## Abstract

Purpose: The purpose of this paper is to investigate the optimal geometrical design of concentric tube robots (CTR) for intracerebral hemorrhage (ICH) evacuation, with a focus on minimizing the risk of damaging white matter tracts and cerebral arteries. Methods: To achieve our objective, we propose a parametrization method describing a general class of CTR geometric designs. We present mathematical models that describe the CTR design constraints and provide the calculation of a path risk value. We then use the genetic algorithm to determine the optimal tube geometry for targeting within the brain. Results: Our results show that a multi-tube CTR design can significantly reduce the risk of damaging critical brain structures compared to the conventional straight tube design. However, there is no significant relationship between the path risk value and the number and shape of the additional inner curved tubes. Conclusion: Considering the challenges of CTR hardware design, fabrication, and control, we conclude that the most feasible geometry for a CTR path in ICH treatment is a straight outer tube followed by a planar curved inner tube. These findings have important implications for the development of safe and effective CTRs for ICH evacuation by enabling dexterous manipulation to minimize damage to critical brain structures.

## 1 Introduction

Intracerebral hemorrhage (ICH) is the most common subtype of hemorrhagic stroke with a 1-year mortality rate of 54% and 5-year mortality rate of 71% [1]. ICH is caused by bleeding from injured blood vessels that form a hematoma within the brain parenchyma. The expansion of this hematoma compresses the surrounding brain tissues, reducing oxygen delivery and inducing cellular damage, collectively known as a mass effect. It is hypothesized that evacuating a hematoma can reduce its mass effect [2], but studies have repeatedly shown that craniotomy and other conventional surgical interventions are not effective in treating this condition [3,4]. This is primarily due to the damage induced to healthy brain tissues along the path to deep-seated hematomas, offsetting any benefit surgery might provide [3].

To mitigate this damage and improve patient outcomes, clinicians have investigated minimally invasive approaches as an alternative to open surgery. These include stereotactic thrombolysis [5–12], endoscopic surgery [13–17], and stereotactic aspiration [18,19]. These treatment approaches have indicated promising results by minimizing disruption and scarring, reducing recovery time, and lowering medical costs [2]. However, despite these potential benefits, current minimally invasive surgery instruments lack adequate tool dexterity and workspace visualization. For example, current neuroendoscopes (e.g., Artemis™, BrainPath®) consist of a stiff metal cannula with limited dexterity. This setup requires surgeons to tilt and torque the cannula through a burr hole to visualize and evacuate the hematoma [20,21]. This approach leads to brain tissue damage, creating what is colloquially referred to as a “cone of destruction” [21,22]. Further, the endoscopic view is subject to visibility constraints caused by obstructing anatomy or bleeding. Poor visualization increases the risk of breaching the hemorrhagic cavity interface and suctioning normal tissue or surrounding vessels which could cause re-hemorrhage [23]. To improve endoscopic visualization, infusion of saline or gas is used to permit direct visualization of the cavity; however, this has the deleterious effect of concurrently increasing intracranial pressure and inducing aberrant neurotransmitter release [24,25].

Concentric tube robots (CTR) [26], a class of needle-sized robots made of multiple pre-curved elastic tubes [27,28], are a promising candidate for ICH evacuation due to their minimally invasive structure, their ability to navigate tortuous anatomy, and their pre-existing internal lumen for evacuation [29]. Additionally, CTRs are capable of deploying in a “follow-the-leader” fashion (i.e., the CTR traces the path of the tip throughout space over time), minimizing off-axis forces applied to surrounding healthy tissue and mitigating the cone of destruction [30,31]. Several groups have paired CTRs with appropriate imaging modalities to improve ICH evacuation and visualization. Burgner et al. developed a concentric tube robot for CT-guided ICH evacuation and validated the feasibility in a phantom model [32]. However, CT can involve significant radiation exposure, and thus, image feedback should be minimized during the procedure [33]. For example, the number of CT scans performed within one procedure is usually no more than 4–5 [34], precluding real-time intraoperative evacuation monitoring. Alternatively, magnetic resonance imaging (MRI) guidance can provide real-time imaging feedback without radiation exposure and has been used for intraoperative navigation/monitoring in various settings [35–38]. Recently, we proposed an MRI-compatible concentric tube robot for ICH evacuation. We developed the MRI-compatible robot hardware and performed a feasibility study in a 3 T MRI scanner. The results indicated that accurate targeting and intraoperative evacuation monitoring could be achieved with real-time MRI feedback [39,40]. However, despite these promising results, the optimal geometry design and path planning for concentric tube robot-enabled ICH evacuation have yet to be fully developed. Although various methods for needle path planning in neurosurgery have been proposed, these approaches are mostly designed for straight tools [41,42]. Furthermore, even when non-linear paths are considered for CTRs, existing methods oversimplify the follow-the-leader deployment and limit the needle segments to planar curves [43–45]. To address these limitations, we propose a novel approach that models and optimizes CTR shapes with mechanics-based follow-the-leader constraints for CTR needle path planning in neurosurgery.

