Problem 85 - Calculus for Biology and Medicine

Question

Ultrasound is often used to make images of developing fetuses. In particular, by measuring the size of a fetus ultrasound technicians can estimate its age and then predict its birthdate. To do this requires formulas for fetus size as a function of age. Verburg et al. (2008) fit data from over 6000 fetal ultrasounds. They measured the femur length, \(L\), (in \(\mathrm{mm}\) ) as a function of the fetus age, \(t\), (in weeks) and found the following formula: $$ L=-37.50+3.71 t-6.33 \times 10^{-4} t^{3} $$ Calculate the rate of growth, \(d L / d t\), at \(t=15,20\), and 30 weeks. Does the rate of growth of the fetus increase or decrease as it ages?

Step-by-Step Solution

Answer

The rate of fetal growth decreases as the fetus ages.

Step 1: Understand the problem

We need to calculate the rate at which the femur length, \(L\), changes with respect to time \(t\), in other words, \(\frac{dL}{dt}\). This requires us to differentiate the given formula for \(L(t)\).

Step 2: Differentiate the formula

The formula given is \(L = -37.50 + 3.71t - 6.33 \times 10^{-4} t^{3}\). We need to find its derivative, \(\frac{dL}{dt}\).Differentiate each term separately:- The derivative of \(-37.50\) is 0.- The derivative of \(3.71t\) is \(3.71\).- The derivative of \(-6.33 \times 10^{-4} t^3\) is \(-3 \times 6.33 \times 10^{-4} t^2 = -1.899 \times 10^{-3} t^2\).Thus,\[ \frac{dL}{dt} = 3.71 - 1.899 \times 10^{-3} t^2. \]

Step 3: Calculate rate of growth at t=15 weeks

Substitute \(t = 15\) into the derived rate of growth formula:\[ \frac{dL}{dt}(15) = 3.71 - 1.899 \times 10^{-3} \times (15)^2. \]Perform the calculation:\(\frac{dL}{dt}(15) = 3.71 - 1.899 \times 10^{-3} \times 225 = 3.71 - 0.427275 = 3.282725\, \text{mm/week}.\)

Step 4: Calculate rate of growth at t=20 weeks

Substitute \(t = 20\) into the rate of growth formula:\[ \frac{dL}{dt}(20) = 3.71 - 1.899 \times 10^{-3} \times (20)^2. \]Perform the calculation:\(\frac{dL}{dt}(20) = 3.71 - 1.899 \times 10^{-3} \times 400 = 3.71 - 0.7596 = 2.9504\, \text{mm/week}.\)

Step 5: Calculate rate of growth at t=30 weeks

Substitute \(t = 30\) into the rate of growth formula:\[ \frac{dL}{dt}(30) = 3.71 - 1.899 \times 10^{-3} \times (30)^2. \]Perform the calculation:\(\frac{dL}{dt}(30) = 3.71 - 1.899 \times 10^{-3} \times 900 = 3.71 - 1.7091 = 2.0009\, \text{mm/week}.\)

Step 6: Analyze the results

From the calculated rates of growth, we have:- At \(t=15\) weeks, \(\frac{dL}{dt}\) is \(3.282725\, \text{mm/week}\).- At \(t=20\) weeks, \(\frac{dL}{dt}\) is \(2.9504\, \text{mm/week}\).- At \(t=30\) weeks, \(\frac{dL}{dt}\) is \(2.0009\, \text{mm/week}\).The rate of growth decreases as the fetus ages.

Key Concepts

Rate of Change

When examining fetal growth, understanding the rate of change is crucial. The rate of change tells us how quickly a fetus's body part, like the femur, is growing over time. By using differentiation in calculus, we can determine the rate at which the femur length, denoted as \( L \), changes with respect to time \( t \). This is expressed as \( \frac{dL}{dt} \).
Differentiation allows us to find the instantaneous rate of change at a specific time point. For the exercise given, the femur length equation was:

• \( L = -37.50 + 3.71t - 6.33 \times 10^{-4} t^3 \)

The derivative of this equation, \( \frac{dL}{dt} \), tells us how fast the fetus's femur is growing at any given time \( t \). Upon solving, we found \( \frac{dL}{dt} = 3.71 - 1.899 \times 10^{-3} t^2 \). This gives us a concrete way to calculate and compare growth rates at different weeks.
By substituting the weeks, such as 15, 20, and 30 into the derivative formula, we discovered that the growth rate decreases as the fetus ages. Understanding these rates of growth is crucial for monitoring normal fetal development.

Fetal Growth

Fetal growth is a key indicator of a developing baby's health. During pregnancy, doctors and technicians measure different parts of the fetus to ensure it is growing at a healthy rate. One such measurement is the femur length, which is used because it correlates well with gestational age.
The growth rate of the fetus is not constant. In the initial weeks, rapid growth is typical. However, as the fetus matures, the rate typically decreases, which was evident in our example exercise. At 15 weeks, the growth rate was observed to be faster at 3.282725 mm/week and slowed to 2.0009 mm/week by 30 weeks.

• This decline in growth rates can indicate normal maturation processes.
• It is important to track these changes to ensure the fetus is developing as expected.

By understanding and measuring fetal growth patterns, healthcare providers can predict birth outcomes and intervene when necessary to aid healthy development.

Ultrasound Measurements

Ultrasound is a safe and non-invasive method used to monitor fetal development. Ultrasounds use sound waves to create images of the fetus inside the womb. This method is pivotal for obtaining key fetal measurements, such as the femur length, head circumference, and abdominal circumference.
Ultrasound measurements help healthcare professionals estimate the fetus's age and predict the due date. This estimation is more reliable than using the mother's last menstrual period alone, as every pregnancy is unique. Additionally, by having accurate sizes and rates of growth from ultrasounds:

• Doctors can assess the health and development of the fetus more comprehensively.
• Potential issues like growth restrictions can be identified and addressed early.

Using ultrasounds to gather data for functions like the one given in the exercise allows for better predictions of fetal development patterns. As technology advances, ultrasound measurements continue to be essential tools in prenatal care.