1.1.

Theory

Pickwell ¹³ used a FDTD model, utilizing Maxwell’s equations and Debye theory, to describe the propagation of terahertz pulses through water and tissue. For completeness, in this work we outline the fundamental principles of the model and then use it to explain the differences in reflected pulses from carcinoma and normal tissue. References 12, 15, 16 conclude that in the terahertz regime the response of water can be fitted by a double Debye model characterized by a slow relaxation mode $\left({\tau }_1\right)$ and a fast relaxation mode $\left({\tau }_2\right)$ . Debye theory couples the relaxation of the local polarization $P$ to the local electric field $E$ , characterized by time constant $\tau$ and coupling strength $\gamma$ :

The key concept in the simulation is to incorporate the time dependence of the polarization using double Debye theory in Maxwell’s equations (as in Ref. 13). From these equations, we derived iterative equations to model the polarization and the local $E$ and $H$ fields, thus enabling the simulation of the propagation of a terahertz pulse through the discretized region in the time domain. In this work, we study how the double Debye parameters influence the shape of simulated reflected waveforms and which parameters are dominant, and identify reasons for the differences between the waveforms reflected off normal skin tissue and BCC.

3.1.

Complex Refractive Index and Double Debye Parameters of Normal Skin and Basal Cell Carcinoma

We calculated the refractive index and absorption coefficient from the spectral data for normal skin and BCC from the same site by using the mean data from all ten patients. These data are plotted with 95% confidence limits in Fig. 1 . From Fig. 1(a) we see that the refractive index $n$ of the tissue containing BCC is higher than the normal tissue throughout the frequency range, and that the difference is more pronounced between 0.25 and $0.75\mathrm{THz}$ . The absorption coefficient $\alpha$ [Fig. 1(b)] for BCC is also consistently higher than for the normal tissue at $0.5\mathrm{THz}$ . There is a difference of $16{\mathrm{cm}}^{-1}$ , which increases to $25{\mathrm{cm}}^{-1}$ at $2\mathrm{THz}$ . Using this data in the model, the Debye parameters were extracted and are given in Table 1 . For comparison, we consider in vivo measurements of $n$ and $\alpha$ of tissue from Ref. 14. This was calculated from reflection data from the volar forearm. We find that the $n$ and $\alpha$ calculated from the forearm are much lower than $n$ and $\alpha$ from the normal tissue ex vivo data, and thus there are differences in the Debye parameters given in Table 1. We expect there to be some differences between in vivo and ex vivo data, because tissue changes after excision, e.g., interstitial fluid, is present around the tissue due to slight compression and body site variation. Due to the differences between ex and in vivo measurements, we consider the data separately, and it is noted that ex vivo spectroscopy gives higher values for $n$ and $\alpha$ when compared to in vivo reflection data.

Fig. 1

(a) The refractive index and (b) the absorption coefficient of normal tissue and that containing BCC. The error bars represent 95% confidence limits.

Table 1

Double Debye parameters for ex vivo normal tissue and that containing BCC compared to our parameters previously published for in vivo normal tissue (epidermis) and water.

The complex refractive index from the ex vivo data between 0.2 and $2\mathrm{THz}$ was fitted to the frequency-dependent expression for the complex dielectric coefficient in Eq. 2, and the double Debye parameters for the normal tissue and the BCC were extracted. These parameters are given in Table 1, alongside our parameters for in vivo normal tissue (epidermis) and water.

The double Debye parameters are dependent on the complex permittivity which, in turn, is dependent on $n$ and $\alpha$ . By changing $n$ and $\alpha$ separately, we look at how the double Debye parameters affect the corresponding waveform properties. We have changed $n$ for normal tissue from $-0.06$ and 0.25 in steps of 0.01 while keeping $\alpha$ fixed. Then we changed $\alpha$ from $-10$ and $40{\mathrm{cm}}^{-1}$ at intervals of $10{\mathrm{cm}}^{-1}$ while keeping $n$ fixed. For each new complex refractive index, the double Debye parameters were calculated. The resulting graphs are illustrated in Figs. 2(a), 2(b), 2(c), 2(d), 2(e) .