In this paper, we systematically investigate the geometrical design of concentric tube robots and trajectory planning for safe ICH evacuation by minimizing path-induced risk. Here, the risk is predominately determined by the damage to the surrounding white matter fiber tracts and intracerebral blood vessels. Both can be clearly visualized in customized MR imaging sequences, such as Diffusion Weighted Imaging (DWI) [46,47], and intracranial vessel wall (IVM) imaging [48,49]. We first propose a parametrization method to describe the CTR path. Then, a mathematical model is formulated to evaluate the risk value caused by the surgical path. Finally, we implement a convenient and highly accurate optimization algorithm [50], genetic algorithm, to perform the optimization study that investigates the optimal CTR geometry for arbitrary intracranial targeting. The paper is arranged as follows: Sec. 2 describes the path parameterization; Sec. 3 introduces the risk values and risk map generation for path optimization; Sec. 4 presents the path optimization method to determine the best geometry for CTR design; Sec. 5 concludes this article.

## 2 Concentric Tube Robots Path Parametrization

In this section, we present our proposed CTR path parameters and compute the geometry of the CTR for targeting. The components of our CTR include a non-deformable straight tube as the outer tube and multiple pre-curved super-elastic inner tubes as shown in Fig. 1. The non-deformable straight tube is designed to penetrate through the burr hole in the skull, while the pre-curved inner tubes add dexterity to avoid high-risk regions. Herein, we intentionally choose helical tubes, which permit follow-the-leader deployment [30]. Consequently, the resultant shape of our CTR is a straight line followed by successive helices. In this section, we present the method of using one straight line *S* and *n* − 1 helices *H* to represent a configuration of successive helices SH_{n−1}, providing a total of *n* path segments to construct a group of three-dimensional paths from the starting point to the target point. Since straight lines and planar curves are special forms of helices, the parametrization of *n*-connected helices (*H*_{n}) is proposed in this section.

### 2.1 Geometry Description of *H*_{n}.

Before introducing our proposed parametrization method for *H*_{n} (i.e., *n* end-to-end helices, including the first straight tube), we begin by describing a single helix in space. Traditionally, a helical shape can be parameterized in the form *x* = *a* cos *t*, *y* = *a* sin *t*, *z* = *bt*, *a* > 0, *t* ∈ [0, *T*] which lies on a cylinder *x*^{2} + *y*^{2} = *a*^{2} [51]. Specifically, the helix is right-handed with *b* > 0, left-handed with *b* < 0, and circular with *b* = 0. However, this traditional expression only specifies the shape (radius, pitch, and length) of the helical arc, while omitting the helix origin and orientation. Thus, to fully define a series of connected helices in space, a new parametrization method that can directly specify the start position and orientation of the helix is proposed, as shown in Fig. 2.

For the *i*th helical segment of *H*_{n}, 5 parameters (*x*_{i}, *R*_{i}, *l*_{i}, *r*_{i}, and *η*_{i}) are needed to fully describe it, where *x*_{i} ∈ ℝ^{3} and *R*_{i} ∈ SO(3) are the position and orientation of the *i*th helical segment starting point. *l*_{i} and *r*_{i} are the length and radius of the *i*th helical segment, and *η*_{i} is the slope angle of the helical segment. Besides, the sixth parameter *α*_{i} is a scalar to describe the relative rotation between two adjacent segments as shown in Fig. 1. The detailed definitions for these parameters are listed in Table 1.

