Diese Studie geht der mehrskaligen Modellierungsanalyse poroelastischer Eigenschaften von Gehirngewebe nach und konzentriert sich dabei auf die komplizierten Kapillarnetzwerke. Die Forschung zielt darauf ab, Gehirnfunktionen auf einer mehrstufigen Ebene zu verstehen, insbesondere die Wechselwirkung von rtPA mit umgebendem Gehirngewebe und die Bildung von Gehirnödemen. Die Studie nutzt fortgeschrittene Techniken wie die mehrskalige Homogenisierung und Finite-Elemente-Methoden, um effektive Parameter wie die Durchlässigkeit des Blutes, die Durchlässigkeit des Flüssigkeitsinterstitials, die Biot-Koeffizienten, den Young-Modul und das Poisson-Verhältnis abzuschätzen. Die Mikrovaskulatur-Geometrie wird durch den Vergleich der effektiven Blutpermeabilitätstensoren validiert, die durch die Gleichungen von Poiseuille und Stokes gewonnen werden. Die Ergebnisse unterstreichen den signifikanten Einfluss der Zunahme des Mikrovaskularisationsvolumens auf bestimmte effektive Parameter und liefern wertvolle Einblicke in die Deformation von Hirngewebe und die Ödeme. Die Studie endet mit einem Benchmark-Problem, der Lösung der makroskaligen Gleichungen auf einer idealisierten Gehirngeometrie, um Gehirnödeme nach Ischämie zu simulieren. Diese umfassende Analyse bietet ein tieferes Verständnis der Mechanismen des Gehirngewebes und ihrer Auswirkungen auf neurologische Störungen.
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Abstract
The cerebral microvasculature plays a key role in determining the blood perfusion and oxygen diffusion to surrounding tissue. Multiscale models have thus been developed to incorporate the effect of the microvasculature into overall brain function. Moreover, brain tissue poroelastic properties are also influenced by the microvasculature. This study aims to determine the pororelastic properties of brain tissue using multiscale modeling on microvasculature networks described by the following effective parametric tensors: blood flow permeability \({\varvec{K}}\), interstitial fluid flow permeability \({\varvec{G}}\), Biot’s coefficients for blood \({\alpha }_{c}\) and interstitial fluid \({\alpha }_{t}\), Young’s modulus \(\overline{E }\), and Poisson’s ratio \(\overline{v }\). The microvasculature networks are built from a morphometric data of brain capillary distribution, which is represented using 1D lines. To allow for solving the microscale cell equations using finite element method, the microvasculature is modified into 3D shapes. The modifications resulted in 15% increment of the microvasculature volume. Validation is then performed by comparing the permeability tensor \({\varvec{K}}\) obtained using Poiseuille’s and Stokes’ equations, which resulted in the value of \({\varvec{K}}\) obtained through solving Stokes’ equation to be about 70% less than through solving Poiseuille’s equation. Based on these results, the other effective parameters have been estimated by considering the microvasculature volume increment due to the geometry modification. The volume increment significantly affects the parameter \({\alpha }_{c}\) but not the other parameters. The effective parameters are then used in a benchmark simulation, which further demonstrates the model value in describing the effects of brain capillary morphology in cerebrovascular diseases.
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1 Introduction
Ischemic stroke is a neurological disorder characterized by the reduction of blood supply to the brain due to a blockage in brain blood vessels. The main goal of stroke treatment is to restore blood supply to the brain. The most common treatments are thrombectomy and thrombolytic recombinant plasminogen activator (rtPA) to break the blood clot blocking the blood vessels [1]. However, the benefit of thrombectomy and rtPA is time dependent. Therefore, it is crucial to treat the patients as quickly as possible [2]. For example, in the case of late rtPA treatment, it has been associated with increased chances of permanent disability, developing hemorrhagic stroke, and death [3]. In addition, rtPA must be given up to 4.5 h after the stroke onset [4]. Administration of rtPA outside of this time window could cause the blood–brain barrier (BBB) to break down and initiate brain edema formation [5]. The relationship between the late application of rtPA with brain edema formation requires the understanding of brain functions at a multiscale level as it involves blood flow and the interaction of rtPA with surrounding brain tissues [6].
The brain microvasculature plays an important role in the determination of blood perfusion and oxygen transport into the surrounding tissue [7, 8]. The morphometric data of brain microvasculature acquired by [7] have enabled the understanding of blood perfusion and oxygen transport to the brain tissue. In addition, several capillary network generation algorithms have been developed to allow for this analysis. For examples, the constrained constructive optimization (CCO) [9] and fractal growth technique [10] have been used to investigate blood perfusion in myocardium wall and brain tissue, respectively. Furthermore, a network generation algorithm based on arterial tree angiogenesis principle has been applied to investigate blood perfusion [11]. Improvement of the arterial tree algorithm through the principle of the minimum spanning tree by [12] has been used to build brain microvasculature geometry that matches the statistical distribution of the brain microvasculature. This geometry has been used in various mathematical modeling frameworks to estimate various brain transport properties such as the blood flow permeability using multiscale homogenization technique [13], oxygen transport through the surrounding brain tissue using residue function estimation [14‐16], and microembolism transport [17]. Extending the application of the microvasculature toward the understanding of brain edema formation will be valuable as it can estimate the extent of brain tissue deformation in severe ischemic stroke conditions.
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Brain tissue deformation has been modeled using poroelastic theory for the studies of various neurological disorders, for example brain edema [18‐20]. The poroelastic theory describes the deformation of a material that is made up of a solid matrix embedded in a fluid. Since the human brain is made up of brain cells, interstitial fluid, and blood vessels, it is possible to model the brain as a poroelastic material to evaluate the interactions between the various phases in the brain in many conditions. For example, brain tissue swelling after rtPA treatment has been investigated computationally using poroelastic theory [18]. In this model, the brain is assumed to comprise homogeneous elastic brain cells surrounded by interstitial fluid and the cerebral vasculature. However, recent modeling developments based on asymptotic expansion homogenization (AEH) have considered the effect of microvasculature morphometry on the poroelastic properties of the overall tissue, for example, in the works by [21‐25]. The AEH technique relates the homogenized macroscale model parameters to the microscale geometrical properties. For example, the work by [21] utilized AEH on vascularized poroelastic materials to obtain four homogenized macroscale model parameters, namely the effective blood flow permeability \({\varvec{K}}\), effective interstitial fluid flow permeability \({\varvec{G}}\), effective Biot’s coefficients for blood \({\alpha }_{c}\) and interstitial fluid \({\alpha }_{t}\), and effective Young’s modulus \(\overline{E }\), and Poisson’s ratio \(\overline{v }\) of the tissue. These parameters have been calculated in various works using three-dimensional (3D) microscale geometries of tissue with the cerebral microvasculature [23, 24, 26].
