Pith. sign in

REVIEW 4 major objections 6 minor 64 references

Insights into experimental evaluation of the non-fourier heat transfer model in biological tissues

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Bovine skin experiments show the TPL heat model's phase lags are under 0.5 seconds, much smaller than many values used in bioheat simulations.

desk verdict Real experimental effort, but the phase-lag bounds are not yet quantitatively supported; as written, the paper overclaims its headline result. read the letter →

arxiv 2507.17627 v1 pith:U6AP7RA6 submitted 2025-07-23 physics.bio-ph

Mohammad Azhdari , Ghader Rezazadeh , Raghav Pathak , Hans-Michael Tautenhahn , Franziska Tautenhahn , Tim Ricken , Seyed Morteza Seyedpour This is my paper · ORCID
classification physics.bio-ph
keywords bioheatingnon-Fourierheattransferthermalwavepropagationdualphaselag(DPL)three(TPL)experimentalvalidationbovineskinparameterextraction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to pin down the four phase-lag parameters of the Three-Phase Lag (TPL) bioheat model by measuring how bovine skin actually warms and cools under a 450 nm laser. Using three non-contact infrared sensors and a decoupled experimental protocol, the authors find that the heat-flux phase lag $\tau_q$ is below 0.5 s (best matched at 0.1 s), the temperature-gradient lag $\tau_\theta$ is below 0.25 s, the thermal displacement coefficient $k^*$ is negligible, and $\tau_v$ is zero. These values are far smaller than the 15–32 s phase lags used in many earlier simulations, which the paper argues are thermodynamically ill-posed and produce non-physical temperature undershoots. If correct, the results refine bioheat models used in laser therapy, hyperthermia, and cryosurgery, and they show that parameter isolation matters as much as model complexity.

What carries the argument

The central object is the Two-Dimensional Three-Phase Lag (TPL) bioheat equation, a constitutive law relating heat flux and temperature gradient through four parameters: the heat-flux phase lag $\tau_q$, the temperature-gradient phase lag $\tau_\theta$, the thermal displacement coefficient $k^*$, and the thermal displacement phase lag $\tau_v$ (thermal displacement $v$ has time derivative equal to temperature). The paper solves this equation with an implicit finite-difference scheme using central spatial and forward temporal differences, couples it to a Beer-Lambert Gaussian laser heat source, and evaluates each parameter by comparing simulated temperature histories to experimental infrared-sensor readings. The load-bearing mechanism is the decoupled experiment: two laser pulses of equal energy but different delivery rates isolate $\tau_q$; pulsed irradiation with varying periods isolates $\tau_\theta$ via heating/cooling slope asymmetry; a 3 s pulse with two sensors monitors for the sensor-sign reversal that would signal $k^*$ wave propagation; and Equation (13) shows $k^* \tau_v$ acts as an additive conductivity, motivating $\tau_v = 0$.

What would settle it

Measure the thermal conductivity and heat capacity of the actual bovine sample (e.g., with a guarded hot-plate or differential scanning calorimeter) and rerun the two-rate laser experiment; if the $\tau_q$ value that matches the measured temperature difference shifts outside the reported <0.5 s range, the phase-lag estimates are artifacts of the assumed properties. Alternatively, repeat the irradiation with a temperature sensor sampling faster than 0.1 s; if the maximum temperature difference between the two laser cases keeps shrinking, $\tau_q$ is below 0.1 s and the current setup cannot resolve it.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the phase-lag parameters governing non-Fourier heat conduction in bovine skin are sub-second and partially redundant. By matching a two-dimensional axisymmetric finite-difference TPL simulation to temperature histories measured with MLX90614 infrared sensors, the authors estimate $\tau_q < 0.5$ s (simulation at $\tau_q = 0.1$ matches the measured maximum temperature difference between two laser delivery rates), $\tau_\theta < 0.25$ s, $k^* \approx 0$ (Sensor 1 never falls below Sensor 2, so no thermal-wave reversal appears), and $\tau_v = 0$ (because $k^* \tau_v$ merely adds to effective thermal conductivity). The paper concludes that several large phase-lag values reported in the literature are unrealistic and can produce thermal instabilities and undershoots of tens of kelvin, and it positions these experiments as a correction toward more precise, parameter-isolated validation.

Load-bearing premise

The extracted phase lags rest on treating the bovine skin as a thermally homogeneous material with literature values for thermal conductivity (0.45 W/m·K), specific heat (3500 J/kg·K), density (1000 kg/m³), and convection coefficient (15 W/m²·K) rather than values measured on the actual sample, and the paper states that no thorough experimental studies have exactly ascertained these characteristics.