Symbol | Field | Range | Description |
---|---|---|---|

x_{i} | $R3$ | N/A | Start position |

R_{i} | $SO(3)$ | N/A | Start orientation |

l_{i} | $R$ | $(0,\u221e)$ | Helix length |

r_{i} | $R$ | $(0,\u221e)$ | Helix radius |

η_{i} | $R$ | $(\u2212\pi /2,\pi /2)$ | Slope angle |

α_{i} | $R$ | $(0,\pi )$ | R_{i} Rotation angle |

Symbol | Field | Range | Description |
---|---|---|---|

x_{i} | $R3$ | N/A | Start position |

R_{i} | $SO(3)$ | N/A | Start orientation |

l_{i} | $R$ | $(0,\u221e)$ | Helix length |

r_{i} | $R$ | $(0,\u221e)$ | Helix radius |

η_{i} | $R$ | $(\u2212\pi /2,\pi /2)$ | Slope angle |

α_{i} | $R$ | $(0,\pi )$ | R_{i} Rotation angle |

*x*_{i+1}and

*R*_{i+1}, i.e., the starting position and orientation of the next helical segment, are fully determined as shown in Fig. 2. Thus, to obtain the starting pose (

*x*_{i+1},

*R*_{i+1}) of a helical segment according to preceding segment starting pose (

*x*_{i},

*R*_{i}), we have

**.**

*n**ϕ*

_{i}is the enwinding angle (i.e., the angle corresponding to the number of rotations) of the

*i*th helical segment. The connected vectors

*v*_{i,1},

*v*_{i,2}, and

*v*_{i,3}(

*v*

_{i,j}is the

*j*th connected vector of the

*i*th helical segment) are shown in Fig. 2.

The aforementioned equations show that given *x*_{i}, *R*_{i}, *α*_{i}, *l*_{i}, *r*_{i}, and *η*_{i}, the shape of the *i*th helical segment and the starting position and orientation of the *i* + 1 helical segment are fully determined. Thus, to parametrize a *H*_{n} path consisting of *n* helices, 4*n* + 2 parameters are needed: *x*_{1}, *R*_{1}, {*l*_{1}, *l*_{2}, …*l*_{n}}, {*r*_{1}, *r*_{2}, …, *r*_{n}}, {*η*_{1}, *η*_{2}, …, *η*_{n}}, and {*α*_{1}, *α*_{2}, …, *α*_{n}}.

*x*_{end}is given by the treatment target. To ensure the target

*x*_{end}is within the workspace of the tube path, the path parameters must be properly selected. In this paper, we replace the constraints {

*η*

_{n},

*r*

_{n},

*l*

_{n}} with

*x*_{end}without changing the dimension of parameters. {

*η*

_{n},

*r*

_{n},

*l*

_{n}} and

*x*_{end}can be related to each other based on the following equation:

The detailed derivation and solution of Eq. (3) are given in the Appendix in Sec. 7. Thus, the parametrization method for a path defined by *H*_{n} segments that reaches target point *x*_{end} is expressed by *x*_{1}, *R*_{1}, *x*_{end}, {*l*_{1}, *l*_{2}, …*l*_{n−1}}, {*r*_{1}, *r*_{2}, …, *r*_{n−1}}, {*η*_{1}, *η*_{2}, …, *η*_{n−1}}, and {*α*_{1}, *α*_{2}, …, *α*_{n}}. Specifically, a CTR has the shape of SH_{n−1} meaning that the first segment of the needle path is a straight line *S*, and therefore, *η*_{1} = 0, which leads to *r*_{1} = ∞. To avoid introducing ∞, the curvature *κ*_{i} = 1/*r*_{i} is used throughout the remainder of this paper in lieu of the radius *r*_{i}. Hence, the parameters for a SH_{n−1} path are *x*_{1}, *R*_{1}, *x*_{end}, {*l*_{1}, *l*_{2}, … *l*_{n−1}}, {*κ*_{2}, *κ*_{3}, …, *κ*_{n−1}}, {*η*_{2}, *η*_{3}, …, *η*_{n−1}}, and {*α*_{1}, *α*_{2}, …, *α*_{n}}.

