**1. Introduction**

218 Pulmonary Embolism

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beam CT for detection of emboli and assessment of pulmonary blood flow.

When anticoagulants are contraindicated for the venous thromboembolism therapy and thromboembolism recurs, a vena cava filter is inserted percutaneously into a major vein in order to prevent blood clots from entering the lungs (Ando & Kuribayashi, 2000; Streiff, 2000) (Fig. 1). The inferior vena cava filter is a mesh structure designed to capture blood clots while not impeding blood flow. This filter is inserted using an introducer catheter from either the femoral or jugular vein and is fixed by a hook attached to the tip of a wire. Several filters have been designed to lyse captured clots.

Fig. 1. Placement of the inferior vena cava filter.

Ideally, the filter should capture blood clots efficiently while not impeding the blood flow. Moreover, the filter should not move in the blood vessel after deployment and should pass smoothly through the slim introducer catheter. Although insertion appears to be a safe and effective method by which to prevent the recurrence of pulmonary embolism, patients in whom the filter has become tilted may experience pulmonary embolism recurrence because of a decrease in the thrombus-trapping performance of the filter (Nara et al., 1995; Rogers et

Numerical Analysis of the Mechanical Properties of a Vena Cava Filter 221

1991). The total length of the GF is 47 mm (Greenfield et al., 1991). We used a head without a hole and determined the diameter (2*R*h) and length (*L*h) of the head by measurement. The over-the-wire filter has a head with a small hole and inserted over the guidewire to optimize

This model of the filter is constructed with viscoelastic springs and *n* segments for the dynamic deformation of a flexible structure. The viscoelastic springs express the bending and torsional stiffness of the wire. The motion of the filter model is represented by the

> T ( ) ( , ) ( ) - ( - )- <sup>r</sup> <sup>i</sup> o w *n*

where *q* is the joint displacement vector, *q*o is the joint displacement vector when no load is applied, *M*(*q*) is the inertia matrix, *Vqq* (, ) is the centrifugal force (Coriolis force), *F*i is contact force vector of each segment (*i*), and *K* and *D*w are elastic and viscous coefficients of a mobile joint, respectively. The second and third terms of the right-hand side in this equation correspond to the viscoelastic forces of a mobile joint. The head and the wire of the filter consist of one and three segments (diameter: 2*R*w, length: *L*w), respectively. Each segment of the wire has a mobile joint in the center. The elastic coefficient of the mobile joint (*K*) with respect to bending resistance (*K*wx) and torsional resistance (*K*wz) are approximated

 *K*wx = *E*w*I*x/*L*w (2)

 *K*wz = *G*w*I*p/*L*w (3) where *E*w and *G*w are Young's modulus and the modulus of transverse elasticity of the wire of the filter, respectively, *I*x and *I*p are the area moment of inertia and the polar moment of

*<sup>M</sup> qq Vqq q Kq q q* + = *<sup>J</sup> <sup>F</sup> <sup>D</sup>* (1)

*i*

the alignment of the filter with the inferior vena cava (Kinney et al., 1997).

Fig. 2. The Greenfield inferior vena cava filter.

Newton-Euler equations of motion as follows:

as follows (Yamamura et al., 2003):

al., 1998). In order to satisfy these requirements, several types of filters, such as permanent, temporary, and retrievable filters, have been proposed. For example, several types of vena cava filters, such as Greenfield filter, Vena Tech filter, Bird's nest filter, Simon-Nitinol filter, TrapEase filter, Günther tulip filter, Antheor filter, Neuhaus protect filter, and Recovery-Nitinol filter, have been developed (Ando & Kuribayashi, 2000; Boston Scientific, 2007; Greenfield et al., 1990, 1991; Kinney et al., 1997; Nara et al., 1995; Rogers et al., 1998; Streiff, 2000; Swaminathan et al., 2006). However, there are few quantitative data on the mechanical properties of these filters. In particular, although Swaminathan reported the blood clot capturing efficiency from the viewpoint of computational fluid dynamics (Swaminathan et al., 2006), there are few quantitative data concerning, for example, the ease of filter delivery. Therefore, in the present study, we evaluated through numerical analysis the mechanical properties of a Greenfield filter deployed into a vein. In particular, since the filter expands rapidly, the surgeon cannot perceive the expanding motion or the transition of the contact force applied to the blood vessel wall. Therefore, a complete understanding of the mechanical properties of the filters must be determined on not only input from doctors but also on the results of numerical analysis regarding the expansion. These methods are expected to be useful for analyzing the structure of filters and may help to guide the design

of new filters. Based on the above considerations, we herein evaluate by numerical analysis the dynamic motion of a deployed filter using the evaluation standard of the incline and misalignment between the filter and the blood vessel. First, we evaluated whether the filter tilts when the catheter tilts or is misaligned or when the filter cannot expand. Second, we evaluated the migration of the deployed filter under a constant force.