Editorial extensions

If this is right

  • Published phase-lag values of tens of seconds are likely unphysical for skin; bioheat simulations should use sub-second lags to avoid non-physical temperature undershoots of tens of kelvin.
  • The thermal displacement coefficient $k^*$ is negligible at the mm-to-cm spatial scale and second timescales probed here, so the TPL model effectively reduces to a dual-phase-lag model for this setting.
  • The thermal displacement phase lag $\tau_v$ is redundant: the product $k^* \tau_v$ adds to the measured thermal conductivity, so it can be set to zero whenever $k^*$ is negligible.
  • A two-rate laser protocol (same total energy, different delivery rates) gives an operational estimate of $\tau_q$, but only down to the sensor's 0.1 s temporal resolution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the same protocol were applied to perfused human skin in vivo, the extracted lags could shift because blood perfusion and metabolic heating are absent from the excised sample.
  • The 0.1 s resolution floor means $\tau_q$ could be near zero; higher-bandwidth thermal imaging would reveal whether the non-Fourier lag terms are physically needed at all or whether a Fourier model with corrected surface boundary conditions matches the data.
  • Because $\tau_\theta$ and $\tau_v$ both mimic increased thermal conductivity, temperature-only inversions are systematically confounded; jointly estimating conductivity and phase lags from the same data (for instance with a Bayesian approach) would tighten the reported ranges.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. This paper implements a two-dimensional axisymmetric finite-difference solution of the Three-Phase Lag (TPL) heat conduction model and combines it with 450 nm laser experiments on excised bovine skin monitored by three infrared sensors. The stated goal is to extract four model parameters: the heat flux phase lag τ_q, the temperature gradient phase lag τ_θ, the thermal displacement coefficient k*, and the thermal displacement phase lag τ_v. The extraction strategy is a decoupled protocol: τ_q is inferred by matching the maximum temperature difference between two heating-rate cases (Cases 1 and 2), τ_θ is proposed to be inferred from heating/cooling slope asymmetry under periodic pulses, k* is inferred from the presence or absence of a wave-like reversal between Sensor 1 and Sensor 2, and τ_v is argued to be redundant because k*τ_v acts as an additional effective conductivity. The headline conclusion (Section 7, Table 3) is that the parameters are substantially lower than many literature values: τ_q < 0.5 s, τ_θ < 0.25 s, k* ≈ 0, and τ_v = 0.

Significance. If the quantitative claims were supported, the paper would provide useful experimental constraints on non-Fourier bioheat parameters, which are scattered and often weakly justified in the literature. The strengths are genuine: the idea of decoupling the four parameters is sound and clearly motivated by the model structure; repeated trials (10-20 per condition) are reported; the sensor field-of-view is modeled rather than treated as point measurements; and a thermal-conductivity sensitivity study is initiated. However, only τ_q is subjected to a quantitative comparison, and even that comparison lacks uncertainty quantification. The entries for τ_θ, k*, and τ_v in Table 3 are proposed protocols, qualitative null observations, or explicit assumptions rather than measured values. The paper is therefore best viewed as a promising protocol with pilot data, not as a validated parameter set; the requested revisions are primarily analysis and reporting tasks rather than new experiments.

