pith. sign in

arxiv: 2604.06225 · v1 · submitted 2026-03-27 · ⚛️ physics.bio-ph · physics.optics

An Analytical Framework for Frequency-Dependent Electromagnetic Power Absorption in Biological Tissues

Hongyun Wang , Shannon E. Foley , Hong Zhou This is my paper

Pith reviewed 2026-05-14 23:10 UTC · model grok-4.3

classification ⚛️ physics.bio-ph physics.optics
keywords electromagnetic absorptionbiological tissuespenetration depthMaxwell equationsdielectric permittivityfrequency dependencepower reflectance
0
0 comments X

The pith

Closed-form electromagnetic field expressions enable frequency-dependent power absorption calculations in biological tissues

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

This paper develops an analytical framework from Maxwell's equations to describe how a normally incident plane electromagnetic wave propagates through homogeneous lossy dielectric biological tissues. Closed-form solutions for the electric and magnetic fields inside the tissue allow exact computation of power reflectance and transmittance at the air-tissue boundary along with the absorption coefficient and penetration depth. Applying measured complex permittivity values across six tissue types from 1 MHz to 100 GHz reveals that tissues with higher water content experience stronger dielectric losses and shallower penetration while lower-water tissues allow deeper wave travel. Frequency emerges as a key factor that moves the absorbed power from reflection-dominated at low frequencies to surface-limited absorption at high frequencies. The results supply a quantitative basis for evaluating electromagnetic exposure risks and guiding the development of related medical and communication technologies.

Core claim

The paper derives closed-form expressions for the electric and magnetic fields of a normally incident plane wave in lossy dielectric biological tissues, enabling exact determination of power reflectance and transmittance at the interface as well as the absorption coefficient and penetration depth as functions of frequency.

What carries the argument

Closed-form solutions to Maxwell's equations for plane wave propagation in a homogeneous isotropic lossy dielectric medium using the complex wave number from the tissue relative permittivity

If this is right

  • Tissues with high water content exhibit greater power absorption near the surface and reduced penetration depth
  • Low-water tissues such as fat show lower attenuation and allow greater penetration
  • Increasing frequency shifts the power budget toward more superficial absorption and higher reflection
  • The framework supports quantitative assessment of electromagnetic exposure across frequency bands

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The model could be extended to layered or heterogeneous tissues to approximate real anatomical structures more closely
  • Penetration depth results could guide selection of operating frequencies for electromagnetic medical devices
  • The plane-wave normal-incidence assumption may require modification when applied to near-field sources from wearable electronics

Load-bearing premise

Biological tissues behave as homogeneous isotropic lossy dielectrics with complex permittivity values taken from prior literature under normal plane wave incidence

What would settle it

Experimental measurement of reflection coefficient or penetration depth in a uniform tissue sample at one frequency that deviates substantially from the value given by the closed-form expressions

Figures

Figures reproduced from arXiv: 2604.06225 by Hongyun Wang, Hong Zhou, Shannon E. Foley.

Figure 1
Figure 1. Figure 1: The coordinate system for an electromagnetic wave normally incident on a flat layer of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Dielectric properties of dry skin as functions of frequency (1 MHz–100 GHz). Top left: [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Dielectric properties of wet skin as functions of frequency (1 MHz–100 GHz). Top left: [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Dielectric properties of lens cortex as functions of frequency (1 MHz–100 GHz). Top left: [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Dielectric properties of non-infiltrated fat as functions of frequency (1 MHz–100 GHz). [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Dielectric properties of liver as functions of frequency (1 MHz–100 GHz). Top left: Real [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Dielectric properties of muscle as functions of frequency (1 MHz–100 GHz). Top left: Real [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Frequency-dependent dielectric constants (top) and dielectric losses (bottom) for the six [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the power reflectance (top) and power transmission coefficient (bottom) [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the power absorption coefficient (top) and the electromagnetic penetra [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
read the original abstract

