Brain cancer is a devastating ailment with few long-term survivors. In the United States, over 18,000 people contract intraparenchymal brain tumors each year, while over 13,000 succumb to it each year, not including the many more who suffer from intracranial metastases. Because these tumors are characterized by rapid invasive growth through the surrounding tissue, full surgical resection is generally impossible without risking further neurological disorders. Combining surgical resection with chemotherapy and radiotherapy is a popular method for treatment, though these treatments are generally not curative despite advances in imaging, radiotherapy, and chemotherapeutic particles.
Given the state of brain cancer treatment, interest has arisen in compartmental therapy for brain tumors. This technique refers to the application of therapeutic agents directly to the target zone, therefore bypassing the blood-brain barrier and reducing the risk of damage to other tissue. The most well known minimally-invasive example of this is convection-enhanced delivery. Currently, in order to minimally invasively access a deep zone of the brain, standard stereotactic practices can be utilized. However, current practices have their limitations and risks due to the limited degrees of freedom inherent in a straight needle. For example, there is no ability to steer around critical structures or vessels. Further, if the needle is off-target, it needs to be fully retracted and reinserted starting from the cortical surface, creating an entirely new channel through the brain. In addition to these limitations, two developments have become apparent in convection-enhanced delivery. The first is that catheter placement into the surrounding tissue around a tumor is more favorable for drug distribution than placement directly into the tumor. The second, more intuitive development is that the accuracy of catheter placement directly correlates with patient survival. Given current stereotactic practices, in order to optimally deliver therapeutic particles into a tumor, multiple insertions have to be made, reducing the minimally invasive nature of the procedure. Coupled with the potential inaccuracy of catheter placement, catheter-based delivery for compartmental therapy has a number of inherent risks. A system which could address these potential risks by being able to steer around critical structures or vessels and alter its trajectory without full retraction could improve the precision, versatility, and safety of local therapy for lesions. In addition, this system might also make resection more feasible in areas of the brain which are currently inaccessible, because manipulation of tissue could be performed on an awake patient through a burr hole craniostomy.
The proposed technique involves exploiting the tendency of a bevel tip needle to bend during tissue insertion. When a bevel tip needle is inserted into tissue, a deflection force causes the needle to bend in the direction of its tip, with a curvature dependent on the stiffness of the needle relative to the surrounding tissue. A straight trajectory can also be achieved with a bevel tip needle by spinning the needle during insertion, with a rotational velocity relatively larger than the insertion velocity. Combining these two trajectories, other arbitrary trajectories can be envisioned in 3-D by combining short segments, some straight and some curved, as desired. Consequently, any curvature between these two limits, namely the maximal natural curvature and the minimal zero curvature, can be achieved by spinning the needle with a duty cycle, provided the angular orientation of the needle tip remains the same during all non-spinning segments. Longer non-spinning segments produce curvatures closer to the maximal natural curvature, whereas longer spinning segments produce curvatures closer to the minimal zero curvature. Further, the duty cycle period, or the time of one non-spinning segment in addition to the time of one spinning segment, can be appropriately minimized in order to create a smooth curve as opposed to a concatenation of straight and curved trajectories. Thus, this technique provides proportional control over both the direction of the needle and the amount of curvature. The goal is to build a needle driving robot providing control over both insertion and rotational speed of the needle, and to experimentally derive and validate the forward and inverse kinematics of the needle trajectory through tissue.
Given both insertion and rotation speeds of the needle as control inputs into a kinematic nonholonomic system, a needle trajectory not incorporating duty-cycled spinning can be modeled as a variant of a kinematic bicycle, with constant front wheel angle, phi, wheel to wheel distance, l1, back wheel to needle tip distance l2, insertion speed u1, and rotation speed u2.
Let e1, e2, and e3 represent the three dimenional unit vectors correponding to the positive x-, y-, and z- directions. Using the homogeneous matrix representation, let
represent the rigid tranformation between frames A and B. Assuming l1 does not equal 0 and phi ranges between 0 and pi/2 radians, the nullspace of A can be characterized by two directions,
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such that V1 corresponds to pure needle insertion, V2 corresponds to pure needle rotation, and kappa corresponds to
.This leads to the kinematic relation,
and consequently,
where ^ represents the se(3) twists of the 6x1 twist coordinates, i.e.
.
Solving for gab yields
.
Given the homogeneous transformation matrix between frames A and B, the needle tip coordinates are represented by,
.In order to incorporate duty-cycled spinning during insertion, the insertion and rotational velocities need to become time dependent. Given a duty cycle of
where tau is the duration of the spinning portion and T is the duty cycle period, the insertion and rotation speeds can be represented as,
where j is the set of non-negative integers {0, 1, 2, 3, ...}. Substituting the new time dependent velocities into the homogeneous transformation between frames A and B yields
As a result, given the non-spinning 0% duty cycle bicycle model parameters, kappa and l2, the duty cycle parameters, tau and T, and the insertion and rotation speeds, u1 and u2, all possible needle trajectories incorporating duty cycled spinning can be simulated and planned.
In order to validate the model a needle steering robot was first constructed. The base of the robot consists of a McMaster-Carr 6730K1 Precision Ball Bearing Slide with a stroke length of 147 mm and a screw lead of 0.1 inches. Mounted onto the slide is a rotational subassembly consisting of a rotation motor attached to a luer-lock. The luer-lock is used to hold the needle in place. The slide itself is actuated by an insertion motor. Both insertion and rotation motors consist of Faulhaber 2342-S-012-CR DC-Micromotors, Series 30/1 Planetary Gearheads with a 3.71:1 reduction ratio, and HEDM-5500-B Optical Encoders. Altogether, both of the insertion and rotation motor/encoder/gearhead systems provide 14,480 counts per revolution. The motors are controlled via Faulhaber LC3002-A Linear 4-quadrant Servo Amplifiers, and a National Instruments PCI-7344, 4 axis Servo/Step Motion Controller card. User programming and control is achieved through NI LabVIEW.