major comments (4)
  1. [§6.1, Table 3] The upper bound τ_q < 0.5 s is not quantitatively established. The extraction is a parameter search: simulations with different τ_q are compared to the experimental maximum temperature difference between Cases 1 and 2, but the manuscript never reports the measured value of that maximum difference, the standard deviation across the 10-20 repeats, or a fitting metric. Figure 22 shows two measured temperature curves without a difference trace, and the text only states that τ_q = 0.1 'yields results most consistent' with the data. Because the sensor sampling interval is 0.1 s, the claimed best-fit value sits at the resolution limit, so the statement that τ_q values below 0.1 cannot be resolved does not by itself justify an upper bound of 0.5 s. Please report the experimental ΔT_max with its uncertainty and show the mismatch (e.g., residual or RMS error) for τ_q = 0, 0.1, 0.25, 0.5, and 1 s.
  2. [§3 and §5.5] The promised sensitivity propagation is missing. Section 3 states that simulations with 'a range of various thermal conductivity values' will ensure that uncertainty in the thermal properties 'does not compromise the accuracy of the derived phase lag estimates,' but Section 5.5 only displays temperature profiles for different k values; the τ_q extraction is never re-run under perturbed k. This is load-bearing because k = 0.45 W/m·K, c = 3500 J/kg·K, and ρ = 1000 kg/m³ (Table 1) are literature values admitted to be unmeasured (Section 3). The inferred τ_q is derived from differences between simulated temperature curves, and k uncertainty shifts those curves directly. Please recompute the Case 1/Case 2 comparison for k in, say, the range 0.3-0.6 W/m·K and report how the inferred τ_q range changes.
  3. [§6.2, Table 3] The value τ_θ < 0.25 s is not derived anywhere in the manuscript. Section 6.2 describes a proposed periodic-pulsed-laser protocol based on heating/cooling slope asymmetry, but no experimental data, fitted value, uncertainty, or comparison to literature is presented. The text itself notes that τ_θ effects closely resemble variations in k (Figures 7-9 versus 16-18), so without quantitative results the bound in Table 3 is unsupported. Please either remove the numerical range and state that τ_θ remains to be measured, or provide the promised protocol data with a fitting analysis that separates τ_θ from k.
  4. [§6.3 and §6.4] The conclusions k* ≈ 0 and τ_v = 0 are presented as results in Table 3 and Section 8, but they are not established at that level. The k* conclusion rests entirely on the qualitative observation that 'Sensor 1 was never observed to fall below that of Sensor 2' (Section 6.3); no trial count, sensor noise floor, or detection threshold for a wave-like crossing is provided, so the null observation can at most bound k* below an unreported detection limit. Section 6.4 argues that τ_v is redundant because k*τ_v adds to the effective conductivity, but that argument sets τ_v to zero rather than measuring it; Table 3 labels τ_v as 'assumed value,' yet the conclusion lists it as a finding. Please present a detection-threshold calculation for k* and explicitly label τ_v = 0 as an assumption rather than a measurement.
minor comments (6)
  1. [General] There are numerous figure-number mismatches; for example, Section 5.3 refers to 'Figure 16' for the k* time-history, while the corresponding caption is Figure 10, and it cites Figures 17 and 18 for depth and radial profiles that are captioned Figures 11 and 12. Please correct all cross-references.
  2. [Figure 3] The caption of Figure 3 states τ_q = 0.01, but the text describes it as the baseline Fourier case with all phase-lag coefficients zero; the caption should be made consistent with the text.
  3. [Equations (4) and (10)] Equations (4) and (10) are labeled ℛ1 and ℛ2 without definition; please use conventional equation numbering or define the operator notation explicitly.
  4. [Figure 22] The experimental temperature response in Figure 22 shows negative values during the first roughly 0.5 s; if this is sensor settling or ambient drift, please state this, since the maximum-difference metric used for τ_q is sensitive to early-time artifacts.
  5. [§5.4 and §6.4] Section 6.4 states that τ_v 'does not introduce a distinct physical meaning' and is redundant, but Section 5.4 and Figures 13-15 describe distinct, if similar, effects of τ_v compared with τ_θ; reconcile these descriptions so the modeling and the redundancy argument are consistent.
  6. [Data availability] The paper reports 10-20 repeats per condition but provides no raw data, no per-trial statistics, and no data availability statement; please add representative raw traces and trial statistics or deposit the data in a public repository.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the phase-lag estimates are inverse fits or qualitative comparisons against external experiments, not reductions of the model to its own inputs.

full rationale

The paper derives tau_q by an explicit inverse-matching protocol: two laser-energy delivery profiles are simulated under several tau_q values and compared with a measured maximum temperature difference between the two cases (Section 6.1). This is parameter estimation against an external observable, not a quantity fixed by construction. The k* estimate is a null result from experiment (Sensor 1 was never observed to fall below Sensor 2, Figs. 24-25), and tau_v is explicitly marked "assumed value" in Table 3 after showing that k*tau_v is algebraically degenerate with k (Eqs. 13-15). The constitutive equation is cited to the authors' Ref. [30], but the equation is written out and follows standard TPL/Green-Naghdi forms also attributed to Tzou [33] and Podio-Guidugli [34], so the self-citation is not load-bearing. The paper does have non-circular weaknesses: the tau_theta<0.25 entry in Table 3 has no shown experimental derivation; the promised sensitivity study of thermal conductivity (Section 3) is never propagated into re-extraction of tau_q; and a sentence states that phase-lag terms were "considered negligible (i.e., zero) during all the modeling," which contradicts the parameter variations in Sections 5-6. These are evidence-quality and consistency problems, not construction-level circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on four effective parameters that are fitted or set by hand, on a homogeneous-tissue assumption, and on the choice of TPL model. No new physical entities are introduced.

