## Inflation of an Artery with reference to Zero-Stress State

http://www.youtube.com/watch?v=KrCHzj4rOr0

Contents

### Description

- Many older biomechanical studies utilized the no-load state as a reference. However, vascular tissue experiences residual strain in the no-load state. This is evident when a vessel or ring of tissue opens up when radially cut; this new state is called the zero-stress state. This tutorial explores the difference between the no-load state and zero-stress state by modeling and comparing different arteries.

### Create Mesh

Select

**cylindrical polar**in the`Global Coordinates:`pop-up menuClick

**OK**to submit`Coordinate Form`

Mesh→Edit→Material Coordinates...

Use the equation below to write

**dY_dMatl**, the material coordinate transformation from cylindrical polar to rectangular cartesian.- dY_dMatl = Matrix([[cos(X[1]), -sin(X[1]),0],[sin(X[1]),cos(X[1]),0],[0,0,1]])
Click

**OK**to submit the`Material Coordinate`model

Choose

**Lagrange Basis Function→3D→Linear-Linear-Linear**Click

**OK**to submit`Basis Form`

Click

**Insert Node**in the left panel to create a total of 8 nodesIn the

**Value**fields under`Coordinate 1`,`Coordinate 2`, and`Coordinate 3`enter the following (R,Theta,Z) nodal coordinates:- Node 1: (0.99, 0, 0)
- Node 2: (0.99, 0, 1)
- Node 3: (1.16, 0, 0)
- Node 4: (1.16, 0, 1)
- Node 5: (0.99, 59, 0)
- Node 6: (0.99, 59, 1)
- Node 7: (1.16, 59, 0)
- Node 8: (1.16, 59, 1)

Replace all

`default`node labels with the appropriate as shown below.- Node 1: inner_proximal1...
- Node 2: inner_distal1...
- Node 5: inner_proximal2...
- Node 6: inner_distal2...

Enable a

`Field Variable`by selecting the**Field Vector 1**tab and choosing**Linear-Linear-Linear Lagrange 3*3*3**from the`Select Basis Number`menu below`Field variable 1`.

Enter

**6, 2, 5, 1, 8, 4, 7, 3**in the`Global Node Numbers`boxes.Click

**OK**to submit`Element Form`Next,

**Render**the nodes and elements.- The current model should look similar to the one shown below.

### Refine the Mesh

To get a sufficiently converged result using linear elements, it is necessary to use multiple elements.

Decrease the

`New Element per old element in``xi1`and`xi2`to 1. Increase the`New Element per old element in``xi3`to 10Click

**OK**to refine the mesh**Render**the new set of nodes and elements.

### Add Biomechanics Data

A biomechanics problem requires material properties and boundary conditions before it can be solved. The boundary conditions will define how the biological testing is preformed.

##### Incorporate a Published Constitutive Model

Several arteries including the thoracic, abdominal, femoral, carotid, and pulmonary arteries will be modeled (*Am J Physiol Heart Circ Physiol* 282:H622-H629, 2002). An exponential constitutive model template from the continuity database will serve as the basis for the model. However, specific values, constants, and the strain energy function will be used from the literature.

Select model 1097:

**BM_InflateTube_ConstitutiveModel_Basic**In the toolbar above the models select

**Biomechanics→Constitutive Model**Or download the model here: BM_InflateTube_ConstitutiveModel_Basic.con6

Select

**Export**and save the tabled material coordinates model (File→Save...)

Biomechanics→Edit→Constitutive Model...

*Import the constitutive model saved from the Continuity database***Import...**the constitutive model and update- Edit the exponential term of the strain energy function.
**Q = <a1>*E[1,1]*E[1,1] + <a2>*E[2,2]*E[2,2] + <a4>*2.0*E[1,1]*E[2,2]**

Next output variables need to be added. Right-click on the variable stress_out in the left panel of the Edit Equations tab and select

**Insert variable here...**Add any output variables that may be informative using the type

`temporary variable`. Some examples are provided below:**F_out = F#***- Deformation Gradient Tensor wrt material coordinates**C_out = C#***- Right Cauchy-Green Deformation Tensor wrt material coordinates**J_out = J#***- Determinant of Deformation Gradient Tensor**E_out = E#***- Lagrangian Green's Strain Tensor wrt material coordinates**T = (F*stress*F.T)/J#***- Cauchy Stress Tensor**N = stress*F.T#***- Nominal Stress Tensor

**Compile**the modelClick the

**Set parameters**tab to change the`Value`for each coefficient. See literature for specific values.**a1**- x**a2**- y**a4**- z**C**- 8.92 (4.46 -**stress_scaling_coefficient**)**bulk_modulus**- 350

**Compile**the model

#### Define the Boundary Conditions

The boundary conditions for this model will first deform and close the open blood vessel and then apply a pressure on the inner wall causing inflation of the tube. We will need to make sure to account for rigid body motion.

