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Estimation of the Blood Velocity for Nanorobotics Matthieu Fruchard, Laurent Arcese, Estelle Courtial

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Estimation of the Blood Velocity for Nanorobotics Matthieu Fruchard, Laurent Arcese, Estelle Courtial Estimation of the blood velocity for nanorobotics Matthieu Fruchard, Laurent Arcese, Estelle Courtial To cite this version: Matthieu Fruchard, Laurent Arcese, Estelle Courtial. Estimation of the blood veloc- ity for nanorobotics. IEEE Transactions on Robotics, IEEE, 2014, 30 (1), pp.93-102. 10.1109/TRO.2013.2288799. hal-01071275 HAL Id: hal-01071275 https://hal.archives-ouvertes.fr/hal-01071275 Submitted on 3 Oct 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Estimation of the blood velocity for nanorobotics Matthieu Fruchard, Laurent Arcese, Estelle Courtial Abstract—The paper aims at estimating the blood velocity to enhance the navigation of an aggregate in the human vascu- lature. The considered system is a polymer binded aggregate of ferromagnetic nanorobots immersed in a blood vessel. The drag force depends on the blood velocity and specially acts on the aggregate dynamics. In the design of advanced control laws, the blood velocity is usually assumed to be known or set to a constant mean value to achieve the control objectives. We provide theoretical tools to on-line estimate the blood velocity from the sole measurement of the aggregate position and combine the state estimator with a backstepping control law. Two state estimation approaches are addressed and compared through simulations: a high gain observer and a receding horizon estimator. Simulations illustrate the efficiency of the proposed approach combining on- line estimation and control for the navigation of an aggregate of nanorobots. Fig. 1. Aggregate of nanorobots in a blood vessel. (a) Aggregate binded by Index Terms—Blood velocity, high gain observer, receding ligand, including a payload; (b) aggregate binded by magnetic and interaction horizon estimator, magnetic nanorobots. forces. I. INTRODUCTION Whatever the proposed design, these systems are subject There has been a growing interest in the development of to different forces whom modelling is necessary in order to therapeutic microrobots and nanorobots for some years [1]. estimate their dynamic behaviors in a fluidic environment Such systems have the potential to perform complex surgical [15], [16]. The blood velocity is usually assumed to be known procedures without heavy trauma for the patients [2], [3], or set to a constant mean value whilst it is a key nonlinear [4], [5], [6]. Among the numerous prototypes developed, parameter of the drag force which prevails at a small scale. those possessing a deported actuation are the most promising. The measurement of the blood velocity is a difficult task that Indeed these robots do not embed any energy source thus is often assigned to ultrasonic sensors, at least in vessels inducing smaller sizes. Untethered robots thus benefit from a close to the sensor. This solution calls for an end-effector possible better therapeutic targeting since they can access to servoing so as to track the aggregate position. Another hard-to-reach body’s area and also lessen medical side effects solution relies on a priori knowledge of the blood velocity. linked to surgical operations. Works related to the numerical resolution of the velocity profiles using Computational Fluid Dynamics software had The kind of deported actuators required to control such been reported in the literature [17], [18]. But these studies robots depends on the propulsion strategy. A variable magnetic can not be used for real-time purposes because they are field is necessary for robots with elastic flagellum [7], [8] based on the Navier Stokes equations whom resolution is or helical flagella [9]. The control of bead pulled robots or computational time consuming. In [15], the authors proposed swarm of robots [10], [11], [12] is provided by the magnetic analytical expressions of the blood velocity profiles. These field gradients of either Magnetic Resonance Imaging (MRI) expressions are valid in the neighborhood of a bifurcation devices or magnetic setups (e.g. from Aeon Scientific or but require an accurate knowledge of the vessel geometry. Stereotaxis). So as to benefit both from a large motive force The blood velocity could also be considered as a disturbance in the macrovasculature and from a possible break up in and be rejected in the control law, like in [19] for breathing the microvasculature (so as to avoid undesired thrombosis compensation. However the knowledge of the blood velocity and to improve the targeting), a promising approach is to is relevant for the navigation since the main force depends on consider aggregates. Such aggregates of magnetic nanorobots it. Finally the on-line estimation of the blood velocity