Structural disorder and anomalous diffusion in random packing of spheres

A new perspective of molecular diffusion by nuclear magnetic resonance

The diffusion-weighted NMR signal acquired using Pulse Field Gradient (PFG) techniques, allows for extrapolating microstructural information from porous materials and biological tissues. In recent years there has been a multiplication of diffusion models expressed by parametric functions to fit the experimental data. However, clear-cut criteria for the model selection are lacking. In this paper, we develop a theoretical framework for the interpretation of NMR attenuation signals in the case of Gaussian systems with stationary increments. The full expression of the Stejskal-Tanner formula for normal diffusing systems is devised, together with its extension to the domain of anomalous diffusion. The range of applicability of the relevant parametric functions to fit the PFG data can be fully determined by means of appropriate checks to ascertain the correctness of the fit. Furthermore, the exact expression for diffusion weighted NMR signals pertaining to Brownian yet non-Gaussian processes is also derived, accompanied by the proper check to establish its contextual relevance. The analysis provided is particularly useful in the context of medical MRI and clinical practise where the hardware limitations do not allow the use of narrow pulse gradients.

Subdiffusion equation with Caputo fractional derivative with respect to another function in modeling diffusion in a complex system consisting of a matrix and channels

Anomalous diffusion of an antibiotic (colistin) in a system consisting of packed gel (alginate) beads immersed in water is studied experimentally and theoretically. The experimental studies are performed using the interferometric method of measuring concentration profiles of a diffusing substance. We use the g-subdiffusion equation with the fractional Caputo time derivative with respect to another function g to describe the process. The function g and relevant parameters define anomalous diffusion. We show that experimentally measured time evolution of the amount of antibiotic released from the system determines the function g. The process can be interpreted as subdiffusion in which the subdiffusion parameter (exponent) α decreases over time. The g-subdiffusion equation, which is more general than the "ordinary" fractional subdiffusion equation, can be widely used in various fields of science to model diffusion in a system in which parameters, and even a type of diffusion, evolve over time. We postulate that diffusion in a system composed of channels and a matrix can be described by the g-subdiffusion equation, just like diffusion in a system of packed gel beads placed in water.

Transient Anomalous Diffusion MRI in Excised Mouse Spinal Cord: Comparison Among Different Diffusion Metrics and Validation With Histology

Neural tissue is a hierarchical multiscale system with intracellular and extracellular diffusion compartments at different length scales. The normal diffusion of bulk water in tissues is not able to detect the specific features of a complex system, providing nonlocal, diffusion measurement averaged on a 10-20 μm length scale. Being able to probe tissues with sub-micrometric diffusion length and quantify new local parameters, transient anomalous diffusion (tAD) would dramatically increase the diagnostic potential of diffusion MRI (DMRI) in detecting collective and sub-micro architectural changes of human tissues due to pathological damage. In DMRI, the use of tAD parameters quantified using specific DMRI acquisition protocols and their interpretation has often aroused skepticism. Although the derived formulas may accurately fit experimental diffusion-weighted data, the relationships between the postulated dynamical feature and the underlying geometrical structure remains elusive, or at most only suggestive. This work aimed to elucidate and validate the image contrast and information that can be obtained using the tAD model in white matter (WM) through a direct comparison between different diffusion metrics and histology. Towards this goal, we compared tAD metrics extracted from pure subdiffusion (α-imaging) and super-pseudodiffusion (γ-imaging) in excised mouse spinal cord WM, together with T2 and T2* relaxometry, conventional (normal diffusion-based) diffusion tensor imaging (DTI) and q-space imaging (QSI), with morphologic measures obtained by optical microscopy, to determine which structural and topological characteristics of myelinated axons influenced tAD contrast. Axon diameter (AxDiam), the standard deviation of diameters (SD_ax.diam), axonal density (AxDens) and effective local density (ELD) were extracted from optical images in several WM tracts. Among all the diffusion parameters obtained at 9.4 T, γ-metrics confirmed a strong dependence on magnetic in-homogeneities quantified by R2* = 1/T2* and showed the strongest associations with AxDiam and ELD. On the other hand, α-metrics showed strong associations with SD_ax.diam and was significantly related to AxDens, suggesting its ability to quantify local heterogeneity degree in neural tissue. These results elucidate the biophysical mechanism underpinning tAD parameters and show the clinical potential of tAD-imaging, considering that both physiologic and pathologic neurodegeneration translate into alterations of WM morphometry and topology.

