Multiscale modeling analysis of poroelastic properties of brain tissue with capillary networks

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 $${\varvec{K}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> , interstitial fluid flow permeability $${\varvec{G}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> , Biot’s coefficients for blood $${\alpha }_{c}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> and interstitial fluid $${\alpha }_{t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> , Young’s modulus $$\overline{E }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>E</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , and Poisson’s ratio $$\overline{v }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>v</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> . 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 $${\varvec{K}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> obtained using Poiseuille’s and Stokes’ equations, which resulted in the value of $${\varvec{K}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> 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 $${\alpha }_{c}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> 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.