Biomechanics→Update→Initial conditions with undeformed nodes

Biomechanics→Edit→Boundary Conditions...

Click on the

**Deformed Coordinate 2**tabClick the

**Insert Nodes**buttonEnter

**.*proximal1**under`Nodes(s):`and**0.0**under`Value:`Click the

**Insert Nodes**buttonEnter

**.*distal1**under`Nodes(s):`and**0.0**under`Value:`Click the

**Insert Nodes**buttonEnter

**.*proximal2**under`Nodes(s):`and**θ**under`Value:`Click the

**Insert Nodes**buttonEnter

**.*distal2**under`Nodes(s):`and**θ**under`Value:`

Click on the

**Deformed Coordinate 3**tabClick the

**Insert Nodes**buttonEnter

**.*proximal1**under`Nodes(s):`and**0.0**under`Value:`Click the

**Insert Nodes**buttonEnter

**.*distal1**under`Nodes(s):`and**0.0**under`Value:`Click the

**Insert Nodes**buttonEnter

**.*proximal2**under`Nodes(s):`and**0.0**under`Value:`Click the

**Insert Nodes**buttonEnter

**.*distal2**under`Nodes(s):`and**0.0**under`Value:`

Click on the

**External Pressure**tabChange

`Select Pressure Type`to**Incremental**Specify the pressure increment to be

**x**on the**Inner Surface**. See literature for specific values.- The units of the pressure increment depend on your mesh scale and constitutive equation parameters.

- It is a good idea to now go back through the Boundary Conditions Form to double check the parameters you just set up
Click

**OK**

### Solve Biomechanics Problem

Click on

**Apply Marked Recommendations**

Biomechanics→Solve Nonlinear...

For

`Time Step`, enter**0.1**For

`Number of Steps`, enter**10**Click the

**Solve**button, and wait for the solver to finish its jobNote that the time counter (

`Initial time`) updates to 1.0 after the solve. For ever 1.0 increase in the time counter, the boundary conditions are implemented once. For example, the pressure is incrementally increased to 0.4 during this simulation. If a displacement boundary condition of 0.1 had been given and the simulation run to a total time of 3.0 (`Time Step`*`Number of Steps`), a total displacement of 0.3 would be applied.

### Calculate and Render Strains

Click the

**lines**radio buttonThis time select the

**deformed**radio buttonClick

**Render**to display deformed mesh lines- The mesh should look like the screenshot below

In the

`Element List`, enter**1**to render the inner surface of the elementClick the

**surfaces**radio buttonClick the

**deformed**radio buttonClick

**Render**to display inner mesh surfaces (at`Xi(3)`=0.)

Biomechanics→Render→Surface...

Change the pop-up menu choice after

`At Xi`to**2**and change`Location`to**0.5**Select the

**deformed**radio buttonunder

`Variables`select**T - Cauchy Stress Tensor**Click

**OK**to render the fiber (circumferential) strain

- The mesh should now look similar to the screenshot below

Biomechanics→List→Stress and Strain...

**Unselect All Variables**and reselect**X**and**T**In the

**Locations**tab, select**Number of Points**and change`xi1`,`xi2`and`xi3`to 1. This will export the solutions at the center of each element (`xi1`,`xi2`and `xi3' = 0.5) and provide the radial stress distribution. Since this model uses only a small number of linear finite elements, the center of each element has the most accurate solutionClick

**OK**to display a listing of the selected`Output Variables`in the Table Manager- This table can be saved (File→Save...) and graphed using external software

- Biomechanics→List→Nodal Solutions...
- The nodal solutions table contains the final node locations and the residual values
- Since Continuity minimizes these residuals when solving the equilibrium equations, they will be very small. However, if a displacement boundary condition was enforced, the residual will be non-zero. The residual will be the force experienced by the body to achieve that displacement
- In this model, the residuals in the Z direction are non-zero. This is a result of the zero displacement boundary conditions in the Z direction
- The opposite of the residuals could be applied as a force boundary condition to achieve the same displacement