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REVIEW 5 major objections 6 minor 27 references

An analytical model for gold nanoparticle radiosensitisation

T0 review · 5 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that gold-nanoparticle radiosensitisation is set by concentration, beam quality and nucleus size, with closed-form survival curves.

desk verdict A clean derivation of DER = 1 + Kc c that is honestly admitted to match Brown's interpolation, but the survival-curve section has a load-bearing voxel-to-nucleus switch that breaks the closed forms. read the letter →

arxiv 2506.06671 v7 pith:33HCKHX6 submitted 2025-06-07 physics.med-ph

Pedro Teles This is my paper
classification physics.med-ph
keywords goldnanoparticlesradiosensitisationlocaleffectmodeldoseenhancementratiolog-normaldistributionlinear-quadraticclosed-formsurvivalcurvesAugercascade
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

The paper tries to establish that gold-nanoparticle radiosensitisation is governed by one linear relation, $DER = 1 + K_c(R,E)\,c$, where $c$ is intracellular concentration and $K_c$ depends only on beam quality and cell/nucleus geometry. If the enhanced dose is log-normally distributed, this relation fixes the distribution width via $\sigma = \sqrt{2\ln(1+K_c c)}$, and averaging a linear-quadratic survival response over that fluctuation yields three closed-form survival curves for low, intermediate and high concentrations. If correct, the model predicts AuNP-enhanced cell survival analytically from a few measurable quantities, with no voxel-level Monte Carlo. It reproduces published BAEC survival fractions within about 2.5% and concludes that the enhancement is mostly $\alpha$-driven, a departure from the earlier LEM interpretation.

What carries the argument

The central object is the variance-driven Local Effect Model ($\sigma$-LEM), which replaces voxel-level Monte Carlo dose scoring with a log-normal ansatz for the enhanced dose, $D_{\mathrm{enh}} = D_0 \exp(\sigma Z)$ with $Z \sim \mathcal{N}(0,1)$. The load-bearing identity is $DER = 1 + K_c(R,E)\,c$, obtained by counting photo-ionisations in a gold nanoparticle ($N_{\mathrm{ion}} \propto V_{\mathrm{NP}} \propto R^3$), weighting by a nucleus dose-efficiency factor $\Xi(R)$, and canceling the $R^3$ dependence against the concentration formula. This identity converts the stochastic width into a deterministic function of measurable quantities, $\sigma = \sqrt{2\ln(1+K_c c)}$, and the machinery then carries the argument by Gaussian averaging of the linear-quadratic survival response, yielding the three closed-form curves for the low-, mid-, and high-concentration regimes.

What would settle it

Run a nanoscale track-structure Monte Carlo (or a microdosimetric measurement) around individual AuNPs under 100 kVp irradiation, construct the per-voxel histogram of $D_{\mathrm{enh}}$, and check whether $\ln(D_{\mathrm{enh}}/D_0)$ is Gaussian with variance $2\ln(1+K_c c)$ using the same $K_c$ that reproduced the BAEC data; a non-Gaussian shape or a variance that departs from this relation would falsify the closed-form survival predictions.

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Extended reading notes

Core claim

The central discovery is that the variance of the stochastic dose enhancement is not a separate free parameter: once the dose-enhancement ratio is written as $DER = 1 + K_c(R,E)\,c$ from the $V_{\mathrm{NP}} \propto R^3$ scaling of gold photo-ionisations, the width of the log-normal fluctuation is fixed as $\sigma = \sqrt{2\ln(1+K_c c)}$. Combining this with the linear-quadratic survival model and averaging over the Gaussian variable $Z$ gives three closed-form survival formulas, the variance-only, mixed-term and second-order forms (Eqs. 42, 44 and 51), each adapted to a different concentration regime. The paper calibrates the single energy-independent product $\langle \varepsilon_{\mathrm{cas}} N_{\mathrm{Auger}}\rangle$ against monoenergetic synchrotron DER data and then predicts DER and survival for BAEC cells under a 100 kVp beam, reporting survival-fraction deviations below 2.5%. In this studied case the fitted LQ coefficients show the enhancement is mostly $\alpha$-driven, with $\beta$ playing a secondary role until the highest concentration, contrary to the standard LEM picture.

