Question

Suppose that the amount of a drug in a patient's system diminishes by a multiplicative factor \(r<1\) each hour. Suppose that a new dose is administered every \(N\) hours. Find an expression that gives the amount \(A(n)\) in the patient's system after \(n\) hours for each \(n\) in terms of the dosage \(d\) and the ratio \(r\). (Hint: Write \(n=m N+k\), where \(0 \leq k

Short answer

The amount of drug after \( n \) hours is: \[ A(n) = d \frac{r^k}{1-r^N} \]

Step-by-step solution

Define the Parameters and Initial Values

Let's start by defining the variables: the drug diminishes by a factor of \( r \) each hour (\( r < 1 \)), and a new dose \( d \) is administered every \( N \) hours. We want to find the amount \( A(n) \) in the patient's system for each \( n \) in terms of \( d \) and \( r \).

Express Time in Terms of Doses

Express \( n \), the total hours since the treatment began, as \( n = mN + k \) where \( m \) is the number of full intervals \( N \) after which doses are administered, and \( k \) is the number of hours since the last dose. \( k \) satisfies \( 0 \leq k < N \).

Calculate Residual Drug from Previous Doses

The drug from each dose diminishes by the factor \( r \) every hour. After the first dose, which was given \( m \) intervals ago, the amount remaining is \( d \, r^{mN} \). Similarly, after the second dose (\( m-1 \) intervals ago), the remaining amount is \( d \, r^{mN-N} \).

Sum the Contributions from All Doses

Each dose contributes \( d \, r^{mN-iN} \) where \( i \) is the number of complete \( N \) intervals prior to the dose \( m-i \) (counting from the most recent back to the first). Sum these contributions for all \( m+1 \) doses, from \( i=0 \) to \( m \):\[ A(n) = \sum_{i=0}^{m} d \, r^{iN+k} = d r^k \sum_{i=0}^{m} r^{iN} \]

Calculate the Sum of the Geometric Series

The sum of the remaining amounts is a geometric series. Use the formula for the sum of a geometric series, \( S_n = a \frac{1-r^n}{1-r} \), where \( a \) is the first term and \( r \) the common ratio:\[ A(n) = d r^k \frac{1 - r^{(m+1)N}}{1 - r^N} \, \] simplifying since \( r^{(m+1)N} \) is essentially zero as \( r < 1 \).

Final Expression for A(n)

Finally, simplifying the expression further, considering the small value of \( r^{(m+1)N} \), we obtain:\[ A(n) = d \frac{r^k}{1-r^N} \] which is the expression for \( A(n) \), the amount of drug in the system after \( n = mN + k \) hours.

Key concepts

Geometric Series

Understanding geometric series is vital in modeling drug elimination. A geometric series is a sequence of numbers where each term after the first is obtained by multiplying the previous one by a fixed, non-zero number called the common ratio. In our context, every dose administered diminishes in concentration by the factor \( r \), which is less than 1, as time progresses.
If we look at the sequence of drug concentrations from repeated doses, it mimics a geometric series. The factor \( r \) diminishes each component that makes up the series of drug amounts over time. To calculate the sum of this series, we employ the geometric series formula \( S_n = a \frac{1-r^n}{1-r} \).
- \( a \) is the initial dose amount, \( d \), times any reduction factors before the series starts summing.- \( r^n \) is how many times this multiplicative factor impacts the concentration after multiples of the dosage intervals.This representation allows us to understand and predict how drugs are cleared from the body over time.

Dosage Interval

The dosage interval \( N \) represents the fixed time gap between consecutive doses of a drug. This interval is crucial in managing the amount of a drug in the patient's system, as it dictates how often the drug is administered.
In modeling drug elimination, \( N \) establishes the time difference between doses, which indirectly affects how much of each dose remains active in the system after elimination rates are applied. Mathematically, it helps define our variable \( n = mN + k \), partitioning time into multiples of dosage times and residue hours after the last dose.
- \( N \) ensures the drug reaches effective levels without becoming toxic.- Enables the calculation of cumulative drug levels.Adjusting \( N \) can optimize treatment by maintaining the desired drug concentration in the body sustainably.

Multiplicative Factor

The multiplicative factor \( r \) is pivotal in drug elimination modeling. It represents the proportion by which the drug concentration decreases over time, specifically every hour. This factor is always less than 1, reflecting how the drug is metabolized or excreted by the body.
For instance, if \( r = 0.8 \), this means each hour only 80% of the drug remains from the previous hour, continuously reducing the drug's system presence. Over multiple hours and doses, this factor recursively applies, allowing the study of accumulation and eventual stabilization of drug levels.
- Plays a critical role in determining steady-state levels.- Provides insight into how fast a drug clears from the system.Understanding this factor helps in adjusting doses and intervals to achieve therapeutic drug levels safely.

Administration Schedule

An administration schedule outlines the timing and frequency of drug doses based on medical necessity and drug properties. This includes the specific times and amounts administered, crucial in maximizing the drug's efficacy while minimizing potential side effects.
It involves planning the initial dose and subsequent repeated doses at each dosage interval \( N \). For instance, a dose might be given every 6 hours (\( N = 6 \)), maintaining a controlled drug level in the patient's system.
- Synchronizes with biological rhythms to ensure effective use.- Essential for chronic conditions requiring consistent drug levels.Through careful schedule planning, drug therapy is optimized, ensuring continuous therapeutic levels without overdose or subtherapeutic dips.