free parameters (4)
  • tau_q (heat flux phase lag) = 0.1 s (best match), < 0.5 s range
    Chosen to match the measured maximum temperature difference between Case 1 and Case 2 laser pulses (Section 6.1, Figure 22-23).
  • tau_theta (temperature gradient phase lag) = < 0.25 s (proposed range)
    Rounded upper bound from the proposed pulsed-irradiation asymmetry method; the paper does not show a quantitative fit (Section 6.2).
  • k* (thermal displacement coefficient) = ≈ 0 (negligible)
    Set to zero because no reversal of Sensor 1 below Sensor 2 was observed (Section 6.3, Figure 25).
  • tau_v (thermal displacement phase lag) = 0 (assumed)
    Set to zero because k* tau_v is analytically equivalent to added thermal conductivity in Eq. (13), making it redundant (Section 6.4).
assumptions (5)
  • domain assumption The 2D axisymmetric representation adequately captures the 3D thermal response of the skin sample (Section 2).
    The paper assumes symmetry about the laser axis and ignores circumferential variation; no convergence check of 2D vs 3D is shown.
  • domain assumption Bovine skin is thermally homogeneous with literature values k, c, rho, h (Table 1).
    The paper admits no direct measurement of bovine skin thermal properties and aggregates values from meat, leather, and human skin sources (Section 3).
  • domain assumption Thermal convection and evaporation heat loss are neglected at the tissue surface except for a constant convection coefficient h (Section 3).
    The experiment keeps samples in deionized water and wipes the surface dry, but evaporative and convective losses during laser exposure are neglected.
  • domain assumption The MLX90614 sensor FOV can be modeled as an arc-shaped area average in the simulation (Section 6).
    The paper says the FOV averaging 'has been accounted for and modeled' but gives no FOV geometry or calibration data.
  • domain assumption The TPL constitutive Eq. (3) is the correct model for skin tissue, including the Green-Naghdi thermal displacement term (Section 2).
    The paper adopts the TPL model from prior literature and does not test alternative models against the data.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Insights into experimental evaluation of the non-fourier heat transfer model in biological tissues." pith.science (2026). https://pith.science/paper/U6AP7RA6

@misc{pith2026250717627,
  author       = {Pith},
  title        = {Pith review of: Insights into experimental evaluation of the non-fourier heat transfer model in biological tissues},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U6AP7RA6}},
  note         = {Machine review of arXiv:2507.17627}
}
abstract

A comprehensive understanding of heat transfer mechanisms in biological tissues is essential for the advancement of thermal therapeutic techniques and the development of accurate bioheat transfer models. Conventional models often fail to capture the inherently complex thermal behavior of biological media, necessitating more sophisticated approaches for experimental validation and parameter extraction. In this study, the Two-Dimensional Three-Phase Lag (TPL) heat transfer model, implemented via the finite difference method (FDM), was employed to extract key phase lag parameters characterizing heat conduction in bovine skin tissue. Experimental measurements were obtained using a 450 nm laser source and two non-contact infrared sensors. The influence of four critical parameters was systematically investigated: heat flux phase lag ($\tau_{q}$), temperature gradient phase lag ($\tau_{\theta}$), thermal displacement coefficient ($k^*$), and thermal displacement phase lag ($\tau_{v}$). A carefully designed experimental protocol was used to assess each parameter independently. The results revealed that the extracted phase lag values were substantially lower than those previously reported in the literature. This highlights the importance of high-precision measurements and the need to isolate each parameter during analysis. These findings contribute to the refinement of bioheat transfer models and hold potential for improving the efficacy and safety of clinical thermal therapies.

Figures

Figures reproduced from arXiv: 2507.17627 by the authors.

Figure 1
Figure 1. Two-dimensional axisymmetric schematic representation of the skin tissue model [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. The sample of bull skin tissue was used, and the cut section of it was prepared [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Two-dimensional simulation results showing temperature distribution profiles [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (23 more)
Figure 4
Figure 4. Figure 4: Examining the temperature time history at different locations from the two [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 5
Figure 5. Figure 5: Investigation of temperature profiles in skin tissue depth for different [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: Investigation of temperature profiles in the radial direction of skin tissue for [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: Examining the temperature time history at different locations from the two [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: Investigation of temperature profiles in skin tissue depth for different [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]
Figure 9
Figure 9. Figure 9: Investigation of temperature profiles in the radial direction of skin tissue for [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]
Figure 10
Figure 10. Figure 10: Examining the temperature time history at different locations from the two [PITH_FULL_IMAGE:figures/full_fig_p028_10.png]
Figure 10
Figure 10. Figure 10: This figure illustrates the temperature profiles along the depth [PITH_FULL_IMAGE:figures/full_fig_p029_10.png]
Figure 11
Figure 11. Figure 11: Investigation of temperature profiles in skin tissue depth for different [PITH_FULL_IMAGE:figures/full_fig_p030_11.png]
Figure 12
Figure 12. Figure 12: Investigation of temperature profiles in the radial direction of skin tissue for [PITH_FULL_IMAGE:figures/full_fig_p031_12.png]
Figure 13
Figure 13. Figure 13: Examining the temperature time history at different locations from the two [PITH_FULL_IMAGE:figures/full_fig_p032_13.png]
Figure 14
Figure 14. Figure 14: Investigation of temperature profiles in skin tissue depth for different [PITH_FULL_IMAGE:figures/full_fig_p033_14.png]
Figure 15
Figure 15. Figure 15: Investigation of temperature profiles in the radial direction of skin tissue for [PITH_FULL_IMAGE:figures/full_fig_p034_15.png]
Figure 16
Figure 16. Figure 16: Examining the temperature time history at different locations from the two [PITH_FULL_IMAGE:figures/full_fig_p035_16.png]
Figure 17
Figure 17. Figure 17: Investigation of temperature profiles in skin tissue depth for different thermal [PITH_FULL_IMAGE:figures/full_fig_p036_17.png]
Figure 18
Figure 18. Figure 18: Investigation of temperature profiles in the radial direction of skin tissue for [PITH_FULL_IMAGE:figures/full_fig_p037_18.png]
Figure 19
Figure 19. Figure 19: The experimental system employed for conducting 450 nm laser irradiation [PITH_FULL_IMAGE:figures/full_fig_p039_19.png]
Figure 20
Figure 20. Figure 20: Irradiation energy profiles corresponding to Cases 1 and 2, as recorded from [PITH_FULL_IMAGE:figures/full_fig_p040_20.png]
Figure 21
Figure 21. Figure 21: Simulated thermal responses for Cases 1 and 2 (with energy profiles shown in [PITH_FULL_IMAGE:figures/full_fig_p041_21.png]
Figure 22
Figure 22. Figure 22: Experimentally measured temperature responses of skin tissue under laser [PITH_FULL_IMAGE:figures/full_fig_p042_22.png]
Figure 23
Figure 23. Figure 23: Simulated temperature responses under periodic pulsed laser irradiation for [PITH_FULL_IMAGE:figures/full_fig_p043_23.png]
Figure 24
Figure 24. Figure 24: Simulated thermal responses to 3-second laser pulses with different [PITH_FULL_IMAGE:figures/full_fig_p044_24.png]
Figure 25
Figure 25. Figure 25: Temperature measurements obtained from Sensors 1 and 2 during a 3-second [PITH_FULL_IMAGE:figures/full_fig_p045_25.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

