Transient Radiative Transfer Equation Research Articles

Numerical Analysis of Non-Fourier Model-Based Bio-Heat Transfer in the Laser-irradiated Axisymmetric Living Tissue.

The current work is related to the numerical investigation of non-Fourier heat transfer inside the short-pulsed laser-irradiated axisymmetric soft tissue phantom. It utilizes the modified discrete ordinate method to solve the transient radiative transfer equation (TRTE) for determining the intensity field. The laser energy absorbed by the soft tissue phantom behaves like a source in the Fourier/non-Fourier heat conduction model based-bio-heat transfer equation (BHTE), which is solved by employing the finite volume method (FVM) to determine the temperature distribution. Despite the prevalent use of non-Fourier BHTE for this purpose, a second law analysis is considered crucial to detect any potential anomalies. Equilibrium entropy production rates (EPR) are initially computed based on classical irreversible thermodynamics (CIT), which may yield negative values, possibly contravening the second law. Consequently, the EPR based on CIT is adjusted using the extended irreversible thermodynamics (EIT) hypothesis to ensure positivity. After that, the current research findings are compared with the results from the literature, and found good agreement between them. Then, the independent study is performed to select the optimum grid size, control angle size, and time step. A comparative analysis of results between the traditional Fourier and non-Fourier models has been performed. The impact of different parameters on the temperature fields and EPRs, is discussed. The effect of the optical properties of the inhomogeneity on the temperature distribution has been investigated. This study may help to enhance the effectiveness of the laser-based photo-thermal therapy.

Thermal relaxation parameters estimation of non-Fourier heat transfer in laser-irradiated living tissue using the Levenberg-Marquardt algorithm

The present work is concerned with solving the inverse heat transfer problem (IHTP) to estimate the thermal relaxation parameters corresponding to the dual phase lag (DPL) model-based bio-heat transfer in laser-irradiated living tissue. This problem is divided into two parts: direct and inverse problems. The solution of the inverse problem has been obtained using the Levenberg–Marquardt algorithm. On the other hand, the solution to the direct problem is obtained by solving the DPL model-based bio-heat transfer equation (BHTE) with the transient radiative transfer equation (TRTE). So, the TRTE has been numerically solved by the modified discrete ordinate method (DOM) to find the intensity field in the living tissue subjected to short-pulsed laser radiation, and it acts as a heat source in the DPL model-based BHTE. So, the DPL model-based BHTE has been analytically solved by employing the finite integral transform technique to get the temperature variation inside the laser-irradiated living tissue. First, the computer code developed for estimating unknown thermal relaxation parameters using the Levenberg–Marquardt algorithm has been verified against the data given in the literature, and found good agreement between them. Then, the thermal response of laser-irradiated living tissue has been investigated. After that, the influence of sensor position on the sensitivity of the IHTP solution has been discussed. The impact of the total transient readings and the measurement error on estimating the unknown parameters have been investigated. The present study’s findings can significantly contribute to various parameter estimation problems where the conventional methods for direct measurements of parameters are impossible or very difficult. Furthermore, the current results may considerably impact the study of the thermic response of living tissue during laser-based photothermal therapy.

Thermodynamic stability analysis of non‐Fourier model‐based bio‐heat transfer in laser‐irradiated cylindrical‐shaped multilayered biological tissue

AbstractThe present research focuses on examining the thermic response of living tissue in the form of a triple‐layered cylindrical structure when subjected to laser light and the compatibility analysis of non‐Fourier heat transfer with thermodynamics second law. The temperature field in the triple‐layered cylindrical living tissue subjected to laser light is determined by numerically solving the transient radiative transfer equation in conjunction with the dual phase lag (DPL) based bioheat equation. Once the temperature field is known, the equilibrium and nonequilibrium entropy production rate (EPR) is calculated based on the hypothesis of classical irreversible thermodynamics and extended irreversible thermodynamics, respectively. The present results are verified against the data from the literature and found a good match between them. A comparative analysis of the Fourier and non‐Fourier models is accomplished. The equilibrium and nonequilibrium EPR values for the Fourier model are found to be positive. While the equilibrium EPR is negative for non‐Fourier heat conduction and does not satisfy the thermodynamics second law, nonequilibrium EPR is always a positive value for Fourier, DPL, and hyperbolic models and satisfies thermodynamics second law. It has been investigated how thermal relaxation times affect the temperature field and EPRs in tissue are subjected to laser light.

