Problem 7 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

There is 'conventional wisdom' among SCUBA divers that if you are going to make a dive that involves two depths, 'do the deep part first'. This problem and the next explores rationale for that wisdom. To be concrete, assume that \(K=0.071 / \mathrm{min}\) which corresponds to approximately 10 minute half- life for the compartment \(((\ln 2) / 0.07=9.9 \mathrm{~min})\). a. Assume a diver (d1) descends immediately to 10 meters and stays there for 15 minutes, then descends to 30 meters and stays there for 10 minutes. Let $$d_{1}(t)=\left\\{\begin{array}{ll}10 & \text { for } 0 \leq t \leq 15 \\\30 & \text { for } 15

Step-by-Step Solution

Answer

Compute and compare \(N_1(25)\) and \(N_2(25)\) to determine which profile accumulates less nitrogen; this supports the rationale of deep diving first.

Step 1: Understanding the Problem

We are tasked to calculate nitrogen uptake in a diver's body for two distinct diving profiles over a 25-minute period. Each diver has a different depth sequence, and the goal is to evaluate the nitrogen content based on the given mathematical models.

Step 2: Compute N1 (Dive Profile d1)

For diver (d1):1. Compute the integral: \[\int_{0}^{25} e^{0.07 s} \times d_{1}(s) \, ds \] is split into two parts: - From 0 to 15: \[ \int_{0}^{15} e^{0.07 s} \times 10 \, ds \] - From 15 to 25: \[ \int_{15}^{25} e^{0.07 s} \times 30 \, ds \]2. Calculate each integral separately using the formula: \[ \int e^{kx} \cdot C \, dx = \frac{C}{k}e^{kx} + C \] - For 0 to 15: \[ \int_{0}^{15} e^{0.07s} \times 10 \, ds = 10 \left[ \frac{e^{0.07s}}{0.07} \right]_{0}^{15} \] \[ = \frac{10}{0.07}(e^{0.07 \times 15} - e^{0}) \] - For 15 to 25: \[ \int_{15}^{25} e^{0.07s} \times 30 \, ds = 30 \left[ \frac{e^{0.07s}}{0.07} \right]_{15}^{25} \] \[ = \frac{30}{0.07}(e^{0.07 \times 25} - e^{0.07 \times 15}) \]3. Sum the integrals and plug it into the formula for \(N_1(25)\): \[ N_1(25) = 0.79 + 0.007 e^{-0.07 \times 25} \times \left(\text{sum of integrals}\right) + 0.79 e^{-0.07 \times 25} \]

Step 3: Compute N2 (Dive Profile d2)

For diver (d2):1. Compute the integral: \[\int_{0}^{25} e^{0.07 x} \times d_{2}(x) \, dx \] is split into two parts: - From 0 to 10: \[ \int_{0}^{10} e^{0.07 x} \times 30 \, dx \] - From 10 to 25: \[ \int_{10}^{25} e^{0.07 x} \times 10 \, dx \]2. Calculate each integral separately: - For 0 to 10: \[ \int_{0}^{10} e^{0.07x} \times 30 \, dx = 30 \left[ \frac{e^{0.07x}}{0.07} \right]_{0}^{10} \] \[ = \frac{30}{0.07}(e^{0.07 \times 10} - e^{0}) \] - For 10 to 25: \[ \int_{10}^{25} e^{0.07x} \times 10 \, dx = 10 \left[ \frac{e^{0.07x}}{0.07} \right]_{10}^{25} \] \[ = \frac{10}{0.07}(e^{0.07 \times 25} - e^{0.07 \times 10}) \]3. Sum the integrals and plug it into the formula for \(N_2(25)\): \[ N_2(25) = 0.79 + 0.007 e^{-0.07 \times 25} \times \left(\text{sum of integrals}\right) + 0.79 e^{-0.07 \times 25} \]

Step 4: Interpret Results

Compare the results of \(N_1(25)\) and \(N_2(25)\). The calculation will show which diving strategy leads to less nitrogen buildup, making it 'safer'. This analysis supports the idea of diving deeper first and ascending later.

Key Concepts

Nitrogen Uptake

Nitrogen uptake describes how nitrogen is absorbed into a diver's body while underwater. During a dive, the pressure increases with depth, and more nitrogen dissolves into the diver's tissues. This happens because the body is exposed to a higher partial pressure of nitrogen, which comes from the breathing gas.

• Pressure vs. Depth: As a diver descends, the ambient pressure increases roughly by 1 atmosphere every 10 meters in saltwater. This means that at 10 meters, the pressure is approximately 2 atmospheres.
• Nitrogen Solubility: The amount of nitrogen absorbed depends on the pressure and the time spent at that pressure. The deeper a diver goes, the more nitrogen is absorbed over time.
• Dive Profiles: Different dive profiles expose divers to varying depths for different durations, impacting nitrogen uptake. Diving deeper first minimizes nitrogen absorption during subsequent, shallower phases.

Understanding nitrogen uptake is crucial because excessive nitrogen in tissues can lead to decompression sickness when ascending too rapidly.

Mathematical Modeling

Mathematical modeling in dive planning allows us to predict how much nitrogen a diver will absorb during a dive. This involves using equations that take into account time spent at different depths and the rate at which nitrogen is absorbed.

• Exponential Functions: We use exponential functions to model how nitrogen concentration changes over time and depth. The model considers both the accumulation and release (off-gassing) of nitrogen.
• Variables: Key variables in the model include:
  • Time spent at each depth
  • Depth of the dive
  • Rate of nitrogen absorption ( K ) which is given in the problem as 0.07/min.
• Application: Mathematical models are used to create dive profiles that ensure divers ascend safely by minimizing excess nitrogen in their bodies, thereby preventing decompression sickness.

Mathematical models are a critical aspect of safe diving practices as they help in planning dives that consider both safety and efficiency.

Integral Calculus

Integral calculus is essential in calculating the cumulative nitrogen uptake over a dive's duration. It allows for determining the total amount of nitrogen absorbed by a diver's body during various dive profiles.

• Definite Integrals: In this context, a definite integral calculates the total nitrogen uptake over time given varying depths. The integral sums the nitrogen absorption rate over the intervals spent at each depth.
• Splitting Intervals: For dives involving multiple depths, the problem is split into intervals where conditions remain constant, making it easier to apply calculus.
  For example, splitting the problem into 0-15 minutes and 15-25 minutes for a dive to calculate the nitrogen uptake correctly.
• Solving the Integral: Use the formula \( \int e^{kx} \cdot C \, dx = \frac{C}{k}e^{kx} + C \) to solve each interval. Here \( k \) is the absorption rate and \( C \) is the constant representing depth.

Mastering integral calculus helps divers and planners in analyzing and optimizing dive plans to enhance safety by accurately forecasting nitrogen uptake.