64 extracted references · 64 canonical work pages

  1. [1]

    Sudár, G

    Á. Sudár, G. Futaki, R. Kovács, Continuum modeling perspectives of non-fourier heat conduction in biological systems, Journal of Non- Equilibrium Thermodynamics 46 (4) (2021) 371–381

    work page 2021

  2. [2]

    Shomali, R

    Z. Shomali, R. Kovács, P. Ván, I. V. Kudinov, J. Ghazanfarian, Lagging heat models in thermodynamics and bioheat transfer: a critical review, Continuum Mechanics and Thermodynamics 34 (3) (2022) 637–679

    work page 2022

  3. [3]

    Y. Zhang, Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues, International Journal of Heat and Mass Transfer 52 (21-22) (2009) 4829–4834

    work page 2009

  4. [4]

    Łapka, P

    P. Łapka, P. Furmański, T. S. Wiśniewski, Comparison of different bio- heat transfer models for assessment of burns injuries, in: AIP Conference Proceedings, Vol. 1790, AIP Publishing, 2016. 48

    work page 2016

  5. [5]

    E. G. Cravalho, L. R. Fox, J. C. Kan, The application of the bioheat equation to the design of thermal protocols for local hyperthermia, An- nals of the New York Academy of Sciences 335 (1) (1980) 86–97

    work page 1980

  6. [6]

    Huang, T.-L

    H.-W. Huang, T.-L. Horng, Bioheat transfer and thermal heating for tumor treatment, in: Heat transfer and fluid flow in biological processes, Elsevier, 2015, pp. 1–42

    work page 2015

  7. [7]

    C. Li, J. Miao, K. Yang, X. Guo, J. Tu, P. Huang, D. Zhang, Fourier and non-fourier bio-heat transfer models to predict ex vivo tempera- ture response to focused ultrasound heating, Journal of Applied Physics 123 (17) (2018)

    work page 2018

  8. [8]

    Kumar, S

    A. Kumar, S. Kumar, V. Katiyar, S. Telles, Dual phase lag bio-heat transfer during cryosurgery of lung cancer: Comparison of three heat transfer models, Journal of Thermal Biology 69 (2017) 228–237

    work page 2017

Show all 64 references
  1. [9]

    de Monte, A

    F. de Monte, A. Haji-Sheikh, Bio-heat diffusion under local thermal non-equilibrium conditions using dual-phase lag-based green’s functions, International journal of heat and mass transfer 113 (2017) 1291–1305

    2017

  2. [10]

    J. Zhou, J. Chen, Y. Zhang, Dual-phase lag effects on thermal damage to biological tissues caused by laser irradiation, Computers in biology and medicine 39 (3) (2009) 286–293

    2009

  3. [11]

    M. G. Salem, A. E. Abouelregal, F. Alsharari, H. M. Sedighi, Ther- momechanical response of biological tissues to sudden temperature rise induced by laser beam: insights from three-phase lag theory, Physica Scripta 100 (1) (2024) 015290

    2024

  4. [12]