Conjugate heat transfer analysis of laser-irradiated cylindrical-shaped biological tissue embedded with the optical inhomogeneity

The current study is associated with the numerical investigation of the thermal behavior of cylindrical-shaped biological tissue during laser-based photo-thermal therapy. The light-tissue interaction has been modeled using the transient radiative transfer equation (TRTE). The TRTE in a cylindrical coordinate system is solved using the modified discrete ordinate method to obtain the intensity distribution inside the biological tissue. The solution of TRTE is coupled with Pennes bio-heat transfer equation (BHTE) to understand the thermic response of biological tissue. The Pennes BHTE is solved using the finite volume method (FVM). The two different types of optical inhomogeneity (absorption and scattering inhomogeneity) are considered in the current study. The inhomogeneity's optical and thermophysical properties may be the same/different from the homogeneous biological tissue. This conjugate heat transfer problem (CHTP) is solved using the harmonic mean technique. First, the present result is verified with the data available in the literature. Subsequently, the effect of inhomogeneity's optical properties on the temperature distribution is investigated. A comparative study between the same and different thermophysical properties of the biological tissue and inhomogeneity is performed. This study will help to accurately model the thermal characteristics of the laser-irradiated biological tissue having different thermophysical properties than the inhomogeneity.

Solving steady and transient radiative transfer problems with strong inhomogeneity via a lattice Boltzmann method

A complete lattice Boltzmann method (LBM) is developed to solve the steady and transient radiative transfer problems with strong inhomogeneity. The transient radiative transfer equation (RTE) is regarded as a hyperbolic conservation law equation and then solved by the LBM. A forward difference scheme is applied to the time derivative of the source term in the evolution equation, which makes the LBM accurate and stable for radiative transfer problems with strong inhomogeneity. The correctness and accuracy of the proposed LBM is verified first. Afterward, the LBM is applied to solve the one-dimensional radiative transfer problems involving strong inhomogeneous radiative intensity and discontinuous medium system, including a planar slab subjected to Gaussian-shaped source, an infinite slab with an inclusion layer, and a vacuum gapped medium system. The time-resolved radiation is presented and analyzed as well as the steady results. Finally, the present LBM is extended to multidimensional radiative transfer problems.

Three-dimensional FTn finite volume solution of short-pulse laser propagation through heterogeneous medium

In this paper, 3-D heterogeneous medium, containing small inhomogeneous zones, subjected to a short-pulse laser has been examined by solving the transient radiative transfer equation. Both curved-line advection method and STEP schemes of the FTn finite volume method have been applied. The curved-line advection method predictions proved that a decrease of the false scattering and ray effects are obtained. In fact, there was a good agreement between the curved-line advection method and the Monte Carlo method. However, the STEP results are slightly mismatching the predictions of the aforementioned reference method. Then, the effects of the absorption coefficient, the size, the number and the position of inhomogeneous zone on the transmittance and reflectance signals have been analyzed. The predictions showed that the increase of the size of the inhomogeneity reduces the intensity of radiation. For both homogenous and heterogonous medium, the change of the detector position varies both the broadening of the signal pulse-width and the time with peak reflectance and/or transmittance. That is can be explained by the effects of the distance and the medium property between the laser-incident source and the detector position. Thus, these both parameters are the main factors for determining the peak position and the pulse broadening. Finally, the effects of the absorption coefficient in the inhomogeneity zone on the absolute values of logarithmic slope has been discussed. The results proved that the absolute values of logarithmic slope may be a perfect indicator for detecting any abnormal absorbing zones in the medium.

Experimental research on noninvasive reconstruction of optical parameter fields based on transient radiative transfer equation for diagnosis applications

The noninvasive reconstruction of optical parameter fields in participating phantom is investigated experimentally based on the measured time-resolved radiative signals, which are detected by the time-correlated single-photon counting system. The discrete ordinate method is employed to solve the forward transient radiative transfer equation to simulate the time-resolved radiative transfer in the physical phantom exposed to ultra-short pulse laser irradiation. On top of that, the sequential quadratic programming algorithm based on the generalized Gaussian Markov random field is applied to solve the inverse problem. To improve the computational efficiency, an adjoint equation technique is adopted to determine the gradient of objective function. Good agreement was obtained between the predicted optical parameter fields and realistic ones. All the reconstruction results demonstrate that the proposed reconstruction methods are proved to be accurate and efficient experimentally.

