Cardiovascular Physiology Calculations

Cardiovascular System: Pressure and Flow Dynamics

This document explores fundamental concepts of cardiovascular physiology, focusing on blood flow, pressure, and vessel compliance, alongside practical calculations.

1. Terminology Translation

• The expression for "blood flow" in French is débit sanguin.
• The expression for "stroke volume" in French is volume d'éjection systolique.

2. Blood Flow Calculation

Blood flow refers to the volume of blood moving through an artery per unit time. To calculate blood flow in this context, we need the velocity of blood and the cross-sectional area of the aorta. However, given the information, we will calculate the cardiac output in section 3, which is a measure of total blood flow from the heart.

3. Cardiac Output (CO) Calculation

Cardiac Output is the volume of blood the heart pumps per minute. It is calculated as:

CO = Stroke Volume (SV) × Heart Rate (HR)

• Stroke Volume (SV): 70 mL/beat = $70 \times 10^{-3}$ L/beat
• Heart Rate (HR): 70 beats/min
• CO = ($70 \times 10^{-3}$ L/beat) × (70 beats/min) = 4.9 L/min

The cardiac output is 4.9 L/min.

4. Total Blood Volume Ejected Over One Day

To find the total volume of blood ejected from the heart in one day, we multiply the cardiac output by the number of minutes in a day.

• Cardiac Output (CO): 4.9 L/min
• Minutes in a day: 24 hours/day × 60 minutes/hour = 1440 minutes/day
• Total Volume = 4.9 L/min × 1440 min/day = 7056 L/day

The total volume of blood ejected from the heart during one day is 7056 L.

5. Body Weight Calculation

Weight (W) is calculated using the formula $W = mg$, where:

• m: mass (70 kg for the healthy man)
• g: acceleration due to gravity ($9.81 \, m/s^2$)
• W = 70 kg × $9.81 \, m/s^2$ = 686.7 N

The weight of the 70-kg man is 686.7 N.

6. Approximation of Foot Surface Area

An approximation for the surface area of two feet for an adult of average height is about $0.035 \, m^2$.

• This value can vary significantly based on body size.
• For example, one foot might be approximately $0.15 \, m \times 0.12 \, m = 0.018 \, m^2$. So two feet would be $2 \times 0.018 \, m^2 = 0.036 \, m^2$.

We will use a value of $0.036 \, m^2$ for calculations.

7. Pressure Exerted by Feet on the Ground

Pressure (P) is calculated as force (F) divided by area (A): $P = F/A$.

• Force (F): This is the man's weight, 686.7 N.
• Area (A): Surface area of two feet, $0.036 \, m^2$.
• P = 686.7 N / $0.036 \, m^2$ $\approx 19075 \, Pa$

The pressure exerted by the feet on the ground is approximately 19.08 kPa.

8. Comparison with Systolic Blood Pressure (SBP)

• Systolic Blood Pressure (SBP): 16 kPa
• Pressure from feet: 19.08 kPa

The pressure exerted by the feet on the ground (19.08 kPa) is slightly higher than the systolic blood pressure (16 kPa). This comparison helps to contextualize the magnitude of blood pressure in a tangible way.

9. Compliance Calculation for a Young Person

Compliance (C) is defined as $C = \Delta V / \Delta P$.

• $\Delta V$: 2 mL = $2 \times 10^{-6}$ L
• $\Delta P$: 10 mmHg

First, convert $\Delta P$ from mmHg to Pascals (Pa). We know that 1 atm = 760 mmHg = 101325 Pa. So, 1 mmHg = 101325 Pa / 760 $\approx$ 133.322 Pa.

• $\Delta P$ in Pa: 10 mmHg × 133.322 Pa/mmHg = 1333.22 Pa
• C = ($2 \times 10^{-6}$ L) / (1333.22 Pa) $\approx 1.5 \times 10^{-9}$ L/Pa

The compliance for a young person is approximately $1.5 \times 10^{-9}$ L/Pa (to two significant digits).