As exposure to electromagnetic waves becomes increasingly widespread, it is important to quantify how incident fields couple into biological tissue and where absorbed energy is deposited. This work presents an analytical, physics based framework derived from Maxwell's equations to model the propagation of a normally incident electromagnetic plane wave within homogeneous, lossy dielectric biological tissues. Closed-form expressions for the electric and magnetic fields are derived, enabling the determination of frequency-dependent power reflectance and transmittance at the air-tissue interface, as well as the power absorption coefficient and penetration depth within the medium. Using complex relative permittivity data from the literature, we examine six tissue types across a broad frequency range (1 MHz to 100 GHz). The results demonstrate that higher water content significantly increases dielectric loss and reduces penetration depth. Conversely, low-water tissues (e.g., non-infiltrated fat) exhibit lower attenuation and deeper penetration. Frequency is shown to be a dominant driver of this behavior, with higher frequencies shifting the power budget from reflection-limited coupling toward highly superficial absorption. These findings provide a foundation basis for exposure assessments and the design of emerging electromagnetic technologies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The manuscript presents an analytical framework derived from Maxwell's equations for the propagation of a normally incident plane electromagnetic wave in homogeneous, lossy, isotropic dielectric biological tissues. Closed-form expressions are derived for the electric and magnetic fields, from which frequency-dependent power reflectance and transmittance at the air-tissue interface, the power absorption coefficient, and penetration depth are obtained. Using complex relative permittivity data from the literature, the framework is applied to six tissue types across 1 MHz to 100 GHz, demonstrating that higher water content increases dielectric loss and reduces penetration depth while frequency shifts absorption toward superficial layers.

Significance. If the central derivations hold, the work supplies a compact analytical tool for rapid computation of EM power deposition in tissues, useful for exposure assessment and technology design without numerical simulation. The closed-form results and broad frequency/tissue coverage highlight clear physical trends grounded in standard electromagnetics. The approach strengthens reliability by avoiding empirical fitting and relying on external literature inputs for material properties.

minor comments (4)
  1. [§2.1] §2.1: The propagation constant is written as k = ω√(μϵ) with complex ϵ; explicitly state the time-harmonic convention (e^{jωt} or e^{-jωt}) and sign convention for the imaginary part of ϵ to prevent ambiguity in the derived field expressions.
  2. [Results section] Results section, Figure 3: Penetration-depth curves for high-water-content tissues are plotted on linear scales that compress the low-frequency regime; a log-frequency axis or inset would improve readability of the frequency dependence.
  3. [§3.2] §3.2: The power absorption coefficient α is obtained from the imaginary part of k, but the text does not show the explicit algebraic reduction from the field expressions to α; adding one intermediate step would clarify the link to the central claim.
  4. [Table 1] Table 1 and §4: Specific literature sources for the complex permittivity values of each tissue are referenced only generically; listing the exact references (e.g., Gabriel et al. or similar) next to each tissue entry would aid reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, including recognition of the closed-form derivations from Maxwell's equations and the utility of the framework for rapid EM power deposition calculations across tissues and frequencies. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation begins from Maxwell's equations for a normally incident plane wave at an air-tissue interface and proceeds to closed-form expressions for the reflected/transmitted fields, reflection coefficient, propagation constant k = ω√(μϵ*), time-averaged Poynting vector, absorption coefficient, and penetration depth. All steps follow standard boundary-condition and wave-propagation algebra with no internal fitting of parameters; the only external inputs are tabulated complex permittivity values drawn from independent literature. No self-citations are invoked to justify uniqueness or to close any loop, and no quantity is redefined in terms of itself or renamed as a prediction. The central results are therefore independent of the paper's own outputs and remain self-contained against textbook electromagnetism.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on Maxwell's equations for linear media and the assumption of homogeneous tissues; no new free parameters are fitted in the paper and no new entities are postulated.

