Bayesian Optimization Based Preprocedural Planning For Robotic Left Atrial Appendage Occlusion

Abstract

Left atrial appendage occlusion is a procedure to reduce the risk of thromboembolism in atrial fibrillation patients by blocking the left atrial appendage ostium using an occlusion device implanted by an intra-vascular delivery catheter. The preprocedural planning of the left atrial appendage occlusion procedure aims to identify an optimal implantation trajectory for a successful occlusion implant delivery from a structural understanding of the left atrial appendage. In this paper, a Bayesian Optimization based preprocedural planning approach is proposed for the robotic left atrial appendage occlusion procedure. The preprocedural planner efficiently samples transseptal puncture positions over the fossa ovalis and sequentially optimizes the transseptal puncture location. The iterative linear-quadratic-regulator is employed by the Bayesian Optimization planner for locally optimizing the occlusion trajectory for a given transseptal puncture location. The performance of the proposed Bayesian Optimization based preprocedural planner is evaluated in a simulation environment using a real cardiac anatomy model.

I. Introduction

The left atrial appendage (LAA) is a small, hollow, and pouch-like structure in the top left chamber of the heart. Blood clots formed in LAA leading to thromboembolism is known to be a major risk for patients with atrial fibrillation (AFib) [1]. An emerging approach used to prevent stroke in AFib is to eliminate the LAA site of blood clot formation, using a procedure called LAA occlusion. In the LAA occlusiong procedure, LAA ostium is blocked using an occlusion device implanted by an intra-vascular delivery catheter. In this procedure, the delivery catheter is introduced through the femoral vein and threaded through the vena cava into the right atrium, followed by a transseptal puncture through the fossa ovalis to gain access to the left atrium. Positioning of the transseptal puncture (TSP) is critical for success of the LAA occlusion procedure since the TSP location determines if and how easily the catheter can access the LAA [2], [3]. Furthermore, the selection of the TSP location must be done on a patient-specific basis, as there is substantial variability in LAA shapes and locations [4], [5].

The preprocedural planning of the robotic left atrial appendage occlusion procedure has two planning objectives: finding an optimal TSP location and finding an optimal collision-free robotic catheter trajectory for the delivery of the occlusion device. Evaluation of a catheter trajectory from a TSP location is computationally costly. In this paper, we present a Bayesian Optimization based planning framework to efficiently search for the optimal TSP location and optimize the catheter trajectory. Bayesian Optimization provides a derivative-free framework for sequentially optimizing the objective functions that are computationally expensive to evaluate [6], [7], where an acquisition function is defined and optimized to trade off the exploration and exploitation of the search space [7]. For the left atrial appendage preprocedural planning problem, the objective function is defined as the cost of the catheter trajectory given a TSP location. The Bayesian Optimization based preprocedural planner employs the iterative linear quadratic regulator (iLQR) [8], [9] algorithm to optimize the catheter trajectory given sampled TSP positions, and sequentially selects the next TSP location in the optimization process by maximizing the acquisition function.

The rest of this paper is organized as follows. Related work is presented in Section II. The problem formulation of the left atrial appendage preprocedural planning problem is introduced in Section III. The proposed Bayesian Optimization based left atrial appendage catheter planning approach is presented in Section IV. The simulation results of the left atrial appendage preprocedural planning using the proposed approach are presented in Section V, followed by conclusions in Section VI.

II. Related Work

Bayesian Optimization algorithm provides an efficient search strategy for finding the global maximum of objective functions where each evaluation of the objective function is computationally expensive, greatly reducing the exploration time. As such, Bayesian Optimization has been applied in the medical robotics domain, where such optimization problems are frequently encountered. Ayvali et al. [10] use Bayesian Optimization algorithm to guide the tissue palpation for simultaneous registration and stiffness estimation. In [11], the authors develop a Bayesian Optimization guided probing strategy for tissue abnormalities. Yan and Pan [12] propose a simultaneous tumor localization and segmentation approach based on Bayesian Optimization guided probing. In [13], Morere et al. propose a Bayesian Optimization based optimal trajectory search for partially observable Markov decision process (POMDP).

