# Human cortical bone intrinsic permeability distribution based on 3D canalicular morphology

Rémy Gauthier, Hélène Follet, Cécile Olivier, Thibault Lemaire, David Mitton, Francoise Peyrin

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Rémy Gauthier, Hélène Follet, Cécile Olivier, Thibault Lemaire, David Mitton, et al.. Human cortical bone intrinsic permeability distribution based on 3D canalicular morphology. BONE, 2025, 194, pp.117441. 10.1016/j.bone.2025.117441. hal-04972956

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Full Length Article

# Human cortical bone intrinsic permeability distribution based on 3D canalicular morphology

Remy Gauthier<sup>a,*</sup>, Hélène Follet<sup>b</sup>, Cécile Olivier<sup>c</sup>, Thibault Lemaire<sup>d</sup>, David Mitton<sup>e</sup>,
Françoise Peyrin<sup>f</sup>

ARTICLE INFO

Keywords:

Local permeability

Human cortical bone

Lacuno-canalicular network

Heterogeneity

A B S T R A C T

Bone permeability is a key parameter that drives osteocyte-based mechanobiological modelling and remodelling. While previous experimental and numerical studies have estimated bone permeability based on the morphology of the lacuno-canalicular network, these studies often relied on simplified geometries. In the current study, bone permeability was characterized using more realistic canalicular geometry for the morphological data. Bone samples harvested from 27 human femoral bones were investigated using synchrotron radiation-based nano-computed tomography with a voxel size of 100 nm. After segmenting the canaliculi and lacunae, each canaliculus was investigated individually by applying a distance map and watershed algorithms. Bone permeability based on canalicular morphology was then assessed using the Kozeny relation, which defines the permeability of a porous medium with capillary-like pores. An averaged intrinsic permeability value of $8.8 \ 10^{-18} \ m^2$ was obtained. It should be noted that this study considered an empty canalicular network, however *in vivo*, both cellular and peri-cellular matrices decrease space for interstitial fluid flow and thus permeability. Furthermore, a voxel size of 100 nm does not allow for the detection of smaller canaliculi, which may also modify average permeability. With the current data set and the analytic process applied, the results showed a heterogeneous permeability distribution within bone tissue, both when comparing osteonal and interstitial tissues and within an individual osteon. A difference was observed between male and female samples, and permeability appeared to significantly decrease with age. Finally, a significant correlation was found between permeability and canalicular length density, defined as canalicular length per unit bone volume. This study proposes a new form of the Kozeny law to express bone canalicular permeability as a proportional relationship with the canalicular length density. Importantly, this parameter can be directly quantified through confocal fluorescence microcopy, which is more convenient than synchrotron radiation-based nano-computed tomography. In conclusion, the current study confirms that confocal microscopy can be used as a reliable tool to estimate bone permeability. However, the permeability values calculated here are solely based on canalicular morphology and do not consider cellular and peri-cellular intra-canalicular features.

1. Introduction

Bone permeability is a determinant factor in human bone modelling and remodelling mechanisms. It is assumed that bone mechanosensitive cells, the osteocytes, answer specifically to their interactions with the surrounding physiological fluid [1]. Under external mechanical loading,

fluid flow is promoted within a bone multiscale porous network due to its poroelastic behavior. This fluid flow induces a shear stress on the cellular membrane that is then transduced into a bone remodelling mediator [2]. This mechanotransduction pathway is assumed to characterize osteocyte behavior, thus orchestrating the bone remodelling mechanism [3]. Within the bone structure, osteocytes are closely

<footnote><sup>a</sup> Univ Lyon, CNRS, INSAS Lyon, Université Claude Bernard Lyon 1, MATEIS UMR 5510, 69621 Villeurbanne, France</footnote>

<footnote><sup>b</sup> Univ Lyon, Université Claude Bernard Lyon 1, INSERM, LYOS UMR1033, Lyon, France</footnote>

<footnote><sup>c</sup> Université Grenoble Alpes, INSERM UA7 Synchrotron Radiation for Biomedicine, Saint-Martin d'Hères, France</footnote>

<footnote><sup>d</sup> MSME UMR 8208, Univ Paris Est Créteil, Univ Gustave Eiffel, CNRS, 94010 Créteil, France</footnote>

<footnote><sup>e</sup> Univ Lyon, Univ Gustave Eiffel, Université Claude Bernard Lyon 1, LBMC UMR T9406, 69622 Lyon, France</footnote>

<footnote><sup>f</sup> Univ Lyon, INSAS Lyon, Université Claude Bernard Lyon 1, UJM-Saint Etienne, CNRS UMR 5220, Inserm U1206, CREATIS, 69621 Lyon, France</footnote>

<footnote>* Corresponding author.
E-mail address: remy.gauthier@cnrs.fr (R. Gauthier).</footnote>
trapped within the submicrometric lacuno-canalicular network (LCN). Cells bodies are located in lacunae embedded within the bone mineralized matrix (collagen-apatite structures), and are all connected together and with the vascular network (Haversian porous system) through canaliculi that are 300–400 nm in diameter [4,5]. Cell dendrites located within the canaliculi have long been assumed as where bone mechanotransduction occurs [6]. Considering fluid mass conservation and the canalicular submicrometric dimensions, the fluid velocity levels induced by the solid matrix deformation are higher than in larger vascular canals, hence enhancing the shear stress applied on osteocyte dendrites. More specifically, according to Darcy's law describing fluid flow in porous media, intrinsic permeability is directly involved in the relationships between fluid pressure gradient and interstitial flow velocity. This textural parameter depends on the morphological and textural properties of the porous medium. Hence, characterizing bone intrinsic permeability at the LCN scale appears relevant to a better understanding of bone mechanobiology.

