IRP Suction Calculator / annular gap model

How the maths works

1 · Poiseuille's Law — the foundation

All fluid flow here obeys Poiseuille's Law for laminar flow through a tube:

Q = (π × r⁴ × ΔP) / (8 × η × L)

Where Q = flow rate, r = radius, ΔP = pressure difference, η = fluid viscosity (saline ≈ 0.001 Pa·s), L = tube length. The r⁴ term is critical — doubling the radius increases flow 16×. A tiny change in gap has an enormous effect.

2 · Inflow (through the scope working channel)

Saline pumped through the scope's working channel behaves as flow through a tube or annular space (if a laser is present). The working channel diameter is user-configurable (default 3.6 Fr); most instruments have effective diameters 3.0–3.6 Fr. If a laser occupies part of the channel, fluid flows in the annular gap around it. The driving pressure is pump pressure minus whatever IRP already exists in the kidney:

Q_in = K_in × (P_pump − IRP) K_in = (π × r_wc⁴) / (8 × η × L_scope) [or annular formula if laser present]

3 · Outflow (through the annular gap)

Fluid escapes through the ring-shaped gap between the scope outer wall and the sheath inner wall. This is not a simple circle — it's an annulus, so the flow factor uses the exact annular Poiseuille formula:

Q_out = K_out × (IRP − (−P_suction)) K_out = (π × F_annular) / (8 × η × L_sheath) F_annular = (r_o⁴ − r_i⁴) − (r_o² − r_i²)² / ln(r_o/r_i)

Outflow is driven by the pressure difference: kidney pressure (IRP) minus the negative pressure from the suction pump (−P_suction). This is equivalent to IRP plus the suction magnitude. The annular factor F is orders of magnitude smaller than r⁴ for a solid tube of the same outer radius — this is why the gap is the bottleneck, not the sheath size.

4 · Solving for required suction

At equilibrium inflow must equal outflow. Setting Q_in = Q_out and solving for the suction pressure that achieves your IRP target:

K_in × (P_pump − IRP) = K_out × (IRP − (−P_suction)) P_suction = K_in × (P_pump − IRP) / K_out − IRP

The calculator solves this in real time. The result is then multiplied by the clinical safety factor (default 1.5×) to account for non-ideal real-world conditions (pulsatility, scope deflection, debris). If the theoretical P_suction (before safety factor) comes out negative, passive drainage alone is sufficient — no Neptune needed.

5 · Why sheath size matters so much

Even a 1Fr increase in sheath ID widens the gap by ~0.17mm on each side. Because of the r⁴ relationship inside F_annular, this small change can double or triple outflow capacity, dramatically cutting the Neptune pressure required. This is the physical basis for using 12–14Fr sheaths with 9Fr scopes.

Assumptions & limitations

This model assumes laminar, steady-state flow of Newtonian fluid (saline). Key limitations:

Working channel diameter — Most flexible ureteroscopes have effective hydraulic diameters slightly smaller than nominal spec. Default is 3.6 Fr; try reducing to 3.0–3.2 Fr for conservative estimates.

Scope position — Scope deflection, eccentric positioning, and debris reduce real-world outflow. The annular flow factor may be 20–40% higher in practice.

Pulsatile flow & transients — This model solves steady-state equilibrium. Real pump surges and outflow lag during irrigation bursts mean peak IRP spikes above the equilibrium prediction.

For these reasons, the clinical safety factor (default 1.5×) multiplies the calculated suction to provide margin. Verify margins empirically under real operating conditions.