Non-destructive testing of ceramic materials using mid-infrared ultrashort-pulse laser

The non-destructive testing (NDT) of ceramic materials using mid-infrared ultrashort-pulse laser is investigated in this study. The discrete ordinate method is applied to solve the transient radiative transfer equation in 2D semitransparent medium and the emerging radiative intensity on boundary serves as input for the inverse analysis. The sequential quadratic programming algorithm is employed as the inverse technique to optimize objective function, in which the gradient of objective function with respect to reconstruction parameters is calculated using the adjoint model. Two reticulated porous ceramics including partially stabilized zirconia and oxide-bonded silicon carbide are tested. The retrieval results show that the main characteristics of defects such as optical properties, geometric shapes and positions can be accurately reconstructed by the present model. The proposed technique is effective and robust in NDT of ceramics even with measurement errors.

Transient/time-dependent radiative transfer in a two-dimensional scattering medium considering the polarization effect.

Transient/time-dependent radiative transfer in a two-dimensional scattering medium is numerically solved by the discontinuous finite element method (DFEM). The time-dependent term of the transient vector radiative transfer equation is discretized by the second-order central difference scheme and the space domain is discretized into non-overlapping quadrilateral elements by using the discontinuous finite element approach. The accuracy of the transient DFEM model for the radiative transfer equation considering the polarization effect is verified by comparing the time-resolved Stokes vector component distributions against the steady solutions for a polarized radiative transfer problem in a two-dimensional rectangular enclosure filled with a scattering medium. The transient polarized radiative transfer problems in a scattering medium exposed to an external beam and in an irregular emitting medium are solved. The distributions of the time-resolved Stokes vector components are presented and discussed.

Thermal analysis of laser-irradiated tissue phantoms using a novel separation of the variables-based discrete transfer method

ABSTRACTThermal analysis of tissue phantoms subjected to short pulse laser irradiation has been presented. The transient radiative transfer equation (RTE) has been solved using a novel separation of variables-based discrete transfer method (DTM) recently developed by the present authors (Nirgudkar et al. [21]). As an advancement, the solution of RTE has been coupled with Pennes’ bioheat transfer equation for determining the temperature distribution. Homogenous as well as phantoms embedded with optical inhomogeneity have been considered. The numerical model has been verified against the results available in the literature. This study clearly reveals the influence of the nature of the embedded inhomogeneity and its relative contrast on the resultant temperature distribution inside the body of the tissue phantom.

Analysis of polarized pulse propagation through one-dimensional scattering medium

This paper analyzes the polarized light propagation in a one-dimensional scattering medium with the upper surface subjected to an oblique incident short-pulsed laser beam using the natural element method (NEM). The NEM discretization scheme for the transient vector radiative transfer equation (TVRTE) is presented in detail. The accuracy of the natural element method for transient vector radiative transfer in the scattering medium is assessed. Numerical results show that the NEM is accurate, and effective in solving transient polarized radiative problems. We examine a square short-pulsed laser transport firstly in the atmosphere with Mie scattering and then within aerosol scattering medium. We then investigate the transient polarized radiative transfer problem in the atmosphere-ocean system. The time-resolved signals and the polarization state of the Stokes vector are presented and analyzed. It is found that the scattering types of the medium make greatly influence on the transient transportation of the polarized light. Critically, the polarization states of the backward and forward scattered photons show significantly different time varying trends. For the two-layer system with dissimilar refractive index distributions, due to the total-reflection effect, the existence of a Fresnel interface significantly changes the polarization state of the light, and discontinuous distribution features are observed on the interface.

Transient Radiative Transfer in a Graded Index Slab with Fresnel Surfaces

One-dimensional transient radiative transfer in a graded index slab with Fresnel surfaces subject to a short-pulse laser irradiation is investigated by the lattice Boltzmann method. In the scattering medium having Fresnel surfaces, the radiative intensity includes two parts: the collimated intensity, and scattered or diffuse intensity. The contribution of the collimated pulse to the transient process is considered as a source term in the transient radiative transfer equation. First, lattice Boltzmann method solutions for time-resolved signals are validated by comparison with results obtained by Monte Carlo method. The investigations are mainly focused on the medium with linearly changing graded index, and the commonalities and differences of the time-resolved signals resulting from the slab with diffuse reflected surfaces or Fresnel surfaces are discussed. The effects of the gradient of the linearly varying refractive index on transmittance and reflectance signals are examined. Then, transient radiative transfer with two graded index distributions with a singular point is analyzed. Finally, by setting the right boundary of the slab as a diffusely reflecting surface, the transient radiative transfer in the graded index medium under the combined optical boundary conditions is examined.

