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Numerical Simulation of Electrical Properties in Cortical and Trabecular Bone: A Simplified Model
María José Cervantes, Catalina A. Cely-Ortíz, C. Manuel Carlevaro, Ramiro M. Irastorza
TL;DR
Bone electrical models often simplify tissue as homogeneous, despite microstructural and directional variation that matters to electrical characterization. This study combines finite-element simulations with Bruggeman-based homogenization for trabecular and cortical bone. The models reproduce reported trabecular conductivity behavior and cortical anisotropy, while the modified tibia scenario shows lower impedance than the reference.
Problem
Bone is often treated as homogeneous, and simplified models linking trabecular and cortical microstructure to effective electrical properties while accounting for cortical anisotropy remain limited.
Source: §1 · p. 2Method
The study combines finite-element simulations with Bruggeman mixing rules, using microCT-derived trabecular geometries and an anisotropic cortical canal-network formulation.
Source: §1 · p. 2, §2.1 · p. 3, §2.2 · p. 4Results
Approximately 39% lower impedance than the reference model was observed for the tibia scenario with lower trabecular and cortical BV/TV and reduced cortical thickness.
Source: §5 · p. 11Takeaways & Limitations
The models relate bone composition and selected microstructural features to effective conductivity and the electrical response of the modeled tibia.
Source: §5 · pp. 10–11Changes in BV/TV were represented through bone and fluid phase fractions, without explicitly resolving remodeling-related changes in pore size, shape, connectivity, or spatial distribution.
Source: §4.1 · p. 10
Abstract
from arXiv · showhide
The electrical properties of biological tissues depend on their composition and microstructure and determine their response to applied electric fields. In bone tissue, these properties are closely related to its composition, porosity, and microstructural organization. However, simplified models that link bone microstructure (trabecular and cortical) to its effective electrical properties while accounting for the electrical anisotropy of cortical bone remain scarce. In this work, a multiscale approach was developed, combining finite element simulations with Bruggeman-based homogenization models. Trabecular bone was modeled using microCT derived geometries, and its effective conductivity was analyzed as a function of BV/TV and free water content. Cortical bone was represented by a simplified three-dimensional network of Haversian canals (axial = z) and Volkmann canals (transverse), incorporating their directional organization through an anisotropic Bruggeman formulation. The models captured the conductivity range reported experimentally for trabecular bone (approximately 25-200 mS/m) and reproduced the anisotropic behavior observed in the cortical FEM simulations, with higher conductivity in the axial direction. In the generated cortical geometries, 65.5% of the total porosity was associated with axial Haversian canals. At the macroscale, the tibia model with lower trabecular and cortical BV/TV and reduced cortical thickness showed an approximately 38-39% reduction in impedance relative to the reference model. These results indicate that microstructural variations in bone tissue are reflected in its effective electrical properties and in the macroscopic impedance of the model, providing a physical and computational basis for future investigations into the sensitivity of electrical measurements to changes in bone microstructure.
1 Introduction
Bone electrical properties are often represented by bulk values despite bone’s hierarchical structure and cortical anisotropy. This work addresses the need for simple mixing rules linking trabecular and cortical microstructure to effective electrical properties.
1 Introduction
- Electrical studies commonly treat bone as homogeneous, while exposure guidelines and standards give limited attention to tissue heterogeneity and anisotropy.This is especially relevant below 1 MHz, where tissue dielectric properties may themselves be anisotropic.Source: §1 · p. 2
- Trabecular microCT geometries support relating microstructure to dielectric properties and calibrating effective-medium models against BV/TV and free-water content.These approaches quantify bone volume fraction and free-water content.Source: §1 · p. 2
- Cortical bone is difficult to model at microCT resolution, and measured conductivity and permittivity vary across axial, radial, and tangential directions.The directional behavior is likely associated with canal-network organization.Source: §1 · p. 2
- The study proposes fitted mixing rules for both trabecular and cortical bone that explicitly account for porosity and cortical anisotropy.The stated objective is simple mixing rules for macroscale electrical simulations.Source: §1 · p. 2
2 Materials and Methods
The methods combine microCT-derived trabecular geometries and a synthetic cortical canal network with finite-element simulations and Bruggeman homogenization. The cortical formulation represents directionality through canal-family depolarization tensors, with material parameters fitted from literature and simulations.