This study aims to determine the poroelastic properties of brain tissue described by the following effective macroscale parameters \({\varvec{K}}\), \({\varvec{G}}\), \({\alpha }_{c}\), \({\alpha }_{t}\), \(\overline{E }\), and \(\overline{v }\). Furthermore, the brain tissue is represented using the brain microvasculature geometry, which was developed by [12] and has been used by [13] to estimate blood permeability using Poiseuille’s equation and multiscale modeling. However, the Poiseuille’s equation only requires the capillary dimensions, such as the radius and length, to determine the blood permeability. To determine the other effective macroscale parameters, finite element method (FEM) must be used, and thus, the capillary in the microvasculature must be converted into 3D shapes so that it can be meshed and solved using FEM [26]. The permeability values obtained through this study will be compared and validated with the values reported by [13]. Since the remaining parameters are not reported by [13], we will then estimate the parameters using a correction factor from the differences in the blood permeability. This estimation is to evaluate the dependency of these parameters on the dimension of the microvasculature geometry.
The paper is organized as follows: (1) Sect. 2 describes the mathematical model employed, which consists of the macroscale governing equation and the associated microscale cell equations to determine the effective macroscale parameters, the model physiological parameters, and the numerical procedure; (2) Sect. 3 describes the processes to develop the microvasculature geometries, starting from the description of the microvasculature data, modifications required, and validation procedures; (3) Sect. 4 presents the results, starting from mesh sensitivity analysis, microvasculature geometry validation, the estimations of other effective parameters, and example of benchmark problem using the macroscale governing equations; (4) Sect. 5 discusses the results and potential application of the methodology; and (5) Sect. 6 concludes the findings from the research.
2 Multiscale modeling
This section will discuss the multiscale modeling of the brain tissue. Two sets of equations will be discussed, namely the homogenized macroscale governing equations and the respective microscale cell equations.
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2.1 Homogenized macroscale governing equations
The homogenized macroscale governing equations used in this study are taken from [21]. In this model, the subscript \(c\) and \(t\) represent the blood capillary and brain tissue region, respectively. Figure 1 shows a schematic of the macroscale, microscale, and pore-scale (not modeled in this work) of a typical brain tissue. The derivation begins by first modeling the brain as a porous brain tissue with interstitial fluid surrounding the blood in microvasculature. The porous brain tissue can be modeled using poroelastic theory with Darcy’s equation describing the interstitial fluid. The brain tissue is also assumed to be linear and isotropic. The blood is then assumed to be an incompressible Newtonian fluid. The fluid transport between the interstitial and blood is assumed to obey Starling’s law. These introduce several field variables, \(f\), such as the brain tissue displacement and the fluid pressures, which are dependent on the spatial and time variables.
In order to homogenize the brain microvasculature and tissue, it is required that the intervascular distance \(d\) is much smaller than the average length of brain tissue \(L\), such that a small parameter can be defined as:
$$\epsilon =\frac{d}{L}\ll 1$$
(1)
Defining \(\epsilon\) allows for the length scale separation between the microvasculature and brain tissue, thus creates the dependency between the macroscale parameters with the microstructure. Further, two spatial variables \({\varvec{y}}\) and \({\varvec{x}}\) are defined for the microscale and macroscale, respectively, that obey:
$${\varvec{x}}=\epsilon {\varvec{y}}$$
(2)
Here, all the field variables are assumed to be periodic in \({\varvec{y}}\). Then, the multiscale homogenization is performed in four steps: (1) all the field variables are now defined in term of two spatial variables, i.e.: \(f=f\left({\varvec{x}},{\varvec{y}},t\right)\); (2) power series expansion is applied in terms of \(\epsilon\); (3) the macroscale equations are determined by comparing the coefficients for \(\epsilon =0\); and (4) the averaging operator is applied on all field variables over \({\varvec{y}}\) using the following definition:
where \(\left|\Omega \right|\) refers to the volume of the brain domain and the subscript \(k=t, c\) refers to the regions in the brain. Applying the AEH technique produces the homogenized macroscale governing equations, in which the effective parameters are dependent on the microscale cell equations, which must then be solved on a microstructure.
The homogenized macroscale governing equations in non-dimensional form are analogous to the double Darcy’s model with the homogenized stress balance equation given as:
$${\nabla }_{x}\cdot \sigma =0$$
(4)
where \(\sigma\) is the homogenized effective stress, which is given by:
Here \({\varvec{u}}\), \({P}_{c}\), and \({P}_{t}\) represent the tissue displacement, capillary fluid pressure, and interstitial fluid pressure field respectively. The terms \({\alpha }_{c}\) and \({\alpha }_{t}\) represent the effective Biot’s coefficient tensors for capillary and interstitial fluid, respectively. Meanwhile, the effective elasticity tensor \(\overline{C }=\overline{C }(\overline{E },\overline{v })\) can be defined in terms of the effective Young’s modulus \(\overline{E }\) and Poisson’s ratio \(\overline{v }\). These tensors are further defined as:
where \({\varvec{I}}\) is the identity tensor, \({\phi }_{c}\) and \({\phi }_{t}\) are the volume fractions for the capillary and interstitial fluid, respectively, and \(C=C\left(E,v\right)\) is the brain tissue elasticity tensor, which can be defined using the non-dimensional Young’s modulus \(E\) and Poisson’s ratio \(v\). For this study, \(E=584\) Pa and \(v=0.35\), following the work by [18]. Meanwhile, the effective tensors \({\varvec{L}}\) and \({\varvec{Q}}\) are determined from the microscale cell equations.
Moreover, the macroscale fluid pressures \({P}_{c}\) and \({P}_{t}\) are defined using mass balance equations as:
Here, the terms \(\left|\Gamma \right|\), \(\left|\Omega \right|\), and \({\overline{L} }_{p}\) are the total area of the interface, the volume of the microstructure geometry, and the non-dimensional hydraulic conductivity through capillary walls. The average relative velocities \({\langle {{\varvec{w}}}_{c}\rangle }_{c}\) and \({\langle {{\varvec{w}}}_{t}\rangle }_{t}\) over the respective fluid compartments are related with their respective pressure fields through Darcy’s law:
where the terms \({\varvec{K}}\) and \({\varvec{G}}\) are the effective hydraulic permeability for the blood in microvasculature and interstitial fluid, respectively, with \(\overline{k }\) is the non-dimensional brain microscopic interstitial fluid permeability. The tensors \({\varvec{K}}\) and \({\varvec{G}}\) are also determined by solving the microscale cell equations.
2.2 Microscale cell equations
The effective parametric tensors \({\varvec{G}}\), \({\varvec{K}}\), \({\varvec{Q}}\), and \({\varvec{L}}\) described previously can be determined by solving the following microscale cell equations on a microvasculature geometry described below. In the following equations, the terms \({\Omega }_{c}\), \({\Omega }_{t}\), and \(\Gamma\) are defined as the blood microvasculature region, the brain tissue region, and the interface between these regions. It should be noted that in this work, all the microscale cell equations are solved for the different coordinate basis \(x\)-, \(y\)-, and \(z\)-directions due to the microstructure geometry not necessarily being non-rotationally invariant.
