Instrumented Needle Sensors Positioning Method based on Experimental Data

Pierre-Loup SCHAEFER, Gr égory CHAGNON and Alexandre MOREAU-GAUDRY Univ. Grenoble Alpes, CNRS, Grenoble INP Institute of Engineering Univ. Grenoble Alpes, TIMC-IMAG, F38000 Grenoble France Contact: pierre-loup.schaefer@univ-grenoble-alpes.fr I nstrumented needle are interesting tools to use during minimally invasive surgery procedures. Their use allows to reconstruct the shape of the needle during the procedure by using the data of the embedded sensors of the needle. Thus the shape of the needle is accessible to the physician in real time and without further use of medical imaging. This papers focuses on the reconstruction process of the shape of the needle and specially on the positioning of the sensors by proposing a novel method based on experimental data.

Needle Shape Reconstruction

Sensor data processing

An instrumented needle is composed by a standard medical needle and a certain number of sensors. This article focuses on needles with strain sensors embedded on its surface. The technologies of the sensors themself are beyond the scope of this article [START_REF] Abayazid | 3d flexible needle steering in soft-tissue phantoms using fiber bragg grating sensors[END_REF][START_REF] Park | Real-time estimation of 3-d needle shape and deflection for mriguided interventions[END_REF]. We just consider that the data retrieved from the sensors give the strain on the surface of the needle. The sensors are placed following the geometry presented in the Fig. 1. Thus the sensors are placed by groups of three on sections of the needle and with 120 degrees between them. Each of this group is called a triplet. These positions allows for each triplet of sensors to retrieve two scalar values representatives of the bending of the needle at the triplet locations: the curvature κ and the bending angle θ. Eventually the values of all triplets of sensors constitute a discretization of the curvature function and the bending angle functions.

S 1 S 2 S 3 120 •

Geometric reconstruction

The values of curvature (κ i ) i=1,n and bending angle (θ i ) i=1,n retrieved from the sensor triplets are interpolated to give the curvature estimate κ est and the bending angle estimate θ est . These estimates are then used to reconstruct the shape of the needle itself. According to the hypothesis that the mechanical torsion of the needle is negligible the material frame of the needle is a rotation minimizing frame (RMF) [START_REF] Rida | Rational approximation schemes for rotation-minimizing frames on pythagorean-hodograph curves[END_REF][START_REF] Wang | Computation of rotation minimizing frames[END_REF], which is a special type of moving frame. Let (T, N 1 , N 2 ) be the RMF of the needle, we then have the following differential system:

d ds   T(s) N1(s) N2(s)   = κ(s)   0 cosθ(s) -sinθ(s) -cosθ(s) 0 0 sinθ(s) 0 0     T(s) N1(s) N2(s)  
Solution of this system represents the evolution of the RMF of the needle all along its length. By integration it is then possible to obtain the full shape of the needle.

Strain Sensors Locations

The reconstruction of the shape of the needle raises the question of the sensor locations: What locations give the most accurate needle shape reconstruction? These locations are defined as optimal locations. The optimal locations are defined as the locations that minimize the error of reconstruction of the needle shape. Thus the problem of sensor locations can be considered as a minimization problem. To evaluate the relevancy of sensor locations, existing needle shapes are used. For any locations the sensor data are simulated with these reference needle shapes and are then used to obtain the reconstructed needle shape. The comparison of those two shapes then gives the reconstruction error. On the difference of the methods available in the literature [START_REF] Abayazid | 3d flexible needle steering in soft-tissue phantoms using fiber bragg grating sensors[END_REF][START_REF] Park | Real-time estimation of 3-d needle shape and deflection for mriguided interventions[END_REF], the method of sensor location optimization presented in this article is based on experimental data and is fully three-dimensional.

Study of a experimental needle set

The sensor locations method is based on experimental data coming from a needle insertion study. The study was carried out on a set of CT scans of needles insertion into pig shoulder. The choice of pig tissue is due to its similar properties with human tissues.

The study consists in retrieving the property of the needle shape from the CT scans. The different steps include:

• Segmentation • B-spline smoothing

• Rotation minimizing frame computation

Finally for each needle is associated an estimate of its curvature and an estimate of its bending angle.

Minimization problem

The research of the sensor optimal locations is a problem of dimension n, where n is the number of sensor triplet. It falls under the category of Mixed Integer Nonlinear Programming category of minimization problem. Thus special algorithms have been deployed to obtain the solutions [START_REF]MATLAB and Global Optimization Toolbox[END_REF][START_REF] Pauer | Non-linear compensation of production inaccuracies and material drift by adjusting the sensor data fusion algorithms for shape sensing based on fbg-optical fibers[END_REF].

Results

Table 1 expose the results of the optimal locations computation from the experimental set of pig data for a 200 mm needle.

Number of sensor triplets

Sensor triplets optimal locations (mm) Fig. 2 presents the reconstruction error with the optimal sensors locations obtained compared to arbitrary equidistant sensor locations. We observe from Fig. 2 that the use of the optimal locations over equidistant locations decreases both the mean and the maximum of the reconstruction error. Notably the mean of reconstruction error decrease by 17 % to 57%.