### 2.2 Follow-the-Leader Constraints for Concentric Tube Robots.

*x*_{end}is reached if the aforementioned equations are solvable. In this paper, we only consider CTR with

*n*≤ 3, which are SH

_{0}, SH

_{1}, and SH

_{2}. This is because more tubes could lead to significant challenges in robot hardware design, leading to limited feasibility in the practical operation scenarios. In addition, we assume that the outer straight tube is sufficiently rigid, such that we only consider the follow-the-leader deployment between two helical tubes for SH

_{2}configuration. According to Ref. [30], SH

_{2}configuration can satisfy this constraint if the path ensures the last two tubes are (1) planar circular curves that lie in the same plane or (2) helical segments that share a same torsion

*τ*, defined as

*τ*

_{i}is the torsion of the

*i*th helix. Torsion of a helix with the above parameterization is [51]

Here, we divide the parametrization method into two conditions, *η*_{2} = *η*_{3} = 0 or *η*_{2} ≠ 0, *η*_{3} ≠ 0, meaning that the CTR shapes are a straight line followed by (1) two planar curved tubes or (2) two helical tubes, respectively. To distinguish them, we use SA_{2} to denote the planar curved arc configuration. Accordingly, we use SA_{1} to denote the CTR configuration with one straight tube followed by a single planar curved tube. We do not consider the situations that *η*_{2} = 0, *η*_{3} ≠ 0 or *η*_{2} ≠ 0, *η*_{3} = 0 in this paper.

*α*

_{2}can be directly computed

*α*

_{3}can be obtained according to Eq. (4). For helical arc configurations,

*α*

_{3}can be solved from Eq. (3). However, a simultaneous solution for Eqs. (3) and (4) is not always feasible, and an alternative equation is proposed to minimize the error of follow-the-leader deployment and to replace Eq. (4) with a numerical solution

## 3 Path Risk Value Calculation

In this paper, two critical structures are considered for avoidance during the path-planning stage: the nerve fibers in white matter tracts and the cerebral arteries. The overarching goal is to mitigate the damage to selected nerve fibers and cerebral arteries during the procedure. In this section, we will detail the method to calculate the path risk value of a given path to evaluate the safety of a path.

### 3.1 Data Description and Visualization.

Two critical structures, namely, the white matter tracts and cerebral arteries, are considered in the path planning algorithm. As shown in Table 2, eight tracts containing the most concerned nerve fibers are identified to be avoided. They were selected due to the direct correlation between subcortical strokes and these selected fibers [52–54]. However, more fibers can be considered per the physician's suggestions. The spatial representation of the tracts was adopted from the latest version of the tractography atlas developed by Yeh et al. [55]. The statistical atlas of Mouches et al. was used for the cerebral arteries [56]. The hemorrhages are segmented from the MRI scans of anonymous patients provided by the clinical collaborators. Figure 3 presents a visualization of the nerve fibers in the eight selected tracts and the cerebral arteries. The central block represents the intracerebral hemorrhage.

Tract number | Tract name |
---|---|

1 | Anterior thalamic radiation |

2 | Corticospinal tract |

3 | Optic radiation |

4 | Arcuate fasciculus |

5 | Corpus callosum |

6 | Superior longitudinal fasciculus |

7 | Frontal aslant |

8 | Cingulum |

Tract number | Tract name |
---|---|

1 | Anterior thalamic radiation |

2 | Corticospinal tract |

3 | Optic radiation |

4 | Arcuate fasciculus |

5 | Corpus callosum |

6 | Superior longitudinal fasciculus |

7 | Frontal aslant |

8 | Cingulum |

### 3.2 Concentric Tube Robots Path Risk Value.

The risk value of a CTR path is obtained via the following four steps: (1) Calculate RV_{fiber}, RV_{artery}, which are the risk value of a point ** p** ∈ ℝ

^{3}to an individual nerve fiber and artery, respectively; (2) Calculate RV

_{fibers}, RV

_{arteries}, which are the risk value of the point

**to all fibers and arteries; (3) Calculate RV**

*p*_{point}, which is the risk value of the point

**by weighted summing RV**

*p*_{fibers}and RV

_{arteries}, and the tissue damage risk term RV

_{tissue}; (4) Calculate RV

_{path}, which is the risk value of a CTR path summing RV

_{point}of all sampled points along the path.