2.1 Model geometry
- Trabecular bone was modeled from bovine microCT samples in 5 mm cubes, while the cortical model used a parameterized 1 mm^3 domain with quasi-random axial Haversian and transverse Volkmann canals.Lacunae and canaliculi were not modeled explicitly; their contribution was assumed to be included in effective bone-matrix properties.Source: §2.1 · pp. 2–3, Figure 1 · p. 3
2.2 Numerical model
- Both tissue models solved the quasi-static electric-potential problem with FEM; effective conductivity was derived by equating simulated dissipated power to that of a lumped resistor.Electrodes were assigned voltages φ0 and 0 V, with zero-current conditions on the remaining outer surfaces.Source: §2.2 · pp. 3–4
2.3 Mixing-rules
- The trabecular Bruggeman rule treats constituent phases symmetrically, while the cortical rule combines three oriented canal families with bone matrix using tensorial depolarization factors.For trabecular bone, the phases were free water, marrow, and bone matrix; cortical anisotropy is encoded in the depolarization tensor.Source: §2.3 · pp. 4–5
- For Haversian canals with H/R around 30, the model treats the axial depolarization component as negligible, consistent with simulations finding it negligible near H/R 10.This is a geometric assumption for the cortical mixing rule.Source: §2.3 · p. 5
2.4 Material properties
- Trabecular material properties were fitted at 100 kHz, and cortical fitting used reported directional conductivity values plus assigned conductivities for hydrated matrix and canal-network domains.The selected trabecular marrow and matrix values were 300.0 and 20.1 mS/m; cortical matrix and porous-domain values were 3.8 and 267 mS/m, respectively.Source: §2.4 · p. 5
3 Results
The models reproduce trabecular conductivity trends and cortical directional behavior, while the non-healthy tibia model has lower impedance than the healthy model. The cortical model assigns most canal porosity to axial Haversian canals.
3.1 Trabecular bone
- Approximately 25–200 mS/m conductivity was reported for bovine and human samples; Bruggeman captured bovine BV/TV behavior and human free-water dependence.The free-water relationship should be interpreted cautiously because free-water content and BV/TV are coupled and both influence effective conductivity.Source: §3.1 · pp. 5–6, Figure 2 · p. 6
3.2 Cortical bone
- 65.5% of total porosity was assigned to axial Haversian canals, and the anisotropic Bruggeman model reproduced FEM's higher axial than transverse conductivity.The remaining 34.5% was divided equally between transverse canals oriented along x and y; these fractions remained constant as BV/TV varied.Source: §3.2 · pp. 6–7, Figure 3 · p. 8
3.3 Macroscale model
- Approximately 39% lower impedance was reported for non-healthy versus healthy tibia models across both electrode configurations.The non-healthy setup lowered trabecular and cortical BV/TV and reduced cortical thickness by 2 mm; horizontal–vertical differences were 2.0% in healthy and 0.9% in non-healthy models.Source: §3.3 · pp. 7–8, Table 4 · p. 8
4 Discussion
The tissue-scale models captured reported conductivity behavior, including trabecular dependence on BV/TV and free water and cortical anisotropy. In the tibia model, the non-healthy condition reduced impedance, while electrode-orientation differences remained small; several modeling assumptions and the macroscale prediction require caution.
