Contact Stability and Contact Safety of a Magnetic Resonance Imaging-Guided Robotic Catheter Under Heart Surface Motion

Robotic catheter ablation technology has been proposed to improve treatment of atrial fibrillation [1–4] by providing the catheter with dexterous manipulation and precise navigation capabilities. A successful atrial fibrillation ablation procedure requires the physician to apply radio frequency energy with the catheter to create gap-free transmural lesions in the left atrium to stop the abnormal electrical signals in the heart that cause the rhythm problems. In order to create a gap-free ablation lesion, the robotic catheter needs to maintain stable contact with the substrate tissue surface with appropriate contact force during the radio frequency application.

Rapid heart motion is one of the major disturbances for cardiac ablation procedures, which can result in inaccurate lesion formation, potentially leading to incomplete treatment and recurrence of arrhythmia. Rapid heart beating motions make it difficult to keep adequate contact between the catheter tip and the tissue surface. Unstable contact between the catheter tip and the tissue surface increases the possibility of sliding and positional errors [4]. As suggested in Ref. [5], the cardiac motion causes the displacement of desired ablation target point up to 1 cm, leading to inadequate or excessive contact force on the tissue surface. As such, maintaining stable and safe contact while being subjected to the beating heart motions is critical. Therefore, enhancing contact stability has become the cornerstone for the development of catheter ablation technologies [6–8].

The aim of this paper is the development of methods for providing robust contact stability and maintaining safe contact force for a novel magnetically actuated robotic intravascular cardiac catheter system. The magnetic resonance imaging (MRI)-actuated robotic catheter, developed in Refs. [9–11], is equipped with two sets of current-carrying microcoils and operated inside the bore of an MRI scanner to perform ablation, as shown in Fig. 1. Each set of coils (or actuators) contains one axial coil, and two orthogonal side coils [9]. The robotic catheter is actuated by the magnetic torque generated by the MRI scanner's magnetic field on the coils [10,12–14].

In this paper, the contact stability and contact safety of the MRI-actuated robotic catheter during heart motion are studied. Specifically, stable catheter-tissue contact guarantees no slippage between the catheter-tip and the tissue surface under the heart motion disturbances, and safe catheter-tissue contact ensures the normal contact force to remain within the desired force limits under the heart motion disturbances. First, the contact model for the pseudo-rigid-body (PRB) model of the robotic catheter is introduced. The contact force-actuation Jacobian, which approximates the relationship between the changes of the contact force and the changes of the actuation currents, is then formulated. Two contact force optimization algorithms that use this Jacobian are then proposed. The first method is a quasi-static contact force optimization algorithm, which iteratively calculates the actuation needed to achieve a desired contact force for an instantaneous surface configuration. The second method is a least-squares-based contact force optimization algorithm, which provides stable and safe contact forces against predicted heart motion trajectories over a given prediction horizon. In this method, the contact force residuals are minimized over the estimated heart motion trajectory provided by the heart motion prediction method. Four contact force control schemes are then proposed building on these contact force optimization algorithms to achieve safe and stable catheter-tissue contact under heart motion disturbances. Specifically, the first control scheme proposed is an instantaneous heart position feedback-based method. The second scheme applies a constant actuation level based on a quasi-periodic heart motion prediction. In Refs. [15–17], the experimental analysis has shown that the generalized adaptive filter-based heart motion prediction was able to provide the estimated motion trajectories with high accuracy and better robustness for heart motion tracking problems under both regular and arrhythmia heart motions, compared with other state-of-the-art motion prediction methods. In the third contact force control scheme, a prediction feedback-based method employing the generalized adaptive filter-based heart motion prediction [16] is proposed. The same heart motion prediction method is then employed in a receding horizon controller in the last contact force control scheme proposed. The effectiveness and robustness of the proposed contact force control schemes are evaluated in a simulation environment using prerecorded in vivo heart motion data.

The rest of this paper is organized as follows: Related studies in the literature are presented in Sec. 2. The PRB model of the MRI-actuated robotic catheter is reviewed and presented in Sec. 3. The catheter-tissue contact model based on the PRB model is then provided in Sec. 4. The contact force control schemes are presented in Sec. 5. The contact stability and safety analysis and simulation-based validations of the contact force control schemes are presented in Sec. 6. Finally, discussions and conclusions are presented in Secs. 7 and 8, respectively.