2.2.1 Laplace’s cell equation
The effective interstitial fluid permeability tensor \(\mathbf{G}\) can be calculated by:
Here, the auxiliary interstitial fluid pressure vector \({{\varvec{P}}}_{t}\) can be determined by solving the following Laplace cell equation over the microvasculature geometry together with the periodicity constraint:
$$\left.\begin{array}{c}{\nabla }_{y}^{2}{{\varvec{P}}}_{t}=0 \ \ in \ \ {\Omega }_{\text{t}}\\ \left({\nabla }_{y}{{\varvec{P}}}_{t}\right)n=n\ \ on \ \ \Gamma \\ {\langle {{\varvec{P}}}_{t}\rangle }_{t}=0 \ \ in \ \ {\Omega }_{\text{t}}\end{array}\right\}$$
(14)
where the term \({\varvec{n}}\) is the normal vector to the interface \(\Gamma\). The cell equation must be solved three times for \(x\)-, \(y\)-, and \(z\)-directions to obtain the complete components of \({{\varvec{P}}}_{t}\). Examples of solving the Laplace cell equations can be found in [26].
2.2.2 Stokes’ cell equation
The effective blood permeability tensor \({\varvec{K}}\) can be calculated by averaging the auxiliary blood velocity \({\varvec{W}}\) over the capillary volume:
where the auxiliary blood velocity tensor, \({\varvec{W}}\) and its pressure vector, \({{\varvec{P}}}_{c}\) can be calculated by solving the Stokes cell equation over the microvasculature geometry:
$$\left.\begin{array}{c}{\nabla }_{y}^{2}{{\varvec{W}}}^{T}-{\nabla }_{y}{{\varvec{P}}}_{c}+I=0 \ \ in \ \ {\Omega }_{c}\\ {\nabla }_{y}\cdot {{\varvec{W}}}^{T}=0 \ \ in \ \ {\Omega }_{c}\\ {{\varvec{W}}}^{T}n=0 \ \ on \ \ \Gamma \\ {\langle {{\varvec{P}}}_{c}\rangle }_{\text{c}}=0 \ \ in \ \ {\Omega }_{c}\end{array}\right\}$$
(16)
The cell equation must be solved three times for \(x\)-, \(y\)-, and \(z\)-directions to obtain all the components of \({\varvec{W}}\) and \({{\varvec{P}}}_{c}\). The work by [26] described the procedure to solve the cell equations using a meshed microstructure. In addition, the effective blood permeability can also be calculated using Poiseuille’s equation [13], which will be used for microvasculature geometry validation and is described below.
2.2.3 One-elastic cell problem
The effective parametric tensor \({\varvec{Q}}\) is used to calculate the effective Biot coefficient tensors for capillary \({\alpha }_{c}\) and interstitial fluids \({\alpha }_{t}\). The tensor \({\varvec{Q}}\) can be calculated by:
where the auxiliary displacement vector \({\varvec{a}}\) can be determined by solving the following one-elastic cell equations over the microstructural geometry:
$$\left.\begin{array}{c}{\nabla }_{y}\cdot \left(C{\nabla }_{y}{\varvec{a}}\right)=0 \ \ in \ \ {\Omega }_{\text{t}}\\ \left(C{\nabla }_{y}{\varvec{a}}\right)n=-n\ \ on \ \ \Gamma \\ {\langle {\varvec{a}}\rangle }_{\text{t}}=0 \ \ in \ \ {\Omega }_{t}\end{array}\right\}$$
(18)
The cell equation is solved three times for \(x\)-, \(y\)-, and \(z\)-directions to obtain the complete \({\varvec{a}}\) vector. Examples of solving this cell equation can be found in [23, 24, 27].
2.2.4 Six-elastic cell problem
The effective parametric tensor \({\varvec{L}}\) is a fourth-order tensor and is defined as:
where \(\text{A}\) is a third-rank displacement tensor that can be calculated by solving the following six-elastic cell equations, written in component form, as:
The subscripts \(i\), \(j\), \(k\), \(l\), \(m\), and \(n\) have values of 1, 2, or 3 representing the Einstein summation. The equations can be solved following the examples previously given by [23, 24, 27]. Once these have been computed, \({\varvec{L}}\) can be used to calculate the effective elasticity tensor \(\overline{C }=\overline{C }(\overline{E },\overline{v })\) to relate the effects of the microvasculature on the effective brain tissue mechanical properties.
2.3 Model physiological parameters and numerical procedure
The homogenized macroscale Eqs. (3), (8), and (9) are written in non-dimensional form. Table 1 lists all the physiological parameters used to non-dimensionalize the equations, which are taken from previous studies on brain tissue. Moreover, there are other parameters in the macroscale equations that must be computed, as listed in Table 2, which are related to the physiological parameters.
Table 1
Characteristic parameters of the fluid transport in brain tissue
Non-dimensional parameters as functions of dimensional parameters and their typical values
Non-dimensional parameter
Relationship
Value
\(\epsilon\)
\(\frac{d}{L}\)
\(5.39\times 1{0}^{-4}\)
\(\overline{k }\)
\(\frac{k}{{d}^{2}}\)
\(1.09\times {10}^{-6}\)
\({\overline{L} }_{p}\)
\(\frac{\mu {L}_{p}{L}^{2}}{{d}^{3}}\)
\(1.80\times {10}^{-3}\)
The characteristic time, \(T=\frac{L\mu }{ \widehat{P}{d}^{2}}=2.59\) s, velocity, \(U=\frac{ \widehat{P}{d}^{2}}{\mu }=0.154\) m s−1, and pressure, \(P= \widehat{P}L=1330\) Pa, which were derived in similar manner as [21] are used to obtain all of the non-dimensional parameters listed in Table 2. Furthermore, the effective parametric tensors \({\varvec{K}}\) obtained from solving the Stokes’ cell equation are in non-dimensional form. To dimensionalize them for validation purposes, the parameters will be multiplied with a scaling factor of \(\frac{{d}^{2}}{\mu }=1.23\times {10}^{-6}\) m3 s kg−1, following the derivation in [26].
The 3D microvasculature geometry is then constructed using a MATLAB-COMSOL interface. The microscale Eqs. (12)–(19) are solved separately on the microvasculature geometry to determine the effective parametric tensors \({\varvec{G}}\), \({\varvec{K}}\), \({\varvec{Q}}\), and \({\varvec{L}}\) and the associated homogenized macroscale model parameters. The microvasculature geometry is meshed depending on the cell equations to be solved. For the Laplace’s, one-elastic, and six-elastic cell equations, the region \({\Omega }_{t}\) is discretized using the quadratic Lagrange mesh elements. Meanwhile, for the Stokes’ cell equation, the region \({\Omega }_{c}\) is discretized using the P2 + P1 mesh elements. The geometry is meshed using the default coarsest mesh setting in COMSOL, which produced approximately 450,000 quadratic Lagrange and 600,000 P2 + P1 mesh elements. Mesh sensitivity analysis for the microvasculature geometry is presented in the Results section.