To the best of our knowledge, there are no earlier studies in the literature which present an automated method for patient specific selection of TSP location. The TSP location selection methods primarily provide general guidelines based on the specific procedure to be performed [14] or based on broad anatomical variations [15], [16]. In [17] Liu et al. employ 3D printed models of the LAA to help physicians plan and select the right size of appendage closure device beforehand.

In this paper, we present a Bayesian Optimization based preprocedural planning approach for robotic left atrial appendage occlusion procedure. The preprocedural planner provides an efficient search strategy for the optimal TSP position and optimizes the catheter trajectory for the occlusion device implantation.

III. Problem Description

During the implantation of the occlusion device, the robotic catheter performs a TSP on the fossa ovalis [15] and advances to the left atrium until it reaches to the LAA ostium, as shown in Fig. 1. The position and orientation of the catheter tip in relation to the LAA ostium is crucial for a successful implantation of the occlusion device. Specifically, the position of the catheter tip should provide sufficient space for the occlusion device to be successfully and safely unfolded, and the orientation of the catheter tip should align with the proximal segment of the LAA [15]. The location of the TSP is essential for achieving proper alignment between the robotic catheter and the LAA. The choice of the TSP site on fossa ovalis depends on the anatomy of the LAA, e.g., the LAA location and angulation relative to the left atrium [15].

Fig. 1.

Illustration of the left atrial appendage occlusion scheme [18].

A. Model of the MRI-Actuated Robotics Catheter

In this paper, a novel magnetically-actuated robotic in-travascular cardiac catheter is employed for performing the LAA occlusion procedure. The MRI-actuated robotic catheter [19]–[21] operates inside the bore of a MRI scanner and is magnetically actuated by the magnetic torque generated by two sets of electromagnetic micro-coils embedded on the catheter, where the actuation of the catheter is controlled by the currents passing through the coils. Each set of coils (or actuators) contains one axial coil, and two orthogonal side coils [21]. The kinematic models of the MRI-actuated robotic catheter are provided in [19] and [21], including the continuum model based on finite differences approach and beam theory, and the pseudo-rigid-body model (PRBM), respectively. In this study, the catheter kinematics is modeled using a Cosserat rod model (CRM) [22], [23]. In the formulation employed, the relationship between the catheter shape and the actuation currents is described by a quasistatic equilibrium, as the catheter moves with low velocity and acceleration during the LAA occlusion procedure.

Specifically, let $u\in {\mathbb{R}}^6$ denote the actuation currents and $k\in \mathbb{R}$ denote the insertion length of the catheter. The forward kinematics map of the catheter, $f:{\mathbb{R}}^7\to {\mathbb{R}}^6$, maps the concatenated vector of the actuation currents and insertion length, $x_t={\left[u^T,k\right]}^T\in {\mathbb{R}}^7$, to $z_t=f\left(x_t\right),z_t={\left[p_t^T,n_t^T\right]}^T$, where $p_t\in {\mathbb{R}}^3$ is the catheter tip position and $n_t\in {\mathbb{R}}^3$ is the normal direction of the catheter tip.

B. Planning Objectives

Given the geometric model of the left atrium obtained from preoperative medical imaging, and the desired implant delivery location $p_d$ and axial direction $n_d$ provided by the physician, the LAA occlusion preprocedural planning aims to find an optimal catheter trajectory for the occlusion device implantation by selecting an optimal transseptal puncture location. In this paper, a Bayesian Optimization based preprocedural planning framework is proposed, as presented in Section IV, sequentially optimizing the TSP location over the fossa ovalis and the catheter trajectory.

IV. Bayesian Optimization Based Left Atrial Appendage Occlusion Planner

In this section, the proposed Bayesian Optimization (BO) based left appendage occlusion planning framework is presented. The method efficiently samples the TSP positions over the fossa ovalis while optimizes the catheter trajectories for the sampled TSP positions. Bayesian Optimization algorithm iteratively samples TSP position by maximizing an acquisition function [7], where the cost of the catheter trajectory is used as the objective function for the BO algorithm. The iterative-Linear-Quadratic-Regulator (iLQR) [8] algorithm is employed for the catheter trajectory optimization, locally optimizing the catheter trajectory for a given TSP position sample.