*Q*_{k}, we define the following risk value at a point

**as the sum of**

*p**N*

_{k}(

*N*

_{k}is the number of sampled points of the

*k*th fiber, decided by the length of the fiber) point-to-point risk values by uniformly sampling the

*k*th fiber with

*N*

_{k}points $qk,1,qk,2,\u2026,qk,Nk$

*δ*

_{i}= ‖

**−**

*p*

*q*_{k,i}‖ is the distance between

**and the**

*p**i*th point (

*q*_{k,i}) along the fiber (

*Q*_{k}).

*δ*

_{max}is the threshold distance of zero risk value. If $\delta i>\delta max$,

*rv*(

*δ*

_{i}) = 0, meaning that

*q*_{k,i}is too far from

**to influence its risk value. If**

*p**δ*

_{i}≤

*δ*

_{max}, the real distance

*δ*

_{r}between

**and**

*p*

*q*_{k,i}is estimated to be of a normal distribution and risk value is the probability of $\delta r<r$. We use probability to estimate risk value for two reasons: (1) rv(

*δ*) can be interpreted as the probability of hitting the fiber with distance

*δ*; (2) The changing trend of rv is monotonically decreasing.

*r*is the cross-section radius of the CTR, and

*δ*

_{max}is manually set to be 25

*r*in this paper. The standard deviation

*σ*of the normal distribution of the distance between

**and**

*p*

*q*_{k,i}is taken as

*r*for simulation, which can be substituted per physician's preference and patient-specific images. Hence, the risk value is

*F*(

*r*;

*δ*

_{i},

*r*) −

*F*(−

*r*;

*δ*

_{i},

*r*), representing the probability of the point

**damaging the fiber point**

*p*

*q*_{k,i}.

*F*(

*x*;

*μ*,

*σ*) is the cumulative distribution function of normal distribution $X\u223cN(\mu ,\sigma )$.

*N*

_{k}points sampled from the

*k*th fiber.

*f*

_{k}is the weight of fibers. Note that all weights are 1 in this paper, i.e.,

*f*

_{1}=

*f*

_{2}= · · · =

*f*

_{M}= 1, indicating that all the fibers have equal importance, but can be changed per surgeon's input and patient-specific hematoma location.

_{arteries}at a point is computed in a similar manner to the nerve fiber risk calculation. However, to emphasize the priority of protecting arteries over the fibers, the weighted sum of RV

_{fibers}and RV

_{arteries}is introduced

*w*is a scalar weighting RV

_{fibers}and RV

_{arteries}and is chosen to be $w<0.5$ in this paper, meaning that the risk of damaging arteries is more important than the fibers as transecting arteries can potentially cause new hemorrhages.

_{tissue}, determined by the mean value of RV

_{fa}(

**) for all the**

*p**n*sample points in the brain

_{tissue}, which represents the inherent risk associated with all sample points. $\epsilon $ should be small $(\epsilon <0.1)$, indicating the damage to other tissues is trivial compared to fibers or arteries. Thus, the final risk value of a point of interest is

*p*_{1},

*p*_{2}, …,

*p*_{L}are

*L*points sampled from the parameterized CTR path. The arc length between two sample points is constant, which is equal to tube radius

*r*in the preliminary study.

It should be noted that without the introduction of the term RV_{tissue} (i.e., $\epsilon =0$), the optimization of the CTR path based on RV_{fa}(** p**) does not provide a reasonable solution, as all path points with the distance to its nearest fiber point neighbor larger than

*δ*

_{max}share the same risk value 0. Thus, the length of the path cannot be controlled without RV

_{tissue}. This is further explained in Fig. 5, where Path I and Path II have the same RV

_{path}value 0.