10. Compliance Calculation for an Elderly Person and Decrease Percentage

The formula for age-related compliance decline is: $C_{age} = C_{young} \times (1 - 0.01 \times (age - 20))$

• $C_{young}$: $1.5 \times 10^{-9}$ L/Pa
• Age: 60 years
• $C_{60}$ = ($1.5 \times 10^{-9}$ L/Pa) × $(1 - 0.01 \times (60 - 20))$
• $C_{60}$ = ($1.5 \times 10^{-9}$ L/Pa) × $(1 - 0.01 \times 40)$
• $C_{60}$ = ($1.5 \times 10^{-9}$ L/Pa) × $(1 - 0.4)$
• $C_{60}$ = ($1.5 \times 10^{-9}$ L/Pa) × $0.6$ = $0.9 \times 10^{-9}$ L/Pa

The compliance for a 60-year-old person is $0.9 \times 10^{-9}$ L/Pa.

Decrease in Percentage:

Decrease % = $( (C_{young} - C_{60}) / C_{young} ) \times 100\%$

• Decrease % = ($ (1.5 \times 10^{-9} - 0.9 \times 10^{-9}) / (1.5 \times 10^{-9}) ) &times; 100%</li> <li><b>Decrease %</b> = ( (0.6 \times 10^{-9}) / (1.5 \times 10^{-9}) $) × 100%
• Decrease % = $(0.6 / 1.5) \times 100\% = 0.4 \times 100\% = 40\%$

The decrease in compliance between 20 and 60 years old is 40%.

11. Osmotic Pressure Calculation and Comparison with SBP

The van 't Hoff equation for osmotic pressure ($\pi$) is: $\pi = i \times C \times R \times T$

• i (Number of particles): 2 (for NaCl, representing major plasma solutes)
• C (Molar concentration of sodium): 140 mmol/L = 0.140 mol/L
• R (Ideal gas constant): 0.0821 L atm K⁻¹ mol⁻¹
• T (Temperature in Kelvin): Body temperature is $37^\circ C = 37 + 273.15 = 310.15 \, K$

Calculate $\pi$ in atmospheres (atm):

• $\pi$ = 2 × 0.140 mol/L × 0.0821 L atm K⁻¹ mol⁻¹ × 310.15 K
• $\pi$ $\approx$ 7.14 atm

Convert $\pi$ to Pascals (Pa):

• 1 atm = 101325 Pa
• $\pi$ = 7.14 atm × 101325 Pa/atm $\approx 723460 \, Pa$
• $\pi$ $\approx 723.5 \, kPa$

Convert $\pi$ to mmHg:

• 1 atm = 760 mmHg
• $\pi$ = 7.14 atm × 760 mmHg/atm $\approx 5426.4 \, mmHg$

Summary of Osmotic Pressure:

• Osmotic Pressure ($\pi$): $\approx$ 7.14 atm
• Osmotic Pressure ($\pi$): $\approx$ 723.5 kPa
• Osmotic Pressure ($\pi$): $\approx$ 5426.4 mmHg

Comparison with Systolic Blood Pressure (SBP):

• Systolic Blood Pressure (SBP): 16 kPa
• Osmotic Pressure ($\pi$): 723.5 kPa

The calculated osmotic pressure (723.5 kPa) is vastly higher than the systolic blood pressure (16 kPa). This highlights that osmotic pressure in this context refers to the force required to prevent the movement of water across a semipermeable membrane due to solute concentration differences, a mechanism crucial for fluid balance, rather than the mechanical pressure exerted by blood flow.

Key Takeaways

• Cardiac Output is a vital measure of heart efficiency, calculated from stroke volume and heart rate.
• Pressure can be understood and compared across different biological and physical contexts (e.g., blood pressure vs. standing pressure).
• Vascular Compliance significantly decreases with age, impacting cardiovascular health and function.
• Osmotic Pressure, driven by solute concentrations, is a powerful force critical for fluid balance and significantly different in magnitude and function from mechanical blood pressure.