#### **2. Methods**

Catheters and guidewires are used in the treatment of infarctions and aneurysms. In a previous study, in order to make intravascular treatment safer, we developed a computerbased surgical simulation system in order to simulate a catheter placed inside blood vessels for treatment of the brain (Takashima et al., 2006, 2007b, 2009). In the present study, we applied the simulation methods for the guidewire to the inferior vena cava filter because both the guidewire and the inferior vena cava filter are flexible structures. Using a similar model, we can easily combine the filter model with our catheter simulator. Actually, a catheter simulator has been developed as a training environment for inferior vena cava filter placement (Hahn et al., 1998). Ease of percutaneous filter delivery provides numerous advantages to both doctors and patients, including improved operational efficiency and reduced cost. Therefore, it is important that the filter be easy to use in combination with a catheter.

#### **2.1 Filter model**

In the present study, we used a Greenfield filter (GF) constructed of titanium (Fig. 2) (Ando & Kuribayashi, 2000; Boston Scientific, 2007; Greenfield et al., 1990, 1991; Kinney et al., 1997; Nara et al., 1995; Rogers et al., 1998; Streiff, 2000; Swaminathan et al., 2006). This permanent filter is commonly used in Japan. The GF is constructed of either titanium or stainless steel and come in various shapes. The GF used herein is cone-shaped and consists of six wires connected to a head. The diameter of the wires is 0.45 mm (Swaminathan et al., 2006), and each leg has a "zigzag" pattern. The bottom diameter of the cone is 38 mm (Greenfield et al.,

al., 1998). In order to satisfy these requirements, several types of filters, such as permanent, temporary, and retrievable filters, have been proposed. For example, several types of vena cava filters, such as Greenfield filter, Vena Tech filter, Bird's nest filter, Simon-Nitinol filter, TrapEase filter, Günther tulip filter, Antheor filter, Neuhaus protect filter, and Recovery-Nitinol filter, have been developed (Ando & Kuribayashi, 2000; Boston Scientific, 2007; Greenfield et al., 1990, 1991; Kinney et al., 1997; Nara et al., 1995; Rogers et al., 1998; Streiff, 2000; Swaminathan et al., 2006). However, there are few quantitative data on the mechanical properties of these filters. In particular, although Swaminathan reported the blood clot capturing efficiency from the viewpoint of computational fluid dynamics (Swaminathan et al., 2006), there are few quantitative data concerning, for example, the ease of filter delivery. Therefore, in the present study, we evaluated through numerical analysis the mechanical properties of a Greenfield filter deployed into a vein. In particular, since the filter expands rapidly, the surgeon cannot perceive the expanding motion or the transition of the contact force applied to the blood vessel wall. Therefore, a complete understanding of the mechanical properties of the filters must be determined on not only input from doctors but also on the results of numerical analysis regarding the expansion. These methods are expected to be useful for analyzing the structure of filters and may help to guide the design

Based on the above considerations, we herein evaluate by numerical analysis the dynamic motion of a deployed filter using the evaluation standard of the incline and misalignment between the filter and the blood vessel. First, we evaluated whether the filter tilts when the catheter tilts or is misaligned or when the filter cannot expand. Second, we evaluated the

Catheters and guidewires are used in the treatment of infarctions and aneurysms. In a previous study, in order to make intravascular treatment safer, we developed a computerbased surgical simulation system in order to simulate a catheter placed inside blood vessels for treatment of the brain (Takashima et al., 2006, 2007b, 2009). In the present study, we applied the simulation methods for the guidewire to the inferior vena cava filter because both the guidewire and the inferior vena cava filter are flexible structures. Using a similar model, we can easily combine the filter model with our catheter simulator. Actually, a catheter simulator has been developed as a training environment for inferior vena cava filter placement (Hahn et al., 1998). Ease of percutaneous filter delivery provides numerous advantages to both doctors and patients, including improved operational efficiency and reduced cost. Therefore, it is important that the filter be easy to use in combination with a

In the present study, we used a Greenfield filter (GF) constructed of titanium (Fig. 2) (Ando & Kuribayashi, 2000; Boston Scientific, 2007; Greenfield et al., 1990, 1991; Kinney et al., 1997; Nara et al., 1995; Rogers et al., 1998; Streiff, 2000; Swaminathan et al., 2006). This permanent filter is commonly used in Japan. The GF is constructed of either titanium or stainless steel and come in various shapes. The GF used herein is cone-shaped and consists of six wires connected to a head. The diameter of the wires is 0.45 mm (Swaminathan et al., 2006), and each leg has a "zigzag" pattern. The bottom diameter of the cone is 38 mm (Greenfield et al.,

migration of the deployed filter under a constant force.

of new filters.