3 3D microvasculature geometry development
This subsection will discuss the process to develop the 3D microvasculature geometry from capillary distribution data.
3.1 Microvasculature data
The microvasculature geometry is developed from the data obtained from [12]. The data consist of 50 nodes distributed randomly within a cube of 180 μm length. They also have the information about capillary interconnection between points, and values of radius and length. Each capillary vessel is represented by lines connected between two nodes according to the rules developed using the modified spanning tree method [12]. The microvasculature data have been modified to include structural periodicity so that the multiscale homogenization can properly be applied [13]. In total, for a 50-node data, there are seven microvasculature cubes with different capillary distributions that will be used in this study, and these are named as RAN1 to RAN7. Figure 2 shows the seven microvasculatures in 1D line form.
Although the microvasculature geometries shown in Fig. 1 are purely constructed out of lines to solve the microscale cell Eqs. (12)–(19), the capillary must have 3D shapes so that the meshing procedure and finite element analysis can be performed. Therefore, reconstructing the microvasculature from 1D lines into 3D capillary shape must be performed. The capillary in 1D line form is thus rebuilt into 3D cylindrical shape as shown in Fig. 3a. For each capillary, the radius of the cylinder follows the radius provided by the morphometric data.
Fig. 3
a 3D microvasculature geometry, b Part of cylinders outside the cube are shown in red color
However, the 3D microvasculature geometry developed has several problems. Firstly, during the conversion from 1D line to 3D cylindrical structures, there are some parts of the cylinders that will be out of the cube boundary of the tissue as shown in Fig. 3b. Secondly, the out of bound cylinders cause the microvasculature to become non-periodic, thus prevent the solving of the cell equations.
These problems are solved by adding small cylinders at the end of each cylinder located at the cube surfaces. These small cylinders are placed perpendicular to the cube surfaces to maintain the periodicity, as shown in Fig. 4. The perpendicular distance of the cylinder section that is out of the cube boundary, \({D}_{cyl}\) can be calculated from the radius of the cylinder, \({r}_{cyl}\) and the angle measured from the cylindrical cross-section to the normal of the cube boundary, \(\theta\) as:
Fig. 4
Determination of the length of the additional cylinders
It is expected that there are many cylinders that are out of the cube boundary, which depends on the number of nodes on the cube surfaces. In addition, adding the perpendicular cylinders will also increase the microstructure volume. Therefore, to minimize the volume increment of the microstructure, the length of the additional small cylinders perpendicular to the cube surface, \({L}_{add,cyl}\) is determined as:
where it can be calculated by finding the maximum \({D}_{cyl}\) from all the cylinders near the cube surfaces. This is also to ensure that all the cylinders that are out of the cube boundary will be within the new microstructure volume.
Figure 5 shows the modified microvasculature geometry with the additional perpendicular cylinders are marked by red, green, and black circles. Each surface of the cube is symmetric to its opposite surface, maintaining the microstructural periodicity. The detail of the volume increment for the 7 microvasculature geometries will be presented in the Results section.
Fig. 5
Modified microvasculature geometry in 3D cylindrical form for RAN0. The additional cylinders are placed perpendicular to the surfaces of the cube as shown by the red, green, and black circles
The microvasculature geometry validation is performed by comparing the effective blood permeability tensor obtained via the Stokes’ cell Eqs. (15) with the results from the method proposed by [13], in which all of these are denoted here as \({{\varvec{K}}}^{Stokes}\) and \({{\varvec{K}}}^{Poiseuille}\), respectively. It is expected that the effective blood permeability tensor obtained from these two methods are of similar order.
Briefly, the method proposed by [13] uses Poiseuille’s equation and average Darcy’s law. Firstly, the volumetric flow through each capillary \(m\) in the \(j\)-th Cartesian direction within the cube, \({q}_{m}^{j}\) is determined using Poiseuille’s equation:
Here, the terms \({r}_{m}\) and \({L}_{m}\) are the radius and length of the capillary \(m\), \({\mu }_{m}\) is the blood viscosity, and \(\Delta {P}_{m}^{j}\) is the local pressure difference across the capillary length in the \(j\)-direction.
Then, the elements of the Poiseuille’s permeability tensor, \({{\varvec{K}}}^{Poiseuille}\) can be computed by summing the directional fluxes over all capillaries crossing a voxel surface using:
where \({\Gamma }^{j}\) is the outflow surface area normal to the \(j\)-direction and \(\nabla {p}_{i}\) is the macroscale pressure gradient in the \(i\)-direction. To determine the tensor in complete form, the calculation must be done using three independent pressure gradients.
4 Results
This section will describe the microvasculature geometry obtained, the validation by comparing the effective blood permeability, and the calculations of other effective parametric tensors.
4.1 3D microvasculature geometry and mesh sensitivity analysis
Figure 6 shows the 3D microvasculature geometries developed from the seven different capillary distribution data. Each geometry has a different distribution of capillaries. The original cube volume before the geometry modification is 5.83 \(\times {10}^{-12}\) m3. The modification of the geometry resulted in an average cube volume of (6.71 ± 0.02) \(\times {10}^{-12}\) m3, which is about 15% increment from original cube volume. The increase in volume will significantly affect the effective parametric tensors calculated using the modified geometry.
Fig. 6
Seven different capillary models with different capillary distributions
These geometries are then meshed to solve the microscale cell Eqs. (12)–(19). Mesh analysis is performed to investigate the effects of mesh size variation on the simulation outcome. Laplace’s cell equation is used to perform the mesh analysis. This is because it is faster, and it requires less computational resources than the Stokes’ cell equations. In addition, the same mesh type is also used for the one-elastic and six-elastic cell equations. The mesh is varied by changing the number of mesh elements from coarsest (around 450,000 mesh elements) to finest (around 6,000,000 mesh elements) and the variable \({\varvec{G}}\) is calculated. Figure 7a shows the variation of the first element of \({\varvec{G}}\) against the number of mesh elements. The percentage difference between \({\varvec{G}}\) calculated using normal mesh and finest mesh is about 0.56%.
Fig. 7
a Variation in G11 against number of elements (in log10 scale); b RAM against number of elements (in log10 scale)
In addition, Fig. 7b shows the amount of random-access memory (RAM) needed to solve the cell equation using the different number of meshes. Solving the equation with a coarse mesh requires about 20 times a lower amount of RAM than solving with finer mesh. Therefore, the coarse mesh is used throughout the rest of this work so that simulations can be performed with minimum computational resources.
4.2 Effective blood permeability tensor and microvasculature geometry validation
Table 3 shows the comparison between the effective blood permeability tensor \({{\varvec{K}}}^{Poiseuille}\) and \({{\varvec{K}}}^{Stokes}\). The diagonal terms are in similar order of magnitude, but with the percentage difference in mean of \({{\varvec{K}}}^{Stokes}\) are less than \({{\varvec{K}}}^{Poiseuille}\) by about 71 to 73%. Meanwhile, the off-diagonal terms are on average one order of magnitude smaller than the diagonal terms. The differences between \({{\varvec{K}}}^{Stokes}\) and \({{\varvec{K}}}^{Poiseuille}\) are due to the extra volume added after the microvasculature geometry modification. The tensors are calculated by dividing with the volume of the microvasculature geometry, thus resulting in the \({{\varvec{K}}}^{Stokes}\) to have smaller values due to the geometry used being bigger than the original geometry.