Estimation of bone LCN permeability is difficult due to its small dimensions. Experimentally, LCN permeability has been estimated via measurements of bone pressure relaxation coupled to theoretical poroelastic models [7,8]. Values around $10^{-22}$ and $10^{-23} m^2$ were obtained. Alternatively, Galli and Oyen derived the permeability parameter from nanoindentation tests associated with poroelastic laws, with an estimated permeability on the order of $10^{-24}$ to $10^{-23} m^2$ [9,10]. While these experimental analyses are of great importance and provide a global estimated value of LCN permeability, it appears that such low levels of permeability are not relevant in the context of continuum media description [11]. Moreover, bone architecture is complex and local morphological variations of the LCN can be observed [12–16]. In particular, several studies show that osteocyte lacunar density and porosity decrease with age in various bones [5,17–21], even while osteocyte lacunar size is less well understood [22]. Variations in LCN morphology are known to induce variations in the fluid flow pattern during loading [23]. Hence, it would be interesting to implement this local architecture into models of bone LCN fluid flow such as the ones developed by Goulet et al. [24,25]. In these finite element models, the hydrostatic pressure developed within the bone matrix under mechanical deformation is estimated using poroelasticity and Darcy's law. However, in such studies, the LCN is typically represented by assigning a unique permeability value to each element of the model, neglecting the effects of local heterogeneity.

Meanwhile, LCN permeability has been estimated from microstructural analyses, with values ranging from $10^{-22}$ to $10^{-18} m^2$ [26–28]. Morphological data in these previous studies were obtained through confocal fluorescence microscopy. This method presents several advantages in the context of bone LCN characterization, such as the light acquisition process and the possibility to conveniently manually segment an osteon on a low number of slices. Confocal microscopy has been used to directly estimate canicular length density, defined as the length of canaliculi in a defined volume of interest (VOI) [29,30], which is assumed to be an essential parameter representing bone permeability. Nevertheless, confocal microscopy cannot be directly used to estimate network permeability, as it does not provide accurate assessments of real canicular dimensions and has an anisotropic spatial resolution, with a planar resolution being at least twice better than the axial resolution [31]. In contrast, synchrotron radiation-based nano-computed tomography (SR-nCT) has the capability to characterize large VOIs of $2048 \times 2048 \times 2048$, while confocal microscopy is limited in its optical axis. Moreover, the isotropic spatial resolution provided by SR-nCT enables measurement of realistic geometries of 3D sub-micrometric features in bone tissue [32,33]. Finally, unlike confocal microscopy, which relies on the penetration of a fluorescent marker within the whole LCN, there is no need to add any contrast agent in SR-nCT analyses.

As such, synchrotron radiation-based nano-computed tomography (SR-nCT) was used to analyze the LCN from 27 human femoral diaphysis samples with an isotropic voxel size of 100 nm. The segmentation

protocol and characterization of this data set in terms of LCN morphology has already been published and discussed elsewhere [5]. The aim of the current study is to use this data set to estimate the distribution of bone intrinsic permeability based on a realistic LCN geometry using this previously acquired data set [5].

## 2. Materials and methods

### 2.1. Sample preparation

Samples from the lateral side of the femoral diaphysis, harvested from 15 females (50–95 y.o.) and 12 males (66–89 y.o.) (Centre du Don des Corps, Université Paris 5 and Department of Anatomy Rockefeller, Université de Lyon), were prepared following a previous protocol [5]. Rectangular samples, $0.4 \times 0.4 \times 4 mm^3$ with the long axis aligned to the long axis of the diaphysis were prepared using a water-cooled diamond precision saw (Presi Mecatome T210, Struers Diamond Cut-off Wheel EOD15).

### 2.2. Synchrotron radiation nano-computed tomography

The 27 samples were imaged at beam line ID16B (ESRF, Grenoble, France). Each sample was scanned at 4 different sample-to-detector distances, with 3400 projections recorded over a rotation of 360°, resulting in a total scan duration of approximately 3 h. The X-ray beam energy was set to 29.6 keV and the obtained voxel size was 100 nm. Phase retrieval was performed using the multi-distance Paganin algorithm followed by an iterative step of nonlinear conjugate gradient according to previous work [34]. The standard filtered back-projection reconstruction algorithm was then applied to obtain the 3D phase maps. The sizes of the reconstructed phase volumes correspond to a cylinder, 256 µm in diameter and 216 µm in height.