seems are binded either by a biodegradable ligand [13] or by to be an interesting approach to avoid the drawbacks of the self-assembly properties [14] as shown on Figure 1. aforementioned methods. The purpose of this paper is to propose an on-line state M. Fruchard and E. Courtial are with PRISME EA 4229, Univ. Orleans,´ 63 Av de Lattre de Tassigny, F18020, Bourges, estimation of the blood velocity. The latter is modeled by a France. [email protected], truncated Fourier series. The model of the nanorobot dynamics [email protected] taking into account the blood velocity behavior is proved to L. Arcese is with CReSTIC EA 3804, Univ. Reims Champagne- Ardenne, Moulin de la Housse, BP 1039, 51687 Reims cedex 2, France. be observable. Two different state observers are then designed [email protected] and combined with a backstepping law to achieve a reference TABLE I PHYSIOLOGICAL PARAMETERS[20], ACTUATION AND IMAGING. X STANDS FOR A COMPATIBLE DEVICE. Physiological Parameters (mean values) System Design Actuator (∇B in mT.m−1) Imager resolution in m Blood Clinical Pre-clinical Maxwell Cardiovascular Network Diameter Required µ-MRI MRI µ-MRI CT1 µ-CT2 OCT3,4 US5,4 velocity MRI MRI Coils −3 −4 −3 −5 −6 −3 (m) −1 gradient (≈ 450) < 10 < 10 < 10 < 510 < 10 < 10 (m.s ) (< 80) (≈ 100) Setup −2 −1 Arteries 10 2.510 60 X X X X X X X X X X Small 10−3 210−2 90 - X XX X X X X X X Arteries Arterioles 10−4 10−3 250 - - XX - X - XX - Capillaries 10−5 510−5 2800 - - -X - - - - X - 0.12 trajectory tracking. The paper consists in five sections. Section Blood velocity v0 =<v> II details the need for a blood velocity estimation and proposes 0.10 v1 a generic blood velocity model. Then, in Section III, the v2 v ) 0.08 3 nonlinear model of an aggregate navigating in a blood vessel is 1 presented. Section IV is dedicated to the design of observers: − m.s a high gain observer and a receding horizon estimator are ( 0.06 addressed to on-line reconstruct the blood velocity. Finally, simulations illustrate the efficiency of the proposed approach 0.04 combining control and estimation in Section V. Conclusions 0.02 and discussions on open issues are summarized in Section VI. Blood velocity 0 II. WHY AND HOW TO ESTIMATE THE BLOOD VELOCITY? −0.02 0 0.20.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Magnetically actuated robots are classically described Time (s) [11] by the motive magnetic force induced by the magnetic field or gradient depending on the robot design [1], the Fig. 2. Blood velocity v (black solid line), mean value v0 (gray solid line) and nth order truncated Fourier series vn: v1 (blue dots), v2 (black dashes) apparent weight and the drag force induced by a non and v3 (green crosses). pulsatile blood flow. Hydrodynamic wall effects show that a partial vessel occlusion by the particles results in an optimal ratio between the microrobot and the vessel radii, the vasculature, the most disturbed by external time-varying denoted r and R respectively. As suggested in [14], a perturbations is the drag force because of the pulsatile first approximation for dimensioning the actuators relies blood flow. Several possibilities to deal with this periodic on the study of the optimal ratio of motive over external perturbation might be considered: direct measurement, forces. Table I summarizes these technical constraints and disturbance rejection or blood velocity estimation. the consequences on the choice of both actuators and imagers. Physiological periodic perturbations are usual issues in Optimality requires that the robot goes smaller as it medical robotics and are often addressed using periodic enters smaller vessels. Recently the idea of using microscale signal rejection: either using filtering of the output or using a carrier made up of a biodegradable aggregate of iron-cobalt precompensation in the control law, like in [19] for breathing nanoparticles and doxorubicin has been rendered possible rejection. For the present case, both solutions require the blood [13]. In large vessels, a millimeter-sized robot has an velocity measurement. However the pulsatile blood velocity important propelling force making possible to resist to the is a local information and is consequently more difficult to blood drag. In small vessels where the blood flow is lower, be measured. Studies have been carried out to measure the a micrometer-sized robot can ensure a sufficient thrust while blood velocity using the Doppler effect with ultrasound or avoiding any injury caused by embolism or thrombosis. MRI devices [21], [22]. The quality of the measurement Furthermore the motive force is volumetric whilst the drag depends on the ultrasound transducer orientation during the force is —at best— surfacic which implies an increasing acquisition and it would require to ensure the servoing of the prevalence of the drag force as the vessel radius decreases.
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