**2. Methods** 

catheter.

**2.1 Filter model** 

1991). The total length of the GF is 47 mm (Greenfield et al., 1991). We used a head without a hole and determined the diameter (2*R*h) and length (*L*h) of the head by measurement. The over-the-wire filter has a head with a small hole and inserted over the guidewire to optimize the alignment of the filter with the inferior vena cava (Kinney et al., 1997).

Fig. 2. The Greenfield inferior vena cava filter.

This model of the filter is constructed with viscoelastic springs and *n* segments for the dynamic deformation of a flexible structure. The viscoelastic springs express the bending and torsional stiffness of the wire. The motion of the filter model is represented by the Newton-Euler equations of motion as follows:

$$\mathbf{M}(\boldsymbol{q})\ddot{\boldsymbol{q}} + \mathbf{V}(\boldsymbol{q}, \dot{\boldsymbol{q}}) = \sum\_{i}^{n} \mathbf{J}\_{\rm r}^{\rm T}(\boldsymbol{q}) \mathbf{F}\_{i} \cdot \mathbf{K}(\boldsymbol{q} \cdot \boldsymbol{q}\_{\rm o}) \cdot \boldsymbol{D}\_{\rm w} \dot{\boldsymbol{q}} \tag{1}$$

where *q* is the joint displacement vector, *q*o is the joint displacement vector when no load is applied, *M*(*q*) is the inertia matrix, *Vqq* (, ) is the centrifugal force (Coriolis force), *F*i is contact force vector of each segment (*i*), and *K* and *D*w are elastic and viscous coefficients of a mobile joint, respectively. The second and third terms of the right-hand side in this equation correspond to the viscoelastic forces of a mobile joint. The head and the wire of the filter consist of one and three segments (diameter: 2*R*w, length: *L*w), respectively. Each segment of the wire has a mobile joint in the center. The elastic coefficient of the mobile joint (*K*) with respect to bending resistance (*K*wx) and torsional resistance (*K*wz) are approximated as follows (Yamamura et al., 2003):

$$K\_{\rm wrx} = E\_{\rm w} I\_{\rm x} / L\_{\rm w} \tag{2}$$

$$K\_{\rm wxz} = C\_{\rm pw} I\_{\rm p} / L\_{\rm w} \tag{3}$$

where *E*w and *G*w are Young's modulus and the modulus of transverse elasticity of the wire of the filter, respectively, *I*x and *I*p are the area moment of inertia and the polar moment of

Numerical Analysis of the Mechanical Properties of a Vena Cava Filter 223

The simulation models used in the present study are shown in Fig. 4. The centerline of the

blood vessel is along the *z*-axis.

Fig. 3. Dimensions of the filter model (unit: mm).

**2.3 Calculation of contact force** 

Fig. 4. Simulation model (vena cava filter, catheter, and blood vessel).

circumferential direction of the cross-section (*f*ri). *f*ni is expressed as follows:

each joint and nearest point on the central curve of the blood vessel.

In order to determine whether contact occurs between the filter and the vessel, the distances between the joints and the tip of the filter model (*p*i) and the centerline of the vessel, were calculated. Considering the clearance of the filter and the blood vessel, we calculated the contact force according to the distance. Here, *F*i is decomposed into three components along the centerline of the blood vessel (*f*ti), the normal direction to the centerline (*f*ni), and the

 *f*ni = −*K*v(|*l*i|+*R*w−*R*v)3/2 *l*i/| *l*i| (7) where *K*v is the elastic coefficient of vessel deformation and *l*i is the distance vector between

inertia of area, respectively. In the present study, since the wire is a rod (radius: *R*w), *K*wx and *I*p are expressed as follows:

$$K\_{\rm wx} = \rm \rm \rm \rm \rm \, \_{m} \rm \, R\_{\rm w} \rm \, ^{4} / \, L\_{\rm w} \tag{4}$$

$$I\_{\mathbf{p}} = \mathbf{2} I\_{\mathbf{x}} \tag{5}$$

Assuming an isotropic material, *G*w is expressed using Poisson's ratio (*ν*) as follows:

$$G\_{\rm w} = E\_{\rm w} / \mathcal{D} (1 + \nu) \tag{6}$$

In the present study, *ν* = 0.3 (Petrini et al., 2005).