    Tzou, Lagging behavior in biological systems, in: International Con- ference on Micro/Nanoscale Heat Transfer, Vol

    D. Tzou, Lagging behavior in biological systems, in: International Con- ference on Micro/Nanoscale Heat Transfer, Vol. 43918, 2009, pp. 523– 535

    2009

  5. [13]

    Liu, H.-T

    K.-C. Liu, H.-T. Chen, Analysis for the dual-phase-lag bio-heat transfer during magnetic hyperthermia treatment, International Journal of Heat and Mass Transfer 52 (5-6) (2009) 1185–1192

    2009

  6. [14]

    Afrin, Y

    N. Afrin, Y. Zhang, J. Chen, Thermal lagging in living biological tissue based on nonequilibrium heat transfer between tissue, arterial and ve- nous bloods, International Journal of Heat and Mass Transfer 54 (11-12) (2011) 2419–2426. 49

    2011

  7. [15]

    Liu, H.-T

    K.-C. Liu, H.-T. Chen, P.-J. Cheng, Inverse investigation of non-fourier heat conduction in tissue, Journal of thermal biology 62 (2016) 123–128

    2016

  8. [16]

    Lin, C.-Y

    S.-M. Lin, C.-Y. Li, Analytical solutions of non-fourier bio-heat conduc- tions for skin subjected to pulsed laser heating, International Journal of Thermal Sciences 110 (2016) 146–158

    2016

  9. [17]

    Dinda, J

    A. Dinda, J. Acharya, D. Bhanja, S. Nath, Local thermal non- equilibrium bioheat transfer model for interstitial hyperthermia treat- ment of tumour cell: A numerical approach, Journal of Thermal Biology 110 (2022) 103368

    2022

  10. [18]

    Kumar, K

    D. Kumar, K. Rai, Three-phase-lag bioheat transfer model and its val- idation with experimental data, Mechanics Based Design of Structures and Machines 50 (7) (2022) 2493–2507

    2022

  11. [19]

    Singh, P

    S. Singh, P. Saccomandi, R. Melnik, Three-phase-lag bio-heat transfer model of cardiac ablation, Fluids 7 (5) (2022) 180

    2022

  12. [20]

    Liu, J.-S

    K.-C. Liu, J.-S. Leu, Heat transfer analysis for tissue with surface heat flux based on the non-linearized form of the three-phase-lag model, Jour- nal of Thermal Biology 112 (2023) 103436

    2023

  13. [21]

    Zhang, Y

    Q. Zhang, Y. Sun, J. Yang, Thermoelastic responses of biological tissue under thermal shock based on three phase lag model, Case Studies in Thermal Engineering 28 (2021) 101376

    2021

  14. [22]

    Zhang, Y

    Q. Zhang, Y. Sun, J. Yang, Bio-heat response of skin tissue based on three-phase-lag model, Scientific Reports 10 (1) (2020) 16421

    2020

  15. [23]

    Zhang, X

    L. Zhang, X. Shang, An integral transform solution for bioheat transfer in skin tissue subjected to surface laser irradiation, International Journal of Heat and Mass Transfer 180 (2021) 121706

    2021

  16. [24]

    Gupta, A

    P. Gupta, A. Srivastava, Non-fourier transient thermal analysis of bio- logical tissue phantoms subjected to high intensity focused ultrasound, International Journal of Heat and Mass Transfer 136 (2019) 1052–1063

    2019

  17. [25]

    B. Du, G. Xu, D. Xue, J. Wang, Fractional thermal wave bio-heat equa- tion based analysis for living biological tissue with non-fourier neumann boundary condition in laser pulse heating, Optik 247 (2021) 167811. 50

    2021

  18. [26]

    Y. Yin, M. Li, Y. Li, J. Song, Skin pain sensation of epidermal electronic device/skin system considering non-fourier heat conduction, Journal of the Mechanics and Physics of Solids 138 (2020) 103927

    2020

  19. [27]

    Azhdari, G

    M. Azhdari, G. Rezazadeh, L. Lambers, T. Ricken, H.-M. Tautenhahn, F. Tautenhahn, S. M. Seyedpour, Refining thermal therapy: Tempera- ture distribution modeling with distinct absorption in multi-layered skin tissue during infrared laser exposure, International Communications i...

    2024

  20. [28]

    Azhdari, G

    M. Azhdari, G. Rezazadeh, T. Ricken, R. Pathak, H.-M. Tautenhahn, F. Tautenhahn, S. M. Seyedpour, Temperature distribution in multi- layered skin tissue during laser irradiation considering epidermis sub- layers: Virtual element method approach, Thermal Science and Engi- neeri...