Transient Anomalous Diffusion MRI Measurement Discriminates Porous Polymeric Matrices Characterized by Different Sub-Microstructures and Fractal Dimension

Considering the current development of new nanostructured and complex materials and gels, it is critical to develop a sub-micro-scale sensitivity tool to quantify experimentally new parameters describing sub-microstructured porous systems. Diffusion NMR, based on the measurement of endogenous water's diffusion displacement, offers unique information on the structural features of materials and tissues. In this paper, we applied anomalous diffusion NMR protocols to quantify the subdiffusion of water and to measure, in an alternative, non-destructive and non-invasive modality, the fractal dimension d_w of systems characterized by micro and sub-micro geometrical structures. To this end, three highly heterogeneous porous-polymeric matrices were studied. All the three matrices composed of glycidylmethacrylate-divynilbenzene porous monoliths obtained through the High Internal Phase Emulsion technique were characterized by pores of approximately spherical symmetry, with diameters in the range of 2-10 μm. Pores were interconnected by a plurality of window holes present on pore walls, which were characterized by size coverings in the range of 0.5-2 μm. The walls were characterized by a different degree of surface roughness. Moreover, complementary techniques, namely Field Emission Scanning Electron Microscopy (FE-SEM) and dielectric spectroscopy, were used to corroborate the NMR results. The experimental results showed that the anomalous diffusion α parameter that quantifies subdiffusion and d_w = 2/α changed in parallel to the specific surface area S (or the surface roughness) of the porous matrices, showing a submicroscopic sensitivity. The results reported here suggest that the anomalous diffusion NMR method tested may be a valid experimental tool to corroborate theoretical and simulation results developed and performed for describing highly heterogeneous and complex systems. On the other hand, non-invasive and non-destructive anomalous subdiffusion NMR may be a useful tool to study the characteristic features of new highly heterogeneous nanostructured and complex functional materials and gels useful in cultural heritage applications, as well as scaffolds useful in tissue engineering.

Anomalous diffusion in a bench-scale pulsed fluidized bed

We present our analysis on microrheology of a bench-scale pulsed fluidized bed, which represents a weakly confined system. Nonlinear gas-particle and particle-particle interactions resulting from pulsed flow are associated with harmonic and subharmonic modes. While periodic structured bubble patterns are observed at the mesoscale, particle-scale measurements reveal anomalous diffusion in the driven granular medium. We use single-particle tracks to analyze ergodicity and ageing properties at two pulsing frequencies having remarkably different mesoscale features. The scaling of ensemble-averaged mean-squared displacement is not unique. The distribution of time-averaged mean-squared displacements is non-Gaussian, asymmetric, and has a finite trivial contribution from particles in crowded quasistatic surroundings. Results indicate weak ergodicity breaking, which along with ageing characterizes the nonstationary and out-of-equilibrium dynamics.

Modeling of Gas Exchange in the Lungs

This overview presents the recent progress in our understanding of gas transfer by the lungs during the respiratory cycle and during breath holding. Different phenomena intervene in gas transfer, convection and diffusion in the gas, dissolution, diffusion across the alveolar-capillary membrane, diffusion across blood plasma, and finally diffusion and reaction with hemoglobin inside blood cells. The different gases, O₂ , CO, and NO, have very different reaction times with hemoglobin ranging from a few microseconds to tens of milliseconds. This is leading to different outcomes. For O₂ , the solutions to the coupled nonlinear gas and blood equations are obtained at the acinus level. They include the fact that the acinar internal ventilation is strongly heterogeneous due to the arborescent structure. Also, in the dynamic calculation, one takes care of the delay between the start of inhalation and arrival of fresh air in the acinus. This "dead" time is the dynamic equivalent of the dead space ventilation. The question of the dependence of Vo₂ on ventilation and perfusion takes a different form. The results show that Vo₂ is not only a function of the ventilation/perfusion ratio but also depends on the variables: acinar ventilation VE_ac and perfusion Q_ac . The ratio VE_ac /Q_ac roughly determines arterial O₂ saturation and arterial and alveolar O₂ partial pressure. The classic Roughton-Forster interpretation of DLCO (separation between independent membrane and blood resistance) was a mathematical conjecture. It was shown recently that this conjecture was violated. This article presents an alternative interpretation that uses time concepts instead of resistance. © 2021 American Physiological Society. Compr Physiol 11:1289-1314, 2021.

