A Microstructure-Based Mechanistic Approach to Detect Degeneration Effects on Potential Damage Zones and Morphology of Young and Old Human Intervertebral Discs

Abstract

There is an increasing demand to develop predictive medicine through the creation of predictive models and digital twins of the different body organs. To obtain accurate predictions, real local microstructure, morphology changes and their accompanying physiological degenerative effects must be taken into account. In this article, we present a numerical model to estimate the long-term aging effect on the human intervertebral disc response by means of a microstructure-based mechanistic approach. It allows to monitor in-silico the variations in disc geometry and local mechanical fields induced by age-dependent long-term microstructure changes. Both lamellar and interlamellar zones of the disc annulus fibrosus are constitutively represented by considering the main underlying microstructure features in terms of proteoglycans network viscoelasticity, collagen network elasticity (along with content and orientation) and chemical-induced fluid transfer. With age, a noticeable increase in shear strain is especially observed in the posterior and lateral posterior regions of the annulus which is in correlation with the high vulnerability of elderly people to back problems and posterior disc hernia. Important insights about the relation between age-dependent microstructure features, disc mechanics and disc damage are revealed using the present approach. These numerical observations are hardly obtainable using current experimental technologies which makes our numerical tool useful for patient-specific long-term predictions.

Appendix A

In the continuum mechanics framework, the deformation gradient tensor ${\mathbf{F}} = {{\partial {\varvec{x}}} \mathord{\left/ {\vphantom {{\partial {\varvec{x}}} {\partial {\varvec{X}}}}} \right. \kern-0pt} {\partial {\varvec{X}}}}$ maps a material point from its initial position ${\varvec{X}}$ to the current position ${\varvec{x}}$. The time derivative is ${\dot{\mathbf{F}}} = {\mathbf{LF}}$ in which ${\mathbf{L}} = {{\partial {\varvec{v}}} \mathord{\left/ {\vphantom {{\partial {\varvec{v}}} {\partial {\varvec{x}}}}} \right. \kern-0pt} {\partial {\varvec{x}}}}$ is the spatial velocity gradient with ${\varvec{v}} = {{\partial {\varvec{x}}} \mathord{\left/ {\vphantom {{\partial {\varvec{x}}} {\partial t}}} \right. \kern-0pt} {\partial t}}$. Using the conceptual sequence of configurations, the deformation gradient tensor ${\mathbf{F}}$ can be decomposed via multiplicative forms:

$${\mathbf{F}} = {\mathbf{F}}_{{{\text{mech}}}} {\mathbf{F}}_{{{\text{chem}}}} \;{\text{and}}\;{\mathbf{F}}_{{{\text{mech}}}} = {\mathbf{F}}_{{{\text{mech}}}}^{e} {\mathbf{F}}_{{{\text{mech}}}}^{v} ,$$

(A1)

where ${\mathbf{F}}_{{{\text{mech}}}}$ is the mechanical part related to the stress-producing contribution, in turn multiplicatively decomposed into an elastic part ${\mathbf{F}}_{{{\text{mech}}}}^{e}$ and a viscous part ${\mathbf{F}}_{{{\text{mech}}}}^{v}$, and ${\mathbf{F}}_{{{\text{chem}}}} = J^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}} {\mathbf{I}}$ is the chemical-induced part related to the volumetric expansion in which the term ${\mathbf{I}}$ is the identity tensor and $J = \det {\mathbf{F}}_{{{\text{chem}}}} > 0$ is the Jacobian.

The spatial velocity gradient ${\mathbf{L}}$ is described by:

$${\mathbf{L}} = \underbrace {{{\dot{\mathbf{F}}}_{{{\text{mech}}}} {\mathbf{F}}_{{{\text{mech}}}}^{ - 1} }}_{{{\mathbf{L}}_{{{\text{mech}}}} }} + \underbrace {{{\mathbf{F}}_{{{\text{mech}}}} {\dot{\mathbf{F}}}_{{{\text{chem}}}} {\mathbf{F}}_{{{\text{chem}}}}^{ - 1} {\mathbf{F}}_{{{\text{mech}}}}^{ - 1} }}_{{{\mathbf{L}}_{{{\text{chem}}}} }}\;{\text{and}}\;{\mathbf{L}}_{{{\text{mech}}}} = \underbrace {{{\dot{\mathbf{F}}}_{{{\text{mech}}}}^{e} {\mathbf{F}}_{{{\text{mech}}}}^{{e^{ - 1} }} }}_{{{\mathbf{L}}_{{{\text{mech}}}}^{e} }} + \underbrace {{{\mathbf{F}}_{{{\text{mech}}}}^{e} {\dot{\mathbf{F}}}_{{{\text{mech}}}}^{v} {\mathbf{F}}_{{{\text{mech}}}}^{{v^{ - 1} }} {\mathbf{F}}_{{{\text{mech}}}}^{{e^{ - 1} }} }}_{{{\mathbf{L}}_{{{\text{mech}}}}^{v} }}$$

(A2)

in which ${\mathbf{L}}_{{{\text{mech}}}}$ is the stress-producing mechanical part, in turn decomposed into an elastic part ${\mathbf{L}}_{{{\text{mech}}}}^{e}$ and a viscous part ${\mathbf{L}}_{{{\text{mech}}}}^{v}$, and ${\mathbf{L}}_{{{\text{chem}}}}$ is the stress-free chemical-induced volumetric part.

The viscous spatial velocity gradient ${\mathbf{L}}_{{{\text{mech}}}}^{v}$ may be decomposed as the sum of symmetric and skew-symmetric parts:

$${\mathbf{L}}_{{{\text{mech}}}}^{v} = {\mathbf{F}}_{{{\text{mech}}}}^{e} {\dot{\mathbf{F}}}_{{{\text{mech}}}}^{v} {\mathbf{F}}_{{{\text{mech}}}}^{{v^{ - 1} }} {\mathbf{F}}_{{{\text{mech}}}}^{{e^{ - 1} }} = {\mathbf{D}}_{{{\text{mech}}}}^{v} + {\mathbf{W}}_{{{\text{mech}}}}^{v}$$

(A3)

in which ${\mathbf{D}}$ is the stretching rate (symmetric part) and ${\mathbf{W}}$ is the spin (skew-symmetric part):

$${\mathbf{D}} = \frac{1}{2}\left( {{\mathbf{L}} + {\mathbf{L}}^{T} } \right)\;{\text{and}}\;{\mathbf{W}} = \frac{1}{2}\left( {{\mathbf{L}} - {\mathbf{L}}^{T} } \right)$$

(A4)

It is assumed that the inelastic flow is irrotational,¹⁹ i.e. ${\mathbf{W}}_{{{\text{mech}}}}^{v} = {\mathbf{0}}$. The viscous deformation gradient ${\mathbf{F}}_{{{\text{mech}}}}^{v}$ is then extracted from:

$${\dot{\mathbf{F}}}_{{{\text{mech}}}}^{v} = {\mathbf{F}}_{{{\text{mech}}}}^{{e^{ - 1} }} {\mathbf{D}}_{{{\text{mech}}}}^{v} {\mathbf{F}}_{{{\text{mech}}}}^{e} {\mathbf{F}}_{{{\text{mech}}}}^{v}$$

(A5)

The elastic component ${\mathbf{F}}_{{{\text{mech}}}}^{e} = {\mathbf{F}}_{{{\text{mech}}}} {\mathbf{F}}_{{{\text{mech}}}}^{{v^{ - 1} }}$ is then computed.

The tensor ${\mathbf{D}}_{{{\text{mech}}}}^{v}$ defining the specificity of the model is defined by the following general flow rule:

$${\mathbf{D}}_{{{\text{mech}}}}^{v} = \dot{\gamma }_{v} {\mathbf{N}}_{v} \;{\text{with}}\;{\mathbf{N}}_{v} = \frac{{{{\varvec{\uptau}}}_{v} }}{{\left\| {{{\varvec{\uptau}}}_{v} } \right\|}}$$

(A6)

in which $\dot{\gamma }_{{\text{v}}}$ is the accumulated viscous strain rate and ${\mathbf{N}}_{{\text{v}}}$ is the direction tensor of viscous flow aligned with the viscous Kirchhoff stress tensor ${{\varvec{\uptau}}}_{{\text{v}}}$.