Table 3
Mean and standard deviation for \({{\varvec{K}}}^{Poiseuille}\) and \({{\varvec{K}}}^{Stokes}\)
\(K\) components
\({K}^{Poiseuille}\)
(\(\times {10}^{-13}\) m3 s kg−1)
\({K}^{Stokes}\)
(\(\times {10}^{-13}\) m3 s kg−1)
Percentage difference in mean \(\frac{{K}^{Poiseuille}-{K}^{Stokes}}{{K}^{Poiseuille}}\) (%)
\({K}_{11}\)
\(5.76\pm 2.18\)
\(1.63\pm 0.36\)
\(71.63\)
\({K}_{12}\)
\(0.06\pm 0.80\)
\(0.05\pm 0.22\)
\(16.67\)
\({K}_{13}\)
\(0.49\pm 0.54\)
\(0.12\pm 0.15\)
\(75.51\)
\({K}_{21}\)
\(0.21\pm 0.60\)
\(0.04\pm 0.22\)
\(80.95\)
\({K}_{22}\)
\(5.92\pm 1.81\)
\(1.55\pm 0.29\)
\(73.83\)
\({K}_{23}\)
\(-0.3\pm 0.77\)
\(-0.10\pm 0.16\)
\(66.67\)
\({K}_{31}\)
\(0.40\pm 0.29\)
\(0.12\pm 0.15\)
\(70.00\)
\({K}_{32}\)
\(-0.23\pm 0.86\)
\(-0.10\pm 0.16\)
\(56.52\)
\({K}_{33}\)
\(5.04\pm 1.24\)
\(1.44\pm 0.29\)
\(71.34\)
4.3 Estimation of effective interstitial fluid permeability, biot’s coefficient, and elasticity tensors
It is expected that the other effective parametric tensors \({\varvec{G}}\), \({\varvec{Q}}\), and \({\varvec{L}}\) calculated using Eqs. (12), (16), and (18) will also result in smaller values because of the microvasculature volume increment. To estimate the actual values, these equations are then adjusted to include the average percentage difference, \({\Delta }_{avg}\) calculated by taking the average of the percentage difference in mean of the diagonal terms of \({\varvec{K}}\) from Table 3, as follows:
This is expected to be accurate to leading order, although more detailed analysis will be required to test the accuracy of this approximation.
The Laplace’s cell Eqs. (12) and (13) are solved to obtain the effective interstitial fluid permeability tensor, \({\varvec{G}}\). Table 4 shows the tensor \({\varvec{G}}\) obtained for microvasculature geometry developed. Here, the off-diagonal terms of \({\varvec{G}}\) have about two orders of magnitude lower than the diagonal terms and thus, can be approximated to zero. In addition, the \({{\varvec{G}}}^{estimate}\) calculated using Eq. (24) is 0.61% smaller than \({\varvec{G}}\) for all the diagonal terms. This estimation shows that the change in volume of the microvasculature does not affect the term effective interstitial fluid permeability, in which \({\Delta }_{avg}\) is considered for the calculation of the second right-hand side term of Eq. (24), with the first term being dominant.
Table 4
The effective parametric tensor \({\varvec{G}}\) and \({{\varvec{G}}}^{estimate}\)
\(G\) components
\(G\)
\({G}^{estimate}\)
Percentage difference in mean \(\frac{G-{G}^{estimate}}{G}\)(%)
\({G}_{11}\)
\(0.978\pm 0.004\)
\(0.972\pm 0.003\)
\(0.61\)
\({G}_{22}\)
\(0.981\pm 0.002\)
\(0.975\pm 0.003\)
\(0.61\)
\({G}_{33}\)
\(0.979\pm 0.002\)
\(0.973\pm 0.003\)
\(0.61\)
On the other hand, the effective Biot’s coefficient tensors \({\alpha }_{t}\) and \({\alpha }_{c}\) can be calculated using Eqs. (6) and (7). Tables 5 and 6 show the tensors calculated and estimated considering the adjustment from Eq. (25) for \({\alpha }_{t}\) and \({\alpha }_{c}\), respectively. The tensors obtained have off-diagonal terms with two orders of magnitude lower than the diagonal terms, thus can be approximated to zero. From Table 5, the average percentage difference between \({\alpha }_{t}\) and \({\alpha }_{t}^{estimate}\) is 2.95% for all the diagonal terms with \({\alpha }_{t}>{\alpha }_{t}^{estimate}\). Meanwhile, the average percentage difference between \({\alpha }_{c}\) and \({\alpha }_{c}^{estimate}\) is 55% with \({\alpha }_{c}<{\alpha }_{c}^{estimate}\), as shown in Table 6.
Table 5
The effective parametric tensors \({\alpha }_{t}\) and \({{\alpha }_{t}}^{estimate}\)
The effective parametric tensors \({\alpha }_{c}\) and \({\alpha }_{c}^{estimate}\)
\({\alpha }_{c}\) components
\({\alpha }_{c}\)
\({\alpha }_{c}^{estimate}\)
Percentage difference in mean \(\frac{{\alpha }_{t}-{\alpha }_{t}^{estimate}}{{\alpha }_{t}}\) (%)
\({{\alpha }_{c}}_{11}\)
\(0.052\pm 0.005\)
\(0.080\pm 0.008\)
\(54.0\)
\({{\alpha }_{c}}_{22}\)
\(0.050\pm 0.005\)
\(0.078\pm 0.009\)
\(56.0\)
\({{\alpha }_{c}}_{33}\)
\(0.051\pm 0.005\)
\(0.079\pm 0.008\)
\(55.0\)
Meanwhile, the effective elasticity tensor \(\overline{C }\), which is defined in terms of the effective Young’s modulus \(\overline{E }\) and Poisson’s ratio \(\stackrel{-}{v,}\) can be calculated from Eq. (5). Tables 7 and 8 show the results for \(\overline{v }\) and \(\overline{E }\), respectively. Equation (26) is used to estimate the value of the effective Young’s modulus \({\overline{E} }^{estimate}\) and Poisson’s ratio \({\overline{v} }^{estimate}\). The percentage differences between \(\overline{v }\) and \({\overline{v} }^{estimate}\) from \(v\), and between \(\overline{E }\) and \({\overline{E} }^{estimate}\) from \(E\) are small.
Table 7
The mean effective Poisson’s ratios \(\overline{v }\) and \({\overline{v} }^{estimate}\)
Overall, changing the microvasculature geometry into 3D shapes will increase its volume. The modified microvasculature geometry will only significantly affect the effective blood permeability and Biot’s coefficient for capillary but will not affect other effective parameters.