Crossed versus conventional pseudophakic monovision for high myopic eyes: a prospective, randomized pilot study

BACKGROUND

Aiming at spectacle independence, conventional pseudophakic monovision has been widely used in myopia patients with bilateral monofocal intraocular lens implantation. However, the crossed monovision, which is to correct the dominant eye for near vision and the non-dominant eye for distant vision, has been mentioned preferable for high myopic cataract patients by some studies. We have conducted this study to compare clinical results to assess the feasibility of conventional and crossed monovision for high myopic pseudophakic patients by comparing patient satisfaction, visual function and spectacle independence.

METHOD

Forty-sixth high myopia patients were divided into two groups: 22 in crossed monovision group with patients whose refraction targeted to - 2.00 diopters (D) in the dominant eye and - 0.50D in the non-dominant eye; 24 in conventional monovision group with patients whose refraction targeted to - 0.50D in the dominant eye and - 2.00D in the non-dominant eye. Binocular uncorrected distance visual acuity (BUDVA), binocular uncorrected near visual acuity (BUNVA), binocular corrected distant visual acuity (BCDVA), binocular corrected near visual acuity (BCNVA), contrast visual acuity and stereoacuity were examined at postoperative 2 weeks, 1 month and 3 months. Questionnaires were completed by patients 3 months after binocular surgery to evaluate patients' satisfaction and spectacle independence.

RESULTS

The conventional monovision and the crossed monovision group showed no significant differences of mean BUDVA, BUNVA, BCDVA, BCNVA 2 weeks, 1 month or 3 months postoperatively (P > 0.05). There was no difference in the bilateral contrast sensitivity or stereoscopic function between the convention conventional and crossed monovision groups (P > 0.05). Patient satisfaction with near and distant vision, as well as spectacle dependence did not differ significantly between the two groups (P > 0.05).

CONCLUSION

Crossed pseudophakic monovision exhibited similar visual function when compared with conventional monovision technique, which indicates that it is an effective option to improve the visual functionality and quality of life for high myopic patients who considering bilateral cataract surgery.

TRIAL REGISTRATION

The Institutional Review Board and Ethics committee of the First Affiliated Hospital of Chongqing Medical University, Chongqing, China. The trial registration was submitted in September 2018 and passed on March 18, 2020, and the registration number is: ChiCTR2000030935 .

What are appropriate values of relative krogh diffusion Constant of NO against CO and of theta-NO in alveolar septa?

OBJECTIVES

To propose new physical constants for NO and CO (Krogh diffusion constant ratio (KD_NO/CO) and specific blood conductance for NO (θ_NO)) for calculating DMCO and Vc, according to Roughton-Forster's equation (Roughton and Forster, J. Appl. Physiol. 11: 290-302, 1957) from simultaneous DLNO and DLCO measurements.

RESULTS AND CONCLUSIONS

(1) The Graham's law is unacceptable for determining KD_NO/CO because CO does not fulfil all the conditions of an "ideal" gas. We have re-estimated KD_NO/CO in a new way based on difference in molar volumes of two gases (molar volume theory). The KD_NO/CO thus decided is 2.34. (2) θ_NO measured with rapid-reaction, constant-flow method by Carlsen and Comroe (J. Gen. Physiol. 42: 83-107, 1958) may be underestimated by about 40 % due to unstirred water layer surrounding the erythrocyte. (3) Erythrocyte θ_O2 can be harvested from O₂ release kinetics in presence of high concentration of dithionite, which effectively removes the unstirred water layer-elicited effect. Multiplication of erythrocyte θ_O2 by erythrocyte KD_NO/O2 equals erythrocyte θ_NO, the value of which is 6.2 mL/min/mmHg/(mL⋅blood). According to the concepts of Kang et al. (RESPNB. 241: 62-71, 2017) and Borland et al. (RESPNB. 241: 58-61, 2017), in vitro θ_NO decided from rapid-mixing experiments may mirror bulk absorption of NO by erythrocytes. (4) In pulmonary capillaries, NO uptake takes place predominantly in the surface rim of the erythrocyte. This surface absorption of NO increases the θ_NO 10-fold versus bulk absorption of NO to about 60 mL/min/mmHg/(mL⋅blood).