Several studies have shown that contact force plays a critical role in lesion formation during catheter ablation [18–22]. Adequate contact force is required to transfer the heat energy from the catheter tip to the target tissue surface for an effective lesion formation [20,23]. Low contact force may cause failure of tissue heating and transient lesion formation, increasing the risk of recurrence of arrhythmia [8]. Excessive contact force can lead to severe complications, including steam pop, thrombus formation, and perforation [6].

Force-sensing radio frequency ablation catheters have been investigated in several studies in order to better assess the relationship between contact force and ablation efficacy [6–8,18]. Reddy et al. study the relationship between contact force and clinical recurrences for pulmonary vein isolation using a contact force ablation system [20]. This study shows that the contact force for achieving clinical efficacy in pulmonary vein isolation should be >10 g (0.1 N), and arrhythmia is best controlled with contact force >20 g (0.2 N). In Ref. [22], Wakili et al. show that all patients treated with contact force <10 g experienced arrhythmia recurrence. Similarly, Andrade et al. [24] show that low contact force <10 g is associated with higher rate of arrhythmia recurrence (100%) compared to contact force >20 g (20%). Williams et al. [19] suggest that high contact force >25 g may result in heating and edema of extracardiac structures. In this study, we restrict the desired normal contact force range from 10 g to 25 g (0.1–0.25 N), and investigate the ability of the MRI-actuated robotic catheter to maintain the contact force in this narrow therapeutic range during the heart motion [5].

In Refs. [8,19], and [25], it has been shown that contact stability is essential for safety and long-term efficacy of atrial fibrillation ablation. Dewire and Calkins [25] propose that advanced technologies, including contact force sensing catheters and MRI-guided catheters, are promising to improve the success rate and safety of ablation. In Ref. [26], Biase et al. show that using a remote robotic manipulation system can improve the efficacy and safety of ablation procedures. Jayender et al. [27] present an image guided active robotic catheter with shape memory alloy actuators, using a closed-loop PID controller to regulate the tip position and force with additional sensors. Srimathveeravalli et al. [28] design a teleoperated endovascular robotic system for achieving better performance on positioning and force control. Yuen et al. [29] developed a motion compensation system for contact force control under the beating heart motions. Kesner and Howe [4] propose a three-dimensional ultrasound image guidance-based robotic catheter system with a force sensing end effector in order to keep a constant contact force against the tissue motion. Yip et al. [30] develop an adaptive Jacobian estimation method for closed-loop position control of a catheter under heart motion disturbances, where a force sensor is employed to regulate the contact force. Several sensorless contact force estimation methods, employing pose measurement or distal shaft measurement based on different kinematic models of the catheter, are proposed in Refs. [31–33]. Razban et al. [34] propose an approach estimating multiple contact point forces using contact point tracking and image-based deflection measurement. In Ref. [35], a closed-loop controller is designed based on finite element model of the catheter, where the contact force, modeled by Signorini's contact model, is solved numerically as a constrained optimization problem. Haouchine et al. [36] propose an approach that computes contact force using only visual feedback from stereoscopic camera.

While many studies have proposed the strategies to achieve safe robot-tissue contact for robotic catheters [4,29,30], maintaining a stable robot-tissue contact with a safe contact force for the robotic catheter under the beating heart motions remains a challenge. In this paper, we present the analysis of the contact stability and contact safety of the MRI-actuated robotic catheter under cardiac tissue motions. Specifically, we investigate how to control the contact force for maintaining a stable and safe catheter-tissue contact under heart motion disturbances. Unlike the above studies, the proposed contact force control methods are formulated based on the calculation of the contact force-actuation Jacobian without using additional force sensors.

