A mathematical model of a two-layered flow in a catheterized oesophagus under the influence of peristaltic waves of dilating amplitude

Swallowing in a catheterized oesophagus is mathematically modeled considering the presence of a peripheral layer, and ensuring mass conservation of fluids in both the peripheral and core layers. Mass conservation in those layers is guaranteed by solving the interface as a streamline using a fourth-order algebraic equation, rectifying a previous model by [A. Medhavi and U. Singh, Peristaltic transport of a two-layered fluid in a catheterized tube, Theor. Appl. Mech. 39(4) (2012) 291–311], which violated mass conservation separately. Additionally, we assume progressively dilating wave amplitude, leading to higher pressure near the distal part of the oesophagus. That was experimentally observed by [P. J. Kahrilas, S. Wu, S. Lin and P. Pouderoux, Attenuation of esophageal shortening during peristalsis with hiatus hernia, Gastroenterology 109(6) (1995) 1818–1825], and theoretically modeled by [S. K. Pandey, G. Ranjan, S. K. Tiwari and K. Pandey, Variation of pressure from cervical to distal end of oesophagus during swallowing: Study of a mathematical model, Math. Biosci. 288 (2017) 149–158], based on anatomical measurement by [F. Xia, J. Mao, J. Ding and H. Yang, Observation of normal appearance and wall thickness of esophagus on ct images, Eur. J. Radiol. 72(3) (2009) 406–411]. It is concluded that even in the presence of a catheter, the pressure-flow rate relationship remains linear. Flow rate rises with a thinner and more viscous peripheral layer, especially with higher wave amplitudes and larger amplitude dilation. Pressure and flow rate both increase with a thinner peripheral layer due to catheter’s presence, which agrees with arterial findings [H. Kanai, M. Iizuka and K. Sakamoto, One of the problems in the measurement of blood pressure by catheter-insertion: Wave reflection at the tip of the catheter, Med. Biol. Eng. 8 (1970) 483–496; L. Back, E. Kwack and M. Back, Flow rate-pressure drop relation in coronary angioplasty: Catheter obstruction effect, J. Biomech. Eng. 118 (1996) 83–89]. Catheter length is to be optimized inside the oesophagus to avoid pressure rise due to secondary peristalsis. It is, therefore, advised to refrain from oral feeding at the time of nasal feeding is in progress as catheter insertion elevates pressure, potentially exacerbating the situation due to peristalsis.