Conclusion

This paper presented a novel method to compute the optimal sensor locations of an instrumented needle based on experimental data and fully three-dimensional. The results presented here expose the gain of accuracy obtained with the use of the optimal locations computed on the same set of experimental data. Further process of validation conducted on clinical data (not presented here) also confirm the gain of accuracy made by using the optimal locations. We conclude that the methode developed and presented here offers a gain of accuracy of the needle shape and reliable sensor locations for insertions of an instrumented needle in different type of tissues. The work presented in this paper is supported by the French ANR-12-TECS-0019.

Figure 1 :

 1 Figure 1: Three strain sensors S1, S2, S3 placed on a circular section of a needle with a 120 • angle between them.

Figure 2 :

 2 Figure 2: Boxplot of the reconstruction error of the pig needle set for a number of sensors triplets between 1 and 5 according to sensors locations. Ends of the whiskers represent minimal and maximal reconstruction errors.

Table 1 :

 1 Optimal sensor triplet locations obtained from the pig needle set. Locations are expressed as distances in millimeters from the proximal extremity of the needle.

	1	81
	2	28, 101
	3	25, 77, 136
	4	9, 40, 95, 144
	5	11, 34, 77, 128, 171


Thierry Alonso
email: thierry.alonso@univ-grenoble-alpes.fr

Denis Favier

Grégory Chagnon

How the evolution of the dynamic elastic modulus during isothermal tensile tests can bring new information on mechanisms deformation of a NiTi superelastic wire

How the evolution of the dynamic elastic modulus during isothermal tensile tests can bring new information on mechanisms deformation of a NiTi

superelastic wire

Introduction

Thermomechanical behaviour of NiTi SMA is due to the complex interaction between several deformation mechanisms including lattice elasticity, phase transformations between austenite (A), martensite (M) and R-phase (R), reorientations of R and/or M variants, twinning and detwinning phenomena and usual plasticity due to dislocation movements. These phenomena are difficult to identify by considering only stress-strain curves of tensile test. Classically the Dynamical Mechanical Analysis (DMA) is performed during a temperature sweep under a constant low stress and storage modulus E' is measured as function of temperature. In this presentation, superelastic deformation behaviour of NiTi thin wire was investigated by using a non-conventional DMA which allows to measure the storage modulus as function of strain during isothermal tensile tests at high level of stress. DMA tests were performed during superelastic tensile tests up to 10% of strain and up to failure at three temperatures above Af (the reverse martensitic transformation finish temperature). The goal is to identify and study elasticity, phases transformations, localisation, plasticity, by analysing storage modulus evolution during isothermal superelastic tensile tests. From experimental results a model is proposed to simulate storage modulus evolution during isothermal tensile test. This model and the results of the three isothermal tests are used to determine the Clausius-Clapeyron coefficients for the forward A-M and reverse M-A transformations.

Experimental Results

Sress-strain curve (Fig. 1a) exhibits a classical superelastic behaviour. During loading, E' decreases monotonically between (1) and ( 2) with an initial value of 70 GPa. At point (2), matching with the start of the loading stress plateau, E' reaches 52 GPa. Between (2) and ( 3), E' continues to decrease almost linearly down to E'3=35 GPa. At the end of the plateau in (3), E' starts to increase monotonically to reach 52 GPa in (4) to become constant between (4) and (5). Fig. 3b shows the evolution of E' as function of stress.

Discussion

Identification of deformation phenomena present in NiTi wire from stress and storage modulus E' evolution Analysing both stress E' during a tensile test, allows to identify different zone (I to IV) and associated phenomena. E' evolution brings new information and clarifies the ambiguities we may have by analysing only stress as function of strain. Table 1 shows the evolution of stress and E' to identify elasticity, transformation, localisation and plasticity during loading. To simplify interpretation, arrows are used to code evolution. ,,  mean that the value is constant, increases and decreases with increasing strain successively. For the NiTi wire studied, there is no stage with constant value of the storage modulus ; it means that there is no deformation stage for which the deformation is purely elastic or purely plastic, even for tests performed in the temperature range of superelastic behaviour.

Elastic moduli of austenite and martensite

Elastic moduli of austenite and martensite are usually measured by analysing the slope of the stress-strain curve. However the continuous decrease of the storage modulus E' between ( 1) and ( 2) shows the difficulty to measure elastic modulus of austenite from this initial slope. Estimating the elastic modulus from the initial stress-strain slope leads to underestimate elastic modulus of austenite. Elastic modulus of oriented martensite is often estimated from the slope of the stress-strain curve after the stress plateau The continuous increase of E' between (3) and ( 5) shows the impossibility to measure the elastic modulus of martensite by measuring this slope. A better approximation is obtained by considering the values of the storage moduli E' at high temperature and low stress for the austenite and at low temperature and very high stress for martensite.