Load-bearing premise

The load-bearing premise is that the extra dose in each voxel is log-normally distributed; if the true distribution of Auger-cascade doses around gold nanoparticles is not log-normal, the Gaussian averaging and all three survival formulas lose their basis.

Editorial extensions

If this is right

  • If the linear relation holds, radiosensitisation is controlled by intracellular concentration once beam quality and cell geometry are fixed; particle size enters only through the conversion to molar concentration.
  • Because $\sigma$ is fixed by $K_c$ and $c$, survival curves can be generated analytically and refit to linear-quadratic parameters without any voxel-level Monte Carlo dose map.
  • The three closed forms divide concentration space: the variance-only form for sparse single-cascade hits, the mixed term when simultaneous baseline-plus-Auger hits matter, and the second-order form when double cascades dominate at high concentration.
  • The model predicts predominantly $\alpha$-driven enhancement in the kilovoltage regime, with $\beta$ contributing only at the highest concentrations, a mechanistic conclusion that differs from earlier LEM-based analyses.
  • For megavoltage beams the model predicts $DER$ effectively equal to 1 (about 1.0003 at 6 MV and 1 mM), so its validity is limited to roughly $\lesssim 150$–$200$ keV; MV radiosensitisation must involve physics outside this model.

Reading between the lines

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

  • A direct test of the linearity premise would be to measure DEF80 or clonogenic survival across a wide concentration range (for example 0.1 to 5 mM) in a fixed beam; curvature in the DER-versus-$c$ plot would indicate the $R^3$-cancellation argument is incomplete.
  • The $\alpha$-driven prediction is experimentally cheap to probe: fit survival curves at several concentrations; $\alpha(c)$ should rise linearly with $c$ while $\beta(c)$ stays nearly flat, and a clear $\beta$ growth would refute the picture.
  • A numerical integration of the exact log-normal average $S = E[\exp(-\alpha D_{\mathrm{enh}} - \beta D_{\mathrm{enh}}^2)]$ would reveal how much each closed-form truncation errs at intermediate concentrations and could yield a single interpolation valid across all $c$.
  • The MV result suggests clinically reported AuNP effects at high energy need mechanisms beyond local Auger-cascade dose to the nucleus, such as long-range secondary-electron transport or biological targeting effects; adding a small deterministic shift (a $k<2$ splitting of the gain) is a testable way to recover MV DERs without disturbing the kV results.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. This paper proposes an analytical 'variance-driven Local Effect Model' (σ-LEM) for gold-nanoparticle (AuNP) radiosensitisation. Assuming the number of gold photo-ionisations scales with nanoparticle volume, Section 3.5 derives a linear dose-enhancement relation DER = 1 + Kc(R,E) c, with Kc a beam-quality- and geometry-dependent constant. Assuming the enhanced dose in a target voxel is log-normally distributed (Eq. 9), the width is linked to concentration by σ² = 2 ln(1 + Kc c), and Section 3.6 obtains three closed-form cell-survival curves (variance-only, mixed, second-order; Eqs. 42, 44, 51) by Gaussian averaging of truncated Taylor expansions. The single calibration constant ⟨εcasNAuger⟩ is fitted to synchrotron DERs [21]; Table 1 compares model DERs with those data, and Table 4 compares the survival variants with BAEC 100 kVp data from [5,7], with claimed agreement within about 2.5%. The paper concludes that AuNP radiosensitisation is predominantly α-driven and that the formalism is limited to photon beams below roughly 150–200 keV.

Significance. If valid, the framework would be a useful analytical complement to Monte-Carlo-based LEM approaches, since it reduces AuNP radiosensitisation predictions to beam quality, nucleus size and concentration and produces closed-form survival expressions. The paper merits credit for transparently disclosing its assumptions and limitations (the MV-regime failure is acknowledged and the calibration is explicit), for showing in Appendix A that DER = 1 + Kc c is algebraically identical to Brown's interpolation, and for providing reproducibility notebooks at a Zenodo DOI. The α-dominated enhancement claim is a concrete, falsifiable prediction. The significance is nonetheless limited by load-bearing problems in the central derivation and in the validation: the survival formulas do not follow from the stated voxel-level premise, the log-normal ansatz lacks AuNP-specific track-structure support, and the empirical comparisons are fitted to (or selected among) the very data they purport to test.