### 3.3 3D Risk Map.

For a given cerebral model, the risk values RV_{fa} for all sample points in the space can be precalculated to obtain a three-dimensional risk map, which can serve as a lookup table to simplify the computation of RV_{path} during the path planning and optimization procedure. Here, we create a risk map of RV_{fa} for the brain model by repeating the process from Eqs. (8)–(12) and calculate RV_{fa} for 171 × 278 × 245 sample points in the brain model, with the sampling resolution 1 mm × 1 mm × 1 mm, i.e., *r* × *r* × *r*. Figure 6 depicts the risk map of the left brain in the plane of $z=40mm$. As the figure illustrates, the peaks of the risk value (red) appear near fiber-dense area or near-arterial regions, whereas the regions far from the nerve fibers have significantly low-risk values (blue). The region, which does not intersect with the nerve fibers or arteries, but is sufficiently close to them, has a relatively lower risk value than the higher risk region.

## 4 Optimal Concentric Tube Robots Path Planning

### 4.1 Optimization of Concentric Tube Robots Path.

This section details the method to determine the optimal number and shape of a CTR for reaching an arbitrary target. Here, we limit the number of tubes to three to simplify the follow-the-leader criteria and analyze current common CTR configurations. The corresponding paths for three-tube CTR designs are SH_{0}, SA_{1}, SH_{1}, SA_{2}, and SH_{2} (*S* represents the path shape beginning with a straight tube, and *H*_{i} or *A*_{i} represents *i* number of helical or planar curved tubes, respectively). Here, we would like the emphasize that *H*_{0} is a straight line, which will be used as the control data. We use a genetic algorithm with the same hyperparameters to optimize the paths to reach a total number of 100 target points randomly generated inside half of the brain (*y* ≥ 0, where *y* is the coordinate corresponding to the origin ** O** as shown in Fig. 7, which is the centroid of the skull model). We only generate target points in half of the brain because we assume the geometry of the brain fibers, cerebral arteries, and other structures are approximately symmetrical (about the

*xz*plane) [57]. Moreover, the position and direction of an entry point (

*x*_{1}and

*R*_{1}) is also limited to half of the skull at the same side of the target points. This will rule out infeasible solutions (e.g., a path starting from one side of the brain, and penetrating through the brain to reach the target points on the opposite side).

*x*_{1}can be determined by

*β*and

*γ*, which parameterizes a straight line

*L*starting from the origin

**(center of the skull) to the skull's exterior surface. The point**

*O*

*x*_{1}is the intersection of the line and the surface. The starting direction

*R*_{1}can be determined by the deviation angle

*β*

_{d}and

*γ*

_{d}and the normal direction unit vector

*z*_{nm}, which can be computed from the skull model shown in Fig. 7. To get the insertion orientation,

*R*_{nm},

*x*_{nm}and

*y*_{nm}are introduced

*R*_{1}can be specified by

*x*_{1}and

*R*_{1}are substituted for path parametrization and the final input vectors for the genetic algorithm are

Notice that the target point *x*_{end} does not appear in Eq. (19) since *x*_{end} will be constant once the target is selected. The genetic algorithm only needs to optimize the parameters in Eq. (19) to reduce the risk value of the path. The parameters and constraints for the genetic optimization algorithm are summarized in Table 3.

Items | Parameters | Values/Ranges |
---|---|---|

Path | $r$ | 1 mm |

$\epsilon $ | $0.01$ | |

GA input vector | β | $[0,\pi ]$ |

γ | [0, π/6] | |

β_{d} | $[\u2212\pi ,\pi ]$ | |

γ_{d} | $[0,\pi /4]$ | |

α_{1}, α_{2} | $[0,\pi ]$ | |

κ_{2} | $[\u22120.2,0.2]$ | |

η_{2} | $[\u2212\pi /2,\pi /2]$ | |

l_{1} | $[10mm,80mm]$ | |

l_{2} | $[0,80mm]$ | |

Genetic algorithm | Generations | 25 |

Population size | 1000 |

Items | Parameters | Values/Ranges |
---|---|---|

Path | $r$ | 1 mm |

$\epsilon $ | $0.01$ | |

GA input vector | β | $[0,\pi ]$ |

γ | [0, π/6] | |

β_{d} | $[\u2212\pi ,\pi ]$ | |

γ_{d} | $[0,\pi /4]$ | |

α_{1}, α_{2} | $[0,\pi ]$ | |

κ_{2} | $[\u22120.2,0.2]$ | |

η_{2} | $[\u2212\pi /2,\pi /2]$ | |

l_{1} | $[10mm,80mm]$ | |

l_{2} | $[0,80mm]$ | |

Genetic algorithm | Generations | 25 |

Population size | 1000 |

_{path}. However, when solving Eq. (7) to satisfy the follow-the-leader constraint in the optimization process for an SH

