Blood Flow Dynamics

1. Poiseuille’s Law (Conceptually)

Poiseuille’s law comes from fluid dynamics, specifically Hagen-Poiseuille’s equation. It’s derived from Newton’s law of viscosity and describes steady, laminar flow of a Newtonian fluid (like blood in small vessels) in a tube:

$$Q = \frac{\pi \, \Delta P \, r^4}{8 \, \eta \, L}$$

Where:

• $Q$ = flow rate (volume per time, e.g., mL/s)

• $\Delta P$ = pressure difference across the length of the vessel

• $r$ = radius of the vessel (cm or m)

• $L$ = length of the vessel

• $\eta$ = dynamic viscosity of the fluid

• $\pi$ = constant (≈ 3.1416)

• 8 = comes from the mathematical derivation of flow in a cylinder using calculus

2. Stepwise Origin

1. Start with Newton’s law of viscosity:

   $$\tau = \eta \frac{dv}{dr}$$

   • $\tau$ = shear stress

   • $\eta$ = viscosity

   • $\frac{dv}{dr}$ = velocity gradient (change in velocity with radius)

2. For laminar flow in a tube, the velocity profile is parabolic. Integrating this over the cross-sectional area of the tube gives total flow, $Q$.

3. After integrating, you end up with:

   $$Q = \frac{\pi \, \Delta P \, r^4}{8 \, \eta \, L}$$

3. Why the Characters are What They Are

Symbol	Meaning	Why it appears
$Q$	Flow rate	That’s what we’re trying to calculate — volume/time.
$\Delta P$	Pressure difference	Drives the flow — higher ΔP → faster flow.
$r^4$	Radius to the 4th power	Comes from the integration of the parabolic velocity profile; small changes in radius hugely affect flow.
$L$	Length	Longer tubes → more resistance → slower flow.
$\eta$	Viscosity	Thicker fluid → more resistance → slower flow.
$\pi$ & 8	Constants	From the cylindrical geometry and calculus.

4. Key Takeaways

• Flow is extremely sensitive to radius — doubling radius → 16× increase in flow.

• Viscosity and length are directly proportional to resistance — higher viscosity or longer vessel → lower flow.

• This is why small vessels and thick blood dramatically affect perfusion.

PROBLEM:

A patient has a narrowed artery due to atherosclerosis. The artery is 4 mm in radius and 20 cm long. Blood has a viscosity of 0.04 Poise (4 cP, typical for human blood). The pressure difference across this segment is 50 mmHg.

1. Calculate the blood flow (Q) through this artery using Poiseuille’s Law.

2. If the radius decreases by 25% due to plaque, how does flow change?

Given constants:

• Poiseuille’s Law (conceptual form):

$$Q = \frac{\pi \, \Delta P \, r^4}{8 \, \eta \, L}$$

• Conversion: 1 mmHg = 133.3 dynes/cm² (we will keep consistent units).

Step 1: Write down known values

Variable	Value
ΔP	50 mmHg
r	4 mm = 0.4 cm
L	20 cm
η	0.04 Poise
π	3.1416

Step 2: Convert ΔP to consistent units

• Using dynes/cm²:

$$\Delta P = 50 \times 133.3 = 6665 \, \text{dynes/cm²}$$

Step 3: Apply Poiseuille’s Law

$$Q = \frac{\pi \, \Delta P \, r^4}{8 \, \eta \, L}$$

Substitute:

$$Q = \frac{3.1416 \times 6665 \times (0.4)^4}{8 \times 0.04 \times 20}$$

Step 3a: Compute $r^4$

$$r^4 = 0.4^4$$

Stepwise:

• 0.4² = 0.16

• 0.16² = 0.0256

So, $r^4 = 0.0256 \, \text{cm}^4$

Step 3b: Compute numerator

$$\pi \Delta P r^4 = 3.1416 \times 6665 \times 0.0256$$

Stepwise:

• 6665 × 0.0256 ≈ 170.5

• 170.5 × 3.1416 ≈ 535.5

Numerator ≈ 535.5 dyn·cm³/s

Step 3c: Compute denominator

$$8 \eta L = 8 \times 0.04 \times 20 = 6.4$$

Step 3d: Compute Q

$$Q = \frac{535.5}{6.4} \approx 83.7 \, \text{cm³/s}$$

Flow through artery ≈ 84 mL/s

Step 4: If radius decreases by 25%

• New radius: $r_{new} = 0.4 \times 0.75 = 0.3 \, \text{cm}$

• New $r^4 = 0.3^4$

  • 0.3² = 0.09

  • 0.09² = 0.0081

• New numerator:

$$3.1416 \times 6665 \times 0.0081$$

• 6665 × 0.0081 ≈ 53.94

• 53.94 × 3.1416 ≈ 169.4

• Denominator unchanged = 6.4

$$Q_{new} = \frac{169.4}{6.4} \approx 26.5 \, \text{cm³/s}$$

Flow decreases from 83.7 → 26.5 mL/s (~68% reduction)

Notice how a small decrease in radius dramatically reduces flow, due to the r⁴ relationship.

Step 5: Key Conceptual Takeaways

1. Poiseuille’s Law is highly sensitive to radius: even a 25% decrease reduces flow by more than 2/3.

2. Viscosity and length: higher viscosity or longer vessels also reduce flow linearly.

3. Clinical relevance: Atherosclerosis, vasospasm, or edema can drastically reduce perfusion.