The segmentation process to separately analyze canaliculi and lacunae has already been provided in a previous study [5]. Briefly, lacunae were segmented after applying a 10 voxels median filter and thresholding using an Otsu algorithm followed by a manually defined hysteresis threshold to adjust the segmentation of the lacunar walls [5]. Connected component analysis was performed and used to filter out vascular canals based on component size. Canaliculi were first segmented using Sato's vesselness enhancement method, improving the visibility of the structure, followed by a manually defined hysteresis thresholding to adjust the segmentation of the canicular walls [5]. A final variational region growing algorithm was applied to restore the lost small connections within the network.

It is known that the canicular network presents several ramification [35]. In order to evaluate the distribution of intrinsic permeability over the Volume-of-interest (VOI), an algorithm was applied to analyze each canicula separately. As it has already been applied in the past on confocal fluorescence microcopy data, the canicular network was considered in terms of connectomics [29,36,37]. In this definition, the network is considered as links connected at their nodes, which are the points of canicular ramification or canicular junctions. This multi-step algorithm was based on TINA open-source software provided by (https://bitbucket.org/refelix/tina/). Once the links and nodes were identified, nodes were removed from the mask and the Euclidian distance map of the unconnected canaliculi was calculated. Finally, a watershed algorithm was applied resulting in the tessellation of the cylinder VOI in which each cell contains one canicula and each edge of the cell is at an equal distance from any neighboring canicula (Fig. 1). This tessellation was performed in 3D. Volume rendering was achieved using Dragonfly software (Object Research Systems (ORS) Inc., Montreal, CA (109)). Processing was performed using the Image Processing Toolbox from Matlab.
<figure><img src="image_3.png" /><figcaption>Fig. 1. Different steps to obtain a tessellated volume with unit cells containing one individual canalicula. a. A rendering of the segmented canaliculi [5]. b. Mask of the canaliculi, that is skeletonized and convert into nodes and links. c. Distance map calculated from the links mask. d. Rendering of the tessellated volume after the watershed algorithm applied on the distance map. e. Global 3D rendering of the segmentation canaliculi, lacunae, and vascular canal. f. 3D rendering of the tessellated volume. Each cell of this tessellated volume contains one canalicula. The Kozeny relation is applied on each cell.</figcaption></figure>

## 2.3. Quantitative analysis

The permeability calculation was based on the Kozeny relation for porous media with capillary-like pores, which is a rough representation of the bone matrix and the canaliculi [11]. If we consider the canaliculi as a tortuous cylinder (diameter $\delta$ and tortuosity $\theta$), the permeability in the flow direction can be expressed as:

$$ \kappa = \frac{\eta_f \delta^2}{32 \theta^3}, \qquad (1) $$

with $\eta_f$ being the canaliculi volume fraction (equivalent here to the in situ cellular porosity). This equation is from Lemaire et al. [11].

Hence, the formula from Eq. (1) was applied for each unit cell of the tessellated volume. The local canalicular volume fraction, $\eta_f$, was calculated as the ratio of the total volume of the canaliculi and the total bone volume of the unit cell. Considering fluid mass conservation arguments when comparing the lacunar/vascular volume to the canalicular volume and keeping in mind that the fluid pathway from the different parts of the lacunar/vascular compartment must flow through the canaliculi, the vascular canals and lacunae were not included in bone volume. The canalicular diameter was considered twice the average distance between the canalicular skeleton and the canalicular wall over the canalicular length. Finally, canalicular tortuosity was evaluated as the ratio between the canalicular geodesic and straight length. To estimate geodesic length, the canalicular skeleton was interpolated by a cubic spline and the smoothed length was calculated [30].

Finally, the permeability value obtained for a single unit cell was assigned to the whole volume of this unit cell, and an averaged permeability, as well as the first, second, and third quartiles, and the standard deviation within a VOI were calculated over the bone tissue volume.

In addition, canalicular length density ($\rho_{cani}$, in $\mu m \cdot \mu m^{-3}$), defined as

the length of canaliculi per volume, was also calculated. This parameter is the one that was previously measured using confocal fluorescent microscopy [30].

For each sample, these different parameters were calculated for a large number of unit cells (Fig. 1f). The distribution was then used to calculate the averaged, quartiles values, and standard deviation for each sample. The mean value for these statistical parameters, averaged over the 27 samples, is provided in the results section.

To analyze the evolution of permeability with distance from an osteocyte lacuna, a Euclidian distance map of the lacunae was calculated, providing a volume whose values increase with distance to a lacuna. This distance volume was split in different volumes corresponding to thickness bins of 0.5 $\mu m$. For instance, the first volume gathers the voxels at a distance from 0 to 0.5 $\mu m$ from a lacuna, the second one gathers the voxels at a distance from 0.5 to 1 $\mu m$ from a lacuna, up to 15 $\mu m$. In each distance volume, the average permeability value was calculated and plotted against the distance to a lacuna. Additionally, the normalized value, normalized over maximum permeability, was also calculated and plotted against distance to lacunar walls.

## 2.4. Statistical analysis

Normality of the distribution was first assessed by applying the Shapiro-Wilk test. Permeability and canalicular length density distributions did not follow a normal law. Hence, parametric tests were applied on the logarithmic transformation for these two data that follow a normal distribution. To compare the male and female populations, an unpaired t-test was applied. If no difference between these two populations was observed, an ANCOVA with the age as a covariate was used. Finally, Pearson's correlation between the different parameters and the donor's age was performed. A p-value under 0.05 was considered as significant. Statistical analysis was done using R© software (The R
foundation for Statistical Computing, Austria).

