Model parameters
The model used here is based on the experiments in Burrows et al. (2013), where two adjacent segments of the programmable bevel-tip needle were advanced equally to create a fixed offset that was maintained throughout the physical experiment. This configuration restricts needle steering to a single plane, and so a 2D plane strain simulation is used. The needle has an outer diameter of \(\textit{b}=8\text { mm}\), a total bevel angle of \(2\alpha = 20\)°and tip radius equal to \(\rho _\mathrm {tip}=0.5 \text { mm}\) (Fig. 2b). The offset c represents a programmable parameter which is changed between the simulations. The needle material is modelled as linear elastic, with a Poisson coefficient equal to \(\nu =0.475\), whereas the Young's modulus \(E_{\mathrm {needle}}\) is varied in the simulations, in the range 117–940 kPa. The substrate consists of a block of gelatine phantom, with a concentration of 10% by weight, having dimensions \(\textit{2W}=235\text { mm}\) and \(\textit{H}=245\text { mm}\). The gelatine material is also modelled as linear elastic, with Young’s modulus \(E_{\mathrm {gel}}=14.8\) kPa and a Poisson coefficient \(\nu =0.475\). It should be noted that although a linear elastic material description is used here, as in many other models available in the literature (Vigneron et al. 2011; Tai et al. 2013; Bui et al. 2019), the presence of large deformations is accounted for in the formulation. The mean value of the fracture energy \(G_{\rm c}\) for the gelatine, obtained from wire-cutting tests at small strain rates, is assumed equal to 1.1 J/m2 (Forte et al. 2015). Contact interaction between the needle and the surrounding gelatine is represented by a coefficient of Coulomb’s friction equal to \(f=0.3\), which agrees with the values obtained from cutting experiments in similar gelatine materials (Oldfield et al. 2013b).
A series of simulations were run on the PBN employing the algorithm described in Sect. 3. Separately, a symmetrical needle configuration, equivalent to a ‘zero offset’, was simulated to test whether a straight path could be achieved with the proposed procedure. Each simulation was run sufficiently long to reach a condition of steady state and obtain a stable curvature of the penetration path. Refer to the electronic supplementary material for animations of needle insertion, concerning the symmetric configuration (Online Resource1) and the PBN with offset \(\textit{c}=47\text { mm}\) (Online Resource2). Please note that the animations are related to the last iteration of the algorithm (mesh adaptivity not explicitly shown), when the needle travels within the path obtained from all the previous iterations.
Force–displacement curves
The algorithm has proved able to reproduce different stages of the insertion of the programmable bevel-tip needle. The level of detail might be appreciated in Fig. 8, where a typical force–displacement curve obtained from the simulation is shown with the different stages of penetration. Compared to the overall characteristic shape of a symmetric needle insertion (thin dashed line in the figure) and the experimental results reported in Oldfield et al. (2013a), additional detailed features are captured by our numerical model. (Notice that the depth reached in this and the other simulations falls within the region 2b of Fig. 1.) After an initial stage, with almost linear increase in the reaction force, there is a first relaxation, corresponding to the completion of the bevel penetration. It should be noted that the presence of the initial notch in the gelatine mesh (Fig. 5b) is responsible for the lack of an initial sharp increase, which is usually present in the experimental curves and corresponds to the displacement at which a crack is initiated. After the first relaxation, the curve resumes a slightly increasing trend, up to the point corresponding to the end of tip penetration, i.e. when \(D_\mathrm{e}=c\). At this point, the force rises rapidly to a maximum, corresponding to the complete tip–shaft transition, when the backward needle segment penetrates the material. Afterwards, a strong relaxation is followed by increasing force with the penetration of the needle shaft, which continues until full penetration (not reached in the numerical simulations). The second peak in the force, and the following relaxation, is absent in the symmetrical needle curve and is believed to be peculiar of the PBN insertion. Note that the force is expressed per unit of out-of-plane depth, since we have adopted a plane strain approximation. The values are consistent with those observed in experimental trials of needle insertion and cutting of similar gelatine materials (Oldfield et al. 2013a, b).
Force–displacement curves are known to be affected by various parameters, correlated with the needle geometry, the material elasticity and the characteristics of interaction. For instance, the penetration force has been shown to be sensitive to the bevel angle \(\alpha\), with smaller bevel angles resulting in larger forces (Misra et al. 2008). While certainly worthy to be investigated, that parameter might not be the most relevant to consider when dealing with the peculiar steering mechanism of the PBN, which is controlled by the programmable offset c. For this reason, in this work we have run simulations changing the initial offset of the needle; additionally, we have also investigated the effects of the needle stiffness, keeping the offset fixed. Results of the analyses, in terms of the force profile during needle penetration, are reported in Fig. 9. The plots seem to suggest that changes in the needle–gelatine stiffness ratio have only limited effects on the maximum reaction force, with a 40 % increase following an eightfold growth of the needle Young’s modulus. Analogous behaviour has been observed in bevel-tip needles by other authors (Misra et al. 2008). The force is not influenced so much by changing the needle offset either, with the curves displaying only minor differences in their slopes with respect to the relative penetration. Both these parameters are expected to have a much larger impact on the needle trajectories.
Needle trajectory and curvature
An example of the typical trajectory followed by the needle in the simulations is illustrated in Fig. 10. The images are obtained from different iterations of the algorithm and overall show a smooth progression of the needle, which follows a curved path with an almost constant curvature. In the rightmost image, however, a deflection point is noticed, after which the path continues with a slightly increasing curvature, although smaller than the initial. This point corresponds to the completed transition from the needle tip to the thicker shaft, with an increase in bending stiffness of the needle, and is marked by the peak in the force–displacement curves (Figs. 8, 9). While such a deflection is difficult to capture in the experiments (Burrows et al. 2013), which either look at the needle insertion starting from positions already past the tip–shaft transition or do not have the resolution necessary to capture such effect, this shows that the full curvature evolution might not have stabilised (i.e. become constant) when the insertion is already progressed far into the soft material. This must be accounted for when comparisons with experiments are made.