axioms (2)
  • standard math Maxwell's equations govern electromagnetic wave propagation in linear isotropic media.
    The framework is explicitly derived from Maxwell's equations as stated in the abstract.
  • domain assumption Tissues can be treated as homogeneous lossy dielectrics with frequency-dependent complex permittivity taken from external literature.
    Required for closed-form plane-wave solutions and for the numerical evaluation across tissue types.

pith-pipeline@v0.9.0 · 5489 in / 1344 out tokens · 51521 ms · 2026-05-14T23:10:56.369416+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

  1. [1]

    Zhadobov, N

    M. Zhadobov, N. Chahat, R. Sauleau, C. Quement and Y. Drean, Millimeter-wave interac- tions with the human body: state of knowledge and recent advances, International Journal of Microwave and Wireless Technologies, 3(2), 237-247, 2011

    work page 2011

  2. [2]

    Colombi, B

    D. Colombi, B. Thors, C. Tornevik and Q. Balzano, RF energy absorption by biological tissues in close proximity to millimeter-wave 5G wireless equipment, IEEE Access, 6, 4974-4981, 2018

    work page 2018

  3. [3]

    A. Owda, N. Salmon, A. Casson and M. Owda, The reflectance of human skin in the millimeter- wave band, Sensors, 20, 1480, 2020

    work page 2020

  4. [4]

    Wang et al., Millimeter waves in medical applications: status and prospects, Intelligent Medicine, 4, 16-21, 2024

    H. Wang et al., Millimeter waves in medical applications: status and prospects, Intelligent Medicine, 4, 16-21, 2024

    work page 2024

  5. [5]

    A. Gasmelseed, Parametric analysis of electromagnetic wave interactions with layered biolog- ical tissues for varying frequency, polarization, and fat thickness, Scientific Reports, 16, 3445, 2026

    work page 2026

  6. [6]

    Sacco, S

    G. Sacco, S. Pisa and M. Zhadobov, Age-dependence of electromagnetic power and heat de- position in near-surface tissues in emerging 5G bands, Scientific Reports, 11, 3983, 2021

    work page 2021

  7. [7]

    Vilagosh, A

    Z. Vilagosh, A. Lajevardipour and A. Wood, Computer simulation study of the penetration of pulsed 30, 60 and 90 GHz radiation into the human ear, Scientific Reports, 10, 1479, 2020

    work page 2020

  8. [8]

    Adekola, K

    S. Adekola, K. Amusa and G. Biowei, Impact of 5G MMW radiation on human tissue using skin, cornea (eye) and enamel (tooth) as study candidates, Journal of Engineering and Applied Science, 72, 51, 2025

    work page 2025

  9. [9]

    Alekseev and M.C

    S.I. Alekseev and M.C. Ziskin, Reflection and absorption of millimeter waves by thin absorbing films, Bioelectromagnetics, 21, 264-271, 2000

    work page 2000

  10. [10]

    Alekseev, O.V

    S.I. Alekseev, O.V. Gordiienko, and M.C. Ziskin, Reflection and penetration depth of millime- ter waves in murine skin, Bioelectromagnetics, 29, 340-344, 2008

    work page 2008

  11. [11]

    Gandhi and A

    O.P. Gandhi and A. Riazi, Absorption of millimeter waves by human beings and its biological implications, IEEE Trans Microwave Theory Tech 34, 228–235, 1986

    work page 1986

  12. [12]

    Gabriel, R

    S. Gabriel, R. W. Lau and C. Gabriel, The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues, Phys. Med. Biol., 41, 2271-2293, 1996

    work page 1996

  13. [13]

    Griffiths, Introduction to Electrodynamics, 4th edition, Cambridge University Press, Cam- bridge, 2013

    D. Griffiths, Introduction to Electrodynamics, 4th edition, Cambridge University Press, Cam- bridge, 2013

    work page 2013

  14. [14]

    Yeh, Optical Waves in Layered Media, John Wiley and Sons, New York, 1988

    P. Yeh, Optical Waves in Layered Media, John Wiley and Sons, New York, 1988

    work page 1988

  15. [15]

    Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1998 24

    J.D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1998 24

    work page 1998