Problem 70 - Calculus for Biology and Medicine

Question

The median lifetime is defined as the age \(x_{m}\) at which the probability of not having failed by age \(x_{m}\) is \(0.5 .\) Find the median lifetime if the hazard-rate function is $$ \lambda(x)=\left(3.7 \times 10^{-6}\right) x^{2.7}, \quad x \geq 0 $$

Step-by-Step Solution

Answer

The median lifetime is approximately 72.99.

Step 1: Define the Survivor Function

The survivor function, denoted as \( S(x) \), is defined as the probability of not failing by age \( x \). It is given by the exponential of the negative integral of the hazard-rate function. Thus, \( S(x) = e^{-\int_0^x \lambda(t) \, dt} \).

Step 2: Integrate the Hazard Function

Integrate \( \lambda(x) = (3.7 \times 10^{-6}) x^{2.7} \) from 0 to \( x \). This yields:\[\int_0^x (3.7 \times 10^{-6}) t^{2.7} \, dt = \frac{3.7 \times 10^{-6}}{2.7 + 1} x^{3.7} = \frac{3.7 \times 10^{-6}}{3.7} x^{3.7} = 10^{-6} x^{3.7}\]

Step 3: Write the Survivor Function

Substitute the result from the integration into the expression for the survivor function: \[ S(x) = e^{-10^{-6} x^{3.7}} \].

Step 4: Solve for Median Lifetime Condition

The median lifetime, \( x_m \), satisfies \( S(x_m) = 0.5 \). Thus,\[e^{-10^{-6} x_m^{3.7}} = 0.5 \] Taking the natural logarithm of both sides gives:\[ -10^{-6} x_m^{3.7} = \ln(0.5) \]

Step 5: Compute \( x_m \)

Solving for \( x_m \), we have \[ x_m^{3.7} = -10^6 \ln(0.5) \] Calculate the right-hand side:\[ -10^6 \ln(0.5) = 693147.18 \]Thus, \[ x_m^{3.7} = 693147.18 \]Finally, solve for \( x_m \):\[ x_m = (693147.18)^{1/3.7} \approx 72.99 \]

Key Concepts

Survivor Function

The survivor function, often denoted as \( S(x) \), represents the likelihood that a subject survives beyond a certain age \( x \). Think of it as the probability of "surviving" or "not failing" up to that time. The survivor function starts at 1, indicating 100% survival at age 0. It then decreases toward 0 as age increases, reflecting a higher probability of failure over time.

Here's how it's formally defined: the survivor function is the exponential of the negative integral of the hazard-rate function over time. This is expressed through the formula:

• \( S(x) = e^{-\int_0^x \lambda(t) \, dt} \)

The integral calculates the accumulated hazard from the start to the age \( x \), and the exponential function shapes this into a probability.

A powerful aspect of the survivor function is that it ties together the concepts of hazard and time, offering a comprehensive snapshot of survival likelihood. This idea is extremely valuable in fields like biology and medicine, where understanding survival probabilities can influence research and treatment strategies.

Hazard-rate Function

The hazard-rate function, denoted as \( \lambda(x) \), provides insight into the instantaneous rate of failure at any given age \( x \). It tells us how likely something is to fail at that specific moment, given that it hasn't failed yet. Imagine it as the risk of failure occurring exactly at age \( x \).

• In mathematical terms for our specific exercise, the hazard-rate function is given as: \( \lambda(x) = \left(3.7 \times 10^{-6}\right) x^{2.7} \).

The structure of this function helps highlight a key concept: the failure rate increases as time progresses, since the function grows with \( x \).

This exponential increase is common in many biological processes, where the chance of system failure often rises over time due to wear and tear or other degenerative processes.

Understanding the hazard-rate is crucial in multiple domains of biology, especially in better comprehending how likely it is for a particular biological process or organism to fail at any given time.

Median Lifetime

The median lifetime is a key measure in survival analysis used to evaluate the typical lifespan or duration until failure in a population. It reflects the age \( x_m \) at which there is a 50% probability that the subject will not yet have failed or died. In simple terms, it's the midpoint of the survival distribution.

To find the median lifetime, you need to solve for the age \( x_m \) where the survivor function \( S(x_m) = 0.5 \). This means exactly half of the subjects are expected to survive past this age. Using the survivor function derived from the hazard rate, we compute:

By setting the equation \( e^{-10^{-6} x_m^{3.7}} = 0.5 \) and solving for \( x_m \), we calculate:

\(\begin{align*}-10^{-6} x_m^{3.7} &= \ln(0.5) \x_m^{3.7} &= -10^6 \ln(0.5) = 693147.18 \x_m &\approx 72.99\end{align*}\)This means the median age at which the probability of failure is 50% is approximately 72.99 time units.

The median lifetime is especially useful in public health and epidemiology where determining the "average" expected survival can guide policy and inform healthcare planning.