Embedded Finite Elements for Modeling Axonal Injury

Abstract

The purpose of this paper is to propose and develop a large strain embedded finite element formulation that can be used to explicitly model axonal fiber bundle tractography from diffusion tensor imaging of the brain. Once incorporated, the fibers offer the capability to monitor tract-level strains that give insight into the biomechanics of brain injury. We show that one commercial software has a volume and mass redundancy issue when including embedded axonal fiber and that a newly developed algorithm is able to correct this discrepancy. We provide a validation analysis for stress and energy to demonstrate the method.

Appendix

How Does Our Element Level Corrections Compare to the Transverse Isotropic Constitutive Modeling?

In this section, we will look into some of the implementation details of selected areas of this finite element formulation—that are specific to the embedded element technique.

Transformation Matrix

This matrix can be used to transform a tensor from one coordinate system to another coordinate system. Let’s assume that the first coordinate system is represented by the basis $\left( {m_{1} ,m_{2} ,m_{3} } \right)$ and the second coordinate system is represented by the basis $\left( {e_{1} ,e_{2} ,e_{3} } \right)$, then the transformation matrix to transform a tensor from first coordinate system to the second coordinate system can be defined as:

$${\mathbf{T}} = \left[ {\begin{array}{*{20}c} {m_{1} \cdot e_{1} } & {m_{1} \cdot e_{2} } & {m_{1} \cdot e_{3} } \\ {m_{2} \cdot e_{1} } & {m_{2} \cdot e_{2} } & {m_{2} \cdot e_{3} } \\ {m_{3} \cdot e_{1} } & {m_{3} \cdot e_{2} } & {m_{3} \cdot e_{3} } \\ \end{array} } \right]$$

(21)

i.e., if a tensor in the first coordinate system is ${\varvec{\upsigma}}_{m}$ and the same tensor in the second coordinate system is ${\varvec{\upsigma}}_{e}$, then

$${\varvec{\upsigma}}_{c} = {\mathbf{T\sigma }}_{m} {\mathbf{T}}^{T}$$

(22)

This matrix is frequently used in transforming a tensor of interest from fiber to matrix coordinate systems and vice versa.

Adding Embedded Fiber Mass to the Host Element Mass

For implementation purposes, an element level direct lumped mass matrix was used. The below-listed procedure explains the detailed calculation of global mass matrix using the host as well as the embedded element information.

• 1. Calculate the host element mass using the volume and density information

• 2. Calculate the total fiber volume embedded in this particular host element

• 3. Calculate the fiber mass using its volume (Step-2) and using the material model information.

• 4. Calculate the effective element mass by adding the host and embedded element masses.

• 5. Divide this effective element mass among all the host element nodes equally and form an element level diagonal mass matrix.

• 6. Scatter this element level mass matrix into a global mass matrix and repeat this step for all the elements.

Deformation in the Embedded Elements

Since the displacements of the embedded nodes are assumed to be a function of the displacements of the host nodes, it can be shown that the deformation gradient of the embedded elements will be same as that of the host element.

$${\mathbf{F}}_{host} = \frac{{\partial {\text{x}}_{\text{host}} }}{{\partial {\text{X}}_{\text{host}} }} = \frac{{\partial ({\text{X}}_{\text{host}} + {\text{u}}_{\text{host}} )}}{{\partial {\text{X}}_{\text{host}} }} = {\mathbf{I}} + \frac{{\partial {\text{u}}_{\text{host}} }}{{\partial {\text{X}}_{\text{host}} }}$$

(23)

But, since ${\text{u}}_{\text{embed}} = \left[ {\mathbf{N}} \right]{\text{u}}_{{n,{\text{host}}}}$ and ${\mathbf{B}} = \frac{{\partial \left[ {\text{N}} \right]}}{{\partial {\text{X}}_{\text{host}} }}$,

$$\frac{{\partial {\text{u}}_{\text{host}} }}{{\partial {\text{X}}_{\text{host}} }} = {\mathbf{B}}{\text{u}}_{{n,{\text{host}}}} = \frac{{\partial {\text{u}}_{\text{embed}} }}{{\partial {\text{X}}_{\text{host}} }}$$

(24)

Therefore, we can see that the above Eq. (22) can be extended such that

$${\mathbf{F}}_{\text{host}} = {\mathbf{I}} + \frac{{\partial {\text{u}}_{\text{host}} }}{{\partial {\text{X}}_{\text{host}} }} = {\mathbf{I}} + \frac{{\partial {\text{u}}_{\text{embed}} }}{{\partial {\text{X}}_{\text{host}} }} = {\mathbf{F}}_{\text{embed}}$$

(25)

In the above equations, $\varvec{ }{\mathbf{F}}_{\text{host}}$ represents the deformation gradient of the host element,$\varvec{ }{\mathbf{F}}_{{{\mathbf{embed}}}}$ represents the deformation gradient of the embedded element,$\varvec{ }{\text{x}}_{\text{host}}$ represents the current configuration of the host element, $\varvec{ }{\text{X}}_{{{\mathbf{host}}}}$ represents the reference configuration of the host element, ${\text{u}}_{\text{host}}$ represents the continuum displacement of the host domain,$\varvec{ }{\text{u}}_{{n,{\text{host}}}}$ represents the nodal displacements of the host domain, $\varvec{ }{\text{u}}_{\text{embed}} \varvec{ }$ represents the nodal displacement vector of the embedded elements, ${\mathbf{B}}$ represents the strain displacement matrix and $\varvec{ }\left[ {\mathbf{N}} \right]$ represents the shape function matrix of the host element.

