DO model. However, for this aim, one should know effective masses and charges of dipoles, that contribute to the different relaxation processes and that cannot be estimated based on the THz data. Therefore, in this paper, we used simplified numerical integration of the sample dynamic conductivity within the spectral operation range of our spectrometer. This method resulted in quite reasonable predictions of water content in tissues, which agreed well with the earlier-reported data; thus, justifying correctness of our considerations.

Conclusion
In this study, we analyzed the THz complex dielectric permittivity of the ex vivo intact tissues and WHO grade I-IV gliomas of the human brain using the DD and DO models. It is worth noting that the obtained results are consistent with the previously-reported data, which involved measurements of both glioma models from rats and mice, and few tumors of the human brain [28][29][30][31][32][33][34][35][36][37]. Human brain tumors and edematous tissues feature higher refractive index and absorption coefficient in the THz range that correlates well with the previously-reported differences between intact tissues and tumors of the brain, that were represented in form of the THz time-domain waveforms or tissue reflectivity. It was demonstrated that both models accurately reproduce our spectroscopic measurements of brain tissues in the THz range. We show that the DO model is more physically rigorous, since it satisfies the sum rule. Estimation of water content in tissues relying on the THz pulsed spectroscopy data yields the results that are in agreement with the earlier-reported measurements involving different experimental techniques, which justifies water as a main endogenous label of brain tumors at THz frequencies. Thus developed DD and DO models yield description of THz-wave -human brain tissues interactions, being quite useful for further research and engineering efforts in the area of THz neurodiagnosis.