A. Iterative Linear Quadratic Regulator (iLQR) For Catheter Trajectory Optimization

The derivation of the iterative linear quadratic regulator (iLQR) [8] for catheter trajectory optimization is presented in this subsection. Given a TSP position $P\in {\mathbb{R}}^3$, we consider the following discrete-time dynamics:

$$\begin{gathered} x_{t+1}=x_t+\Delta x_{t,} \\ {y}_t=g\left(x_t\right). \end{gathered}$$ (1)

where the state variable $x_t={\left[u_t^T,k_t\right]}^T\in {\mathbb{R}}^7$ is the concatenated vector of the actuation currents $u\in {\mathbb{R}}^6$ and insertion length $k\in \mathbb{R}$ of the robotic catheter. The control input $\Delta x_t\in {\mathbb{R}}^7$ of the dynamic in (1) is the displacement of the state variable, i.e., the change of actuation currents and insertion length. The output function $g:{\mathbb{R}}^7\to {\mathbb{R}}^4$ returns the positional and normal directional difference between the desired catheter tip configuration and actual tip configuration, and is defined as $g=h\circ f$. Let $y_t=h\left(z_t\right),h:{\mathbb{R}}^6\to {\mathbb{R}}^4$, the output $y_t$ is then computed as:

$$y_t={\left[{\left(p_t-p_d\right)}^T,1-n_t^T{n}_d\right]}^T\in {\mathbb{R}}^4,$$ (2)

where $p_d$ and $n_d$ are the desired catheter tip position and tangential tip normal vector, respectively.

The cost of the catheter trajectory for the implant delivery given the TSP position $P$ and the initial state $x_0$ of the robotic catheter is defined as:

$$\begin{gathered} J\left(P,x_0\right)=c_N\left(x_N\right)+\sum _{t=0}^{N-1} \left(c_t^{\prime }\left(x_t,\Delta x_t\right)+c_t^{\prime \prime }\left(x_t\right)\right), \\ =\frac{1}{2}{y}_N^T{Q}_N{y}_N+\frac{1}{2}\sum _{t=0}^{N-1} \left(x_t^{T}Q{x}_t+\frac{1}{2}\Delta x_t^{T}R\Delta x_t+c_t^{\prime \prime }\left(x_t\right)\right). \end{gathered}$$ (3)

$N$ denotes the terminal state. The cost function weighting matrices $Q_N$, $Q$ and $R$ are symmetric positive semi-definite, and $Q_N\in {\mathbb{R}}^{4\times 4}$, $Q,R\in {\mathbb{R}}^{7\times 7}$. A set of marker points $\{l{\}}^{1:M}$ for collision checking are sampled along the inserted catheter body for the calculation of the cost of collision $c_t^{\prime \prime }=k_c{\sum }_{i=1}^M \mathit{\text{exp}}\left(-d_i\left(x_t\right)\right)$, where $k_c$ is the coefficient of the collision cost, $d_i$ is the closest distance between the $i^{\text{th }}$ marker point to the wall of the left atrial chamber.

The iLQR controller uses value iteration to locally optimize the trajectory, where a backward sweep and a forward sweep are performed in each iteration. Let $c_t=c_t^{\prime }+c_t^{\prime \prime }$ denotes the immediate cost in (3), the cost-to-go function $V(x,\Delta x)$ is then defined as [8]:

$$\begin{gathered} V_N=c_N, \\ {V}_t\left(x_t\right)=\underset{\Delta x_t}{\text{min}} \left\{c_t\left(x_t,\Delta x_t\right)+V_{t+1}\left(x_{t+1}\right)\right\} \\ =\underset{\Delta x_t}{\text{min}} S^t\left(x_t,\Delta x_t\right), \end{gathered}$$ (4)

where $S^t$ denotes value function. Let $\delta x$ and $\delta \Delta x$ denotes the small changes around $x$ and $\Delta x$, the value function $S^t$ is then approximated as [8]:

$$S^t\left(x_t+\delta x,\Delta x_t+\delta \Delta x\right)\approx S^t\left(x_t,\Delta x_t\right)+S_x^t\delta x+S_{\Delta x}^t\delta \Delta x+\frac{1}{2}{\left[\begin{array}{c}\delta x\\ \delta \Delta x\end{array}\right]}^T\left[\begin{array}{cc}{S}_{xx}^t& S_{x\Delta x}^t\\ S_{\Delta xx}^t& S_{\Delta x\Delta x}^t\end{array}\right]\left[\begin{array}{c}\delta x\\ \delta \Delta x\end{array}\right],$$ (5)