A general phenomenological relation for the subdiffusive exponent of anomalous diffusion in disordered media

This work numerically investigates the diffusion of finite inert tracer particles in different types of fixed gels. The mean square displacement (MSD) of the tracers reveals a transition to subdiffusive motion MSD ∼ tα as soon as the accessible volume fraction p in the gel decreases from unity. Individual tracer dynamics reveals two types of particles in the gels: mobile tracers cross the system through percolating pores following subdiffusive dynamics MSDmob ∼ tαmob, while a fraction ptrap(p) of the particles remain trapped in finite pores. Below the void percolation threshold p < pc all the particles get trapped and α → 0. By separately studying both populations we find a simple phenomenological law for the mobile tracers αmob(p) ≈ a ln p + c where c ≈ 1 and a ∼ 0.2 depends on the gel type. On the other hand, a cluster-analysis of the gel accessible volume reveals a power law for the trapping probability ptrap ∼ (p/pc)-γ, with γ ≃ 2.9. This yields a prediction for the ensemble averaged subdiffusion exponent α = αmob(1 - ptrap). Our predictions are successfully validated against the different gels studied here and against numerical and experimental results in the literature (silica gels, polyacrylamide gels, flexible F-actin networks and in different random obstacles). Notably, the parameter a ∼ 0.2 presents small differences amongst all these cases, indicating the robustness of the proposed relation.

Mutual influence of molecular diffusion in gas and surface phases

We develop molecular transport simulation methods that simultaneously deal with gas- and surface-phase diffusions to determine the effect of surface diffusion on the overall diffusion coefficients. The phenomenon of surface diffusion is incorporated into the test particle method and the mean square displacement method, which are typically employed only for gas-phase transport. It is found that for a simple cylindrical pore, the diffusion coefficients in the presence of surface diffusion calculated by these two methods show good agreement. We also confirm that both methods reproduce the analytical solution. Then, the diffusion coefficients for ink-bottle-shaped pores are calculated using the developed method. Our results show that surface diffusion assists molecular transport in the gas phase. Moreover, the surface tortuosity factor, which is known to be uniquely determined by physical structure, is influenced by the presence of gas-phase diffusion. This mutual influence of gas-phase diffusion and surface diffusion indicates that their simultaneous calculation is necessary for an accurate evaluation of the diffusion coefficients.

An efficient algorithm for extracting the magnitude of the measurement error for fractional dynamics

Modern live-imaging fluorescent microscopy techniques following the stochastic motion of labeled tracer particles, i.e. single particle tracking (SPT) experiments, have uncovered significant deviations from the laws of Brownian motion in a variety of biological systems. Accurately characterizing the anomalous diffusion for SPT experiments has become a central issue in biophysics. However, measurement errors raise difficulty in the analysis of single trajectories. In this paper, we introduce a novel surface calibration method based on a fractionally integrated moving average (FIMA) process as an effective tool for extracting both the magnitude of the measurement error and the anomalous exponent for autocorrelated processes of various origins. This method is developed using a toy model - fractional Brownian motion disturbed by independent Gaussian white noise - and is illustrated on both simulated and experimental biological data. We also compare this new method with the mean-squared displacement (MSD) technique, extended to capture the measurement noise in the toy model, which shows inferior results. The introduced procedure is expected to allow for more accurate analysis of fractional anomalous diffusion trajectories with measurement errors across different experimental fields and without the need for any calibration measurements.

Fractional cable equation for general geometry: A model of axons with swellings and anomalous diffusion

Different experimental studies have reported anomalous diffusion in brain tissues and notably this anomalous diffusion is expressed through fractional derivatives. Axons are important to understand neurodegenerative diseases such as multiple sclerosis, Alzheimer's disease, and Parkinson's disease. Indeed, abnormal accumulation of proteins and organelles in axons is a hallmark of these diseases. The diffusion in the axons can become anomalous as a result of this abnormality. In this case the voltage propagation in axons is affected. Another hallmark of different neurodegenerative diseases is given by discrete swellings along the axon. In order to model the voltage propagation in axons with anomalous diffusion and swellings, in this paper we propose a fractional cable equation for a general geometry. This generalized equation depends on fractional parameters and geometric quantities such as the curvature and torsion of the cable. For a cable with a constant radius we show that the voltage decreases when the fractional effect increases. In cables with swellings we find that when the fractional effect or the swelling radius increases, the voltage decreases. Similar behavior is obtained when the number of swellings and the fractional effect increase. Moreover, we find that when the radius swelling (or the number of swellings) and the fractional effect increase at the same time, the voltage dramatically decreases.

