Why does catheter flow rate scale as r⁴?

This is the Hagen-Poiseuille equation for laminar flow: Q = π·r⁴·ΔP / (8·η·L). Flow rate is proportional to the fourth power of inner radius. Practical consequence: doubling the inner diameter multiplies flow rate by 16 at constant pressure. This is why a few tenths of mm of ID make a huge practical difference.

What is the viscosity of iodinated contrast media?

Common iodinated contrast media have viscosities at 37 °C significantly higher than water: Iopamidol 300 around 4.7 mPa·s, Iohexol 350 around 10.4 mPa·s, Visipaque 320 around 11.8 mPa·s. For comparison, saline is 0.69 mPa·s (near water), whole blood 3.5 mPa·s. This is why contrast injection requires high-pressure injectors.

What is the Reynolds number in a catheter?

The Reynolds number Re = ρ·v·D/η characterizes flow regime. Re 4000 = turbulent regime (linear equation no longer valid, required pressure rises sharply). For typical flow rates and biological fluids, catheters generally remain in laminar regime.

How do you compute flow rate in an oval or non-circular lumen?

For a non-circular lumen, use the hydraulic diameter Dh = 4·A/P, where A is the cross-sectional area and P the wetted perimeter. For an elliptical lumen, an exact formula exists: Q = π·a³·b³·ΔP / (4·η·L·(a²+b²)) with a and b the semi-axes. For a free-form lumen (kidney shape, D-shape), generalized Hagen-Poiseuille applies via Dh; deviation from the rigorous calculation can reach 15–30 % for highly eccentric shapes, which justifies experimental characterization in critical cases.

Why does catheter flow rate scale as r⁴?

This is the Hagen-Poiseuille equation for laminar flow: Q = π·r⁴·ΔP / (8·η·L). Flow rate is proportional to the fourth power of inner radius. Practical consequence: doubling the inner diameter multiplies flow rate by 16 at constant pressure. This is why a few tenths of mm of ID make a huge practical difference.

What is the viscosity of iodinated contrast media?

Common iodinated contrast media have viscosities at 37 °C significantly higher than water: Iopamidol 300 around 4.7 mPa·s, Iohexol 350 around 10.4 mPa·s, Visipaque 320 around 11.8 mPa·s. For comparison, saline is 0.69 mPa·s (near water), whole blood 3.5 mPa·s. This is why contrast injection requires high-pressure injectors.

What is the Reynolds number in a catheter?

The Reynolds number Re = ρ·v·D/η characterizes flow regime. Re 4000 = turbulent regime (linear equation no longer valid, required pressure rises sharply). For typical flow rates and biological fluids, catheters generally remain in laminar regime.

How do you compute flow rate in an oval or non-circular lumen?

For a non-circular lumen, use the hydraulic diameter Dh = 4·A/P, where A is the cross-sectional area and P the wetted perimeter. For an elliptical lumen, an exact formula exists: Q = π·a³·b³·ΔP / (4·η·L·(a²+b²)) with a and b the semi-axes. For a free-form lumen (kidney shape, D-shape), generalized Hagen-Poiseuille applies via Dh; deviation from the rigorous calculation can reach 15–30 % for highly eccentric shapes, which justifies experimental characterization in critical cases.

Catheter fluidics suite — Flow, pressure, Reynolds | Protobrix

Fluidics suite

Catheter fluidics suite

Interactive calculation of flow rate, pressure and flow regime in a catheter lumen. Automatic switch between Hagen-Poiseuille (laminar) and Darcy-Weisbach + Blasius (turbulent, smooth tube) based on Reynolds number. Typical use cases: contrast injection, catheter priming, lumen sizing.

Parameters

Working fluid

η = 0.692 mPa·s · ρ = 1005 kg/m³ · at 37 °C

Lumen shape

Circular lumen — classic Hagen-Poiseuille

Target flow (Q) 100mL/min

12000

Inner diameter (ID) 1.00mm

0.203.00

Lumen width (major axis) 1.20mm

0.204.00

Lumen height (minor axis) 0.80mm

0.204.00

Cross-section area (A) 0.750mm²

0.0510.00

Wetted perimeter (P) 3.20mm

0.520.0

Length (L) 100cm

30200

Inlet pressure (ΔP) 50mmHg

5300

Target flow (Q) 10.0mL/min

0.51200

Cross-section
ID = 1.00 mm
A = 0.785 mm²
Dh = 1.00 mm

Computed flow

— mL/min

—

Hydraulic resistance

— mmHg / (mL/min)

—

Flow regime

— Re

Laminar

Internal volume · velocity

— µL

—

Estimated priming time

— s

Theoretical time to fully fill the lumen at the current flow rate. Useful to estimate purge duration.

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Influence of inner diameter on flow rate

at current ΔP · L · fluid

Laminar regime (Hagen-Poiseuille)

Turbulent / transition regime (Darcy-Weisbach)

Formulas used — automatic switch based on Re

Laminar regime (Re < 2300) — Hagen-Poiseuille

Q = π · r⁴ · ΔP / (8 · η · L)

R = 8 · η · L / (π · r⁴) (constant)

Turbulent regime (Re > 2300) — Darcy-Weisbach + Blasius

ΔP = f · (L/D) · (ρ · v² / 2)

f = 0.316 · Re^−0.25 (smooth tube, Blasius)

Always valid

Re = ρ · v · D / η with v = Q / A

V = π · r² · L

Formula switching occurs at Re = 2300. Between 2300 and 4000 (transition zone), Blasius is used but the result is indicative — flow is unstable and the correlation is strictly valid only above Re ≈ 4000. Overall assumptions: rigid, straight, smooth cylindrical conduit, incompressible Newtonian fluid, established flow.

Fluid data sources

Viscosity and density values at 37 °C, from standard biomedical literature and manufacturer datasheets: saline (close to water), iodinated contrast media Iopamidol 300 and Iohexol 350 (Bayer, GE Healthcare, Bracco — public datasheets).

Blasius coefficient: valid for 4·10³ < Re < 10⁵ on smooth tubes; beyond that, use Colebrook-White or Moody for roughness.

Educational tool: indicative information. Output values do not replace experimental validation. Model assumptions: straight, smooth, rigid cylindrical tube, established flow, Newtonian fluid. This tool does not cover singular pressure losses (entry, bends, side holes, fittings), thermal effects, or transients. For balloon design or complex geometry, use CFD or experimental characterization.

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Protobrix · R&D Tools · Prototype v0.2 — Hagen-Poiseuille | Darcy-Weisbach + Reynolds + Volume

Before using this tool

Catheter fluidics suite

Decision-support tool

This tool brings together public data and good engineering practice to support your design and validation decisions.

The proposed values and results are indicative orders of magnitude. It is up to each user to verify their relevance to a specific project, evaluate use conditions, and have any critical decision validated by a qualified reviewer.

This tool does not replace regulatory expertise, experimental validation, or the original normative documentation.

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Catheter fluidics suite

Need results to include in a design file?

Protobrix offers over 480 ready-to-test catheter configurations, in under 7 days.

Decision-support tool

Catheter fluidics suite

Need results to include in a design file?

Protobrix offers over 480 ready-to-test catheter configurations, in under 7 days.

Decision-support tool

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