and

$$\begin{gathered} S_x^t=c_x^t+V_x^{t+1}, \\ {S}_{\Delta x}^t=c_{\Delta x}^t+V_x^{t+1}, \\ {S}_{xx}^t=c_{xx}^t+V_{xx}^{t+1}, \\ {S}_{\Delta x\Delta x}^t=c_{\Delta x\Delta x}^t+V_{xx}^{t+1}, \\ {S}_{x\Delta x}^t=c_{x\Delta x}^t+V_{xx}^{t+1}, \end{gathered}$$ (6)

where the subscripts $x$ and $\Delta x$ denote the partial derivatives with respect to the state and the displacement of the state, respectively. The optimal policy of the state displacement is then computed as [8]:

$$\begin{gathered} \delta \Delta x^{*}=\underset{\delta \Delta x}{\text{argmin\hspace{0.17em}}}S(x+\delta x,\Delta x+\delta \Delta x) \\ =-S_{\Delta x\Delta x}^{-1}\left(S_{\Delta x}+S_{\Delta xx}\delta x\right). \end{gathered}$$ (7)

Let $K=-S_{\Delta x\Delta x}^{-1}{S}_{\Delta xx}$ and $k=-S_{\Delta x\Delta x}^{-1}{S}_{\Delta x}$, the backward value iteration is given as:

$$\begin{gathered} V_x=S_x+K^T{S}_{\Delta x\Delta x}k+K^T{S}_{\Delta x}+S_{\Delta xx}^{T}k, \\ {V}_{xx}=S_{xx}+K^T{S}_{\Delta x\Delta x}K+K^T{S}_{\Delta xx}+S_{\Delta xx}^{T}K. \end{gathered}$$ (8)

The new state trajectory $\{\hat{x}{\}}^{1:N}$ is then propagated given the optimal displacement policy $\{\Delta \hat{x}{\}}^{1:N}$ from (7) in the forward sweep as:

$$\begin{gathered} {\hat{x}}_0=x_0, \\ \Delta {\hat{x}}_t=\Delta x_t-S_{\Delta x\Delta x}^{t-1}\left(S_{\Delta x}^t+S_{\Delta xx}^t\left({\hat{x}}_t-x_t\right)\right) \\ {\hat{x}}_{t+1}={\hat{x}}_t+\Delta {\hat{x}}_{t.}, \end{gathered}$$ (9)

B. Bayesian Optimization For Transseptal Puncture (TSP) Position Selection

Bayesian Optimization (BO) provides an efficient sampling strategy for finding global optima of functions that are computationally expensive to evaluate [7]. In this paper, a Bayesian Optimization based planner is employed for finding an optimal TSP position. The BO planner sequentially samples TSP positions over the fossa ovalis while maximizing an acquisition function, trading off between exploration and exploitation.

Conventionally, Bayesian Optimization solves a maximization problem. In this paper, we define the objective function of the maximization problem as $q\left(P\right)=-J\left(P,x_0\right)$, given the cost of catheter trajectory $J\left(P,x_0\right)$ in (3). BO algorithm defines a Gaussian Process $\left(𝒢𝒫\right)$ over the objective functions $q\left(P\right)~𝒢𝒫\left(m\left(P\right),k\left(P,P^{\prime }\right)\right)$, where $m\left(p\right)$ denotes the mean of the functions and $k\left(P,P^{\prime }\right)$ denotes the covariance function:

$$k\left(P,P^{\prime }\right)=\text{exp}\left(-\frac{1}{2\theta }{∥P-P^{\prime }∥}^2\right),$$ (10)

where $\theta$ is a hyperparameter [7]. Given the observation data $𝒟={\left\{P_i,q_i\right\}}_{i=1:n}$ drawn from the $𝒢𝒫$ priors, Bayesian optimization selects the next TSP position sample by maximizing the acquisition function over the incumbent $q\left(P^{+}\right)$, where

$$P^{+}=\underset{P_i\in P_{1:n}}{\text{argmax\hspace{0.17em}}}EI\left(P_i\right).$$ (11)

