Applications of exponential decay and geometric series in effective medicine dosage ()
1. INTRODUCTION
One of the physician’s responsibilities is to give medicine dosage for a patient in an effective manner. In this research study, the effective medicine dosage and its concentration in bloodstream of a patient are discussed in detail using two mathematical techniques: one is EDM (Exponential Decay Model) and other one is GSF (Geometric Series and its Formula). The EDM is very useful technique for simulating the dose concentration of a drug over time and GSF plays a vital role in modelling the minimum and maximum concentration of a drug administered intravenously.
2. EXPONENTIAL GROWTH AND DECAY
Exponential growth and decay are rates; that is, they represent the change in some quantity over time.
2.1. Exponential Growth Model
A quantity say 
is said to be subject to exponential growth, 
, if the quantity 
increases at a rate proportional to its value over time
. Symbolically, this can be expressed as follows:
That is, 
, which is a differential equation.
where 
is the rate of change of quantity 
over time
, 
is the value of the quantity 
at time
, and 
is a positive number called the growth constant.
Now, we can find solution for the differential equation
By rearranging this equation, we get
and then, by integrating this equation, we have 
where 
is the constant of the integration.
By simplifying this equation, we get
.
We can obtain 
by evaluating the equation 
at 
and 
is the initial value of the quantity 
that is denoted by 
for our convenience.
Therefore, 
, which is called the Exponential Growth Model.
2.2. Exponential Decay Model (EDM)
A quantity 
is said to be subject to exponential decay, 
, if the quantity 
decreases at a rate proportional to its value over time
. Symbolically, this can be expressed as follows:
That is, 
where the negative sign ‘–‘ means the decrease in the quantity 
over time
.
By solving this differential equation, we obtain 
, which is called the Exponential Decay Model (EDM).
Remarks: In general, 
and 
are exponential functions.
2.3. Geometric Series and its Formula (GSF)
Traditionally, geometric series played a key role in the early development of calculus, but today, the geometric series have many key applications in medicine, biochemistry, informatics, etc.
Usually, a geometric series is the sum of the terms of the geometric sequence:
.
Now, the sum of the geometric sequence of n terms is denoted by
where 
denotes the sum, 
the first term, 
the ratio, and 
the number of terms.
.
When
,
(
).
and when 
or
,
where
.
In the geometric series, the first term shows a = 1.
Thus, 
when 
3. EDM AND GSF IN EFFECTIVE MEDICINE DOSAGE
In this section, we discuss about the effective medicine dosage using Exponential Decay Model (EDM) and Geometric Series and its Formulae (GSF). Let us consider a patient is given the same dose of a medicine at equally spaced time intervals. The dose concentration in the bloodstream decreases as the drug is broken down by the body. However, it does not disappear completely before the next dose is given. Let us understand the exponential decay model for the concentration of a drug in a patient’s bloodstream. It is assumed that the drug is administered intravenously and that the concentration of the drug in the bloodstream jumps almost immediately to its highest level, i.e. the concentration of the drug decays exponentially.
Now, we use the function 
to represent a dose concentration at time t and 
to represent the concentration just after the dose is administered intravenously. Then the exponential decay model is formulated by
where k is the decay constant or a property of the particular drug being used.
Now, let us consider that 
be the first dose concentration at time t and that
the concentration at time t = 0 just after the first dose is administered intravenously. Suppose that at t = c, a second dose of the drug is given to the patient. The concentration of the drug in the bloodstream jumps almost immediately to its highest level 
and then the concentration is diffused so rapidly throughout the bloodstream over time (Figure 1).
The expression 
is valid as long as only a single dose is given [1]. However, suppose that, at t = c, a second dose is given and that the amount of thedrug administered is the same as the first dose. Ac-
cording to the exponential decay model, the concentration will jump immediately by an amount equal to 
when the second dose is given. However, when the second dose is given, there is still some of the drug in the bloodstream remaining from the first dose. This means that to compute the concentration just after the second dose, we have to add the value 
to the concentration remaining from the first dose (Figure 1). During the time between the second and third doses, the concentration decays exponentially from this value. To find the concentration after the third dose, the same process must be repeated.
At t = c, the dose concentration is calculated as 
just before the second dose is administered intravenously.
Here,
.
When the second dose is administered intravenously, the concentration jumps by an increment
, i.e. the concentration just after the second dose given is
.
Note that 
denotes ‘just before the new dose is administered’ and 
denotes ‘just after the new dose is administered’.
The concentration then decays from this value according to the exponential decay rule [2], but with a slight twist. The twist is that the initial concentration is at t = c, instead of t = 0. One way to handle this is to write the exponential term as 
so that at t = c, the exponent is 0. If we do this, then we can write the concentration as a function of time as
This function is only valid after the second dose is administered and before the third dose is given. That is, for 
Now, suppose that a third dose of the drug is given at t = 2c. The concentration just before the third dose is given would be
, which is
i.e., 
When the third dose is given, the concentration would jump again by 
and the concentration just after the third dose would be
Now, suppose that a forth dose of the drug is given at t = 3c. The concentration just before the forth dose is given would be
, which is
When the third dose is given, the concentration would jump again by 
and the concentration just after the third dose would be
Let us consider the process is continued up to n-th dosei.e. 
The concentration just before the n-th dose of the drug would be
(1)
The concentration just after the n-th dose of the drug would be
(2)
Let 
Note that
, since k and c are both positive constants.
From the geometric series (1) and (2), we formulate as
(3)
and
(4)
The Eqs.3 and 4 are formulae for the partial sum of a geometric series.
Suppose a treatment for a patient is continued indefinitely. Then the Eq.4 becomes
.
Now, we conclude from the results that the minimum concentration is the concentration just before the second dose is giveni.e. 
and that the maximum concentration is the concentration just after the last dose is given, i.e.
4. DISCUSSION
For example, a patient is injected a particular drug. Just after the drug is injected, the concentration is 1.5 mg/ml (milligrams per milliliter). After four hours the concentration has dropped to 0.25 mg/ml.
Here, 
at t = 4 and 
at t = 0. So,
.
To find k, Maple commands were used [8].
Result: k = 0.4479398673.
A problem facing physicians is the fact that for most drugs, there is a concentration,
below which the drug is ineffective and a concentration, 
above which the drug is dangerous. Thus, the concentration 
must satisfy the condition:
. For example, suppose that for the drug in the experiment [8] the maximum safe concentration is 5 mg/ml, or M = 5, and the minimum effective concentration is 0.6 mg/ml, or m = 0.6. Then the initial dose must not produce a concentration greater than 5 mg/ml.
5. CONCLUSIONS
In the research study, the EDM (Exponential Decay Model) and GSF (Geometric Series and its Formula) discuss in detail for effective medicine dosage. Especially the two techniques have been used for analysis of dose concentration in bloodstream of a patient and modelling of minimum and maximum concentration of a drug administered intravenously.