Adding the Nodal Force Contribution from the Embedded Fibers Back to That of the Host Elements

Adding the nodal force vector of the truss element to the host–parent element is not straight forward and involves a complicated procedure. The overall methodology adopted here can be classified into multiple steps:

• 1. Calculating integration points

• 2. Calculating the internal nodal force vector.

Calculating Integration Points

This step involves a calculating the integration points and moving between coordinate systems. Note that this step in explained in the context of truss elements embedded in hexahedral host elements. Here, we have three coordinate systems. First one is the global coordinate system (coordinate system in which the host and embedded meshes are defined). The second one is the iso-parametric coordinate system of the host element. And the third one is the iso-parametric coordinate system of the embedded fiber. We will designate these systems as G (global system), LH (iso-parametric coordinate system of the host element) and LE (iso-parametric coordinate system of the embedded fiber), respectively for our future discussion. Since we will be integrating over the fiber volume to calculate the internal nodal force vector contribution from the embedded fibers and adding it to the nodal force vector from the host element, we need the integration points of the embedded fibers calculated in the LH coordinate system (iso-parametric coordinate system of the host element). This can be done using an iterative solver. In our library, we have currently implemented the Newton–Raphson solver for this purpose. More information on the mapping of integration points or the Newton–Raphson solving can be obtained from the dissertation work by Hartl.18

Calculating the Internal Nodal Force Vector of the Embedded Fiber

Once the fiber integration points are calculated in the host element coordinate system, the nodal force vector can be calculated by integrating over the fiber volume using the following equation:

$${\mathbf{f}}_{{e,{\text{fiber}}}}^{\text{int}} = \mathop \smallint \limits_{\text{fiber}} {\mathbf{B}}^{T} {\varvec{\upsigma}}d\varOmega_{f} = \mathop \sum \limits_{{n_{\xi } }} {\mathbf{B}}^{T} {\varvec{\upsigma}} *wts\left( \xi \right)*{\text{jacobian}}_{\text{fiber}}$$

(26)

In the above equation, $n_{\xi }$ represents the number of Gauss quadrature points, and $wts\left( \xi \right)$ represents the Gaussian weights of the particular quadrature point.

Calculating the Stress Along the Fiber

As seen in the above sub-section, both the host and embedded elements will have the same deformation gradient. Therefore, the Cauchy stress tensor calculated using the deformation gradient and the embedded fiber material properties will be transformed into the fiber coordinate system (a unit vector along the fiber axis forms the basis of this coordinate system), and the corresponding fiber stress will be calculated.

$${\varvec{\upsigma}}_{fiber} = f\left( {{\mathbf{F}}, {\text{Fiber material model}}} \right)$$

(27)

where F is the deformation gradient and ${\varvec{\upsigma}}_{\text{fiber}}$ is the fiber Cauchy stress tensor calculated in the host element coordinate system.

$${\varvec{\upsigma}}_{{{\text{fiber}} - {\text{axis}}}} = {\mathbf{T}} \cdot {\varvec{\upsigma}}_{\text{fiber}} \cdot {\mathbf{T}}^{T}$$

(28)

Calculating the Strain Along the Fiber

The deformation gradient tensor can be used to calculate the strain projected along the fiber using the following equations:

$${\mathbf{C}} = {\mathbf{F}}^{T} {\mathbf{F}}$$

(29)

$$\lambda_{\text{fiber}} = \sqrt {a_{0} \cdot {\mathbf{C}}a_{0} }$$

(30)

$$\varepsilon_{\text{fiber}} = log\left( {\lambda_{\text{fiber}} } \right)$$

(31)

where C is the right Cauchy-Green deformation tensor, $\lambda_{\text{fiber}}$ is the stretch along the fiber, a₀ is the unit vector along the fiber direction and $\varepsilon_{\text{fiber}}$ is the strain along the fiber.

EEMA Energy Plot

Since embedded element technique has been formulated using the finite element method, it would be essential for the energy to be conserved. Appendix Figure A1 shows a system and its corresponding energy plot. As expected, the total energy of the system remains zero throughout the span of the simulation.

Figure A1

Energy plot for a cube with one fiber (addressing volume redundancy).