The parameters and dimensions of the filter model are shown in Table 1 and Fig. 3, respectively. In this table, subscripts *w* and *h* indicate the parameters of the wire and the head, respectively. Here, *K*wx and *K*wz are obtained by substituting *R*w and *E*w in Table 1 into Eqs. (2) and (3). Assuming a fixed joint, the values of the joints between the wire and the head are 100 times as large as those obtained using Eqs. (2) and (3). The recurved hook is set at an angle of 80° (Greenfield et al., 1991). The actual leg hook consists of a twisted and complicated structure in order to avoid migration of the filter after deployment or penetration of the vein (Greenfield et al., 1990, 1991) (Fig. 2). However, since the penetration of the vein was not evaluated in the present study, the bending number of the hook is one. The change of the bending angle is equal to the change of *q*o, as shown in Eq. (1). In a manner similar to the catheter simulator (Takashima et al., 2006, 2007b, 2009), the viscous term for each movable joint was considered in terms of *D*w. Since we cannot measure the exact values, we assumed *D*w to be smaller than *K*wx and *K*wz. We neglected the "zigzag" pattern and the slight flare of the wire. In the design of the titanium GF, the addition of a slight flare to the legs was made in order to facilitate discharge from the carrier without leg crossing, particularly from the jugular direction where the legs discharge first (Greenfield et al., 1990).


Table 1. Parameters of the filter model used in simulation.

#### **2.2 Blood vessel model**

The vessel is a circular elastic cylinder defined by a centerline and a radius (*R*v). The centerlines are represented by numerical data. The position of the centerline of the vessel is constant. The contact forces between the filter and the vessel are calculated according to the stiffness and the friction of the vessel wall. The friction force are derived from the fixation of the hook. Moreover, the diameter of the blood vessel (2*R*v) is assumed to be 20 mm, which is similar to Swaminathan (Swaminathan et al., 2006). In the instructions for the use of the GF (Boston Scientific, 2007), the maximum diameter of the inferior vena into which the filter can be deployed is 28 mm.

inertia of area, respectively. In the present study, since the wire is a rod (radius: *R*w), *K*wx and

 *K*wx = π*E*w*R*w4/*L*w (4)

 *I*p = 2*I*<sup>x</sup> (5)

 *G*w = *E*w/2(1+*ν*) (6)

The parameters and dimensions of the filter model are shown in Table 1 and Fig. 3, respectively. In this table, subscripts *w* and *h* indicate the parameters of the wire and the head, respectively. Here, *K*wx and *K*wz are obtained by substituting *R*w and *E*w in Table 1 into Eqs. (2) and (3). Assuming a fixed joint, the values of the joints between the wire and the head are 100 times as large as those obtained using Eqs. (2) and (3). The recurved hook is set at an angle of 80° (Greenfield et al., 1991). The actual leg hook consists of a twisted and complicated structure in order to avoid migration of the filter after deployment or penetration of the vein (Greenfield et al., 1990, 1991) (Fig. 2). However, since the penetration of the vein was not evaluated in the present study, the bending number of the hook is one. The change of the bending angle is equal to the change of *q*o, as shown in Eq. (1). In a manner similar to the catheter simulator (Takashima et al., 2006, 2007b, 2009), the viscous term for each movable joint was considered in terms of *D*w. Since we cannot measure the exact values, we assumed *D*w to be smaller than *K*wx and *K*wz. We neglected the "zigzag" pattern and the slight flare of the wire. In the design of the titanium GF, the addition of a slight flare to the legs was made in order to facilitate discharge from the carrier without leg crossing, particularly from the jugular direction where the legs

> *R*w (mm) 0.225 *R*h (mm) 0.875 *E*w (GPa) 22 Density (g/cm3) 5.3 *D*w (N·m·s/rad) 0.005 *L*w (mm) 4, 43 *L*h (mm) 2 *n* 25

The vessel is a circular elastic cylinder defined by a centerline and a radius (*R*v). The centerlines are represented by numerical data. The position of the centerline of the vessel is constant. The contact forces between the filter and the vessel are calculated according to the stiffness and the friction of the vessel wall. The friction force are derived from the fixation of the hook. Moreover, the diameter of the blood vessel (2*R*v) is assumed to be 20 mm, which is similar to Swaminathan (Swaminathan et al., 2006). In the instructions for the use of the GF (Boston Scientific, 2007), the maximum diameter of the inferior vena into which the filter can

Assuming an isotropic material, *G*w is expressed using Poisson's ratio (*ν*) as follows:

*I*p are expressed as follows:

In the present study, *ν* = 0.3 (Petrini et al., 2005).

discharge first (Greenfield et al., 1990).

**2.2 Blood vessel model** 

be deployed is 28 mm.

Table 1. Parameters of the filter model used in simulation.

The simulation models used in the present study are shown in Fig. 4. The centerline of the blood vessel is along the *z*-axis.

Fig. 3. Dimensions of the filter model (unit: mm).

Fig. 4. Simulation model (vena cava filter, catheter, and blood vessel).