## 3. Results

It is worth first noticing that the canalicular network is not homogeneous across the bone volume. Two osteons can distinctly be observed on the 3D rendering of the VOI shown in Fig. 2 (top, green lines). The density of canaliculi appears to be higher in the osteon compared to the interstitial tissue. Due to the large size of the volume segmented in the current study, manual segmentation of the osteon was not feasible, and no algorithm is currently available to automatically segment the osteon from the interstitial tissue. This is why such local analysis cannot yet be performed. Furthermore, if no canaliculi are detected in the interstitial tissue with a voxel size of 100 nm, it is still possible that smaller canaliculi are present but not detected. These points have been discussed in a previous study [5].

While heterogeneity in terms of architecture can be qualitatively observed in the VOI, the permeability of the network appears heterogeneous even within a single osteon (Fig. 2, bottom). The permeability values range from $10^{-21}$ to $10^{-14}$ m². The canaliculi are oriented towards the lacunae within one osteon.

Zooming in on individual lacuna revealed that the permeability distribution around a lacuna remains heterogeneous (Fig. 3b). It is interesting to notice that the single lacuna seen on Fig. 3 is located at the cement line, with the left part of the canaliculi coming from the vascular canals while the right part corresponds to the interstitial tissue.

Descriptive statistics for canalicular volume fraction, canalicular diameter, canalicular tortuosity, intrinsic permeability and canalicular length density are provided for male, female, and the complete population (Table 1). For each donor, the averaged, three quartiles (Q1, Q2, and Q3) and the standard deviation over the population of unit cells (Fig. 1d and Fig. 1f) were calculated in the volume. Table 1 presents the averaged and (SD) of these statistical parameters over the male, female, and the complete population. The *t*-test for unpaired samples did not show a difference between the male and female populations. However, when considering age as a covariate, ANCOVA showed significant differences for the $\eta_f$, $\kappa$, and $\rho_{cani}$ parameters. For these three parameters, the female population shows a significantly higher value for the different statistical descriptors (Fig. 4). For instance, local volume fraction, permeability, and canalicular length density are 6 %, 29 %, 28 % higher, respectively, for the female population. Variabilities within one sample were high for both permeability and canalicular length

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Top: maximum intensity projection over the volume after the tomographic reconstruction. The dotted green lines highlight the cement lines. Middle: 3D rendering of the LCN after the segmentation. Canaliculi are in blue, and lacunae are in grey. Bottom: 3D rendering after the permeability calculation. Canaliculi colors correspond to their permeability values, lacunae are in dark yellow and vascular canals are in white. It has to be noticed that the VOI was cropped to ease its manipulation with the 3D rendering software. The black arrows indicate the side from which is observed the rendering on the right. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Zoom on a lacuna from the VOI seen in Fig. 2. a. 3D rendering of a segmented lacuna and its surrounding canaliculi. Canaliculi are in blue, and the lacuna is in dark yellow. b. 3D rendering after the permeability calculation. Canaliculi colors correspond to their permeability values. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<caption>**Table 1**</caption>

The average and (SD) of the average, Q1, Q2, Q3, and SD of the canaliculi characterized in each individual samples for the female (♀), the male (♂), and the whole population.

