Problem 23 - Calculus for Biology and Medicine

Question

The Gompertz function is used in mathematical models for the rate of growth of certain tumors. The mass \(M(t)\) of a tumor described by Gompertz's equation changes with time according to: $$M(t)=\exp \left(a e^{-t}\right), \quad t \geq 0$$ where you may assume that \(a>0\) is a positive coefficient. (a) Determine where \(M(t)\) is increasing and where it is decreasing. (b) Find and classify any local extrema that the function has. (c) Where is the function concave up and where is it concave down? Find all inflection points of \(M(t)\). (d) Find \(\lim _{t \rightarrow \infty} M(t)\) and decide whether \(M(t)\) has a horizontal asymptote. (e) Sketch the graph of \(M(t)\) together with its asymptotes and inflection points (if they exist). (f) Describe in words how the graph of the function changes if \(a\) is increased.

Step-by-Step Solution

Answer

(a) \(M(t)\) is always decreasing. (b) No local extrema. (c) Always concave up; no inflection points. (d) Asymptote at \(y = 1\). (e) Graph decreases to asymptote \(y=1\). (f) Faster initial growth for larger \(a\).

Step 1: Differentiate the Function

To determine where the function is increasing or decreasing, we need to find the first derivative of \(M(t)\). Using the chain rule, the derivative is:\[M'(t) = \frac{d}{dt}\left(\exp\left(a e^{-t}\right)\right) = \exp\left(a e^{-t}\right) \times \left(-a e^{-t}\right) = -a e^{-t} \exp\left(a e^{-t}\right)\]

Step 2: Analyze the First Derivative

Set \(M'(t)\) equal to 0 to find critical points: \[-a e^{-t} \exp\left(a e^{-t}\right) = 0\]This equation equals zero when either factor is zero. Since \( e^{-t} \) and \( \exp\left(a e^{-t}\right) \) are never zero, \( M'(t) = 0 \) has no solutions for positive \(a\). Therefore, there are no critical points where \( M(t) \) changes from increasing to decreasing or vice versa.

Step 3: Confirm Always Decreasing Behavior

Since \( M'(t) = -a e^{-t} \exp\left(a e^{-t}\right) \) is always negative (as all terms are positive and \(-a\) is negative), \(M(t)\) is decreasing for all \( t\ge 0\).

Step 4: Find the Second Derivative for Concavity

Now, determine concavity by finding \(M''(t)\). Differentiating \(M'(t)\), we get:\[M''(t) = \frac{d}{dt}\left(-a e^{-t} \exp\left(a e^{-t}\right)\right) = a e^{-t} \left(a e^{-t} \exp\left(a e^{-t}\right) + \exp\left(a e^{-t}\right)\right) = a e^{-t}\exp\left(a e^{-t}\right)\left(a e^{-t} +1 \right).\]

Step 5: Determine Concavity and Inflection Points

\(M''(t) = 0\) when \[a e^{-t} + 1 = 0,\]which is impossible since \(a e^{-t}\) is always positive. Thus, there are no inflection points. Since \(M''(t)\) is positive for all \( t \), \(M(t)\) is concave up for all \( t \geq 0\).

Step 6: Calculate Horizontal Asymptote

Find \( \lim_{t \to \infty} M(t) \): \[ \lim_{t \to \infty} \exp\left(a e^{-t}\right) = \exp(0) = 1.\]This means \(M(t)\) has a horizontal asymptote at \(y = 1\).

Step 7: Describe the Graphical Changes for Varied \(a\)

If \(a\) increases, \(\exp\left(a e^{-t}\right)\) initially approaches 1 more rapidly, showing growth at a faster rate initially, but ultimately, due to the form of \(M(t)\), the end behavior and asymptote at \(y = 1\) remains unchanged.

Key Concepts

Tumor Growth Model

The Gompertz function is a mathematical model often used to describe tumor growth. It captures the growth pattern where the rate of cell division decreases as the tumor size increases. This realistic growth behavior is observed because, in biological systems, resources become limiting as cells proliferate, slowing the growth rate.

For the Gompertz model, the function describing the tumor's mass over time is given by \(M(t) = \exp(a e^{-t})\), where \(a\) is a positive parameter influencing the growth rate. This formula reflects how tumor growth slows as time increases. The exponential decay \(e^{-t}\) represents the diminishing growth rate over time.

• The function is specifically used because it describes two stages of growth: rapid initial growth and a slowing phase as the tumor size approaches a limiting size due to constraints like nutrient availability.
• Understanding how \(M(t)\) behaves with time helps medical researchers predict tumor growth patterns and assess intervention strategies.

Calculus Derivatives

Calculus is a powerful tool for analyzing functions like the Gompertz function. The function's growth behavior is examined through its derivatives.

The first derivative, \(M'(t)\), can indicate whether the function is increasing or decreasing by showing the rate of change of \(M(t)\) over time. For \(M(t) = \exp(a e^{-t})\), the chain rule helps us calculate:\[M'(t) = -a e^{-t} \exp(a e^{-t})\]This derivative is negative for all \(t \ge 0\), showing that \(M(t)\) is always decreasing, reflecting the slowing growth as time progresses.

• This unfaltering decrease echoes the concept of growth saturating as a tumor consumes more resources.
• Without critical points found in \(M'(t)\), \(M(t)\) doesn't have local extrema for \(t \ge 0\).

Concavity and Inflection Points

Concavity helps depict how the function's curve bends and captures the nature of growth acceleration. Derivatives also allow us to determine concavity by analyzing the second derivative, \(M''(t)\).

For \(M(t)\), the second derivative is:\[M''(t) = a e^{-t} \exp(a e^{-t})(a e^{-t} + 1)\]This second derivative is positive for all \(t \ge 0\), indicating that the entire graph of \(M(t)\) is concave up.

• A function being concave up means its rate of decrease is slowing down over time, corresponding to biological limits as resources become scarce.
• Inflection points are where concavity changes, but since \(M''(t) > 0\) always, there are no inflection points for this function.

Horizontal Asymptote

The analysis of \(M(t)\) also benefits from identifying its horizontal asymptote, providing insight into the long-term behavior of tumor growth.

As \(t\) approaches infinity in \(M(t) = \exp(a e^{-t})\), the term \(e^{-t}\) trends toward zero. Thus the limit is:\[\lim_{t \to \infty} M(t) = \exp(0) = 1\]Therefore, \(M(t)\) has a horizontal asymptote at \(y = 1\).

• This horizontal asymptote reflects the stable tumor size the model predicts when time is infinitely large, aligning with the physical reality of maximum growth due to resource limitations.
• Understanding the asymptote helps in anticipating the growth plateau, assisting in long-term planning for treatment and management.