COMSOL 6.4 - Biodegradation of a Magnesium Stent

Biodegradable metallic biomaterials such as Magnesium (Mg) are gathering attention for biomedical applications due to their favorable properties. While degrading, magnesium parts undergo geometrical changes, and the dissolution of metal ions may give rise to harmful changes in pH in human tissue. The present model simulates dissolution of a Mg stent placed in a blood vessel.

In the first study of the model, the Level Set interface is used to model the geometrical changes of the stent when subject to a local constant dissolution rate over time.

The resulting stent shapes at various stages of dissolution are then used in a second study in order to calculate the concentration of Mg ions in the tissue and vessel surrounding the stent, and the corresponding pH levels.

Model Definition

Figure 1 shows the full geometry of a biomedical stent. The highlighted region shows the circumferential symmetry in the stent geometry.

Figure 1: Full Mg stent geometry along with minimal symmetry region highlighted.

Due to this symmetry, only one-twelfth of the full stent geometry is considered in the model. The reduced geometry of the Mg stent placed in the tissue and blood vessel is shown in Figure 2.

Figure 2: Reduced Mg stent geometry considered in the model due to symmetry.

Dissolution interface tracking

The Level Set interface is used to keep track of the Mg stent dissolution due to uniform corrosion. The Level Set interface automatically sets up the equations for the movement of the interface between the Mg stent and the tissue. The level set variable varies from 1 in the tissue and blood vessel domains to 0 in the Mg stent domain. The interface is represented by the 0.5 contour of the level set variable ϕ. The transport of the level set variable is given by:

(1)

The ε parameter determines the thickness of the interface and is defined as ε = hstent, where hstent is the mesh element size at the stent surface. The γ parameter determines the amount of reinitialization. A suitable value for γ is the maximum velocity magnitude occurring in the model.

The level-set variable is advected by a constant normal velocity, vn, which is set to a value of 2 mm/y (Ref. 2).

The level set delta function is approximated by

(2)

Aqueous Electrolyte Transport Interface

The interface is used to define material balances accounting for the mass transport of species, i, and various sources in the tissue and blood vessel domains:

(3)

In the equation, εl is the plasma volume fraction, Ji is the diffusion and migration flux, and S any type of source (mol/(m3·s)). The velocity field, u, is set using an analytical expression to mimic Hagen–Poiseuille flow in the blood vessel, and it is set to 0 in the stent and tissue domains.

Carbonic acid chemical equilibria are considered according to Table 1.

Table 1: Carbonic acid chemical equilibria
Reactions	Equilibrium constant, K
	715
	0.32 mol/m3
	5.01·10-8 mol/m3

Phosphoric acid chemical equilibria are considered according to Table 2.

Table 2: phosphoric acid chemical equilibria
Reactions	Equilibrium constant, K
	7.5 mol/m3
	6.2·10-5 mol/m3

Two nodes are used to set the chemical equilibria of carbonic acid and phosphoric acid, respectively. The first two equilibrium reactions in carbonic acid chemical equilibria (Table 1) are clubbed together in the first dissociation step used in the node for carbonic acid. Values of the equilibrium constants are taken from Ref. 2 and Ref. 3. All species involved in chemical equilibria of carbonic acid and phosphoric acid are set as immobile species. The initial concentration of

and

ions are set to 25 mol/m3 and 1 mol/m3, respectively.

Since the net current on the Mg stent surface is zero, the induced electric field is assumed to be negligible. Hence, the electrolyte potential is not solved in this model. This also leads to the assumption of constant total concentrations of the CO2 and H3PO4 buffer solutions in the tissue and blood vessel domains. Since the current densities are low and the transport is considered to be diffusion-convection dominated, only transport of Mg ions is solved for. The change in pH in the tissue and blood vessel is attributed to the change in Mg ion concentration, in combination with the equilibrium speciation of the two weak acids.

The transport of Mg ions is solved using the node. The source term, SMg, representing the dissolution of ions on the stent surface, is set in terms of the dissolution velocity, vn, and the level set delta function according to

(4)

where MMg is the molar mass (24.3 g/mol) and ρMg is the density (1735 kg/m3) of Mg. The and boundary nodes are set at the two non-symmetry exterior boundaries of the blood vessel domain. The initial concentration of Mg ions is set to 1.5 mol/m3.

To account for all remaining ions in the blood, another node is added to the model, also being set as immobile species. The concentration of these auxiliary ions are set in order to reach an inlet pH level of 7.4.

The corresponding pH and speciation of the weak acids vs the magnesium concentration in the buffer system are shown in Figure 3 and Figure 4, respectively.

Figure 3: pH versus magnesium ion concentration for the carbonate and phosphoric acid buffer system for an initial/inlet Mg concentration of 1.5 mM.

Figure 4: Speciation of carbonic (left) and phosphoric (right) acid.

Studies

The first study of the model, where the Level Set interface is solved, uses a time-dependent solver for 5 days. The second study of the model, where the Aqueous Electrolyte Transport interface is solved uses a quasi-stationary approach, where stationary analysis is performed for different stent shapes obtained from the first study of the model.

Results and Discussion

Figure 5 shows an isosurface plot of volume fraction of fluid 1 with a value of 0.5 representing the stent surface for the reduced geometry at 1 d (left) and 4 d (right). The change in Mg stent shape can be seen with time representing its dissolution in the tissue and blood vessel.

Figure 5: The isosurface plot of volume fraction of fluid 1 of value 0.5 for the reduced geometry of Mg stent at t = 1 d (left) and t = 4 d (right).

Figure 6 shows the isosurface plot of volume fraction of fluid 1 with a value of 0.5 representing the stent surface for the full Mg stent geometry at 1 d (left) and 4 d (right). The dissolution of the Mg stent leading to topological changes can be seen at 4 d.

Figure 6: The isosurface plot of volume fraction of fluid 1 of value 0.5 for the full geometry of Mg stent at t = 1 d (left) and t = 4 d (right).

Figure 7 shows the change in the stent relative mass loss against time. It can seen that the dissolution rate is uniform until the Mg stent dissolves completely at around 5 d.

Figure 7: Change in Mg stent mass loss against time.

Figure 8 shows the volume plot of Mg ion concentration and volume arrow plot of velocity in the blood vessel domain over a sector of the model geometry at 1 d (left) and 4 d (right). It can be seen that the Mg ion concentration is lower toward the inflow boundary and it gradually increases toward the outflow boundary. The higher value of Mg ion concentration is also seen over a small region at the stent/tissue interface.

Figure 8: The volume plot of Mg ion concentration and arrow volume plot for velocity in blood vessel at t = 1 d (left) and t = 4 d (right).

Figure 9 shows the volume plot of pH and volume arrow plot of velocity in the blood vessel domain over a sector of the model geometry at 1 d (left) and 4 d (right). It can be seen that pH stays close to 7.4 (desired value) at the stent/blood vessel interface. The higher value of pH is seen over a small region at the stent/tissue interface. It should be noted that if pH gets too high, it could potentially damage the tissue. The higher value of pH at initial times and at the outer stent surface (stent/tissue interface) could be attributed to the longest transport distance to the blood vessel as well as the highest total Mg dissolution rate.