<table><thead><tr><th></th><th></th><th>Average</th><th>Q1</th><th>Q2</th><th>Q3</th><th>SD</th></tr></thead><tbody><tr><td rowspan="4">η<sub>f</sub> (%)</td><td>♀</td><td>0.33<br>(0.13)</td><td>0.01<br>(0.02)</td><td>0.14<br>(0.06)</td><td>0.37<br>(0.17)</td><td>1.11<br>(-)</td></tr><tr><td>♂</td><td>0.26<br>(0.13)</td><td>0.04<br>(0.03)</td><td>0.11<br>(0.07)</td><td>0.28<br>(0.17)</td><td>0.76<br>(-)</td></tr><tr><td>All</td><td>0.30<sup>a</sup><br>(0.14)</td><td>0.05<sup>a</sup><br>(0.03)</td><td>0.13<sup>a</sup><br>(0.07)</td><td>0.33<sup>a</sup><br>(0.18)</td><td>0.96<sup>a</sup><br>(-)</td></tr><tr><td>♀</td><td>0.38<br>(0.01)</td><td>0.35<br>(0.01)</td><td>0.38<br>(0.01)</td><td>0.42<br>(0.01)</td><td>0.06<br>(-)</td></tr><tr><td rowspan="4">δ (μm)</td><td>♂</td><td>0.38<br>(0.01)</td><td>0.34<br>(0.01)</td><td>0.38<br>(0.01)</td><td>0.42<br>(0.01)</td><td>0.06<br>(-)</td></tr><tr><td>All</td><td>0.38<br>(0.01)</td><td>0.35<br>(0.01)</td><td>0.38<br>(0.01)</td><td>0.42<br>(0.01)</td><td>0.06<sup>a</sup><br>(-)</td></tr><tr><td>♀</td><td>1.25<br>(0.01)</td><td>1.02<br>(0.06)</td><td>1.25<br>(0.01)</td><td>1.48<br>(0.07)</td><td>0.34<br>(-)</td></tr><tr><td>All</td><td>1.25<br>(0.01)</td><td>1.06<br>(0.03)</td><td>1.25<br>(0.01)</td><td>1.43<br>(0.04)</td><td>0.27<br>(-)</td></tr><tr><td rowspan="4">θ (no unit)</td><td>♂</td><td>1.25<br>(0.01)</td><td>1.04<br>(0.05)</td><td>1.25<br>(0.01)</td><td>1.46<br>(0.06)</td><td>0.31<br>(-)</td></tr><tr><td>All</td><td>9.82<br>(4.0)</td><td>1.21<br>(0.5)</td><td>3.5<br>(1.6)</td><td>10.2<br>(4.8)</td><td>44.0<br>(-)</td></tr><tr><td>♀</td><td>7.6<br>(4.0)</td><td>0.9<br>(0.7)</td><td>3.5<br>(1.8)</td><td>7.7<br>(4.9)</td><td>29.9<br>(-)</td></tr><tr><td>All</td><td>8.8<sup>a</sup><br>(4.1)</td><td>1.1<sup>a</sup><br>(0.7)</td><td>3.1<sup>a</sup><br>(1.7)</td><td>9.1<sup>a</sup><br>(5.0)</td><td>37.7<sup>a</sup><br>(-)</td></tr><tr><td rowspan="4">κ (10<sup>-18</sup> m<sup>2</sup>)</td><td>♂</td><td>0.023<br>(0.009)</td><td>0.004<br>(0.002)</td><td>0.011<br>(0.005)</td><td>0.027<br>(0.012)</td><td>0.073<br>(-)</td></tr><tr><td>All</td><td>0.018<br>(0.010)</td><td>0.003<br>(0.002)</td><td>0.008<br>(0.006)</td><td>0.021<br>(0.013)</td><td>0.051<br>(-)</td></tr><tr><td>♀</td><td>0.021<sup>a</sup><br>(0.010)</td><td>0.003<sup>a</sup><br>(0.002)</td><td>0.010<sup>a</sup><br>(0.005)</td><td>0.024<sup>a</sup><br>(0.013)</td><td>0.063<sup>a</sup><br>(-)</td></tr><tr><td>All</td><td>0.018<br>(0.010)</td><td>0.003<br>(0.002)</td><td>0.008<br>(0.006)</td><td>0.021<br>(0.013)</td><td>0.051<br>(-)</td></tr><tr><td rowspan="4">ρ<sub>cani</sub> (μm·μm<sup>-3</sup>)</td><td>♂</td><td>0.021<sup>a</sup><br>(0.010)</td><td>0.003<sup>a</sup><br>(0.002)</td><td>0.010<sup>a</sup><br>(0.005)</td><td>0.024<sup>a</sup><br>(0.013)</td><td>0.063<sup>a</sup><br>(-)</td></tr><tr><td>All</td><td>0.021<sup>a</sup><br>(0.010)</td><td>0.003<sup>a</sup><br>(0.002)</td><td>0.010<sup>a</sup><br>(0.005)</td><td>0.024<sup>a</sup><br>(0.013)</td><td>0.063<sup>a</sup><br>(-)</td></tr></tbody></table>

density (Fig. 4).

Donor's age was significantly correlated with canalicular local volume fraction η<sub>f</sub>, permeability κ, and canalicular length density ρ<sub>cani</sub> (Table 2). While canalicular tortuosity was not significantly correlated to donor's age, it was correlated to canalicular local volume fraction η<sub>f</sub> and canalicular diameter δ. Both canalicular permeability and canalicular length density were significantly correlated to volume fraction, the

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Box plots permeability κ (a.) and canalicular length density ρ<sub>cani</sub> (b.), with a distinction between the male and female populations. The points represent the averaged value of the mean, Q1, Q2, Q3, and SD for each individual. For all of these values, a significant difference was found using the ANCOVA statistical test with the donor's age as a covariate (p < 0.05).</figcaption></figure>

tortuosity, and canalicular diameter. Finally, canalicular length density was significantly correlated to permeability, with a p-value <0.00001.

The negative linear regression between donor's age and intrinsic permeability is shown in Fig. 5. The coefficient of determination was quite low, R<sup>2</sup> = 0.32, while the standard error of the estimate (SEE) was close to the value of permeability. Although the linear correlation remained weak, it was still significant (Table 2), suggesting a genuine tendency.