The expected improvement (EI) is chosen as the acquisition function [24]:

$$\begin{gathered} EI\left(P\right)=\left\{\begin{array}{ll}\left(\mu \left(P\right)-q\left(P^{+}\right)\right)\Phi \left(z\right)+\sigma \left(P\right)\varphi \left(z\right)& \sigma \left(P\right)>0\\ 0& \sigma \left(P\right)=0\end{array},\right. \\ z=\frac{\mu \left(P\right)-q\left(P^{+}\right)}{\sigma \left(P\right)}, \end{gathered}$$ (12)

where $\Phi$ and $\varphi$ are the cumulative distribution function and the probability density function of the standard normal distribution, respectively. $𝒢𝒫$ is then updated with the new sample $P_{n+1}$:

$$\begin{gathered} \left[\begin{array}{c}{\mathit{q}}_{1:n}\\ q_{n+1}\end{array}\right]~𝒩\left(0,\left[\begin{array}{cc}\mathit{K}& \mathit{k}\\ {\mathit{k}}^T& k\left(P_{n+1},P_{n+1}\right)\end{array}\right]\right), \\ \mathit{K}=\left[\begin{array}{ccc}k\left(P_1,P_1\right)& \dots & k\left(P_1,P_n\right)\\ \cdot & \ddots & \cdot \\ k\left(P_n,P_1\right)& \dots & k\left(P_n,P_n\right)\end{array}\right], \end{gathered}$$ (13)

where $\mathit{k}=\left[k\left(P_{n+1},P_1\right),k\left(P_{n+1},P_2\right),\dots k\left(P_{n+1},P_n\right)\right]$. The distribution of the objective function of the new sample $P_{n+1}$ can be computed using Sherman-Morrison-Woodbury [7], [25] as $p\left(q_{n+1}\mid 𝒟,P_{n+1}\right)=𝒩\left(\mu \left(P_{n+1}\right),{\sigma }^2\left(P_{n+1}\right)\right)$, where

$$\begin{gathered} \mu \left(P_{n+1}\right)={\mathit{k}}^T{\mathit{K}}^{-1}{\left[q_1,q_2,\dots q_n\right]}^T, \\ {\sigma }^2\left(P_{n+1}\right)=k\left(P_{n+1},P_{n+1}\right)-{\mathit{k}}^T{\mathit{K}}^{-1}\mathit{k}. \end{gathered}$$ (14)

A global search algorithm DIRECT proposed in [26] is employed as the sampling strategy for the BO algorithm. Given the existing TSP position samples, DIRECT divides the fossa ovalis into finer rectangles to decide where to sample next [7].

The complete Bayesian Optimization based left atrial appendage catheter planner is presented in Algorithm 1 [7]. The algorithm takes as input the catheter actuation state $x_0$ as the initial state of the iLQR planner. In Line 1, the BO planner first generates an initial Gaussian Process $𝒢𝒫$ and samples TSP positions uniformly over the fossa ovalis, where the initial observation data ${𝒟}_{1:n}=\left\{P_{1:n},q_{1:n}\right\}$ is collected. In Line 3, the acquisition function (12) is optimized in OptimizeAcquisitionFunction, which returns a new TSP position sample $P_{n+1}$. Line 4 evaluates the objective function as $\left.q_{n+1}=q\left(P_{n+1},x_0\right)\right)$ and collect the catheter trajectory $\{x{\}}_{n+1}^{1:N}$ at the new TSP location using the iLQR presented in Section IV-A. The observation data ${𝒟}_{1:n+1}$ is then augmented in Line 5, and the Gaussian Process $𝒢𝒫$ is updated given the new TSP sample in Line 6. In Line 7, BO proceeds to sub-sample over the fossa ovalis given the new sample. The algorithm converges until the covariance matrix $\mathit{K}$ converges to zero, as given in Line 2, where $\epsilon >0$ is a very small number. The Bayesian optimization LAA preprocedural planner returns the optimal TSP position $P_{n+1}$ and the catheter trajectory $\{x{\}}_{n+1}^{1:N}$ given the optimal TSP position.

