Predictive mechanics-based model for depth of cut (DOC) of waterjet in soft tissue for waterjet-assisted medical applications

Abstract

The use of waterjet technology is now prevalent in medical applications including surgery, soft tissue resection, bone cutting, waterjet steerable needles, and wound debridement. The depth of the cut (DOC) of a waterjet in soft tissue is an important parameter that should be predicted in these applications. For instance, for waterjet-assisted surgery, selective cutting of tissue layers is a must to avoid damage to deeper tissue layers. For our proposed fracture-directed waterjet steerable needles, predicting the cut depth of the waterjet in soft tissue is important to develop an accurate motion model, as well as control algorithms for this class of steerable needles. To date, most of the proposed models are only valid in the conditions of the experiments and if the soft tissue or the system properties change, the models will become invalid. The model proposed in this paper is formulated to allow for variation in parameters related to both the waterjet geometry and the tissue. In this paper, first the cut depths of waterjet in soft tissue simulants are measured experimentally, and the effect of tissue stiffness, waterjet velocity, and nozzle diameter are studied on DOC. Then, a model based on the properties of the tissue and the waterjet is proposed to predict the DOC of waterjet in soft tissue. In order to verify the model, soft tissue properties (constitutive response and fracture toughness) are measured using low strain rate compression tests, Split-Hopkinson-Pressure-Bar (SHPB) tests, and fracture toughness tests. The results show that the proposed model can predict the DOC of waterjet in soft tissue with acceptable accuracy if the tissue and waterjet properties are known.

Appendix A: Procedure for derivation of Eqs. 27 and 28

In this appendix, the procedure for derivation of the Eqs. 27 and 28 is explained.

Equation 26 can be rewritten using the volume change elements as:

$$ \frac{\partial E_{e}}{\partial l_{1}} = {\int}_{r_{1}}^{\infty} \phi 2\pi s_{1} ds_{1} = {\int}_{r_{2}}^{\infty} \phi 2\pi s_{2} ds_{2} $$

(51)

Starting with ${\int \limits }_{r_{2}}^{\infty } \phi 2\pi s_{2} ds_{2}$, and incorporating (25) one can write:

$$ {\int}_{r_{2}}^{\infty} \phi 2\pi s_{2} ds_{2} = {\int}_{r_{2}}^{\infty} 2\pi s_{2} \frac{2\mu}{{\alpha}^{2}}\left[\left( \frac{s_{1}}{s_{2}}\right)^{\alpha} + \left( \frac{s_{2}}{s_{1}}\right)^{\alpha} - 2\right] ds_{2} $$

(52)

In order to make this integral neater, we can define: $\gamma = (\frac {s_{2}}{r_{2}})^{2}$, and thus $d\gamma = \frac {1}{{r_{2}}^{2}} (2s_{2} ds_{2})$. Therefore:

$$ \begin{array}{@{}rcl@{}} \frac{\partial E_{e}}{\partial l_{1}} = {\int}_{r_{2}}^{\infty} \pi {r_{2}}^{2} \frac{2\mu}{\alpha^{2}} \left[ \left( \frac{s_{1}}{s_{2}}\right)^{\alpha} + \left( \frac{s_{2}}{s_{1}}\right)^{\alpha} - 2 \right] \frac{2s_{2}}{{r_{2}}^{2}} ds_{2} = &&\\ {\int}_{1}^{\infty} \pi {r_{2}}^{2} \frac{2\mu}{{\alpha}^{2}} \left[ \left( \frac{s_{1}}{s_{2}}\right)^{\alpha} + \left( \frac{s_{2}}{s_{1}}\right)^{\alpha} - 2 \right] d\gamma = &&\\ \frac{2\pi \mu {r_{2}}^{2}}{{\alpha}^{2}} {\int}_{1}^{\infty} \left[ \left( \frac{s_{1}}{s_{2}}\right)^{\alpha} + \left( \frac{s_{2}}{s_{1}}\right)^{\gamma} - 2 \right] d\gamma&&\\ \end{array} $$

(53)

$(\frac {s_{1}}{s_{2}})^{\alpha }$, and $\frac {s_{2}}{s_{1}})^{\alpha }$ can be rewritten as:

$$ \left( \frac{s_{1}}{s_{2}}\right)^{\alpha} = \frac{{s_{1}}^{\alpha}}{({s_{2}}^{2})^{\frac{\alpha}{2}}} = \frac{{s_{1}}^{\alpha}}{(\gamma {r_{2}}^{2})^{\frac{\alpha}{2}}} = \frac{{s_{1}}^{\alpha}}{{\gamma}^{\frac{\alpha}{2}} {r_{2}}^{\alpha}} $$

(54)

Using the same procedure, $(\frac {s_{2}}{s_{1}})^{\alpha } = \frac {{\gamma }^{\frac {\alpha }{2}}{r_{2}}^{\alpha }}{{s_{1}}^{\alpha }}$.

From volume conservation, we already know that ${s_{1}}^{2} - {r_{1}}^{2} = {s_{2}}^{2} - {r_{2}}^{2}$. Dividing the sides of this equation by ${r_{2}}^{2}$, we can write:

$$ \frac{{s_{1}}^{2} - {r_{1}}^{2} + {r_{2}}^{2}}{{r_{2}}^{2}} = \left( \frac{s_{2}}{r_{2}}\right)^{2} = \gamma $$

(55)

And thus:

$$ \left( \frac{s_{1}}{r_{2}}\right)^{2} = \left( \frac{r_{1}}{r_{2}}\right)^{2} + \gamma - 1 = \left( \frac{d}{D}\right)^{2} + \gamma - 1 $$

(56)

From here, the following equation can be deduced:

$$ \begin{array}{@{}rcl@{}} \left( \frac{s_{1}}{s_{2}}\right)^{\alpha} + \left( \frac{s_{2}}{s_{1}}\right)^{\alpha} - 2 = &&\\ \left( \frac{\gamma + (\frac{d}{D})^{2} - 1}{\gamma}\right)^{\frac{\alpha}{2}} + \left( \frac{\gamma}{\gamma + (\frac{d}{D})^{2} - 1}\right)^{\frac{\alpha}{2}} - 2\\ := f\left( \frac{d}{D},\gamma\right) \end{array} $$

(57)

And thus (27) and (28) can be derived.