4.4 Benchmark problem for macroscale governing equations
The homogenized macroscale governing Eqs. (4) to (12) are solved on an idealized brain geometry proposed in a previous work for brain tissue edema post-ischemia [20]. The equations are in non-dimensional form. Therefore, the equations are dimensionalized to ensure clarity in explaining the results. This is done by multiplying the terms \({\varvec{K}}\) and \({\varvec{G}}\) with \(\frac{{d}^{2}}{\mu }\) and \(\frac{k}{\mu }\), respectively, following the derivation in [26]. In addition, in the brain geometry, a small spherical infarct of 14 mm radius is introduced to model brain edema. In other non-infarct region, the term \({\overline{L} }_{p}=0\), indicating that the BBB breakdown does not occur. The boundary conditions are defined at the lateral ventricle and the outer brain layers. The tissue displacement is fixed at zero displacement at the boundaries to indicate that the layers will not move during the swelling process. Meanwhile, the interstitial pressure and blood pressure are set at the baseline values of \({\overline{P} }_{t}\) and \({\overline{P} }_{c}\), respectively, at the boundaries. Figure 8 shows the idealized brain geometry with a small infarct. The simulations are performed using COMSOL Multiphysics 5.3a. The geometry is meshed using 10-node tetrahedral elements, with a total of 60,000 elements. Equation (5) is solved using the solid mechanics solver, while Eq. (11) and (12) are solved using the general form PDE solver. All equations are solved simultaneously using fully coupled solver.
Fig. 8
Ideal brain geometry for the benchmark macroscale simulation
Figure 9 shows the distribution of the tissue displacement \(u\), interstitial fluid pressure \({P}_{t}\), and blood pressure \({P}_{c}\) using the effective parameters calculated. From the results, the maximum \({P}_{t}\) occurs at the center of the infarct before slowly spreading toward the direction of the infarct radius. Meanwhile, the opposite trend is shown by \({P}_{c}\), where the minimum occurs at the center of the infarct before slowly increasing toward the direction of the infarct radius. On the other hand, \(u\) is higher around the outermost diameter of the infarct and then slowly spreading to the outside of the infarct. The trends for \(u\) and \({P}_{t}\) are comparable with the studies performed by [18, 20, 29, 30]. The maximum \(u\) and \({P}_{t}\) are 12.53 mm and 2293.30 Pa, while the minimum \({P}_{c}\) is 2061.20 Pa.
Fig. 9
Distribution of \(u\), \({P}_{t}\), and \({P}_{c}\) in the idealized brain obtained from solving the macroscale governing equations
Microvascular dysfunction during post-ischemia and BBB disruption are associated with cerebral edema formation and hemorrhagic transformation in patients [31]. The severity of a BBB disruption in ischemia can be characterized by an increase of microvascular permeability, which subsequently may result in the worsening of brain edema [32]. Therefore, the relationship between the microvasculature and the brain mechanics during edema formation has been investigated. It has been shown in various mathematical modeling studies that vascular volume fraction plays a crucial role in the mechanics of porous tissues [33, 34]. The severity of BBB disruption could also be investigated by varying the vascular hydraulic permeability [18, 33].
In this work, the microvasculature based on statistically accurate cerebral capillary distribution has been developed using 3D cylinders to represent the capillary. This is to enable for solving microscale cell equations that require the use of finite element method such as the one-elastic and six-elastic cell equations. The microvasculature is validated by comparing the effective blood permeability tensors obtained through Poiseuille’s flow of Eq. (23) [13], \({{\varvec{K}}}^{Poiseuille}\) and Stokes’ cell Eq. (15) \({{\varvec{K}}}^{Stokes}\). However, it was found that, in terms of the diagonal elements, \({{\varvec{K}}}^{Stokes}<{{\varvec{K}}}^{Poiseuille}\) by about 70%. This is due to the increase in microvasculature volume when using 3D cylinders. Although the value of \({{\varvec{K}}}^{Stokes}\) is smaller than \({{\varvec{K}}}^{Poiseuille}\), the order of magnitude remains the same, which is \({10}^{-13}\). Furthermore, it should be noted that the off-diagonal terms of \({{\varvec{K}}}^{Stokes}\) obtained are on average one order of magnitude smaller than the diagonal terms. Increasing the number of nodes of the microvasculature geometry will further reduce the order of magnitude of the off-diagonal terms, as suggested in [13]. However, our study only limits to microvasculature with 50 nodes due to minimizing the computational resources.
The validation is then used to estimate the values for other effective parameters such as the interstitial fluid permeability tensor, Biot’s coefficient tensors, Poisson’s ratio, and Young’s modulus. All these parameters can be calculated by solving the microscale cell Eqs. (13), (17), and (19) using finite element method, which require the microvasculature to be meshed. These microscale cell equations have been solved on a different microstructure, as found in the works by [16, 26, 35, 36]. Increasing the microvasculature volume does introduce a small difference in the tensor \({\varvec{G}}\). This is consistent with the finding of [22], in which varying the capillary radius or reducing the volume of the interstitial space does not produce a significant changes in the value of the effective parametric tensor \({\varvec{G}}\). On the other hand, changing the microvasculature volume does substantially affect the effective Biot’s coefficient for capillary, but does not significantly affect the Biot’s coefficient for interstitial fluid. In the previous work by [22], changing the capillary radius causes a more significant decrement of \({\alpha }_{c}\) as compared to the increment of \({\alpha }_{t}\). Lastly, it can be concluded that the increasing in the microvasculature volume does not influence the elasticity of the brain tissue. This is also consistent with the findings from [22], which show that changing the microvasculature volume does not significantly affect the effective elasticity tensor.
In the work by [37], the Green’s function has been utilized to estimate the interstitial fluid permeability tensor, \({\varvec{G}}\) on a 1D microvasculature without requiring finite element mesh. Meanwhile, meshless methods such as peridynamics [38], radial point interpolation [39], and solution structure method [40] have been used to solve the elastic equations on a complex microstructure to reduce the computational loads. These methods also have the potential to be explored to nullify the effects of microvasculature volume increment on the estimation of the effective parameters. Another limitation from using the microvasculature conversion from 1D line to 3D cylindrical shapes is the partial overlapping between two capillaries at a common node. This limitation is due to the design complexity, which poses challenges such as potential geometrical errors and increases the mesh generation time [41]. The work done by [42] attempted to produce similar microvasculature geometry using AutoCAD and AutoLISP to reduce the impact of microvasculature volume increment and partial overlapping. However, due to the design complexity, the volume increment is inevitable, but the overlapping has been reduced. The results obtained in [42] for Stokes’ cell equation are comparable with the results presented in this work. The method employed in this work also has been shown to require only a relatively coarse mesh to solve the Laplace cell equation, achieving similar results when compared to a finer mesh size for microvasculature with 50-node. However, in a future work, a microvasculature that is built using more than 50 nodes will be required to improve the accuracy of the effective parameter calculations, in which the microscale cell equations can be solved using a meshless method to reduce the computational cost.