Figure 2(a) shows that ${\epsilon }_{\infty }$ is more dependent on $n$ than $\alpha$ . This is expected, as ${\epsilon }_{\infty }$ is the high frequency limit of the dielectric constant, and $n$ converges as the frequency increases. Therefore, increasing $n$ causes an increase in ${\epsilon }_{\infty }$ . Increasing $\alpha$ also increases ${\epsilon }_{\infty }$ , although to a lesser extent. This is because the imaginary part of the complex dielectric coefficient falls to zero at the high frequency limit. Conversely, the low frequency limit parameter ${\epsilon }_s$ is more dependent on changes in $\alpha$ than $n$ , because at low frequencies terms containing $\alpha$ dominate [Eq. 3]. It is interesting to observe that ${\epsilon }_2$ is more sensitive to changes in $n$ than $\alpha$ over this frequency range. This suggests that the intermediate frequency limit corresponds to nearer the higher frequencies rather than the lower. As expected, both the relaxation times ${\tau }_1$ and ${\tau }_2$ are more sensitive to $\alpha$ than $n$ . Increasing $\alpha$ increases the relaxation times, which means that reorientation processes are slowing down. In tissue, this could be due to the increased presence of structural proteins to which more water molecules are bound. Thus, a material with higher attenuation will have more bound water molecules which, in this case, hinders the propagation of terahertz radiation and will broaden waves passing through.

3.2.

Simulations of the Interaction of Terahertz Radiation with Basal Cell Carcinoma

The ex vivo parameters in Table 1 were entered into the FDTD model to simulate the reflection off a quartz-tissue interface. This was captured for both the normal and BCC tissue. The quartz-tissue interface is modeled, because in vivo images of skin cancer are taken by placement on a z-cut quartz imaging plate, as quartz has a similar refractive index to tissue. Figure 3 depicts reflected waveforms and we see that the wave reflected off the tissue containing BCC has a lower minimum (i.e., it is more greatly reflected) than the reflection off the normal tissue. It is a minimum, because the reflection from a quartz-tissue interface has the opposite phase to the reflection from the quartz-air interface (which was used as the reference). The pulse width is the same for both waves. $NB$ , the pulse width, is calculated for the time-domain waveform in the same way as the full-width half maximum (FWHM), although here it is the minimum. The combined terahertz properties of the quartz window and the tissue give rise to the shape of the reflected waveform. For example, the difference between the refractive index of the quartz and tissue affects the intensity and phase of the reflected pulse. Therefore, we consider how the reflected amplitude $E_{\min }$ and the pulse width change in simulations using the Debye parameters plotted in Fig. 2 from the systematic changes to $n$ and $\alpha$ . From Fig. 4 , we see that increasing both $\alpha$ and $n$ increases the intensity of the reflected waveform, i.e., $E_{\min }$ becomes more negative. Now we consider the effects on $E_{\min }$ and the pulse width when changing $\alpha$ and $n$ together. The results are illustrated in the 3-D contour plots in Fig. 5 . Similarly, the effects of changing both $\alpha$ and $n$ on the pulse width are illustrated in Fig. 6 . The surface in Fig. 5 illustrates how increasing $\alpha$ and $n$ increase $E_{\min }$ . The contour corresponding to the $E_{\min }$ of the wave reflected off BCC is marked where it intersects the surface. This emphasizes the point that by increasing the components of the complex refractive index of normal tissue, we can cause observable changes in the reflected waveform to match those off tissue containing BCC. Similarly in Fig. 6, the contour corresponding to the pulse width of the wave reflected off BCC is marked on the surface, and thus depicts the locus of values of $\alpha$ and $n$ that would provide suitable waveforms. Clearly, the intersection of the loci of $\alpha$ and $n$ from Figs. 5 and 6 will correspond to the deviation of $\alpha$ and $n$ of BCC from normal tissue. Therefore, it may be possible to solve inverse problems and deduce optical properties of tissues from measured reflected waveforms.

Fig. 2

Dependence on the refractive index and absorption coefficient of the double Debye parameters: (a) ${\epsilon }_{\infty }$ , (b) ${\epsilon }_s$ , (c) ${\epsilon }_2$ , (d) ${\tau }_1$ , and (e) ${\tau }_2$ .

Fig. 3

Simulations of the waveforms reflected off a quartz-tissue interface calculated by using the double Debye parameters extracted from the ex vivo normal and BCC tissue data.

Fig. 4

The dependence of $E_{\min }$ on the refractive index and absorption coefficient.

Fig. 5

A 3-D contour plot with isobars of $E_{\min }$ as dependent on $\alpha$ and $n$ . The dashed line indicates the isobar of the minimum of the waveform from the simulation of BCC containing tissue.

Fig. 6

A 3-D contour plot with isobars of pulse width as dependent on $\alpha$ and $n$ . The dashed line indicates the isobar of the pulse width of the waveform from the simulation of BCC containing tissue.