In considering the needle to be used for experimentation, the needle curvature depends on the flexibility of the shaft, which can be maximized by decreasing the diameter. However, in decreasing the diameter, the amount of bevel tip surface area over which the deflection force can act is decreased, causing a decrease in deflection. As a result, a custom needle prototype was created in order to accommodate both a large bevel tip surface area while maintaining a relatively small shaft diameter. The needle consists of a 16-gauge stainless steel hypodermic needle tip with a 10-degree bevel, attached to a 29-gauge nitinol wire. The needle tip is bent a further 15-degrees in order to increase the surface area on which the deflection force can act. Nitinol was chosen as the alloy for the needle shaft due to its superelastic nature, or its ability to return to its original shape after undergoing large deformations.
The needle steering robot was used to insert the custom needle into in vitro phantom material. The material used was Knox gelatin (Kraft Food Global Inc.), mixed at a ratio of 20 cc water to 1.3 cc gelatin. This mixture has been used as a brain tissue substitute due to it having a similar needle insertion force profile to that of in vitro calf brain. The gelatin was prepared in multiple small bowls with holes cut into the sides for insertion of the needle. In each insertion of the needle, a 1 cm2 grid was placed under each bowl in order to provide a quantitative reference. The needle was inserted into each bowl only once so as to avoid any influencing of the needle trajectory by preexisting needle tracks. For each trial, the needle was inserted with a constant insertion speed, u1, of 0.5 cm/s and rotation speed, u2, of 2 rev/s. Trials for duty cycles of 0%, 33%, 67%, and 100% were performed.
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The resulting digital images of the various trails were used to extract twenty points in pixel coordinates of each of the needle trajectories. Nine previously designated calibration points in both pixel coordinates and physical coordinates were also obtained. Registration between pixel space and physical space was performed by use of the Iterative Closest Point (ICP) algorithm. This allowed transformation of the needle trajectory from pixel coordinates to physical coordinates. The insertion point coordinates for each needle trajectory were subtracted from the entire trajectory so that each began at the origin in physical space.
Given twenty points of each needle trajectory in physical space, inverse bicycle kinematics not incorporating duty cycled spinning were used to extract the kappa and l2 model parameters. A single arc generated by the kinematic bicycle model, beginning at the origin, has a center at
,and a radius of
.Utilizing the equation of a circle in the zy coordinate plane,
,in conjunction with the Matlab function solve, the y coordinate prediction value is attainable for any given z coordinate value. Kappa and l2 model parameters were derived for each trial by fitting this model to the experimental data using the Matlab function nlinfit.
Consequently, the bicycle model parameters corresponding to the 0% duty cycle trajectory were imposed on the forward kinematic model in order to provide a model basis. The 33%, 67%, and 100% duty cycle trajectories were simulated using the experimentally derived 0% duty cycle model parameters.
The inverse kinematics were again used in order to derive the model parameters of the 33%, 67%, and 100% duty cycle simulated trajectories. As a result, model parameters were found for all experimental trials and all simulated trials, providing a comparative basis for determining whether or not the bicycle model accurately models needle insertion via duty cycled spinning.
The results of the model parameter derivation for the experimental needle trajectories are shown in Table 1. The results of the model parameter derivation for the simulated needle trajectories are shown in Table 2.
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While the root mean squared error between experimental trajectory and simulated trajectory kappa values is only 0.0063 cm-1, the deviation between experimental and simulated trajectory l2 values is much larger. In single curve trajectories such as those presented here, l2 dictates the z-coordinate of the center of the arc. Assuming l2 is zero implies that the needle is entering the tissue at a point where the derivative of the generated arc is equal to zero. Variations in l2 from zero imply a modification in the initial entry angle of the needle. Conversely, curvature dictates both the arc center y-coordinate and the radius of the trajectory arc, having more of an effect than l2 on generated single curve needle trajectories as the needle travels through tissue. Essentially, the deviation between experimental and simulated trajectory l2 values denotes a variability in initial entry angle which could have been caused by slight kinking outside of the tissue or improper alignment of the needle and the gelatin. As a result, the model is still considered to be a good fit due to the very low root mean squared error between experimental and simulated trajectory kappa values. The model is qualitatively shown to be a good fit due to the strong linear relationship between |kappa| and D for both the simulated and experimental needle trajectories.
An accurate kinematic model that quantitatively describes needle steering via duty-cycled spinning allows for a variety of possibilities in surgical procedures. Peroperative trajectory planning and intraoperative closed-loop systems incorporating duty-cycled spinning become possibilities. Current invasive surgical procedures can be performed minimally invasively, reducing tissue manipulation and exposure. Improvements in current diagnosis and treatment accuracy also become attainable. Ongoing and future research activities include further validation of the kinematic model for needle insertion via duty-cycled spinning, specifically for multiple curve and three dimensional trajectories. A dynamic model will also be pursued, which will allow for robust modeling of needle insertion of arbitrary needle types through various tissues and/or materials. Intraoperative electromagnetic tracking of the needle tip via the microBIRD (Ascension Technology Corp.) system will also be incorporated into the needle steering system, creating a fully closed-loop system. Testing in both human cadaver and animal tissue will follow.
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