Modelling of DMA results to determine Clausius-Clapeyron coefficients

A model is proposed for different zone and especially for zone (III+IV). First, zone (III+IV) at T=60°c was fitting by equation (1) (fitting parameter : K3) and was plotted in Fig. 1b. Then, fitting was extended to other temperature with equation ( 2) thanks to the Clausius Clapeyron coefficient CAM. The coefficient CAM.is chosen to minimize the gap with experimental curves at 40°C and 80°C. Figure 2 shows the result with CAM.=7.9 MPa.C -1 .

             ' 3 ' 3 3 ' 3 ' ' 3 0 tanh ) ( ) 60 , ( ' E E K E E E c T E M M    (1)                ' 3 ' ) 0 3 3 ' 3 ' ' 3 ( tanh ) ( ) , ( ' E E T T C K E E E T E M AM M    (2)

Conclusion

Evolution of E' during isothermal tensile test allows to :

-identify clearly different phenomena (elasticity, localisation, transformation, plasticity) -determine elastic modulus of austenite and martensite without using the slope of the stress-strain curve which is influenced by transformation phenomenon -determine Clausius Clapeyron coefficient of forward transformation without using stress plateau which can be difficult to identify if the transformation occurs without localisation.

Figure 1 :

 1 Figure 1: DMA-SS test at T= 60°C a) stress-strain and storage modulus-, b) storage modulus-stress

Figure 2 :

 2 Figure 2: storage modulus-stress at 40°C,60°C,80°C. Model with CAM.=7.9 MPa.C -1

Table 1 :

 1 Identification of different phenomena from stress and storage modulus evolution during loading

	 ()	E'()	Phenomena
			Elasticity
			Plasticity
			Starting transformation
			End transformation
		mostly linear	Localisation

Table 2 :

 2 Elastic moduli of austenite and martensite determined by the slope of stress-strain curve and by DMA

	method
	Phase	E by slope method	E from E'
	Austenite	62 GPa	70 GPa
	Martensite	25 GPa	60 GPa


Mrs Estephanie Nobre

Dantas Grassi

Pr Denis Favier

Dr Grégory Chagnon
email: gregory.chagnon@univ-grenoble-alpes.fr

Anisotropic Tensile Behavior Of NiTi Tubes And Its Dependence On Temperature

Introduction

NiTi thin walled tubes are highly textured structures used in the fabrication of nearly 60% of self-expanding stents. The presence of texture leads to strong anisotropic mechanical behavior, which has important consequences in the final behavior of devices manufactured from tubes. In this work the anisotropic superelastic mechanical behavior of a 50.8%atNi-Ti thin walled tube was studied through isothermal tensile tests performed in samples cut at five different orientations from the tube drawing direction. The tube was flattened by a one-step shape setting process and dogbone samples were cut by laser. Tests were performed at three temperatures above Af (the reverse martensitic transformation finish temperature). Results show that transformation strains strongly depend on orientation but not on temperature. Transformation stresses also depend on orientation and this dependence is different for forward (A M) and reverse (M A) transformations. Clausius-Clapeyron coefficients also show an important orientation-dependent behavior. The observed difference between forward and reverse Clausius-Clapeyron coefficients are explained through the dependences of dissipated and stored energies with temperature, often neglected in the Clausius-Clapeyron relation.

Experimental Results

. Nominal tensile stress versus DIC logarithmic strain curves plotted for each orientation from the tube drawing direction, at 40°C (thicker line), 50°C (medium line) and 60°C (thinner line).

Figure 2 contains a set of five σ yy -ε yy curves for all orientations θ and testing temperatures T. Perfect superelastic behavior is observed for all orientations and testing temperatures, with negligible residual strain after unloading.

Through the curves of Fig. 2 it is possible to qualitatively infer a strong anisotropic behavior, perceived mainly in the amount of strain achieved at the end of loading in the stress level. Experimental results are used in the evaluation of transformation strains Δε tr , transformations stresses and forward and reverse Clausius Clapeyron coefficients C A-M and C M-A (Figure 3). With a thermodynamic analysis of martensitic phase transformation, other important properties are explored, as Clausius-Clapeyron coefficients and dissipated energy per cycle.

Conclusions

Full transformation strain depends strongly on orientation but not on temperature. Transformation stresses for forward (A M) and reverse (M A) transformations depend on orientation, but to a lesser extend when compared to transformation strain. Stress hysteresis, calculated as the difference between transformation stresses of forward and reverse transformations, is both orientation and temperature dependent. (4) The dissipated energy per unit of transforming material was the property that showed the greater variation with orientation.

  Figure 1. Scheme showing the fabrication process of dog bone tensile samples from the NiTi tube.

Figure 2

 2 Figure 2. Nominal tensile stress versus DIC logarithmic strain curves plotted for each orientation from the tube drawing direction, at 40°C (thicker line), 50°C (medium line) and 60°C (thinner line).

Figure 3 .

 3 Figure 3. Clausius-Clapeyron coefficients for forward and reverse transformations. b) Product of Clausius-Clapeyron coefficients and transformation strain.

Acknowledgments

The author Estephanie Nobre Dantas Grassi would like to acknowledge the financial support conceived by the Brazilian institutions CNPq and Capes through the program Ciências Sem Fronteiras to the realization of her doctoral studies.