In our previous work presented in Ref. [37], the proposed least-squares-based contact control scheme is implemented employing a simple heart motion prediction method, which uses the first cycle of the heart motion as the prediction of the subsequent heart beat motions. However, the robustness of the contact force control method cannot be guaranteed when dealing with large temporal variance of the heart motion. In this paper, two force control methods, which employ recursive least-squares adaptive filter-based heart motion prediction, are presented. Additionally, in Ref. [37], the contact force control methods are implemented assuming zero time lag between the MRI image acquisition for measurement of the heart motion and the control update. However, such an assumption cannot be applied to the MRI-guided robotic catheter in practice because of the time needed for MRI image acquisition, construction, and processing. In this paper, an investigation of how the proposed contact force control schemes perform under different time lags between the measurement and the control update is presented. As such, the validation studies and discussion of the results are updated and expanded in this paper compared to Ref. [37]. Additionally, a more extensive literature review is presented.

The kinematic models of the MRI-actuated robotic catheter are provided in Refs. [10] and [38], including the continuum model based on finite differences approach and beam theory, and the pseudo-rigid-body model, respectively. The continuum model of the robotic catheter is able to describe the catheter deflection with high reproducibility and accuracy [39], but with low computation efficiency. In this paper, the pseudo-rigid-body model is employed as the kinematic model of the robotic catheter due to its computational efficiency.

where $gst(0)$ is the initial configuration, S denotes the spatial frame, and T denotes catheter tip frame. The shape of the catheter under the n + 1-link PRB model can then be described by the joint angle vector $θ=[θ1T,θ2T,…,θnT]T∈C⊂ℝ3n$⁠, where $C$ denotes the configuration space of the robotic catheter [43].

The magnetic moment μ_j generated by the jth actuator embedded on the catheter with currents $uj∈ℝ3$ is computed as $njAjuj$⁠, where $j=1,..,M$⁠, and M is the number of actuators. n_j and A_j are 3 by 3 diagonal matrices whose diagonal elements are the number of winding turns and the cross-sectional areas (in body frame) of the microcoils in the tri-axial actuation coil set of the actuator j, respectively. The static magnetic field B_s of the MRI scanner expressed in the body frame of the jth actuator is given as $Bj=RsjTBs$⁠, where $RsjT$ denotes the rotation matrix of the frame attached to the jth actuator relative to the spatial frame [9]. The resulting Lorentz forces generated on the coils by the static magnetic field is given by $(njAjuj)∧ Bj=−Bj∧ (njAj)uj$⁠, where ^∧ is the cross product operator.

$j=1,..,M$⁠. $τ(θ,u)$ will be denoted as τ_u in the rest of the paper for convenience.

where the wrench basis $Bc=[I3×3,03×3]T, Jsts$ denotes the spatial manipulator Jacobian of the catheter. g_sc denotes the transformation from the contact frame to the spatial frame, and, R_sc and p_sc are, respectively, the rotational and translational components of the contact frame to spatial frame transformation.

where $JCT†$ is the left pseudo-inverse of $JCT$ with $JCT†=(JCJCT)−1JC$⁠.

where $σμ∈ℝ$⁠. The catheter tip remains on the target contact point if the contact force is inside the friction cone FC, or equivalently, $0≤σμ≤μs$⁠, where μ_s is the static friction coefficient between the catheter tip and the tissue surface.

In this section, contact force-actuation Jacobian is derived to approximate the relationship between the changes in contact forces and the changes in actuation currents.

Let $Ψ(x,t)$ denote the parametrization of the tissue surface, x = x₀ be the location of the desired contact point in surface frame, and t denote the time. Let $h(θ)$ denote the forward kinematics of the catheter, and $h:ℝ3n→ℝ3$⁠. As shown in Fig. 2, during heart surface motion, catheter tip position is required to be static in the surface frame, i.e., $h(θt)=Ψ(x0, t)$⁠. The goal of the contact force control is to improve the stability and safety of the catheter tip-tissue surface contact under surface motion, namely, maintaining static catheter tip positioning at the desired location on the tissue surface with appropriate normal contact force, despite heart motion.

In this section, we first introduce a quasi-static contact force optimization algorithm, which computes a set of actuation currents for a desired normal contact force and target tip position under a given instantaneous surface configuration. This algorithm is then generalized using a least-squares formulation to optimize the contact stability and safety for a given estimated heart motion trajectory over a prediction horizon. Four contact force control schemes are then proposed based on these algorithms.