<footnote><sup>a</sup> Statistical difference between the male and female populations after ANCOVA analysis.</footnote>
<table><caption>**Table 2**

Pearson correlation matrix.</caption><thead><tr><th></th><th>Age</th><th>η<sub>f</sub></th><th>δ</th><th>θ</th><th>κ</th><th>ρ<sub>cani</sub></th></tr></thead><tbody><tr><td>Age</td><td>1.00</td><td>-0.56<sup>*</sup></td><td>-0.28</td><td>-0.35</td><td>-0.56<sup>*</sup></td><td>-0.55<sup>*</sup></td></tr><tr><td>η<sub>f</sub></td><td></td><td>1.00</td><td>0.47<sup>*</sup></td><td>0.50</td><td>0.99<sup>**</sup></td><td>0.99<sup>*</sup></td></tr><tr><td>δ</td><td></td><td></td><td>1.00</td><td>0.44<sup>*</sup></td><td>0.52<sup>*</sup></td><td>0.39<sup>*</sup></td></tr><tr><td>θ</td><td></td><td></td><td></td><td>1.00</td><td>0.50<sup>*</sup></td><td>0.46<sup>*</sup></td></tr><tr><td>κ</td><td></td><td></td><td></td><td></td><td>1.00</td><td>0.99<sup>***</sup></td></tr><tr><td>ρ<sub>cani</sub></td><td></td><td></td><td></td><td></td><td></td><td>1.00</td></tr></tbody></table>

* p-Value < 0.05.

** p-Value < 0.001.

*** p-Value < 0.00001.

<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Permeability in function of the donor's age and their linear regression. The equation of the regression, the coefficient of determination (R²) and the standard error of the estimate (SEE) of the regression are provided. κEst is the estimation of the permeability using the linear regression.</figcaption></figure>

There was a decrease in permeability with distance to lacunar walls (Fig. 6). The bone matrix is less permeable further from a lacuna. The difference between permeability at 0.5 µm and 15 µm is significant, with a decrease of 73 % and 75 % for the female and male populations, respectively. No difference was observed comparing male and female groups.

In comparison, the linear regression between permeability and canalicular length density was high, with a coefficient of determination close to 1 (R² = 0.97) and a small standard error of the estimate (Fig. 7).

This enabled to define a new form of the Kozeny relation for bone LCN permeability based on canalicular length density (see Appendix A):

$$ \kappa (10^{-16} m^2) = 4.2 \rho_{\text{cani}} (\mu\text{m}.*\mu\text{m}^{-3}) - 6.6 \times 10^{-4}, \quad (2) $$

$$ \kappa (10^{-16} m^2) = 4.2 \rho_{\text{cani}} (\mu\text{m}.*\mu\text{m}^{-3}) \quad (2\text{bis}) $$

## 4. Discussion

Bone mechanobiology is driven by its intrinsic permeability associated with the lacuno-canalicular network (LCN). Within this sub-micrometric and interconnected network, osteocytes are stimulated through a pressure-driven interstitial fluid flow over their membrane, resulting in the development of shear stress. Characterizing the permeability of the bone LCN is therefore relevant to a better understanding and modelling of osteocytic mechanobiology. The current study utilized SR-nCT with a voxel size of 100 nm to analyze the LCN from human femoral diaphysis in terms of intrinsic permeability. Standard LCN morphological parameters, such as canalicular diameter or canalicular volume fraction have already been published and discussed elsewhere by the authors. For further discussion regarding these morphological parameters, the reader is referred to this previous study [5]. The discussion here focuses on LCN intrinsic permeability.

A significant finding of the current study is that bone permeability

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Plot of the averaged permeability in function of the averaged distance to a lacuna based on the lacunar distance map. a. Absolute permeability averaged over the male (blue) and female (red) populations. The large errors are associated with the large variation inter-donor. b. The averaged normalized value for the male (blue) and female (red) populations. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Permeability in function of the canalicular length density and their linear regression. The equation of the regression, the coefficient of determination (R²) and the standard error of the estimate (SEE) of the regression are provided. κEst is the estimation of the permeability using the linear regression.</figcaption></figure>

based on canalicular morphology is heterogeneous within the bone matrix. This is a critical consideration for studies investigating LCN fluid behavior during bone loading at the macroscale [38]. Interestingly, few canaliculi were observed at the osteon – interstitial tissue frontier, known as the cement line, suggesting a potential barrier for fluid flow in and out of an osteon. Such decreased canalicular density at the cement line has previously been observed [39]. However, with the spatial resolution used in the current study, small canaliculi crossing this barrier could remain undetected. The current results can be used to further
define homogenization laws, allowing implementation of the information on LCN heterogeneity in numerical models of LCN fluid flow. Still, it is important to note, however, that the current study focused on human femoral bone and that the findings may not directly be extrapolated to other anatomical locations due to LCN morphological differences between bones [40,41].