V. Simulation Results

In this section, we present an example scenario of the left atrial appendage occlusion preprocedural planning using the proposed Bayesian Optimization planner.

The parameters of MRI-actuated robotic catheter model used in this paper are based on the parameters identified in [20] of our MRI-actuated robotic catheter prototype. 10 marker points are sampled along the robotic catheter body for the calculation of the collision cost. A realistic left atrium model [27] is used in this paper as presented in Fig. 2 (a), where the atrial septum, i.e., the insertion plane, is marked by the red mesh and fossa ovalis are circled in black with a diameter of 14 mm [28]. The desired catheter tip position and catheter-LAA alignment direction are marked by the red point and arrow at the LAA ostium.

Fig. 2.

(a) The 3D left atrium and left atrial appendage model. The insertion plane is marked by the red mesh, where the black circle marks the fossa ovalis area. The desired catheter tip position and tip normal direction are indicated by the red point and arrow at the LAA ostium. (b) The TSP positions sampled by the BO planner over the fossa ovalis using DIRECT [26].

The terminal state cost weighting matrix $Q_N$ for the iLQR planner is chosen as $Q_N={10}^{-6}\mathit{\text{diag}}\left(\right[1,1,1,100\left]\right)$ where the first 3 terms are the weights of the positional cost, and last term is the weight of angular cost. The state and control cost weighting matrices are chosen to be $Q={10}^{-6}\mathit{\text{diag}}\left(\right[\mathrm{1,1},\mathrm{1,1},\mathrm{1,1},0.0001\left]\right)$ and $R={10}^{-4}\mathit{\text{diag}}\left(\right[\mathrm{1,1},\mathrm{1,1},\mathrm{1,1},0.0001\left]\right)$, respectively, where the first 6 terms are the weights for the cost of actuation currents and last term is the weight for the cost of the insertion motion. As shown in Fig. 2 (b), the sampled TSP positions over the fossa ovalis are presented. The BO algorithm started with 9 sample points uniformly spread over the fossa ovalis, and converged in 9 iterations. Figure. 3 shows two views of the distribution of the cost of insertion over the insertion plane. The BO algorithm converged to the lowest cost TSP sample over the insertion plane.

Fig. 3.

The cost of insertion distribution over the insertion plane in two views (a) and (b). The TSP positions sampled by the BO planner over the fossa ovalis are marked in black. The optimal TSP position sample is marked in red.

The final catheter configuration optimized by the iLQR planner given the optimal TSP position is shown in Fig. 4 (a). Fig. 4 (b) shows the cost of the catheter insertion trajectory $J\left(P^{+}\right)$ of the new TSP position sample over each iteration. Fig. 5 presents 5 final catheter configurations with different TSP position samples in comparison to the final catheter configuration for the case when the catheter is inserted from the optimal TSP position. The optimal configuration provides a smoother catheter shape and a more accurate coaxial alignment with the left atrial appendage. Finally, the complete catheter motion optimized by the BO preprocedural planner is presented in Fig. 6.

Fig. 4.

(a) The TSP positions sampled by the BO planner over the fossa ovalis are marked in black. The final configuration of the robotic catheter inserted from the optimal TSP position is marked in red. (b) The cost of the catheter trajectory of the new TSP position sample in each iteration of the BO algorithm.

Fig. 5.

5 sampled catheter occlusion configurations (in black) with final costs of 0.086 mm, 0.131 mm, 0.079 mm, 0.077 mm, 0.087 mm, in comparison to the optimal catheter occlusion configuration (in red) with final cost of 0.068 mm.

Fig. 6.

The left atrial appendage preprocedural catheter trajectory optimized by the BO planner in two perspectives (a) and (b).

VI. Conclusion

In this paper, we present a Bayesian optimization based preprocedural planning method for left atrial appendage occlusion procedure. The proposed method employs the Bayesian optimization algorithm, which sequentially optimizes the transseptal puncture position over the fossa ovalis. The Bayesian optimization planner iterativelly selects the subsequent TSP position samples by evaluating the catheter trajectories optimized using the iLQR algorithm. Experimental results for the left atrial appendage occlusion preprocedural planning using the proposed method are presented in a simulation environment. Our future works will focus on incorporating the different sources of catheter motion uncertainty to the planning framework.