Under a given instantaneous surface configuration, the proposed quasi-static contact force optimization algorithm iteratively computes the actuation currents to achieve a desired normal contact force and target tip position, as presented in Algorithm 1.

The algorithm assumes that the catheter tip is initially in contact with the target point position on the tissue surface. The initial actuation current u⁰, the desired normal contact force $fcnd$⁠, and the spatial coordinates of the contact point on the tissue surface $Ψ(x0)$ are given as inputs. In this algorithm, because the contact force is controlled for a given instantaneous configuration of the surface (hence the name quasi-static), the time parameterization t in $Ψ(x0,t)$ is excluded. In line 3, the quasi-static equilibrium configuration of the catheter is computed for the given input currents and the tissue surface contact point. In this step, a potential energy minimization based algorithm is performed in the constrained space in order to guarantee the tip position constraint [9,44]. The contact force f_c is then updated given the updated joint angles and actuation currents using Eq. (9). The incremental change of the normal contact force $dfcn$ is then calculated from the desired normal component of the contact force in line 5, where k_n is the step size to adjust the update speed. The incremental change of the contact force $dfc$ is calculated in line 6. A negative feedback $−kf$ is applied on the tangential forces to keep the contact force inside the friction cone during the contact control, while the normal component is driven toward the desired value. In line 7, the contact force-actuation Jacobian J_cu is computed given current joint configuration using Eq. (14). The incremental current update du is computed and applied, respectively, in lines 8 and 9. The algorithm returns the resulting actuation currents once the desired normal force is achieved.

In the least-squares contact force optimization algorithm, the actuation currents are optimized to achieve the desired normal contact force over a control horizon for a given estimated heart motion trajectory. The proposed contact force optimization algorithm iteratively computes the actuation currents that best approximates the desired contact force over a given heart motion trajectory in a least-squares sense. The full least-squares contact force optimization algorithm is given in Algorithm 2.

The algorithm again assumes that the catheter tip is initially in contact with the target point on the tissue surface. The initial actuation current u⁰, the desired normal contact force $fcnd$⁠, and the spatial coordinates of the predicted positions of the contact point on the tissue surface ${Ψ(x0,ti), i=1,..,m}$ are given as inputs. First, the algorithm collects the contact force-actuation Jacobians and the calculated incremental changes of the contact forces for all of the tip positions throughout the estimated heart motion over the horizon. Specifically, for each tip position in the heart beat cycle i, the equilibrium joint angles θ_i under catheter tip position constraint are obtained in line 4. Similar to Algorithm 1, the contact force-actuation Jacobian is calculated and collected by J_i in line 5. The incremental change of the contact force is calculated and collected by $dfci$ in lines 6–8. k_n is the step size to regulate the change of normal contact force, and a negative feedback $−kf$ is applied on the tangential forces to keep the contact force inside the friction cone. Lines 10–12 compute the actuation current update du as a weighted least-squares problem, where W_I is the weight matrix. The actuation currents are then updated in line 13.

Based on the contact force optimization algorithms presented in Algorithms 1 and 2, we propose four contact force control schemes for achieving stable and safe catheter tip–tissue contact under heart surface motions as shown in Fig. 3.

The parameters of catheter robot model used in this paper are based on the parameters identified in Ref. [47] of our MRI-actuated robotic catheter prototype. The mechanical model of the catheter has five pseudo-rigid links, each with 20 mm length, for a total catheter length of 100 mm. In this simulation study, a servocontrol sampling period of 48 ms, approximately matching the targeted servocontrol rate of 20 Hz of the prototype catheter system [48], is used. For evaluation of the control performance, we have used two types of in vivo heartbeat motion data combined with respiratory motion, as shown in Fig. 4. First, the regular heart motion data collected from a swine model in our earlier studies [16], is tested. Specifically, the data used is a 62.5 s long recording of the motion of a POI on the left ventricular of a free beating heart with uniform heart rate, sampled at a 249 Hz sampling rate. In addition, a 128 s long heart motion data with arrhythmia is used for evaluating the performance of the proposed schemes under the presence of arrhythmia motions [17]. The arrhythmia motion data are sampled at a 404.5 Hz sampling rate. The variance of the regular heart motion and the arrhythmia motion data are 6.97 mm and 3.48 mm, respectively.