Thermal analysis of laser-irradiated tissue phantoms using dual phase lag model coupled with transient radiative transfer equation

The present work is concerned with the development and application of dual phase lag (DPL) based heat conduction model for investigating the thermal response of laser-irradiated biological tissue phantoms. The developed heat transfer model has been coupled with the transient form of radiative transfer equation (RTE) that describes the phenomena of light propagation inside the tissue phantom. The RTE has been solved using the discrete ordinate method (DOM) to determine the 2-D distribution of light intensity within the tissue phantom, while finite volume method (FVM) based discretization scheme has been employed for solving the heat transfer model. The developed numerical model has first been verified against the results available in the literature. The results obtained in the form of temperature distribution through DPL model have been compared with conventional Fourier heat conduction model as well as with hyperbolic model. The effects of two phase lags terms in the form of relaxation times i.e. τT and τq associated with DPL model on the resultant thermal profiles have been investigated. Thereafter, the temperature distribution inside the biological tissue phantom embedded with optical inhomogeneities of varying contrast levels have been determined using the DPL-based model. Here the optical inhomogeneities represent the malignant (absorbing inhomogeneity) and benign (scattering inhomogeneity) cells present in an otherwise homogeneous medium. Results of the study reveal that the hyperbolic heat conduction model consistently predicts high temperature values and also the associated thermal profiles exhibit the largest amplitude of oscillations throughout the body of the tissue phantom. The DPL-based model results into relatively lesser oscillations due to the coupled effects of τT and τq. The conventional Fourier model, on the other hand, results into the lowest temperature values without any oscillations in the temperature profiles. The effect of the presence of varying nature of optical inhomogeneities is also brought out quite clearly using the developed DPL-based heat conduction model.

Evaluation of the FTn Finite Volume Method for Transient Radiative Transfer in Anisotropically Scattering Medium

In this paper, we formulated, applied, and tested the FTn Finite Volume Method (FTn FVM) for transient radiative transfer in three-dimensional absorbing, emitting, and anisotropically scattering medium. Both the STEP and the Curved-Line Advection Method (CLAM) are introduced for spatial discretization of the transient radiative transfer equation. The results show that FTn FVM reduces largely the ray effects and it is more accurate than the standard FVM. Also, using both STEP and CLAM schemes, FTn FVM has smaller convergence time than the standard FVM for all cases. On the contrary, the STEP scheme is faster than the CLAM scheme but it has less accuracy. Then, the effects of optical thickness, scattering albedo, and anisotropy factor on incident radiation and radiative flux are presented and discussed.

Fast method of retrieving the asymmetry factor and scattering albedo from the maximum time-resolved reflectance of participating media.

This research presents a parametric study of the time-resolved hemispherical reflectance of a semi-infinite plane-parallel slab of homogeneous, nonemitting, absorbing, and anisotropic scattering medium exposed to a collimated Gaussian pulse. The one-dimensional transient radiative transfer equation was solved by using the finite volume method. The internal reflection at the medium-air interface caused by the mismatch of the refractive indices was considered. In particular, this work focused on the maximum diffuse hemispherical reflectance. Three different optical regions were identified according to the dimensionless pulsewidth βct_p. The correlation between the normalized maximum hemispherical reflectance and βct_p was conformed to the Boltzmann function. The coefficients in the correlating functions of the match and mismatch refractive index cases were fitted as polynomial fitting functions of the single scattering albedo ω and Henyey-Greenstein asymmetric factor g. Thus, ω and g can be simultaneously reconstructed by the semi-empirical correlations without solving the forward model. In conclusion, the proposed method can potentially retrieve the asymmetry factor and single scattering albedo of participating media accurately and efficiently.