major comments (5)
  1. [§3.2, §3.4, §3.6 (Eqs. 8–9, 23, 37, 42, 44, 51)] The stochastic variable Denh is defined at two incompatible scales. Equation (8) defines Denh as the dose deposited in a scoring voxel smaller than the Auger-electron range, and the geometric-central-limit-theorem justification of lognormality applies at that voxel scale. Equation (23), however, sets ⟨Denh⟩ = D0 + Dnuc,extra, which is a whole-nucleus quantity, and Eq. (37) inserts that same Denh into the whole-cell linear-quadratic expression before averaging over a single Z. A nucleus contains many voxels; survival should be a product over voxels (or an average of that product for correlated voxels), and the voxel count never appears in Eqs. (42), (44) or (51). If Denh is genuinely voxel-level, the closed forms omit the voxel-count factor and the correct averaging of a sum of lognormals; if it is the whole-nucleus dose, its lognormality is not established by the voxel-scale argument of §3.2. The three survival formulas therefore do not follow from the stated assumptions, independently of whether the empirical lognormal claim is true; a reformulation in which Denh is the per-cell total nucleus dose could rescue the algebra, but that is not what the text states and it would require a different justification.
  2. [§3.2 (Eq. 9); §3.6 (Eqs. 36, 45)] The log-normality of the enhanced dose is a load-bearing assumption that is asserted rather than established for AuNP-targeted irradiation. The supporting references [14–17] report photoelectron track-length distributions measured in gas detectors and electron-microbeam dosimetry; no AuNP track-structure evidence is cited, and no alternative distribution or sensitivity analysis is tested. In addition, the expansions in Eqs. (36) and (45) are small-σ Taylor expansions, but with the fitted Kc = 2.11 mM⁻¹ the model's own σ values are σ ≈ 0.92 at 0.25 mM and σ ≈ 1.51 at 1.00 mM. The expansion parameter is therefore not small in the regime in which the model is validated, and the accuracy of the truncated Gaussian averages relative to the exact lognormal average is never quantified.
  3. [§2, §3.5, Table 1, Table C.8] The validation is circular: the single energy-independent constant ⟨εcasNAuger⟩ is fitted by minimising the RMSE against the synchrotron DERs of [21], and Table 1 then presents the agreement with those same data as a test. The residuals are large (−50.7% at 60 keV, +132.2% at 30 keV, mean absolute error ≈ 37%), so even as fits the claim of agreement 'within experimental scatter' is not supported. Because the survival data of [5,7] derive from the same BAEC/Rahman experiments and the variant of the model (variance-only, mixed, or second-order) is chosen per concentration in Table C.8, the ≤2.5% survival agreement is also not an out-of-sample prediction of a single model. The fitted constant is additionally quoted inconsistently as 1×10⁻⁵ Gy (§2) and 2.2×10⁻⁴ Gy (§3.5), a factor-22 discrepancy that affects any reproduction of the results.
  4. [§3.5 (Eqs. 29–31) vs §3.6 (Eqs. 36, 45, 42, 44, 51)] The dose distributions actually averaged in the survival formulas are inconsistent with the DER used to set σ. Equation (29) gives the lognormal mean as D0 exp(σ²/2) = D0·DER, but the first-order expansion in Eq. (36) has mean D0 and the second-order expansion in Eq. (45) has mean D0(1 + σ²/2); neither equals D0·DER except for small σ. With σ ≈ 1.51 at 1.00 mM, the second-order mean is about 2.14 D0 whereas DER·D0 = 3.11 D0, so the survival curves of Table C.6 do not encode the dose enhancement the model claims to predict. This mean mismatch is absorbed differently at each concentration and may be the reason why a different model variant has to be selected for each concentration.
  5. [§3.4, Eq. (18)] Equation (18) contains a geometric error: the fraction of AuNPs in the cytoplasm shell within distance Reff of the nucleus is written as a ratio of shell volumes with denominator (Rcell − Rnuc)³, but the volume of the spherical shell Rnuc < r < Rcell is (4π/3)(Rcell³ − Rnuc³). With Rnuc = 5 µm and Rcell = 10 µm, the printed denominator is smaller than the correct one by a factor of about 7, so fhit and hence Ξ(R) in Eqs. (21)–(22) are overestimated. The functional form of Eq. (28) is unaffected because Kc is calibrated, but the claimed first-principles values of the mechanistic constants are not reliable as printed.
minor comments (6)
  1. [Abstract, Table 4, Conclusions] The abstract claims agreement within ≤2.5%, but Table 4 contains deviations of −2.8% (4 Gy, 0.50 mM) and +2.5% (5 Gy, 1.00 mM), and the Conclusions claim 'DEF80% below 5% for all concentration regimes' while Table C.7 lists a 7.7% error for the best model at 0.25 mM; the numerical claims should be made internally consistent.
  2. [§3.5, after Table 1] The sentence beginning 'The two sole free quantities are the product < εcasNAuger >, i.e. the mean lethal dose delivered to nuclear DNA per gold photo-ionisation, Xi(R) = 0.05594...' is grammatically broken and should be rewritten.
  3. [§3.4, Eq. (17)] Equation (17) states the deposition shell as 'Rnuc < r < Reff' and the accompanying text refers to a shell '(Reff − Rnuc ∼ 500 nm)'; since Reff ≈ 500 nm while Rnuc = 5 µm, the intended inequality is almost certainly Rnuc + R < r < Rnuc + Reff, and the phrase 'we can state from that' is a typo.
  4. [§2, Methodology] Stating that the algebra was 'verified using ChatGPT-4o ... and Claude 4.0' is unconventional for a journal article and is better placed in acknowledgements or removed; the subsequent Wolfram/Mathematica and author verification is unproblematic.
  5. [§3.4, Eq. (14)] The quantities D(1)(E, x) and Φ′(1)(E, x) in Eq. (14) are not defined; the superscript and prime notation should be explained at first use.
  6. [Table 1, 70 keV row] At 70 keV the model predicts DER = 0.96, i.e. net radioprotection, while the experimental value is 1.30 (net enhancement); the text's 'clear prediction of the dip at 70 keV' should acknowledge that the model over-predicts the dip into a deficit rather than reproducing the experimental enhancement.