    2025

  21. [29]

    Z. A. Mirza, M. Azhdari, D. Kolomenskiy, G. Rezazadeh, T. Ricken, R. Pathak, H.-M. Tautenhahn, F. Tautenhahn, S. M. Seyedpour, En- hancing laser therapy procedure through surface temperature control in multi-layered skin tissue, Journal of Thermal Biology (2025) 104106

    2025

  22. [30]

    Azhdari, S

    M. Azhdari, S. M. Seyedpour, L. Lambers, H.-M. Tautenhahn, F. Taut- enhahn, T. Ricken, G. Rezazadeh, Non-local three phase lag bio ther- mal modeling of skin tissue and experimental evaluation, Interna- tional Communications in Heat and Mass Transfer 149 (2023) 107146. doi:10....

    2023

  23. [31]

    Strąkowska, G

    M. Strąkowska, G. De Mey, B. Więcek, Experimental verification of non-fourier heat transfer in a multilayer skin tissue struc-ture by ir tem- perature measurement

  24. [32]

    D. Y. Tzou, A unified field approach for heat conduction from macro-to micro-scales (1995)

    1995

  25. [33]

    D. Y. Tzou, A unified field approach for heat conduction from macro- to micro-scales, Journal of Heat Transfer 117 (1) (1995) 8–16. doi:10.1115/1.2822329

    1995 doi

  26. [34]

    Podio-Guidugli, A virtual power format for thermomechanics, Con- tinuum Mechanics and Thermodynamics 20 (2009) 479–487

    P. Podio-Guidugli, A virtual power format for thermomechanics, Con- tinuum Mechanics and Thermodynamics 20 (2009) 479–487. 51

    2009

  27. [35]

    A. N. Bashkatov, E. A. Genina, V. V. Tuchin, Optical properties of skin, subcutaneous, and muscle tissues: a review, Journal of Innovative Opti- cal Health Sciences 4 (01) (2011) 9–38. doi:10.1142/S1793545811001319

    2011 doi

  28. [36]

    Bhatt, K

    M. Bhatt, K. R. Ayyalasomayajula, P. K. Yalavarthy, Generalized beer– lambert model for near-infrared light propagation in thick biological tissues, Journal of Biomedical Optics 21 (7) (2016) 076012–076012. doi:10.1117/1.JBO.21.7.076012

    2016 doi

  29. [37]

    Oshina, J

    I. Oshina, J. Spigulis, Beer–lambert law for optical tissue diagnostics: current state of the art and the main limitations, Journal of biomedical optics 26 (10) (2021) 100901–100901. doi:10.1117/1.JBO.26.10.100901

    2021 doi

  30. [38]

    Saemathong, N

    J. Saemathong, N. Pannucharoenwong, V. Mongkol, S. Vongpradubchai, P. Rattanadecho, Analyzing two laser thermal energy calculation equa- tions: A comparison of beer-lambert’s law and light transport equation, Engineered Science (2023). doi:10.30919/es912

    2023 doi

  31. [39]

    J. E. Hill, J. D. Leitman, S. JE, Thermal conductivity of various meats, Food technology 21 (8) (1967) 1143

    1967

  32. [40]

    Bhattacharya, R

    A. Bhattacharya, R. L. Mahajan, Temperature dependence of ther- mal conductivity of biological tissues, Physiological measurement 24 (3) (2003) 769

    2003

  33. [41]

    J. R. Kanagy, Specific heats of collagen and leather, Journal of Research of the National Bureau of Standards 55 (4) (1955) 191–195

    1955

  34. [42]

    Z. Pan, R. P. Singh, Physical and thermal properties of ground beef during cooking, L WT-Food Science and Technology 34 (7) (2001) 437– 444

    2001

  35. [43]

    J. Xu, A. Psikuta, J. Li, S. Annaheim, R. M. Rossi, Evaluation of the convective heat transfer coefficient of human body and its effect on the human thermoregulation predictions, Building and Environment 196 (2021) 107778

    2021

  36. [44]

    N. N. Mohd Nazali, N. F. Adull Manan, The numerical analysis of hy- perelastic properties in commercial and original aloe vera gels, Journal of Mechanical Engineering 18 (3) (2021) 1–20. 52

    2021

  37. [45]

    P. J. Rodríguez de Rivera, M. Rodríguez de Rivera, F. Socorro, J. A. Calbet, M. Rodríguez de Rivera, A dvantages of in vivo measurement of human skin thermal conductance using a calorimetric sensor, Journal of Thermal Analysis and Calorimetry 147 (18) (2022) 10027–10036

    2022

  38. [46]

    Azhdari, S

    M. Azhdari, S. M. Seyedpour, T. Ricken, G. Rezazadeh, On the thermo- vibrational response of multi-layer viscoelastic skin tissue to laser irra- diation, International Journal of Thermal Sciences 187 (2023) 108160

    2023

  39. [47]

    S. M. Seyedpour, M. Azhdari, L. Lambers, T. Ricken, G. Reza- zadeh, One-dimensional thermomechanical bio-heating analy- sis of viscoelastic tissue to laser radiation shapes, Interna- tional Journal of Heat and Mass Transfer 218 (2024) 124747. doi:10.1016/j.ijheatmasstransfer.2...