In the literature, LCN permeability values range from $10^{-24}$ to $10^{-18}$ m² [9,11,26,27,42]. In the current study, human cortical bone LCN permeability was found around $10^{-18}$ m². While this value corresponds to the largest values found in the literature, it is in accordance with the different studies investigating LCN permeability only based on morphological parameters [11,27,28]. It is well known that canaliculi are not empty and that the cell's matrix is embedded within a pericellular matrix [43–45]. This represents a major weakness in the current study. Both the size of cell dendrites and pericellular matrix properties have a direct influence on LCN permeability [6,26]. In the current study, the LCN was considered as an empty porous network. Hence, permeability values obtained are only based on LCN morphology and do not take into account what is inside canaliculi. Virtually filling canaliculi with a matrix would have decreased the overall permeability of the porous network. For instance, considering an osteocytic process diameter of 100 nm [44] would results in an annulus with an averaged width of 138 nm (canalicular radius – osteocytic process radius). The presence of cellular and pericellular matrices will decrease LCN permeability. Still, the authors decided not to integrate it in the calculations. There is currently no data allowing to rigorously assign a realistic osteocytic process diameter value for each canalicula across the bone matrix. In their study, You et al. did measure osteocyte dendrites geometry using 2D analyses and found large variations from one site to another (from 50 to 710 nm) [44]. Hence, integrating the cellular and pericellular components in calculations would have involved hypotheses on their geometry and density, causing a model parameter bias in addition to measurement errors induced by the present 3D observations. As a result, the interpretation of each part of the problem would quite irrelevant. A future work implementing a sensitivity analysis of interstitial flow within pores partially clogged by added cellular geometrical components is a numerical perspective of this work. This represents a major limitation for the current study. But characterizing LCN permeability parameter based on canalicular morphology is important in itself to understand how fluid flows at the bone matrix scale, allowing the analysis of each osteocyte depending on its location within an osteon or in interstitial tissue.

It was observed that permeability was higher in the vicinity of a lacuna and decreased progressively with the distance to a lacuna. This suggests that, during osteocytogenesis and the formation of the LCN, osteocytes are able to tune their close structural neighborhood to promote circulation of fluid towards the cell's body. Further work on the local distribution of permeability around a lacuna, depending on its location within bone structure, will then be conducted. Furthermore, a voxel size of 100 nm does not allow to detect small canaliculi. Considering these smaller canaliculi in the Kozeny relation may modify the permeability value, depending both on their diameter and their topological characteristics within the network.

Another limitation in the current study relies on the isotropic permeability considered while the LCN is known to be anisotropic and that a permeability tensor can be calculated from morphological data [26,27,30]. In the current study, permeability, measured using the Kozeny relationship, was defined in the fluid flow direction, or the pressure gradient direction. In an osteon, this corresponds to the permeability for fluid flowing from the vascular canals towards the cement line, or inversely. Still, it has to be noticed that canaliculi are investigated separately. In each unit cell of the tessellated volume, there is only one canalicula. In each unit cell, there is only one way in and one way out, and thus only one fluid flow direction. Hence, at the scale of the individual canaliculi, network anisotropy is indirectly considered. But this anisotropy is no longer considered at a higher scale. To define

parameters that could be used to homogenize further models of bone fluid flow at the tissue scale, such anisotropic permeability has to be evaluated. To that end, the current data set appears relevant as it covers the human LCN network investigated on 27 samples on large VOIs, larger than those in the literature [26,27].

In the current study, the diameter measured in the current study is slightly larger than the one obtained in the previous study because of the different calculation methods applied [5]. In the previous one, the method of fitting the sphere was applied [5]. This method was developed to estimate trabecular thickness, with trabecular having a complex geometry [46]. Here, canaliculi were treated as tortuous cylinders, making it reasonable to approximate their diameters by measuring the distance between the cylinder skeleton and walls, a method that is also much less time-consuming.

In addition, canalicular length density value obtained in the current study is higher than what can be found in the literature measured through confocal fluorescent microscopy [30]. This point has already been discussed in a previous study [5]. In previous confocal microscopy studies, the focus is made on osteonal tissue that is denser in terms of canaliculi compared to the interstitial tissue, as observed in Fig. 2. In the current study, bone tissue volumes containing both osteons and interstitial tissue are considered. In that way, canalicular length density values found in the interstitial tissue decrease the averaged value. While focusing on osteonal tissue might be of interest, considering a more complete bone volume containing interstitial tissue is relevant. Even if currently it is not known whether or not osteocytes are biologically active there, fluid flow within the interstitial tissue, thus its permeability, is assumed to be involved in fluid flow within the neighboring osteons [47].

The t-test for unpaired samples did not raise any global difference between the male and female populations. However, by considering age as a covariate, different permeability, density, and canalicular volume fraction were observed. For donors of a same age, these parameters are significantly higher for female than for male. This result is in line with previous ones highlighting a higher canalicular volume fraction for women [5,48]. This was explained by a higher number of canaliculi within female bone compared to male. With a higher number of canaliculi, the local volume fraction increases resulting in a higher permeability. This is an interesting result as it is assumed that women have an altered bone remodelling potential with age due to the menopause and a loss of estrogen [49]. In a previous study, the transport of a fluorescent marker in bone sample under mechanical loading was increased for ovariectomized rats [50]. This previous result was mainly observed in trabecular bone but is in line with an increased LCN permeability for estrogen deficient bones. Similarly, ovariectomized rat presented a higher LCN porosity and number of canaliculi per lacuna [48]. It might be of interest to understand whether this increase in LCN occurs to counterbalance the loss of estrogen and increase the mechanically activated bone remodelling.

Local canalicular volume fraction, permeability, and canalicular length density were significantly correlated with age, with a decreasing value with increasing age. A decreased LCN permeability with age was already observed in mice [51]. This age-dependent decrease is in accordance with a decreased remodelling potential and increased bone fragility. For instance, with age, osteocytes are less mechanically stimulated due to a decreased physical activity from the subject. This reduced mechanical stimulation is associated with osteocytes apoptosis [52,53] resulting in microptosis and the LCN mineralization [54]. Permeability decreases with the hypermineralization of canaliculi. Still, the correlation with age remains weak in the current study, indicating that other parameters are involved in LCN permeability.