Circularity Check

3 steps flagged · score 6.0 of 10

DER validation in Table 1 uses the same synchrotron data that calibrated the single free parameter; survival agreement is reached by post hoc choice of model per concentration, and the survival derivation reuses a voxel-level lognormal as a whole-nucleus dose.

  1. fitted input called prediction [Section 2 (Methodology); Section 3.5 and Table 1]
    "A single energy–independent constant, εcasNAuger ≈ 1 × 10−5 Gy, was obtained by minimising the Root Mean Square Error between model and synchrotron DERs. ... Calibrating the product to the synchrotron dataset yields < ϵcas NAuger >≈ 2.2 × 10−4 Gy, well within the plausible range."

    The only free parameter of the DER model is obtained by least-squares minimization against the synchrotron DERs; Table 1 then presents 'model DER' for those same energies and the same experimental dataset as the validation. The quoted percent differences and RMSE are therefore in-sample fit diagnostics, not out-of-sample predictions. The abstract's claim that the model 'agrees within ≤ 2.5%' with experimental data is partly a restatement of the calibration target.

  2. other [Section 3.6; Appendix C, Tables C.6–C.8]
    "With this analysis, it becomes clear that the optimal models for each concentration are: • for 0.25 mM and 0.50 mM, the first-order approximation including the mixed term; • for 1.00 mM, the second order approximation."

    The survival-curve agreement is obtained by selecting, after seeing the experimental data, which of three derived survival forms (variance-only, mixed, second-order) is used for each concentration. Appendix C.8 labels this 'Best model (minimum error) per concentration'. No single predictive formula is fixed in advance for all concentrations, so the reported ≤2.5% agreement is an ex post selection outcome rather than a prospective validation of one derived curve.