_{2}configuration, it cannot guarantee the difference of torsion |

*τ*

_{3}−

*τ*

_{2}| to be exactly zero; i.e., the path is perfectly following the leader. To compensate for this, the objective function of the genetic algorithm is updated as follows:

*ρ*is a coefficient weighting the difference between the torsions of two helical tubes in SH

_{2}. It is noteworthy that

*ρ*is only introduced for SH

_{2}configurations.

To find the best *ρ* value that can reduce path risk value with relatively low |*τ*_{3} − *τ*_{2}|, we vary *ρ* from 0 to 20 and obtain the results shown in Fig. 8. The results indicate that a reasonable *ρ* value can lead to a smaller difference between *τ*_{2} and *τ*_{3} without significantly affecting path risk value. According to Fig. 8, we choose *ρ* to be 10 for all SH_{2} path optimization configurations in this paper.

## 5 Results

To evaluate the risk values of five different geometrical designs, we compared the mean risk value of 100 optimized paths for the same set of targets with the path shape being SH_{0}, SA_{1}, SH_{1}, SA_{2}, and SH_{2}. The value of *w* (the weight between RV_{fibers} and RV_{arteries}) is set to 0.1, 0.2 and 0.3 to get three data groups as shown in Fig. 9.

From Fig. 8 and Table 4, it is shown that the introduction of planar curved (SA_{1}, SA_{2}) or helical (SH_{1}, SH_{2}) tubes can efficiently reduce the risk value of the path compared to one obtained with SH_{0}. On average, the risk value can be reduced from a range of 83.4–84.2% for different *w* values. This is primarily due to the improved system dexterity provided by additional curved tubes, enabling safe paths toward the target while minimizing the damage to the surrounding obstacles (i.e., white fiber tracts and blood vessels). Comparing the helical tube configurations to the planar curved ones, we can conclude that the robot with helical tubes (SH_{1}, SH_{2}) can achieve a slightly lower risk value. This is likely due to the increased dexterity provided by helical tubes compared to the planar curved tubes. This can be further explained by the mathematical formulation of helical and planar curved arcs, where more parameters are required in the complete definition of the helical arc, providing additional parameters for defining a safer path. For the same reason, multiple inner tube CTRs (SA_{2}, SH_{2}) are more dexterous than single inner tube CTRs (SA_{1}, SH_{1}) and provide additional tunable parameters. Thus, multiple inner tube CTRs can provide lower risk values. Additionally, it is noteworthy that the risk values are lower with smaller *w* values since fibers have a higher density within the brain compared to arteries, and hence, most path risk values are mainly dominated by the fibers.