Due to the processing time required for localization of the heart surface from MRI images, a time lag of $ΔT$ between the data acquisition from the MRI and the update of actuation currents is introduced, where the actuation currents are applied to the heart beat positions $ΔT$ after the measurement is acquired. Specifically, $ΔT$ = 75 ms, 100 ms, 125 ms, 150 ms, and 175 ms are evaluated with the proposed contact force control schemes over the given heart motions.

In the simulation studies, a normal contact force range from 0.10 N to 0.25 N is used based on the values reported in the earlier literature, as discussed in Sec. 2. A desired normal force of 0.15 N is used for the proposed contact force control schemes in order to leave enough tolerance between the desired force and the lower/upper force limits.

One of the key assumptions used in the formulation presented in Sec. 5 is that the cardiac tissue surface is much stiffer than the catheter and hence the surface deformation can be ignored. In order to verify the validity of this assumption, the stiffness values of the catheter and the cardiac tissue were estimated and compared. Silicone rubber is used as the material for the body of the catheter prototype [10] (Fig. 1). The stiffness of the robotic catheter k_c is evaluated to be $kc=0.23$ N/m at the configuration used in the validation studies under the normal contact force of 0.15 N. The stiffness of the heart tissue k_s is estimated using the equation provided in Refs. [49] and [50] for surface stiffness observed by a cylindrical indenter on a flat surface, as $ks=2ER/(1−v2)=423.3$ N/m, where E = 100 kPa is the Young's modulus and $v≈0.5$ is the Poisson's ratio of the heart muscle, and R = 1.58 mm is the outer radius of the catheter. As the estimated stiffness of the heart muscles is much greater than the stiffness of the robotic catheter (⁠$ks≫kc$⁠), a stiff contact between the catheter tip and the heart tissue surface is a valid approximation.

The internal wall of the heart is relatively smooth and well-lubricated due to the presence of blood [51]. The static friction coefficient between the catheter tip and blood vessel depends on the material of the catheter and the type of lubricant used [52,53]. In Ref. [52], the static friction coefficient of a silicone catheter against porcine aorta is reported as 0.1, where distilled water is used as lubricant. In Ref. [53], the static friction coefficients of a silicone catheter against aorta and superior vena cava are reported as 0.67 and 0.56, respectively, where blood is used as lubricant for both cases. In this study, we use a relatively conservative value of $μs=0.2$ as the static friction coefficient between the robotic catheter and the atrial surface [11].

For the evaluation of the IFC and PFC schemes, a desired normal contact force value of $fcnd=0.15$ N is used. The obtained actuation currents are then applied to the catheter for the 48 ms sampling duration, until the next sample time, while the “actual” contact forces and contact ratios are calculated at the full sampling rate of the underlying heart motion data. The adaptive filter-based heart motion prediction used in the PFC (and RHCAF) scheme has a sampling period of 48 ms, matching the servocontrol rate, since there will be a single MRI-based estimation of the target tissue motion at every sample period. The PFC is performed after the first 2.4 s of the heart motion data, for the adaptive filter to converge. The resulting contact ratio and normal contact force over the remaining heart motion data are calculated for evaluation of the proposed schemes.

In the CALC scheme, the desired normal contact force is given as $fcnd=0.15$ N for the least-squares contact force optimization algorithm, with the update rates $kn=0.003$ and $kf=0.005$ for the normal and tangential forces, respectively. The CALC scheme assumes the heart motion is periodic and uses one set of optimized actuation currents throughout the full heart motion; thus, the resulting contact ratio and normal contact force is not affected by time lag $ΔT$⁠.

Finally, for the validation of the RHCAF scheme, similar to the other schemes, the desired normal contact force of $fcnd=0.15$ N is used. The least-squares contact force optimization is performed using the estimated heart motion data. The obtained actuation currents are then applied to the real heart motion data and updated at servocontrol rate. Similar to the PFC scheme, the RHCAF is performed after the first 2.4 s of the heart motion data, for full convergence of the adaptive filter. The resulting contact ratio and normal contact force are calculated for the remaining heart motion data.

