GENERALIDADES DE LAS TIC

The performance of the large deflection beam model for describing the behavior of the catheter tip is evaluated experimentally. It should be noted that the forces applied at the tip of the catheter are equal to those read by the mounted force sensor. This fact, theoretically proved by the static equilibrium equations for the catheter tip, is also validated experimentally. Another force sensor was placed right at the tip of the catheter and it was verified that the difference between the readings of the force sensors does not exceed 0.1 gram-force (gf). The catheter tip mass is estimated to be 1 g and it was verified experimentally that it does not contribute to the tip deflection. Nevertheless, the weight can be easily included in the applied loads without the need for altering the model.

2.4.1

Rigidity of the Catheter Tip

A pseudo-rigid-body model can be used to determine the rigidity of the catheter tip at each deflection angle. Using this model for describing the large deflections of an initially curved beam (Figure 2.5), the deflection angle Θ depends on the applied transverse force Ft through [25]:

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 4 5 6 7 8 9 10 11x 10 4

Tip Angle: Φ (rad)

E*I (g.mm

2 )

Experimental Values

Fitted Curve: (2.5Φ7.2+4.4)×104

Figure 2.6: Relation between the catheter tip deflection Φ and the value of EI for the catheter tip

Θ = L

2

K_ΘEIFt (2.25)

where L is the length of the beam, γ is the characteristic radius, K_Θ is the stiffness coefficient and ρ is a function of γ and the curvature [25].

Using Equation (2.25), the value of the product E ∗ I is determined experi- mentally for a number of bending angles. Figure 2.6 shows the experimental results and the curve to which the points are fitted. For each bending angle at the catheter tip, the corresponding value forEI is used in the beam equations (Equations (2.16)- (2.18)) to predict the final shape of the catheter from the initial bending and force data.

2.4.2

Evaluating Model Performance

Using the experimental setup described in Section 2.2, force and deflection data are collected with three different approaches. These three approaches will be helpful in observing how the previous shape of the catheter affects its current shape:

0 2 4 6 8 0.4 0.6 0.8 1 1.2 Force (gf) Tip Angle: Φ (rad) Experiment

Initial Curvature Approach Moment Approach

(a) Tip angle (Φ) vs. force (P) 0 2 4 6 8 0 5 10 15 Force (gf) Horizontal Deflection: δx (mm) Experiment

Initial Curvature Approach Moment Approach (b) Horizontal deflection (δx) vs. force (P) 0 2 4 6 8 10 15 20 25 30 35 Force (gf) Vertical Deflection: δy (mm) Experiment

Initial Curvature Approach Moment Approach

(c) Vertical deflection (δy) vs. force (P)

Figure 2.7: Estimation of Φ, δx and δy for datasetA, whenP is measured using the force sensor and n= 0.

1. The catheter tip is bent to an initial angle. A concentrated vertical force (n = 0) is applied at the tip and force data and the catheter shape are recorded. The force is released and the catheter restores its initial bent shape before force is applied for the next sample. The data collected using this approach is stored as dataset A.

2. The catheter tip is bent to an initial angle and a concentrated vertical force (n = 0) is applied at the tip. The force and deflection data are recorded consecutively without restoring the catheter to its initial bent shape between each two samples. Moreover, the tip angle may change during data collection using the catheter handle at the proximal end. The dataset B includes the data points collected at different tip angles.

3. The catheter tip is bent to an initial angle and a series of increasing concen- trated axial loads (n = −cot Φ) are applied. Using the catheter’s proximal handle, the catheter tip is returned to its straight position before the tip is bent to reach a new deflection angle. The force and deflection data collected using this approach are stored as dataset C.

For all datasets, the beam model takes the initial bending angle of the tip and the force data to calculate the final deflection angle, which is then compared to the actual value measured experimentally. The results are shown in Figures 2.7-2.9 for both the initial curvature and moment approaches.

0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 Force (gf) Tip Angle: Φ (rad) Experiment

Initial Curvature Approach Moment Approach

(a) Tip angle (Φ) vs. force (P) 0 2 4 6 8 10 −2 0 2 4 6 8 10 12 14 Force (gf) Horizontal Deflection: δx (mm) Experiment

Initial Curvature Approach Moment Approach (b) Horizontal deflection (δx) vs. force (P) 0 2 4 6 8 10 0 5 10 15 20 25 30 35 Force (gf) Vertical Deflection: δy (mm) Experiment

Initial Curvature Approach Moment Approach

(c) Vertical deflection (δy) vs. force (P)

Figure 2.8: Estimation of Φ, δx and δy for datasetB, whenP is measured using the force sensor and n= 0.

0 2 4 6 8 10 12 14 0.2 0.4 0.6 0.8 1 1.2 Force (gf) Tip Angle: Φ (rad) Experiment

Initial Curvature Approach Moment Approach

(a) Tip angle (Φ) vs. force (P) 0 5 10 15 −2 0 2 4 6 8 10 12 14 Force (gf) Horizontal Deflection: δx (mm) Experiment

Initial Curvature Approach Moment Approach (b) Horizontal deflection (δx) vs. force (P) 0 5 10 15 5 10 15 20 25 30 35 Force (gf) Vertical Deflection: δy (mm) Experiment

Initial Curvature Approach Moment Approach

(c) Vertical deflection (δy) vs. force (P)

Figure 2.9: Estimation of Φ, δx and δy for datasetC, whenP is measured using the force sensor andn =−cot Φ.

Table 2.1 provides statistical measures that show the model performance. In this table, MAE and RMSE denote the mean absolute error and the root mean square error. The tip angle was measured in radians and all deflection measurements were in millimeters.

The model predicts the tip deflection angle with an error less than 0.0197 rad (≈ 1.13◦). The error in estimating the horizontal and vertical components of the displacement is less than 0.6 mm (Table 2.1). Considering that the catheter diameter is 1.67 mm, this error is acceptable since the position of the catheter tip is estimated with an error of the order of the tip radius. From Table 2.1, the performance of the analytical beam model is almost the same as that of the FE-based model. Moreover,

Table 2.1: Statistical measures showing model performance for the datasets A, B

and C evaluated for all results shown in Figures 2.7, 2.8 and 2.9 (Appr.1: Initial Curvature Approach; Appr.2: Moment Approach, FEM: FE-based Model)

MAE

Dataset A Dataset B Dataset C

Appr.1 Appr.2 FEM Appr.1 Appr.2 FEM Appr.1 Appr.2 FEM

φ(rad) 0.013 0.013 0.015 0.015 0.015 0.021 0.012 0.013 0.007

δx (mm) 0.184 0.182 0.138 0.227 0.228 0.201 0.228 0.228 0.233

δy (mm) 0.423 0.421 0.216 0.432 0.43 0.388 0.435 0.383 0.326

RMSE

Dataset A Dataset B Dataset C

Appr.1 Appr.2 FEM Appr.1 Appr.2 FEM Appr.1 Appr.2 FEM

φ (rad) 0.017 0.017 0.019 0.02 0.02 0.025 0.017 0.018 0.009

δx (mm) 0.223 0.221 0.167 0.299 0.3 0.259 0.287 0.279 0.283

δy (mm) 0.495 0.493 0.269 0.599 0.597 0.472 0.546 0.483 0.3902

modeling the initial bending of the catheter with a moment that could have caused such a deflection leads to slightly more accurate results. Furthermore, the model performs better in predicting the final tip angle for DatasetsA andC. This is mostly due to the different approaches in collecting data samples and shows that the previous shape of the catheter introduces some error in the results. Finally, the results suggest that the model can predict the final tip angle accurately for axial loads as well as vertical ones.