The current work did not consider the osmotic pressure effect from the molecular and ionic concentrations between the interstitial and blood compartments, as has been done in [18]. Further improvement of the current double Darcy’s and linear elastic equations can be made by including the effect of protein and ionic concentrations from blood plasma that may cause brain edema. In addition, the blood viscosity was assumed constant throughout the modeling process. In fact, blood viscosity depends on several factors, including the hematocrit concentration [43], temperature [44], and shear rate [45], especially for blood flow in small vessels such as the capillary. Including hematocrit into the model has the potential to be used to predict tissue oxygenation surrounding the capillary [46]. The work done by [16] has considered several chemical concentrations including oxygen and glucose and their interactions that may cause fluid pressure imbalance between the interstitial and blood. Meanwhile, the homogenization of heat transfer in complex porous media has been developed by [47] could be used to include the temperature-dependent viscosity into the current model.
The current work can be applied to estimate various poroelastic model parameters of the brain, especially in diseased states. For example, brain stroke will reduce blood permeability in the tissue [48]. However, the effect of stroke for other effective parameters is still not fully understood, and thus, the current work allows for the estimation of these parameters. It has been suggested that the poroelastic parameters such as Biot’s coefficients and Biot’s modulus may play roles in the compressibility of tumor tissue [49]. Besides, the findings of this study can also be used understand drug deliveries in brain diseases [50]. Cell adhesion molecules are activated that may be a potential therapeutic target for stroke treatment [51]. Poroelastic model parameters may be affected by these molecules, as have been investigated in a recent multiscale modeling study [52].
A benchmark problem has also been described, where the macroscale governing equations are solved on an idealized brain geometry. The brain tissue displacement \(u\) obtained is more than 12 mm, which shows a significant shift of brain tissue [53]. The potential reason of \(u\) being very large is because of the boundary conditions posed on the lateral ventricle. In the work by [20], the lateral ventricle is allowed to move during the brain edema process. This allows for the evaluation of brain tissue herniation, which has been used by clinicians as indicator for brain edema severity [54]. It should be noted that the homogenization procedure used in this study requires the assumption of periodic microvasculature. In addition, there is a distinction between the capillary density in the gray matter compared to the white matter [55], which imposes a limitation for the current procedure. However, the brain model used here assumed that there is no distinction between gray and white matter, for the sake of model simplification. A more realistic brain geometry can be used by developing it from a patient-specific brain image obtained from a CT scan or MRI and using capillary density according to the specific regions within the brain.
6 Conclusion
Poroelastic properties of brain tissue with capillary networks have been analyzed. The microvasculature geometry based on the statistical distribution of the cerebral capillary sizes have been modified to ensure that the geometry remains periodic for the use in the asymptotic homogenization procedure. Effective macroscale parametric tensors, namely the blood permeability \({\varvec{K}}\), interstitial fluid permeability \({\varvec{G}}\), Biot’s coefficients \({\alpha }_{c}\) and \({\alpha }_{t}\), Young’s modulus \(E\), and Poisson’s ratio \(v,\) are the poroelastic properties estimated on the microvasculature. Validation was performed by comparing the tensor \({\varvec{K}}\) obtained from solving the Poiseuille’s and Stokes’ equations. The modification of the microvasculature geometry causes an increase in its volume, which resulted in the value of \({\varvec{K}}\) obtained through solving Stokes’ equation to be about 70% less than through solving Poiseuille’s equation. Furthermore, the modification also significantly affects \({\alpha }_{c}\), while do not affect other effective parameters. The effective parameters obtained are then used in a benchmark simulation of the macroscale governing equations for brain edema post-ischemia using an idealized brain geometry.
Acknowledgements
The research is supported by the Fundamental Research Grant Scheme from the Ministry of Higher Education of Malaysia (MOHE Grant Number: FRGS/1/2018/TK03/UMP/02/15). SJP is supported by a Yushan Fellowship from the Ministry of Education, Taiwan (#111V1004-2).
Declarations
Conflict of interest
The authors declare no competing interests.
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Huang, B., Qian, F., Fan, X., Guan, S., Zheng, Y., Yang, J., et al.: Efficacy and safety of intravenous thrombolysis with alteplase for treating acute ischemic stroke at different time windows: a protocol for systematic review and meta-analysis. Medicine (2020). https://doi.org/10.1097/MD.0000000000023620CrossRef
5.
Yang, D.-Y., Pan, H.-C., Chen, C.-J., Cheng, F.-C., Wang, Y.-C.: Effects of tissue plasminogen activator on cerebral microvessels of rats during focal cerebral ischemia and reperfusion. Neurol. Res. 29(3), 274–282 (2007). https://doi.org/10.1179/016164107X159171CrossRef
6.
Karaszewski, B., Houlden, H., Smith, E.E., Markus, H.S., Charidimou, A., Levi, C., et al.: What causes intracerebral bleeding after thrombolysis for acute ischaemic stroke? Recent insights into mechanisms and potential biomarkers. J. Neurol. Neurosurg. Psychiatry 86(10), 1127 (2015). https://doi.org/10.1136/jnnp-2014-309705CrossRef
7.
Cassot, F., Lauwers, F., Fouard, C., Prohaska, S., Lauwers-Cances, V.: A novel three-dimensional computer-assisted method for a quantitative study of microvascular networks of the human cerebral cortex. Microcirculation 13(1), 1–18 (2006). https://doi.org/10.1080/10739680500383407CrossRef
8.
Lauwers, F., Cassot, F., Lauwers-Cances, V., Puwanarajah, P., Duvernoy, H.: Morphometry of the human cerebral cortex microcirculation: general characteristics and space-related profiles. Neuroimage 39(3), 936–948 (2008). https://doi.org/10.1016/j.neuroimage.2007.09.024CrossRef
Reichold, J., Stampanoni, M., Keller, A.L., Buck, A., Jenny, P., Weber, B.: Vascular graph model to simulate the cerebral blood flow in realistic vascular networks. J. Cereb. Blood Flow Metab. 29(8), 1429–1443 (2009). https://doi.org/10.1038/jcbfm.2009.58CrossRef
Park, C.S., Payne, S.J.: A generalized mathematical framework for estimating the residue function for arbitrary vascular networks. Interface Focus 3(2), 20120078 (2013). https://doi.org/10.1098/rsfs.2012.0078CrossRef
16.
Mohamed Mokhtarudin, M.J., Wan Abd. Naim, W.N., Shabudin, A., Payne, S.J.: Multiscale modelling of brain tissue oxygen and glucose dynamics in tortuous capillary during ischaemia-reperfusion. Appl. Math. Model. 109, 358–373 (2022). https://doi.org/10.1016/j.apm.2022.04.001MathSciNetCrossRef
17.