In this section, the results of the simulation-based validation studies evaluating the performance of the proposed contact force control schemes are presented. First, the simulation results of contact ratio and normal contact force over the full 62.5 s of regular heart motions are summarized in Table 1.

As shown in Table 1, all of the proposed contact force control schemes achieved the average normal force of 0.15 N, the desired normal contact force, for all of the time lag values considered, with no violations of the force limits detected. In the CALC scheme, 9.8% of heart motion samples can cause potential slippage between the catheter tip and tissue surface during the given heart motions, with maximum contact ratio of 0.23. In the IFC scheme, when no time lag is considered, 1.4% of the heart motion samples violated the friction coefficient, with maximum contact ratio of 0.21. However, the number of heart position samples that violated the friction coefficient increased drastically as the time lag increased. At $ΔT$ = 175 ms, 13.7% of the motion samples violated the friction coefficient, with maximum contact ratio of 0.25. In contrast, the PFC scheme is able to provide better contact stability with 4.1% of sample violations of the friction coefficient at $ΔT$ = 175 ms. The RHCAF scheme outperforms the other schemes in terms of contact stability for all the given time lags, with 3.9% of the motion samples violating the force limits at $ΔT$ = 175 ms. No loss of contact is detected during the regular heart motion using the four proposed control schemes.

Table 2 presents the simulation results of the contact ratio and normal force of the 128 s long heartbeat motion under the presence of arrhythmia. All four proposed control schemes achieved the average normal force of 0.15 N, with no violation of the force limits. The RHCAF scheme is able to provide better contact stability, compared to IFC and PFC scheme, with maximum of 2.2% motion samples violated the friction coefficient for time lag $ΔT$ = 175 ms. When no time lag is considered, compared to the CALC scheme, the RHCAF scheme is able to provide better catheter-tissue contact stability with 1.7% violations of friction coefficient. However, as time lag increases, the CALC scheme outperforms the other proposed control schemes, with 1.9% violations of the friction coefficient. No loss of contact is detected during the heartbeat motion under arrhythmia for all control schemes.

Finally, the simulation results of contact stability and contact safety achieved without employing any control scheme are provided in Table 3 as baseline performances. As shown in Table 3, five different motion positions sampled from the first heart motion cycle are selected, where at each of the sampled positions, a single actuation current set is obtained with normal force of 0.15 N (the desired normal force used for the proposed control schemes) and applied throughout the rest of heartbeat motions.

The results in Table 3 show that, for both regular and arrhythmia heartbeat motions, without a specific control scheme, the robotic catheter cannot guarantee to achieve the desired average normal force value. Furthermore, in the no control scheme case, the quality of the contact, in terms of the contact stability and mean contact force, is highly dependent on the time point in the heart cycle where the control is applied. In contrast to the baseline performance, both PFC scheme and RHCAF scheme show great improvement on contact stability for the regular heart motions with desired normal contact force achieved. In the case of heartbeat motions under arrhythmia, the proposed PFC, RHCAF, and CALC schemes are able to provide better catheter-tissue contact stability while sustaining the desired normal contact force, as the time lag $ΔT$ increases.

The simulation-based study provides an evaluation of the proposed contact force control schemes on achieving contact stability and contact safety given heart motion disturbances, which paves the way for hardware implementation and validation of the proposed contact force control methods in future research. The validation of the kinematic model and the contact model of the robotic catheter employed in this paper were performed in Refs. [40,43,45,54], and [55]. Specifically, the validation of pseudo-rigid-body model of the MRI-actuated robotic catheter can be found in Refs. [40] and [43], and the contact model used in this paper is verified in Refs. [45,54], and [55].

The presented simulation results indicate that the proposed contact force control schemes are able to improve the contact stability while maintaining the safety of the normal contact force. Due to the lack of prediction, the IFC scheme cannot accommodate large time lags between the measurement acquisition and current update, and fails to provide stable and safe catheter-tissue contact as the time lag increases. On the other hand, the prediction-based contact force control schemes (PFC, RHCAF, CALC) are able to compensate for the potential time lags and optimize the actuation currents using the estimated heart motion trajectories over the future horizon, achieving safer and more stable contact forces when dealing with the rapidly changing heart motions. The performance of the CALC scheme on the arrhythmia motion is significantly better than its performance on the regular motion. This is due to the larger magnitude of the regular heart motion, which is almost twice as large as the magnitude of the arrhythmia motion. The CALC scheme outperforms the RHCAF scheme for the arrhythmia heartbeat motions when time lag $ΔT>$ 75 ms. One of the major reasons is that the prediction error of the employed generalized adaptive filter-based heart motion predictor tends to increase as the prediction horizon increases.