Shape-dependent effective diffusivity in packings of hard cubes and cuboids compared with spheres and ellipsoids

We performed computational screening of effective diffusivity in different configurations of cubes and cuboids compared with spheres and ellipsoids. In total, more than 1500 structures are generated and screened for effective diffusivity. We studied simple cubic and face-centered cubic lattices of spheres and cubes, random configurations of cubes and spheres as a function of volume fraction and polydispersity, and finally random configurations of ellipsoids and cuboids as a function of shape. We found some interesting shape-dependent differences in behavior, elucidating the impact of shape on the targeted design of granular materials.

Diffusion Reaction of Carbon Monoxide in the Human Lung

The capture of CO, a standard lung function test, results from diffusion-reaction processes of CO with hemoglobin inside red blood cells (RBCs). In its current understanding, suggested by Roughton and Forster in 1957, the capture is represented by two independent resistances in series, one for diffusion from the gas to the RBC periphery, the second for internal diffusion reaction. Numerical studies in 3D model structures described here contradict the independence hypothesis. This results from two different theoretical reasons: (i) The RBC peripheries are not equi-concentrations; (ii) diffusion times in series are not additive.

Using Diffusion To Characterize Interfacial Heterogeneity

We report on the use of molecular diffusional motion over a range of length scales to characterize compositional heterogeneity in monolayer structures. This work focuses on the diffusional motion of perylene in two types of films supported on functionalized silica surfaces: single-component (stearic acid) and two-component (hydrocarbon/fluorocarbon) films. Langmuir-Blodgett (LB) monolayers were deposited directly on silica or were bound to surface-modified silica by means of metal ion complexation. The LB films were characterized by their π-A isotherms and by Brewster angle microscopy (BAM) during formation and deposition. Chromophore mobility and monolayer structural heterogeneity were evaluated by comparing rotational diffusion data (fluorescence anisotropy decay imaging) and translational diffusion data (fluorescence recovery after photobleaching) on the same LB films. Our results indicate that the mobility of the chromophore depends sensitively on both metal ion identity and film composition.

Diffusional motion as a gauge of fluidity and interfacial adhesion. Supported alkylphosphonate monolayers

We report on the use of diffusion measurements to gauge the fluidity and surface binding properties of a molecular monolayer. The monolayer film consists of octadecyl-1-phosphonic acid (ODPA) and controlled amounts of a lyso-phosphatidic acid tagged with the fluorescent probe BODIPY (BLPA). The monolayer films were formed using a Langmuir-Blodgett (LB) trough and deposited onto a glass slide. Monolayer morphology was characterized during film formation using Brewster angle microscopy (BAM). Fluorescence Recovery After Photobleaching (FRAP) microscopy was used to measure translational diffusion of BLPA and Fluorescence Anisotropy Decay Imaging (FADI) was used to measure rotational diffusion of the BLPA chromophore. These results provide information on the motional freedom of the probe and, importantly, on the strength of interaction between the probe and the support. Compositional variations in the monolayer give rise to changes in constituent dynamics that reflect intermolecular interactions.

Time-based understanding of DLCO and DLNO

Capture of CO and NO by blood requires molecules to travel by diffusion from alveolar gas to haemoglobin molecules inside RBCs and then to react. One can attach to these processes two times, a time for diffusion and a time for reaction. This reaction time is known from chemical kinetics and, therefore, constitutes a unique physical clock. This paper presents a time-based bottom-up theory that yields a simple expression for DLCO and DLNO that produces quantitative predictions which compare successfully with experiments. Specifically, when this new approach is applied to DLCO experiments, it can be used to determine the value of the characteristic diffusion time, and the value of capillary volume (Vc). The new theory also provides a simple explanation for still unexplained correlations such as the observed proportionality between the so-called membrane conductance DM and Vc of Roughton and Forster's interpretation. This new theory indicates that DLCO should be proportional to the haematocrit as found in several experiments.

Estimating the anomalous diffusion exponent for single particle tracking data with measurement errors - An alternative approach

Accurately characterizing the anomalous diffusion of a tracer particle has become a central issue in biophysics. However, measurement errors raise difficulty in the characterization of single trajectories, which is usually performed through the time-averaged mean square displacement (TAMSD). In this paper, we study a fractionally integrated moving average (FIMA) process as an appropriate model for anomalous diffusion data with measurement errors. We compare FIMA and traditional TAMSD estimators for the anomalous diffusion exponent. The ability of the FIMA framework to characterize dynamics in a wide range of anomalous exponents and noise levels through the simulation of a toy model (fractional Brownian motion disturbed by Gaussian white noise) is discussed. Comparison to the TAMSD technique, shows that FIMA estimation is superior in many scenarios. This is expected to enable new measurement regimes for single particle tracking (SPT) experiments even in the presence of high measurement errors.