Finally, a major result of the current study is the good correlation between bone permeability associated with the canalicular morphology and canalicular length density, defined as length of canaliculi per unit bone volume. This is an interesting result as canalicular length density can be accurately quantified using confocal microscopy [30,55].
Confocal microscopy is more convenient to use and more available compared to synchrotron radiation facilities. Hence, the current study validates that canalicular length density is a good surrogate of bone LCN permeability and confocal microscopy can be further used to assess bone LCN permeability. Interestingly, in Eq. (2) it can be seen that permeability is almost proportional to canalicular length density and the intercept between the regression and the ordinate axis is close to 0. It is worth reminding that the data obtained in the current study are true for human femoral diaphysis, but may not be extrapolated to other anatomical regions, as morphological differences between bones have been observed at the lacunar scale [40,41]. Still this proportionality confirms that canalicular length is a relevant parameter and fluorescence confocal microscopy can be used to estimate bone permeability and to further investigate LCN load-induced fluid flow [28,56,57].

In the current study, the distribution of bone intrinsic permeability based on the morphology of the canalicular network was investigated using 27 bone samples harvested from human femoral samples using SR-nCT. Further work should perform local analysis, considering the location within bone tissue or within the osteon, in order to define some parameters at the tissue scale that can be further implemented in homogenized models of bone load-induced fluid flow [24,25]. By doing so, we can consider the influence of osteocyte location within the bone architecture and estimate a more representative shear stress induced by fluid flow.

## 5. Conclusions

In the current study, bone intrinsic permeability was quantified using 27 bone samples harvested from human femoral diaphysis and characterized through synchrotron radiation nano-computed tomography. An averaged permeability value of 8.8 $10^{-18}$ m$^2$ was obtained. This value has to be considered for an empty porous lacuno-canalicular network. Additionally, smaller canaliculi that could contribute to overall average bone permeability may not have been accounted for, as they fall below the detection threshold of 100 nm. Despite these limitations, the current study highlighted the heterogeneous distribution of intrinsic permeability across bone tissue. Differences were qualitatively observed between osteonal and interstitial tissues, and even within a single osteon. At comparable ages, the female population exhibited a higher permeability compared to the male population. In addition, permeability significantly decreases with age. Finally, the current study validated the use of canalicular length density, defined as the canalicular length per bone volume, as a good estimator of bone permeability. Finally, a new form of the Kozeny relation was defined to express bone canalicular

permeability. This result is of interest as bone canalicular length density can be accurately measured through confocal fluorescence microscopy that is less resource-intensive than synchrotron nano-CT. The present study confirms the potential of confocal microscopy to estimate bone permeability associated with canalicular morphology and to investigate LCN fluid flow. However, it is essential to emphasize that the values reported here are based solely on canalicular morphology. Intra-canalicular cellular and peri-cellular components need to be investigated and integrated within geometrical models to further develop relevant numerical models of LCN fluid flow.

## CRediT authorship contribution statement

**Remy Gauthier:** Writing – review & editing, Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis, Data curation. **Hélène Follet:** Writing – review & editing, Validation, Supervision, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. **Cécile Olivier:** Writing – review & editing, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Thibault Lemaire:** Writing – review & editing, Validation, Investigation. **David Mitton:** Writing – review & editing, Validation, Supervision, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Francoise Peyrin:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

This work was done in the framework of LabEx PRIMES (ANR-11-LABX-0063) of Université de Lyon within the program "Investissements d'Avenir". The data used for this work were funded by the ANR MULTIPS Project (ANR-13-BS09-0006). We thank the ESRF for support through the Long-Term Proposal MD830 and for provision of beam time (experiment MD933) using the ID16B beamline, and Julie Villanova for her support during the experiment. We thank Boliang Yu for his contribution to data acquisition and image segmentation codes.

## Appendix A

Here is the Kozeny relation for media with capillary-like porosity:

$$ \kappa = \frac{\eta_f \delta^2}{32 \theta^2}, \qquad (A1) $$

With the hypothesis that, for the n Voronoi cell i, each canaria of length $l_i$ is characterized by a diameter $\delta_i$, the averaged canalicular volume fraction $\eta$ over the volume V can be estimated:

$$ \eta = \frac{\sum_{i=1}^{n} \frac{\pi \delta_i^2 l_i}{V},}{\qquad (A2)} $$

With the further hypothesis that the local diameters $\delta_i$ can be translated over the n Voronoi cells by an averaged macroscopic value $\delta_m$, we can obtain the relationships between the volume fraction $\eta$ and the canalicular length density $\rho_{cani}$:

$$ \eta = \frac{\pi}{4} \delta_m^2 \rho_{cani}, \qquad (A3) $$

This can then be implemented within the Kozeny relation as follows:
$$ \rho_{cami} = \frac{128\kappa\theta^2}{\pi\delta_m^4}, \qquad (A4) $$

Eq. (A4) highlights the proportionality between canicular permeability and canicular length density.

## Data availability

Data will be made available on request.

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