While the simulation results demonstrate the feasibility of the proposed contact force control schemes, several limitations of the proposed methods are identified. In this work, the robotic catheter is modeled by the pseudo-rigid-body model, which cannot guarantee the smoothness of the catheter shape, for computation efficiency and proof-of-concept. This can be improved by employing the continuum model proposed in Ref. [10]. Avoiding ill-conditioned contact force-actuation Jacobian is also important for the proposed contact force control methods. In this study, both quasi-static and least-squares-based contact force optimization algorithms need to start in a stable configuration in order to avoid abrupt changes in contact force. Since this paper emphasizes on the contact stability and safety under the effect of the positional heart motions, the change of the surface orientation during the heart beat motion is ignored in the simulation experiments. A potential avenue for future work is to include the angle between the catheter tip and the tissue surface normal as an additional optimization objective.

The proposed algorithms were implemented in matlab (R2019a) on Ubuntu 16.04 operating system. The computer is equipped with Intel^® Core^TM i7-8700 CPU at 3.20 GHz and 8.0 GB memory. matlab's constrained optimization function fmincon is employed for computing the constrained equilibrium configuration of the robotic catheter, using the default “interior-point” algorithm with a linear equality constraint and a linear inequality constraint for satisfying the desired tip position constraint on the tissue surface. In the quasi-static contact force optimization algorithm, the computation time of computing one iteration of the actuation update is approximately 40 ms. For least-squares contact force optimization, the computation time of one iteration of the current update for one predicted horizon is approximately 650 ms. The computation time for the least-squares contact force optimization algorithm can be significantly improved by parallelization of solving the constrained equilibrium and the Jacobian computations for the heart beat motions over the expected horizon. The computation time of the algorithms can be further improved by a computationally efficient C/C++ implementation.

Our future work will focus on the hardware implementation and validation of the proposed contact force control schemes. The experimental validation of the proposed methods on hardware will be performed once the algorithms for real-time tracking of the robotic catheter system and the tissue surface from intra-operative MRI imaging are available. These algorithms are currently under development as part of a parallel research study in our research group. Our future work will also focus on the improvement of the computation efficiency of the proposed algorithms, specifically, the parallel computing-based implementation of the least-squares contact force optimization.

In this paper, the contact stability and contact safety of a novel magnetically actuated robotic catheter under regular and arrhythmia heart motions are studied. A catheter-tissue contact model based on pseudo-rigid-body model of the robotic catheter is presented, and the contact force-actuation Jacobian is formulated. Two contact force optimization methods are proposed. The first method is a quasi-static contact force optimization algorithm, which improves the contact stability and safety given an instantaneous surface configuration. Two contact force control schemes, an instantaneous heart position feedback and a predicted position feedback based contact force control scheme, are proposed based on this algorithm. The second optimization method is a least-squares contact force optimization algorithm, which optimizes the actuation currents for the estimated heart motions over a control horizon. A receding horizon control scheme with an adaptive filter-based heart motion prediction and a constant actuation-level contact force control scheme are proposed based on this optimization method.

The simulation-based validations are presented for evaluating the performance of the proposed contact force control schemes. The simulation results show that the contact force control schemes proposed based on least-squares contact force optimization algorithm are able to provide the most stable and safe contact force between the catheter tip and tissue surface under heart motions and sensing-to-actuation time delays.

This paper was presented in part at the 2020 IEEE International Conference on Robotics and Automation (ICRA 2020), Paris, France.

• National Science Foundation (Grant Nos. CISE IIS-1524363, CISE IIS-1563805, and ENG IIP-1700839; Funder ID: 10.13039/100000001).

• National Heart, Lung, and Blood Institute of the National Institutes of Health (Grant No. R01 HL153034; Funder ID: 10.13039/100000050).