Staging the axilla post-NACT (neo-adjuvant chemotherapy) by sentinel lymph node biopsy (SLNB) may allow patients to avoid undergoing axillary lymph node dissection (ALND) 1 ; however, its optimum usage is still questionable. Three prospective clinical trials (ACOSOG Z1071, SENTINA, and SN FNAC) evaluated the accuracy of SLNB post-NACT described identification rates (IR) ranging from 87.6%-92.9%. Overall, false-negative rates (FNRs) were higher than 10%, and all those trials did not achieve their primary end point.Reduction of the FNR (12.6%-9.8%) could be achieved in those patients with a normal axillary ultrasound (US) after NACT and three or more removed SLNs. [2][3][4] Methylene blue (MB) 1% is used for SLNB in node-negative (-ve) early breast cancer with IR and FNR comparable to other blue dyes or radioactive isotopes at a very low cost 5The natural concept that the best LN to evaluate response to NACT is the pathologic proven LN before starting NACT has been previously emphasized. Marking +ve LNs pre-NACT and then removing them together with SLNB post-NACT was described as targeted axillary dissection (TAD). Applying a marker clip for +ve LNs under US guidance and then marking these LNs post-NACT with radioactive I 125 seeds achieved IR of 96.7%, concordance among clipped LNs and SLNs reached 91%. 6 Caudle et al, 2016, used similar technique with IR of clipped nodes of 97.6%. The marked LN was recognized as SLN in 77% of cases. 7 Donker et al, 2015, used only radioactive I 125 seed for marking +ve ALNs pre-NACT with IR of 97%. 8 Tattooing of biopsied ALNs using sterile black carbon suspension (Spot TM ) was tried as a technique for TAD. Spot TM is FDA-approved carbon-based marker for presurgical colonic tattooing and remains in the site for months. An equivalent to Spot TM is also available, BlackEye ink. Choy et al, 2015, used suspended carbon particles (Spot TM ) to mark +ve ALNs in 28 cases. Overall, IR was 96.4%. 9 Park et al, in 2018, used Charcotrace TM for 20 cases pre-NACT, IR was 100% during surgery. Concordance among tattooed LNs and SLNs was described to be 75%. 10

The subject of this study is the solution of a fractional Bernoulli equation and a chaotic system by using a novel scheme for the fractional derivative and comparison of approximate and exact solutions. It is found that the suggested method produces solutions that are identical to the exact solution. We can therefore generalize the strategy to different systems to get more accurate results. We think that the novel fractional derivative scheme that has been offered and the algorithm that has been suggested will be utilized in the future to construct and simulate a variety of fractional models that can be used to solve more difficult physics and engineering challenges.

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