Question

The percentage of patients \(P\) who have survived \(t\) years after initial diagnosis of advanced-stage pancreatic cancer is modeled by the function $$ P(t)=100 \cdot 0.3^{t} $$ (a) According to the model, what percent of patients survive 1 year after initial diagnosis? (b) What percent of patients survive 2 years after initial diagnosis? (c) Explain the meaning of the base 0.3 in the context of this problem.

Short answer

a) 30% b) 9% c) The base 0.3 indicates the survival rate multiplier for each passing year.

Step-by-step solution

Identify the function

The given function for the percentage of patients surviving after t years is: \( P(t) = 100 \times 0.3^t \)

Calculate 1-year survival rate

Plug in \( t = 1 \) into the function: \[ P(1) = 100 \times 0.3^1 \] Simplify: \[ P(1) = 100 \times 0.3 = 30 \]So, 30% of patients survive 1 year after initial diagnosis.

Calculate 2-year survival rate

Plug in \( t = 2 \) into the function: \[ P(2) = 100 \times 0.3^2 \] Simplify: \[ P(2) = 100 \times 0.09 = 9 \]So, 9% of patients survive 2 years after initial diagnosis.

Explain the base 0.3

The base 0.3 indicates the survival rate multiplier for each passing year. It means that each year, only 30% of the patients who survived the previous year will survive the next year.

Key concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to the power of a variable. In the context of the survival rate model for advanced-stage pancreatic cancer, the equation used is \( P(t) = 100 \times 0.3^t \). Here, P(t) represents the percentage of patients surviving t years after diagnosis.
The base, which is 0.3, is crucial. Each year, the survival rate of patients is multiplied by 0.3. This means that there is an exponential decrease in the number of survivors over time.
Why is this an exponential function? Because the survival percentage decreases geometrically, not linearly. Geometric, or exponential, decay happens when a quantity decreases at a rate proportional to its current value. This is common in scenarios involving decay, such as radioactive material and populations of organisms.

Survival Rates

Survival rates are a measure of the percentage of people who continue to live for a certain period of time after diagnosis with a disease. This is particularly important in the context of severe illnesses like advanced-stage pancreatic cancer.
In our model, the function \( P(t) = 100 \times 0.3^t \) illustrates how survival rates drop each year. After 1 year, 30% of the patients survive because \( P(1) = 100 \times 0.3 \). After 2 years, only 9% are still alive since \( P(2) = 100 \times 0.3^2 = 100 \times 0.09 \).
The decay in survival rates is essential information for patients and healthcare providers to understand the nature of the disease progression and to make informed decisions about treatment and care plans.

Advanced-Stage Pancreatic Cancer

Advanced-stage pancreatic cancer refers to a late stage of cancer development where the disease has metastasized or spread to other parts of the body. This stage is typically challenging to treat and has a very poor prognosis.
The function \( P(t) = 100 \times 0.3^t \) helps to quantify survival expectations. The base of 0.3 in our model is particularly telling—it shows just how aggressive and lethal this type of cancer can be. Each passing year drastically reduces the number of survivors by 70%.
Pancreatic cancer in its advanced stage often requires a multi-disciplinary approach, including chemotherapy, radiation therapy, and possibly surgery, aimed at improving quality of life and extending survival for as long as possible.
Understanding these survival rates can provide valuable context for healthcare professionals and families when discussing prognosis and treatment options.