The effect of the needle stiffness and the influence of the programmable offset c are again considered in Fig. 11, with respect to the needle penetration paths. These are defined in terms of the relative tip penetration D/c versus the horizontal deflection of the needle tip \(\varDelta /c\). Figure 11b also illustrates the case of ‘zero offset’ between the needle segments, which as expected results in a straight path and provides a validation for the model. As observed by other authors, the needle deflection \(\varDelta\) seems to be altered by both the needle elasticity (Misra et al. 2010) and the offset (Burrows et al. 2013), although it is not straightforward to quantify the exact dependence.
A better insight is gained by considering the needle curvature 1/R and the angle \(\theta _\mathrm {tip}\) measured at the tip (Fig. 2b). The needle curvature is calculated from the path taken by the needle tip node, when the needle displacement is equal to \(D_\mathrm{e}=2c\). Under the assumption of constant curvature along the whole trajectory, the radius of curvature R is determined as a parameter of the best fit circle, estimated using the hyper-circle algorithm (Chernov 2010). The angle \(\theta _\mathrm {tip}\) is that formed by the local tangent direction at the needle tip and the vertical insertion axis and is computed from the first derivative of the hyper-circle at the specified depth. The results are reported in Fig. 12. The trends seem to suggest a linear dependence of both the curvature and the tip angle with the stiffness ratio and the needle offset. Specifically, Fig. 12b shows a linear increase in the curvature with growing offset. This provides good agreement with the experimental offset–curvature trends shown by Burrows et al. (2013), although the absolute values are lower.
The discrepancy might depend on various aspects. Firstly, the estimated curvature depends on the computation criterion and, in particular, on the depth at which it is calculated. This point might become evident if we explore the curvature variation with respect to the needle displacement (Fig. 13). To obtain this plot, we have removed the assumption of constant curvature and applied the hyper-circle fitting to increasing insertion depths, with a span of 5 mm. It is found that initially the curvature decreases sharply, until the needle has penetrated for a length approximately equal to the offset c. Then, the curvature appears to reach a minimum value at \(D_\mathrm{e}\simeq 2c\) (compare the data reported in Fig. 12b), before starting a slightly increasing trend. The simulations are interrupted due to computational costs, as the increase in simulation time would have not provided any additional insight into describing the evolution of the penetration path, with attention focused on parametric study rather than deeper insertion. However, during experiments the needle is inserted further into the substrate, so it is reasonable to expect a larger curvature.
Other aspects to consider when comparing experiments and simulations are related to the characterisation of the tool–tissue interaction and, in particular, to the margin of error in quantifying the frictional behaviour. In this work, we have used Coulomb’s friction, although other authors have adopted a constant adhesive shear stress to model the frictional contact between soft materials and stiff tools (Atkins et al. 2004). The coefficient of friction has been chosen accordingly to similar experimental studies and has been shown to influence both the maximum force and the force–displacement gradient (Oldfield et al. 2013b), although a quantification of its effect on the curvature is still lacking. Frictional effects are expected to be lower when gelatine is replaced by other brain tissue surrogates, such as composite hydrogels (Leibinger et al. 2016). Lastly, the simplification introduced by the assumption of a planar insertion yields a stiffer behaviour with respect to the 3D case, since our 2D model is infinitely stiff in the out-of-plane direction. In general, a quantitative matching between experiments and simulation should be possible when all the parameters involved are carefully evaluated, but this is beyond the scope of the present paper.
The experimentally observed effects of a linear offset–curvature relationship (Burrows et al. 2013) provide a good point of validation for the modelling approach presented here. Physically, the behaviour seen in the model can be explained by the interaction of needle bending stiffness, substrate stiffness and the stress state around the needle tip during insertion. While not dramatically altering the overall curvature of the needle, kinks in the penetration path can be directly attributed to the change in bending stiffness as both segments of the needle interact with the tissue phantom substrate (Fig. 11). Other effects, such as the evolving curvature for different offsets (Fig. 13), are more complex. They are dependent on a more subtle coupling between the bending forces exerted on the offset bevel tip, the cut path in the substrate and the shaping influence this path has on the overall profile and curvature of the needle as it then passes through.
The nonlinearity caused by the relationship between offset, material stiffnesses and the pre-existing path is indicated when projecting the offset–curvature relationship in Fig. 12b to a zero offset. For the substantial offsets used here, and which would be typically necessary for curvatures with clinically significant benefit, the offset–curvature relationship is linear. However, at very small offsets the linearity breaks down and a more complex offset–curvature relationship would need to be established. It is clear that for no offset, forces around the needle tip are symmetric and a straight path is followed. For ‘small’ offsets, the more withdrawn bevel segment will have an influence on the distribution of stress around the tip. Similarly, the impact of increasing the stiffness of the needle at a fixed offset is highly intuitive yet also deteriorates at physically limiting situations. The linear relationship between needle stiffness and curvature (Fig. 12a) will break down if extrapolated to the assumption that a rigid needle results in no curvature. In practical terms, when there is a bevel present, even if the needle is rigid and undeformed, the shape of the bevel will still create a crack that is projected away from the insertion axis. On the other hand, when the needle stiffness becomes very low in relation to the forces at the tip, buckling is not preventable even with the practical assistance of trocar delivery. The modelling tool developed by the authors can certainly be used to shed light on all the aspects mentioned above and, therefore, to improve the design of PBNs.