Mean(RV_{path}) | |||||
---|---|---|---|---|---|

SH_{0} | SA_{1} | SH_{1} | SA_{2} | SH_{2} | |

w = 0.1 | $20.667$ | $3.650$ | $3.482$ | $3.534$ | $3.442$ |

w = 0.2 | $27.160$ | $5.149$ | $4.800$ | $4.775$ | $4.475$ |

w = 0.3 | $33.403$ | $6.234$ | $6.078$ | $5.844$ | $5.267$ |

Mean(RV_{path}) | |||||
---|---|---|---|---|---|

SH_{0} | SA_{1} | SH_{1} | SA_{2} | SH_{2} | |

w = 0.1 | $20.667$ | $3.650$ | $3.482$ | $3.534$ | $3.442$ |

w = 0.2 | $27.160$ | $5.149$ | $4.800$ | $4.775$ | $4.475$ |

w = 0.3 | $33.403$ | $6.234$ | $6.078$ | $5.844$ | $5.267$ |

By analyzing the actual optimized paths of CTR as shown in Figs. 10 and 11, we note that the risk value improvement depends on the target locations within the brain. As can be seen in Fig. 10, the introduced arcs do not lead to an obvious improvement in the damage ratio (2.4–6.7%) when the targets are located superficially in the brain. When the targets are located deeper inside the brain, the dexterity of the multi-tube designs can contribute to significant damage reduction, as shown in Fig. 10. We selected three scenarios here to show that the improvement of damage ratio can be significant (36% versus 83%). However, Fig. 9 also indicates that the damage reduction from single inner tube designs to multiple inner tube designs is not significant (3.2–6.3% for SA_{1} to SA_{2} and 1.2–6.8% for SH_{1} to SH_{2}). With our cerebral model being sparse in space (most of the cerebral space is not occupied by fibers or arteries), the extra segment of the needle tube is redundant for reaching most targets, and the second segment of the CTR is optimized to be short and neglectable as shown in Fig. 11, leading to the similar risk value obtained between SA_{1} and SA_{2}, SH_{1} and SH_{2}, respectively.

Another important conclusion, we want to highlight is that there is no significant difference between the helical tube and planar curved tube designs. Only a trivial damage reduction can be obtained from SA_{1} to SH_{1} (2.5–6.8%) or from SA_{2} to SH_{2} (2.6–9.9%). The main reason is that the spatial distribution of the chosen neural fibers and cerebral arteries in our model is concentrated in certain regions. Thus, the simple planar curved tube design can enable the CTR to reach most targets among 100 random points safely without increasing the high-risk value by avoiding these congested regions. As shown in Figs. 11(b) and 11(c), the optimized SA_{1} and SH_{1} paths tend to achieve similar performance, indicating that helical tubes behave like planar curved tubes and minimize path length. Further, the extra torsional dimension of helical tubes does not provide any prominent improvement to the damage ratio. Despite the minor improvement in risk values with the use of helical tubes, the planar curved tubes are much easier for fabrication, robot design, and control. Thus, taking all the aforementioned factors into consideration, we can conclude that the CTR with SA_{1} is the most feasible solution compared to other tube geometries mentioned in this paper.

## 6 Conclusion

In this article, we proposed the optimal design and path planning for CTR-enabled ICH removal. The CTR design starts with five potential solutions, which include SH_{0}, SH_{1}, SH_{2}, SA_{1}, and SA_{2}, which represents a straight path, a straight path with one helical segment, a straight path with two helical segments, a straight path with one planar curved segment, and a straight path with two planar curved segments, respectively. A custom-defined risk value is proposed to minimize the damage to nerve fibers, arteries, and total needle insertion distance. The optimal CTR design is searched using the genetic algorithm that minimizes the risk value among 100 targets inside the brain. The results indicate that the introduction of curved tubes (either planar curved tubes or helical tubes) can significantly minimize the risk value compared to the one obtained with the conventional straight tube. There is no significant difference between the number (1 or 2) and shape (planar curved or helical) of the introduced inner tubes. Considering the practical challenges in CTR hardware design, fabrication, and control, we conclude that the CTR with a planar curved inner tube is the preferred design for the proposed ICH evacuation procedure. Our future work will focus on integrating the global path plan method proposed in this paper with our TSGP controller [58] and implementing it on our ICH evacuation robot.

## Acknowledgment

The authors would like to thank Cosmo Xiao for performing the preliminary analysis.

## Funding Data

This project was funded by the Georgia Tech McCamish Blue Sky Grant. Research reported in this publication was also supported by the National Institute of Neurological Disorders and Stroke of the National Institutes of Health under Award Number R01NS116148. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

No data, models, or code were generated or used for this paper.

## Appendix

*x*_{end}is connected to the starting point of the

*n*th tube

*x*_{n}by three intermediate vectors

*r*

_{n}can be eliminated to get the following equation:

Note that solving for Eq. (A7) can lead to multiple solutions. We intuitively hypothesized that the optimized solution is the minimum absolute value of $\varphi n*$ to exclude unreasonable solutions where the helical segment wraps around the axis for multiple revolutions. Thus, *ϕ*_{n} can be numerically solved from Eq. (A7). Then, *η*_{n} and *r*_{n} can be solved according to Eqs. (A6) and (A4).