1 more flagged steps
  1. other [Section 3.2 Eq. (8); Section 3.5 Eqs. (23), (37); Section 3.6 Eqs. (42), (44), (51)]
    "Denh = 1/mvox Σ S(Ei)ℓi ... In this picture, the voxel must be such that it is smaller than the range of Auger electrons. ... ⟨Denh⟩ = D0 + Dnuc,extra. ... Senh = e−αDenh−βD^2enh."

    Denh is introduced as a per-voxel dose whose lognormality follows from the geometric central-limit theorem on track segments; Eq. 23 identifies its mean with the whole-nucleus dose, and Eq. 37 feeds this same scalar Denh into whole-cell LQ survival. A nucleus contains many voxels, so cell survival should involve the sum (or product) of voxel doses, requiring the distribution of a sum of lognormals and a voxel-count factor. The closed forms in Eqs. 42, 44 and 51 therefore presuppose that whole-nucleus dose is itself lognormal—the very quantity Eq. 8 was used to justify at voxel scale—rather than deriving that property from the stated assumptions.

full rationale

The paper's algebraic core—Gaussian moment integrals, truncation corrections, and second-order expansions—is self-contained, and its use of NIST cross-sections and SpekPy spectra is external. However, the headline validations are not independent. The only free parameter <εcasNAuger> is calibrated by RMSE minimization against the synchrotron DERs, so Table 1's 'model DER' is an in-sample fit rather than a prediction; the small deviations quoted in the abstract are partly a restatement of the calibration. Likewise, the survival-curve agreement within ≤2.5% is obtained by choosing the best of three analytic forms separately for each concentration, an ex post selection on the same data, rather than by testing a single prospective formula. The derivation also switches Denh from a sub-Auger-range voxel dose (Eq. 8) to the whole-nucleus dose entering LQ survival (Eqs. 23 and 37) without summing over voxels, so Eqs. 42/44/51 presuppose a lognormal whole-nucleus dose rather than deriving it. These are circular-validation and definitional-reuse issues, but not a self-citation chain: the cited self-work [13] is used only for Taylor-expansion bookkeeping and is not load-bearing. The model does contain non-fitted outputs, such as the near-unity MV DER predictions, which are externally falsifiable; nevertheless, the central 'predictions' against BAEC data are partly forced by the calibration and post hoc selection procedure, so a partial-circularity score of 6 is appropriate.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The model's output is controlled by a single fitted constant (<epsilon_cas N_Auger>), a hand-set variance split (k=2), an unvalidated log-normal ansatz, and a per-concentration choice of survival formula. The linear DER relation is algebraically identical to Brown's prior interpolation (Eq. A4). No new physical entities are introduced.

free parameters (3)
  • <epsilon_cas N_Auger> = 2.2e-4 Gy
    Energy-independent product of mean cascade energy per photo-ionisation and number of Auger electrons; calibrated by RMSE minimisation against synchrotron DERs (Section 2, Section 3.5). This constant sets Kc and hence all DER and survival predictions.
  • k (variance splitting factor) = 2
    Chosen to be 2 so that mu=0 (fully stochastic enhancement), with the only justification being that this 'maintains consistency with our physically motivated scenario' (Section 3.5). The model itself permits 0<=k<=2.
  • Survival model variant per concentration = mixed / mixed / second-order for 0.25/0.50/1.00 mM
    The choice of which of Eqs. 42/44/51 is used for each concentration is made after comparing with the target data (Tables C.6-C.8), adding a discrete overfitting degree of freedom.
assumptions (8)
  • ad hoc to paper Enhanced dose in a voxel is log-normally distributed: Denh = D0 exp(sigma Z), Z ~ N(0,1) (Eq. 9).
    Asserted in Section 3.2, based on the geometric central limit theorem and citations to photoelectron track length measurements [14-17] from unrelated detector contexts. No AuNP-specific track-structure evidence is provided; the entire stochastic derivation rests on this.
  • domain assumption Number of Au photo-ionisations scales strictly as V_NP (N_ion proportional to R^3, Eq. 5).
    Taken from McMahon et al. [8] and the previous paper [13]; standard for small NPs under thin-target approximation. Reasonable but not re-derived in this paper.
  • domain assumption Linear-quadratic (LQ) cell survival model applies (Eq. 37).
    Standard radiobiology assumption, used throughout; no alternative RBE models considered.
  • domain assumption Auger cascade dose is confined within R_eff=500 nm and is additive across NPs (Heaviside cutoff, Eqs. 14, 17).
    Justified by a steep decrease in dose seen in figure 11b of [4]; the sharp cutoff and absence of inter-NP synergy is a simplification.
  • ad hoc to paper The variance splitting factor k=2 (fully stochastic regime) and mu=0.
    Chosen in Section 3.5 to align with 'physically motivated' scenario of purely stochastic enhancement; other values 0<=k<=2 are admitted, so this is a modeling choice, not a derived result.
  • ad hoc to paper Truncation of the Gaussian integral at z = -1/sigma to avoid negative doses (Eq. 41).
    Needed because the first-order expansion Denh = D0(1+sigma Z) goes negative for Z < -1/sigma; the truncation factor Rtrunc is introduced without experimental validation.
  • standard math Charged particle equilibrium (CPE) and spectrum-averaged cross-sections (Eqs. 1-2).
    Standard dosimetry simplification.
  • standard math Gaussian expectation identity E[e^{tZ+bZ^2}] = exp(t^2/(2(1-2b)))/sqrt(1-2b), Eq. 7.4.32 of [20].
    Standard result from Abramowitz and Stegun; used to evaluate the macroscopic survival integrals.