    2024

  40. [48]

    M. Ciesielski, Application of the alternating direction implicit method for numerical solution of the dual-phase lag equation, Jour- nal of Theoretical and Applied Mechanics 55 (3) (2017) 839–852. doi:10.15632/jtam-pl.55.3.839

    2017 doi

  41. [49]

    Majchrzak, B

    E. Majchrzak, B. Mochnacki, Implicit scheme of the finite dif- ference method for 1d dual-phase lag equation, Journal of Ap- plied Mathematics and Computational Mechanics 16 (3) (2017). doi:10.17512/jamcm.2017.3.04

    2017 doi

  42. [50]

    Majchrzak, B

    E. Majchrzak, B. Mochnacki, Implicit scheme of the finite differ- ence method for the second-order dual phase lag equation, Jour- nal of Theoretical and Applied Mechanics 56 (2) (2018) 393–402. doi:10.15632/jtam-pl.56.2.393

    2018 doi

  43. [51]

    Majchrzak, B

    E. Majchrzak, B. Mochnacki, First and second order dual phase lag equation. numerical solution using the explicit and implicit schemes of the finite difference method, in: MATEC Web of Conferences, Vol. 240, EDP Sciences, 2018, p. 05018. doi:10.1051/matecconf/201824005018

    2018

  44. [52]

    Majchrzak, B

    E. Majchrzak, B. Mochnacki, Numerical solutions of the second-order dual-phase-lag equation using the explicit and implicit schemes of the finite difference method, International Journal of Numerical Methods for Heat & Fluid Flow 30 (4) (2020) 2099–2120. doi:10.1108/HFF-11- 20...

    2020 doi

  45. [53]

    S. A. Rukolaine, Unphysical effects of the dual-phase-lag model of heat conduction, International Journal of Heat and Mass Transfer 78 (2014) 58–63

    2014

  46. [54]

    S. A. Rukolaine, Unphysical effects of the dual-phase-lag model of heat conduction: higher-order approximations, International Journal of Thermal Sciences 113 (2017) 83–88

    2017

  47. [55]

    M. A. Ezzat, M. H. Alabdulhadi, Thermomechanical interactions in vis- coelastic skin tissue under different theories, Indian Journal of Physics 97 (1) (2023) 47–60

    2023

  48. [56]

    J. Ma, X. Yang, Y. Sun, J. Yang, Thermal damage in three-dimensional vivo bio-tissues induced by moving heat sources in laser therapy, Scien- tific reports 9 (1) (2019) 10987

    2019

  49. [57]

    X. Li, X. Tian, The thermal injury analysis of skin tissue with a new nonlocal dual phase lag model, Waves in Random and Complex Media (2022) 1–14

    2022

  50. [58]

    Zhang, B

    Y. Zhang, B. Chen, D. Li, Non-fourier effect of laser-mediated thermal behaviors in bio-tissues: A numerical study by the dual-phase-lag model, International Journal of Heat and Mass Transfer 108 (2017) 1428–1438

    2017

  51. [59]

    Tiwari, A

    R. Tiwari, A. Singhal, R. Kumar, P. Kumar, S. Ghangas, Investigation of memory influences on bio-heat responses of skin tissue due to various thermal conditions, Theory in Biosciences 142 (3) (2023) 275–290

    2023

  52. [60]

    A. Sur, S. Mondal, M. Kanoria, Influence of moving heat source on skin tissue in the context of two-temperature memory-dependent heat transport law, Journal of Thermal Stresses 43 (1) (2020) 55–71

    2020

  53. [61]

    X. Li, X. Tian, Fractional order thermo-viscoelastic theory of biological tissue with dual phase lag heat conduction model, Applied Mathematical Modelling 95 (2021) 612–622

    2021

  54. [62]

    Kumar, R

    R. Kumar, R. Tiwari, A. Singhal, S. Mondal, Characterization of ther- mal damage of skin tissue subjected to moving heat source in the purview of dual phase lag theory with memory-dependent derivative, Waves in Random and Complex Media 34 (4) (2024) 3510–3527. 54

    2024

  55. [63]

    A. M. Zenkour, T. Saeed, A. M. Aati, Analyzing the thermoelastic re- sponses of biological tissue exposed to thermal shock utilizing a three- phase lag theory, Journal of Computational Applied Mechanics 55 (2) (2024) 144–164

    2024

  56. [64]

    K. K. Jojare, K. R. Gaikwad, A study of the 3-phase lag model to a two- dimensional isotropic micro-polar thermoelastic medium with memory- dependent properties, Journal of Thermal Stresses 47 (3) (2024) 363– 382. 55

    2024

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.