Atomic layer deposition of metastable β-Fe₂O₃ via isomorphic epitaxy for photoassisted water oxidation

We report the growth and photoelectrochemical (PEC) characterization of the uncommon bibyite phase of iron(III) oxide (β-Fe2O3) epitaxially stabilized via atomic layer deposition on an conductive, transparent, and isomorphic template (Sn-doped In2O3). As a photoanode, unoptimized β-Fe2O3 ultrathin films perform similarly to their ubiquitous α-phase (hematite) counterpart, but reveal a more ideal bandgap (1.8 eV), a ∼0.1 V improved photocurrent onset potential, and longer wavelength (>600 nm) spectral response. Stable operation under basic water oxidation justifies further exploration of this atypical phase and motivates the investigation of other unexplored metastable phases as new PEC materials.

Characterization of maximally random jammed sphere packings: Voronoi correlation functions

We characterize the structure of maximally random jammed (MRJ) sphere packings by computing the Minkowski functionals (volume, surface area, and integrated mean curvature) of their associated Voronoi cells. The probability distribution functions of these functionals of Voronoi cells in MRJ sphere packings are qualitatively similar to those of an equilibrium hard-sphere liquid and partly even to the uncorrelated Poisson point process, implying that such local statistics are relatively structurally insensitive. This is not surprising because the Minkowski functionals of a single Voronoi cell incorporate only local information and are insensitive to global structural information. To improve upon this, we introduce descriptors that incorporate nonlocal information via the correlation functions of the Minkowski functionals of two cells at a given distance as well as certain cell-cell probability density functions. We evaluate these higher-order functions for our MRJ packings as well as equilibrium hard spheres and the Poisson point process. It is shown that these Minkowski correlation and density functions contain visibly more information than the corresponding standard pair-correlation functions. We find strong anticorrelations in the Voronoi volumes for the hyperuniform MRJ packings, consistent with previous findings for other pair correlations [A. Donev et al., Phys. Rev. Lett. 95, 090604 (2005)PRLTAO0031-900710.1103/PhysRevLett.95.090604], indicating that large-scale volume fluctuations are suppressed by accompanying large Voronoi cells with small cells, and vice versa. In contrast to the aforementioned local Voronoi statistics, the correlation functions of the Voronoi cells qualitatively distinguish the structure of MRJ sphere packings (prototypical glasses) from that of not only the Poisson point process but also the correlated equilibrium hard-sphere liquids. Moreover, while we did not find any perfect icosahedra (the locally densest possible structure in which a central sphere contacts 12 neighbors) in the MRJ packings, a preliminary Voronoi topology analysis indicates the presence of strongly distorted icosahedra.

Crossover from anomalous to normal diffusion in porous media

Random walks (RW) of particles adsorbed in the internal walls of porous deposits produced by ballistic-type growth models are studied. The particles start at the external surface of the deposits and enter their pores in order to simulate an external flux of a species towards a porous solid. For short times, the walker concentration decays as a stretched exponential of the depth z, but a crossover to long-time normal diffusion is observed in most samples. The anomalous concentration profile remains at long times in very porous solids if the walker steps are restricted to nearest neighbors and is accompanied with subdiffusion features. These findings are correlated with a decay of the explored area with z. The study of RW of tracer particles left at the internal part of the solid rules out an interpretation by diffusion equations with position-dependent coefficients. A model of RW in a tube of decreasing cross section explains those results by showing long crossovers from an effective subdiffusion regime to an asymptotic normal diffusion. The crossover position and density are analytically calculated for a tube with area decreasing exponentially with z and show good agreement with numerical data. The anomalous decay of the concentration profile is interpreted as a templating effect of the tube shape on the total number of diffusing particles at each depth, while the volumetric concentration in the actually explored porous region may not have significant decay. These results may explain the anomalous diffusion of metal atoms in porous deposits observed in recent works. They also confirm the difficulty in interpreting experimental or computational data on anomalous transport reported in recent works, particularly if only the concentration profiles are measured.