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Cite this review

Pith. "Pith review of An analytical model for gold nanoparticle radiosensitisation." pith.science (2026). https://pith.science/paper/33HCKHX6

@misc{pith2026250606671,
  author       = {Pith},
  title        = {Pith review of: An analytical model for gold nanoparticle radiosensitisation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/33HCKHX6}},
  note         = {Machine review of arXiv:2506.06671}
}
abstract

In this paper, we derive a variance-driven Local-Effect-Model ($\sigma$-LEM) to predict radiosensitization due to gold nanoparticles (AuNP). Assuming that the number of Au photo-ionisations scales strictly with particle volume $V_{\mathrm{NP}}\propto R^{3}$, a linear relation between dose-enhancement ratio and concentration is achieved ($DER = 1 + K_c,c$), in which $K_c$ is a beam-quality and nucleus-size-specific term, and $c$ is the concentration in mM. Furthermore, assuming that the cascade energy deposition is log-normally distributed, the enhanced dose in each target voxel can be written as $D_{\text{enh}} = D_{0}\exp(\sigma Z)$ with $Z \sim \mathcal{N}(0,1)$ and width $\sigma = \sqrt{2\ln(1+Kc)}$. Assuming a linear-quadratic (LQ) dose response, a relation between cell survival and dose can be derived. Despite no closed form for the log-normal distribution, averaging over the entire domain using first- and second-order moments leads to three possible closed forms: variance-only, mixed-term, and second-order. These three variants adapt well to low-concentration, mid-concentration, and high-concentration regimes. The model was tested for Bovine aortic endothelial cells (BAEC) results taken from a Local Effect Model (LEM) and experimental values. The model agrees within $\le 2.5%$ with the experimental and LEM data, but presents significant changes to the conceptual results obtained with the LEM, in particular indicating that AuNP dose enhancement is mostly $\alpha$-driven, as posited previously by other authors. These findings are further developed in the manuscript. The theoretical framework presented here collapses radiobiological outcomes to three experimentally controllable variables -- beam quality, nucleus size, and intracellular concentration $c$ -- while retaining mechanistic fidelity. Additional tests should be made to further confirm the validity of the model.

Figures

Figures reproduced from arXiv: 2506.06671 by the authors.

Figure 1
Figure 1. Dose efficiency to the nucleus as a function of nanoparticle radius [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Graphical comparison of model DER vs syncrhotron experimental DER [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Kc(E) as a function of energy (keV) in log-log scale Taken together, the agreement supports the view that a single, energy￾independent calibration of < εcasNAuger >, combined with known cross￾sections and a realistic cellular dimensionality, is sufficient to predict DER within experimental scatter. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Graphical comparison between the analytical models and the LEM [5] for 0.25 [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]
Figure 5
Figure 5. Figure 5: Graphical comparison between the analytical models and the LEM [5] for 0.00 [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]

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Pith tools

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