P-OGC81 The learning curve for robotic assisted abdominal phase in two stage oesophagectomy

Background Robotic assisted oesophagectomy (RAO) is increasingly being utilised in the management of oesophageal cancer. RAO implementation into practice has an inevitable learning curve. As oesophagectomy usually involves at least 2 stages, a staggered approach to training and introduction of RAO can be done. A major advantage of this is that the surgeon can concentrate on overcoming the learning curve in one phase of the procedure at a time, whilst the remaining phase can be completed by an established technique. This study looks at the learning curve of a robotic assisted abdominal phase for two-stage oesophagectomy compared to an open abdominal phase to achieve parity. Methods This study uses a prospectively maintained database to retrospectively analyse the abdominal phase of the first 17 RAO compared to the previous 20 open abdominal phase procedures. The cases are sequential, done by a single surgeon at a large UK oesophagogastric referral centre. Operating time, nodal count, and R0 rate were reviewed to determine the number of cases on the learning curve to reach parity with the open procedure. Results The open abdominal phase group had a similar age (65.6 vs 65.7), pre-op anaerobic threshold (13.9 vs 14.6 p = 0.3) but a higher BMI (mean 30.6 vs 24.6 p < 0.05) then the RAO group. All cases were T3 adenocarcinoma except for 2 cases in the robotic group (one HGD and one T2 adenocarcinoma). No RAO cases were converted to open. The mean time for the abdominal phase in the open group was 175.4 minutes with an average nodal count of 32.9. After 8 robotic assisted cases the mean operating time decreased from 267 minutes to 197 minutes, which was when a non-significant difference to the open group (p = 0.094) became apparent. The mean nodal count in the first 8 robotic assisted cases was 29.5 and increased to 38.4 in the subsequent cases. All patients had a R0 resection. Conclusions The multi-phase nature of oesophagectomy allows for modular implementation of a robotic programme. We have found that the learning curve for robotic assisted abdominal is around 8 cases. This allows for parity to open abdominal phase to be achieved regarding operative time, nodal count and R0 resection.

Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions

We prove an analogue of the magnetic nodal theorem on quantum graphs: the number of zeros ϕ of the n th eigenfunction of the Schrödinger operator on a quantum graph is related to the stability of the n th eigenvalue of the perturbation of the operator by magnetic potential. More precisely, we consider the n th eigenvalue as a function of the magnetic perturbation and show that its Morse index at zero magnetic field is equal to ϕ −( n −1).

The nodal count {0,1,2,3,…} implies the graph is a tree

Sturm's oscillation theorem states that the n th eigenfunction of a Sturm–Liouville operator on the interval has n −1 zeros (nodes) (Sturm 1836 J. Math. Pures Appl. 1 , 106–186; 373–444). This result was generalized for all metric tree graphs (Pokornyĭ et al. 1996 Mat. Zametki 60 , 468–470 ( doi:10.1007/BF02320380 ); Schapotschnikow 2006 Waves Random Complex Media 16 , 167–178 ( doi:10.1080/1745530600702535 )) and an analogous theorem was proved for discrete tree graphs (Berkolaiko 2007 Commun. Math. Phys. 278 , 803–819 ( doi:10.1007/S00220-007-0391-3 ); Dhar & Ramaswamy 1985 Phys. Rev. Lett. 54 , 1346–1349 ( doi:10.1103/PhysRevLett.54.1346 ); Fiedler 1975 Czechoslovak Math. J. 25 , 607–618). We prove the converse theorems for both discrete and metric graphs. Namely if for all n , the n th eigenfunction of the graph has n −1 zeros, then the graph is a tree. Our proofs use a recently obtained connection between the graph's nodal count and the magnetic stability of its eigenvalues (Berkolaiko 2013 Anal. PDE 6 , 1213–1233 ( doi:10.2140/apde.2013.6.1213 ); Berkolaiko & Weyand 2014 Phil. Trans. R. Soc. A 372 , 20120522 ( doi:10.1098/rsta.2012.0522 ); Colin de Verdière 2013 Anal. PDE 6 , 1235–1242 ( doi:10.2140/apde.2013.6.1235 )). In the course of the proof, we show that it is not possible for all (or even almost all, in the metric case) the eigenvalues to exhibit a diamagnetic behaviour. In addition, we develop a notion of ‘discretized’ versions of a metric graph and prove that their nodal counts are related to those of the metric graph.