Multi-start iterative reconstruction of the radiative parameter distributions in participating media based on the transient radiative transfer equation

Simultaneous reconstruction of the radiative parameter distributions in participating media, based on the transient radiative transfer equation (TRTE), was investigated. The discrete ordinate method was employed to solve the direct TRTE for two-dimensional participating media. A multi-start iterative technique, based on the conjugate gradient (CG) method, was proposed to accelerate the convergence speed of the inverse procedure. The maximum a posteriori (MAP) estimation was used to overcome the retrieval difficulties resulting from the ill-posed nature of the inverse problem. The gradient of the objective function was obtained using adjoint differentiation. All the simulation results show that MAP combined with the multi-start iterative CG technique is effective in simultaneously reconstructing the distributions of absorption and scattering coefficients in two-dimensional inhomogeneous media, even with noisy data.

Lattice Boltzmann method for short-pulsed laser transport in a multi-layered medium

We construct a lattice Boltzmann method (LBM) for transient radiative transfer in one-dimensional multi-layered medium with distinct refractive index in each layer. The left boundary is irradiated normally by a short-pulsed laser. The Fresnel interfaces conditions, which incorporate reflection and refraction, are used at the boundaries and the interfaces. Based on the Fresnel׳s law and Snell׳s law, the interfacial intensity formulas are introduced. The collimated and diffuse intensities are treated individually. At a transient time step, the collimated component is first solved by LBM and then embedded into the transient radiative transfer equation as a source term. To keep the consistency of the directions in all the layers, angular interpolation of the intensities at the interfaces is adopted. The transient radiative transfer in a two-layer medium is first investigated, and the time-resolved results are validated by comparing with those by the Monte Carlo method (MCM). Of particular interest, the angular intensities along the slab at different times are presented to illustrate a variety of interesting phenomena, and the discontinuous nature of the intensity at the interfaces is discussed. The effects of various parameters on the time-resolved signals are examined.

Solution of Radiative Transfer Equation using Discrete Transfer Method for two-dimensional participating medium

The present work reports the development and application of a simplified numerical approach for solving the transient Radiative Transfer Equation (RTE) using Discrete Transfer Method (DTM) in two-dimensional coordinate system. The numerical formulation of the proposed scheme is discussed in detail and its application in the context of understanding light propagation phenomenon in laser-irradiated numerically simulated biological tissue phantoms has been demonstrated. The developed mathematical model has first been benchmarked against the results published in the literature for the same operating conditions. Thereafter, the results of a detailed parametric study have been presented to investigate the effects of optical properties of the biological phantom on the intensity distribution within the two-dimensional tissue phantom, net transmittance and reflectance, etc. The effect of anisotropy of the tissue medium has also been studied to understand the phenomenon of light propagation within the body of the sample. Based on the results of the study, it has been inferred that the developed numerical methodology for two-dimensional Discrete Transfer Method successfully predicts the physics of the phenomena of light propagation within the tissue phantom and compares well with the other conventionally employed numerical models for solving the Radiative Transfer Equation.

Short-Pulse Collimated Radiation in a Participating Medium

A two-dimensional model is presented in this article to study the transient radiative transfer phenomena involved in a participating medium subjected to short pulse collimation. The model assumes the medium bounded by diffusely emitting and reflecting boundaries, one of the boundaries is irradiated with a short pulse collimated beam. The finite volume method is applied to solve the transient radiative transfer equation governing the physical phenomena. The fully implicit scheme is used to discretize the transient term. In the proposed approach, intensity can directly be evaluated by solving the governing transient radiative transfer equation. The effects of scattering albedo and emissivity on the transmitted and reflected flux are studied. The performance of two different spatial schemes: STEP and CLAM are also been tested. It is seen that the CLAM scheme gives results to a greater accuracy and hence, correctly predict the speed of photon whereas STEP scheme over predicts the same.

Short-pulsed laser transport in two-dimensional scattering media by natural element method

The natural element method (NEM) is extended to solve transient radiative transfer (TRT) in two-dimensional semitransparent media subjected to a collimated short laser irradiation. The least-squares (LS) weighted residuals approach is employed to spatially discretize the transient radiative heat transfer equation. First, for the case of the refractive index matched boundary, LSNEM solutions to TRT are validated by comparison with results reported in the literature. Effects of the incident angle on time-resolved signals of transmittance and reflectance are investigated. Afterward, the accuracy of this algorithm for the case of the refractive index mismatched boundary is studied. Finally, the LSNEM is extended to study the TRT in a two-dimensional semitransparent medium with refractive index discontinuity irradiated by the short pulse laser. The effects of scattering albedo, optical thickness, scattering phase function, and refractive index on transmittance and reflectance signals are investigated. Several interesting trends on the time-resolved signals are observed and analyzed.