El-Bouri, W.K., MacGowan, A., Józsa, T.I., Gounis, M.J., Payne, S.J.: Modelling the impact of clot fragmentation on the microcirculation after thrombectomy. PLoS Comput. Biol. 17(3), e1008515 (2021). https://doi.org/10.1371/journal.pcbi.1008515CrossRef
Guo, L., Vardakis, J.C., Chou, D., Ventikos, Y.: A multiple-network poroelastic model for biological systems and application to subject-specific modelling of cerebral fluid transport. Int. J. Eng. Sci. 147, 103204 (2020). https://doi.org/10.1016/j.ijengsci.2019.103204MathSciNetCrossRef
20.
Nadzri, A.N., Nik Mohamed, N.A., Payne, S.J., Mohamed Mokhtarudin, M.J.: Poroelastic modelling of brain tissue swelling and decompressive craniectomy treatment in ischaemic stroke. Comput. Methods Biomech. Biomed. Eng. (2024). https://doi.org/10.1080/10255842.2024.2326972CrossRef
Shabudin, A., Mohamed Mokhtarudin, M.J., Payne, S., Naim, W.N.W.A., Mohamed, N.A.N.:. Multiscale Modelling of 3-Dimensional Brain Tissue Using Ideal Capillary Model. Lecture Notes in Mechanical Engineering, pp. 205–221 (2023). https://doi.org/10.1007/978-981-19-4425-3_19.
23.
Shabudin, A., Mohamed Mokhtarudin, M.J., Payne, S., Mohamed, N.A.N.: Application of asymptotic expansion homogenization for vascularized poroelastic brain tissue. IECBES 2018, 413–418 (2018). https://doi.org/10.1109/IECBES.2018.8626727CrossRef
24.
Shabudin, A., Mohamed Mokhtarudin, M.J., El-Bouri, W.K., Payne, S.J., Naim, W.N.W.A., Mohamed, N.A.N.: Multiscale modelling of 3-dimensional brain tissue with capillary distribution. IECBES 2020, 315–318 (2021). https://doi.org/10.1109/IECBES48179.2021.9398791CrossRef
25.
Penta, R., Miller, L., Grillo, A., Ramírez-Torres, A., Mascheroni, P., Rodríguez-Ramos, R.: Porosity and diffusion in biological tissues. recent advances and further perspectives. In: Merodio, J., Ogden, R. (eds.) Constitutive Modelling of Solid Continua, pp. 311–356. Springer, Cham (2020)CrossRef
Hunziker, O., Abdel’al, S., Schulz, U.: The aging human cerebral cortex: a stereological characterization of changes in the capillary net. J. Gerontol. 34(3), 345–350 (1979). https://doi.org/10.1093/geronj/34.3.345CrossRef
29.
Mohamed Mokhtarudin, M.J.: Mathematical modelling of cerebral ischaemia-reperfusion injury. University of Oxford, Thesis (2017)
30.
Mohamed Mokhtarudin, M.J., Payne, S.J.: The study of the function of AQP4 in cerebral ischaemia–reperfusion injury using poroelastic theory. Int. J. Numer. Methods Biomed. Eng. 33(1), e02784 (2017). https://doi.org/10.1002/cnm.2784CrossRef
31.
Ng, F.C., Churilov, L., Yassi, N., Kleinig, T.J., Thijs, V., Wu, T.Y., et al.: Microvascular dysfunction in blood-brain barrier disruption and hypoperfusion within the infarct posttreatment are associated with cerebral edema. Stroke 53(5), 1597–1605 (2022). https://doi.org/10.1161/STROKEAHA.121.036104CrossRef
32.
Qiu, Y.-M., Zhang, C.-L., Chen, A.-Q., Wang, H.-L., Zhou, Y.-F., Li, Y.-N., et al.: Immune cells in the BBB disruption after acute ischemic stroke: targets for immune therapy? Front. Immunol. (2021). https://doi.org/10.3389/fimmu.2021.678744CrossRef
33.
Mascheroni, P., Penta, R., Merodio, J.: The impact of vascular volume fraction and compressibility of the interstitial matrix on vascularised poroelastic tissues. Biomech. Model. Mechanobiol. 22(6), 1901–1917 (2023). https://doi.org/10.1007/s10237-023-01742-1CrossRef
34.
Shabudin, A., Mohamed Mokhtarudin, M.J., Nik Mohamed, N.A., Mohd Romlay, M.A.: Analysis of brain tissue poroelastic properties using multiscale modelling. IIUM Eng. J. 26(1), 437–449 (2025). https://doi.org/10.31436/iiumej.v26i1.3259CrossRef
Miller, L., Penta, R.: Investigating the effects of microstructural changes induced by myocardial infarction on the elastic parameters of the heart. Biomech. Model. Mechanobiol. 22(3), 1019–1033 (2023). https://doi.org/10.1007/s10237-023-01698-2CrossRef
37.
Sweeney, P.W., d’Esposito, A., Walker-Samuel, S., Shipley, R.J.: Modelling the transport of fluid through heterogeneous, whole tumours in silico. PLoS Comput. Biol. 15(6), e1006751 (2019). https://doi.org/10.1371/journal.pcbi.1006751CrossRef
38.
Weckner, O., Brunk, G., Epton, M.A., Silling, S.A., Askari, E.: Green’s functions in non-local three-dimensional linear elasticity. Proc. R. Soc. A Math. Phys. Eng. Sci. 465(2111), 3463–3487 (2009). https://doi.org/10.1098/rspa.2009.0234MathSciNetCrossRef
39.
Ai, L., Gao, X.-L.: Evaluation of effective elastic properties of 3D printable interpenetrating phase composites using the meshfree radial point interpolation method. Mech. Adv. Mater. Struct. 25(15–16), 1241–1251 (2018). https://doi.org/10.1080/15376494.2016.1143990CrossRef
Kumawat, C., Sharma, B.K., Mekheimer, K.S.: Mathematical analysis of two-phase blood flow through a stenosed curved artery with hematocrit and temperature dependent viscosity. Phys. Scr. 96(12), 125277 (2021). https://doi.org/10.1088/1402-4896/ac454aCrossRef
45.
Trejo-Soto, C., Hernández-Machado, A.: Normalization of blood viscosity according to the hematocrit and the shear rate. Micromachines (2022). https://doi.org/10.3390/mi13030357CrossRef
Eden, M., Freudenberg, T.: Effective heat transfer between a porous medium and a fluid layer: homogenization and simulation. Multiscale Model. Simul. 22(2), 752–783 (2024). https://doi.org/10.1137/22M1541794MathSciNetCrossRef
Dehghani, H., Penta, R., Merodio, J.: The role of porosity and solid matrix compressibility on the mechanical behavior of poroelastic tissues. Mater. Res. Express 6(3), 035404 (2019). https://doi.org/10.1088/2053-1591/aaf5b9CrossRef
Dehghani, H., Holzapfel, G.A., Mittelbronn, M., Zilian, A.: Cell adhesion affects the properties of interstitial fluid flow: a study using multiscale poroelastic composite modeling. J. Mech. Behav. Biomed. Mater. 153, 106486 (2024). https://doi.org/10.1016/j.jmbbm.2024.106486CrossRef
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