4 Discussion
- Trabecular conductivity fell within the bovine experimental range of 25–200 mS/m and depended jointly on BV/TV and free water content.Free-water dependence was compared with human data because simultaneous bovine measurements of free water, BV/TV, and conductivity were unavailable; physiological fluid substitution and trabecular thinning are coupled.Source: §4 · p. 9
- 65.5% of generated cortical porosity belonged to axial Haversian canals, accompanying higher axial than transverse conductivity in the FEM and anisotropic Bruggeman models.The authors attribute the directional behavior to the canal network; its fluid-filled axial pathway facilitates current flow.Source: §4 · p. 9
- Conductivity ranges from the models agreed with ranges obtained by applying Kimel-Naor et al.’s density relationships, while the present model also represents anisotropy.Those relationships yielded 12.48–38.73 mS/m for cortical bone and 58–229 mS/m for cancellous bone.Source: §4 · p. 9
- 38–39% lower total impedance in the non-healthy tibia model resulted from reduced trabecular BV/TV, increased cortical porosity, and 2 mm less cortical thickness.The three modeled changes respectively increase the volume fraction of conductive fluid, increase conductive fluid canals, and shorten current travel through the most resistive bone layer.Source: §4 · p. 10
- Electrode-orientation difference declined from 2.0% in healthy bone to 0.9% in non-healthy bone, making localized or directional damage difficult to detect by rotating these surface electrodes alone.The authors suggest that structural degradation homogenizes the current path and overrides microscopic anisotropy.Source: §4 · p. 10
4.1 Limitations
- The model does not explicitly resolve remodeling-related changes in pore size, shape, connectivity, or spatial distribution, and combines bovine and human parameter data.The macroscale impedance is an in silico prediction requiring independent experimental validation; marrow and electrodes are also simplified, and permittivity is omitted.Source: §4.1 · p. 10
5 Conclusions
The study uses finite element simulations and Bruggeman mixing rules to relate bone composition and selected microstructure to effective conductivity, including cortical anisotropy. In a bovine tibia model, combined bone-volume and thickness changes reduced impedance by approximately 39%.
5 Conclusions
- The multiscale approach combines finite element simulations and Bruggeman mixing rules to link bone composition and microstructure to effective conductivity at 100 kHz.Trabecular conductivity depends on BV/TV and free water under the adopted coupled parameterization; the cortical formulation represents Haversian and Volkmann canal directionality.Source: §5 · p. 10
- Approximately 39% lower impedance occurred in the modified tibia model with lower trabecular and cortical BV/TV and reduced cortical thickness, relative to the reference model.Differences between electrode configurations were 2.0% in the reference model and 0.9% in the modified model.Source: §5 · p. 11
- Homogenization equations incorporated composition and selected microstructural effects into macroscale simulations without explicitly representing the entire bone microstructure.Source: §5 · p. 11
A Orthotropic electrical conductivity
Orthotropic conductivity is represented in a local coordinate system aligned with the principal conduction directions. When those directions do not match global coordinates, the conductivity tensor is transformed using a rotation matrix.
A Orthotropic electrical conductivity
- The conductivity tensor is diagonal in the local coordinate system defined by the three principal directions of electrical conduction.Its diagonal entries, σ1, σ2, and σ3, are conductivities along those three directions; cortical examples use longitudinal, circumferential, and radial directions.Source: A Orthotropic electrical conductivity · p. 13
- When principal material directions do not coincide with global coordinates, the conductivity tensor must be transformed into the global coordinate system.Source: A Orthotropic electrical conductivity · p. 13
- The rotation matrix R has columns formed by the unit vectors defining the local principal directions.Source: A Orthotropic electrical conductivity · p. 14
A.1 Transversely isotropic case
The special orthotropic case has equal conductivity in two directions. Its tensor formulation uses a principal direction and can accommodate spatial variation in that direction.
A.1 Transversely isotropic case
- The special orthotropic case has equal conductivity in two directions.Source: §A.1 · p. 14
- The conductivity tensor is expressed directly using unit vector e3, which defines the principal direction.Source: §A.1 · p. 14
- When e3 varies spatially, the formulation defines the conductivity tensor as a function of position.Source: §A.1 · p. 14