Practical application mathematical models based on the exponential function – the problem interpretation negative values of time functions for differential and integral Calculus

I often deal with mathematical models based on differential and integral Calculus, which are a powerful tool and used in modeling the behavior of systems in various fields: physics, medicine, economy, chemistry, architecture, etc.
I would like to raise an interesting issue of the time function in the context of models based on the exponential function e  – exp(x).

The exponential function ex is one of the most frequently occurring functions in calculus and its applications.

Let’s Consider the use of this function in mathematical models for several applications in medicine:
1. The action of antibiotics
2. increase in blood alcohol level after consumption
3. the rules of working an artificial kidney to remove urea from patient body

ad. 1 The action of antibiotics: After an antibiotic tablet is taken, the concentration of the antibiotic in the bloodstream is modeled by the function

where the time t is measured in hours and G is measured in μg/ml.

Thanks to this function we can calculate the maximum or minimum concentration of the antibiotic during the n hours, by applying the first derivative G’(t) to find globally extremum  (globally maximum or minimum).

ad. 2 increase in blood alcohol level after consumption: After the consumption of an alcoholic beverage, the concentration of alcohol in the bloodstream surges as the alcohol is absorbed, followed by a gradual decline as the alcohol is metabolized. The mathematical model is based on the function:


and models the average BAC, measured in mg/ml, of a group of eight male subjects t hours after rapid consumption of 15 mL of ethanol (one alcoholic drink).

Functions can be applied by using calculus and first derivative H’(t) to calculate extremum function: maximum or minimum average BAC during the n – hours and with pointing when it occurs (x value for found extremum y)

ad. 3 the rules of working an artificial kidney to remove urea from patient body:

Kidneys are paired vital organs located behind the abdominal cavity at the bottom of the ribcage corresponding to the levels T12-L3 of the spine vertebrae. They perform about a dozen physiologic functions and are fairly easily damaged. Some of these functions include filtration and excretion of metabolic waste products, regulation of necessary electrolytes and fluids and stimulation of red blood cell-production.These organs routinely filter about 100 to 140 liters of blood a day to produce 1 to 2 liters of urine, composed of wastes and excess fluid.

Kidney failure results in the slow accumulation of nitrogenous wastes, salts, water, and disruption of the body’s normal pH balance. This failure occurs over a long period of time, and when the patient’s renal function declines enough over the course of the disease, is commonly known as end stage renal disease (ESRD; which is also known as Level 5 or 6 kidney disease, depending on whether dialysis or renal replacement therapy is used). Detecting kidney disease before the kidneys start to shut down is uncommon, with high blood pressure and decreased appetite being symptoms that indicate a problem. Diabetes and high blood pressure are seen as the 2 most common causes of kidney failure. Experts predict that the demand for dialysis will increase as the prevalence of diabetes increases. Until the Second World War, kidney failure generally meant death for the patient. Several insights into kidney function and acute kidney failure were made during the war.

Hemodialysis is a method for removing waste products such as creatinine and urea, as well as free water from the blood when the kidneys are in kidney failure. The mechanical device used to clean the patients blood is called a dialyser, also known as an artificial kidney. Modern dialysers typically consist of a cylindrical rigid casing enclosing hollow fibers cast or extruded from a polymer or copolymer, which is usually a proprietary formulation. The combined area of the hollow fibers is typically between 1-2 square meters. Intensive research has been conducted by many groups to optimize blood and dialysate flows within the dialyser, in order to achieve efficient transfer of wastes from blood to dialysate.

Let’s focus on our mathematical model. Dialysis treatment removes urea from a patient’s blood by diverting some of the bloodflow externally through a machine called a dialyzer. The rate at which urea is removed from the blood (in mg/min) is described by the equation (this is our derivative):

This model is useful to monitoring how many urea has been removed from patient’s body in range of time [a,b] where a..b ∈ (0,+∞) by application standard integral calculus (definite integrals):


Below there is another sample of applying exponential function in Exponential Decay of Polonium. Polonium, a radioactive element discovered by Marie Curie in 1898 and named after her native country Poland, decays exponentially. If y0 grams of polonium are initially present , the number of grams y present after t days is given by function exponential

y = y0 e -0.05t

These four mathematical models have one interesting property, like the other models based on exponential functions (not only based on e – exp). It is a function of time which in its domain takes negative values for the x axis. So how to interpret negative values in this case in relation to hours?

In case of this situations where model function bases on positive domain values we have not problem with interpretation x values in context to time function:

So how do we interpret the negative x-values as time for the first samples shown? Models are often built on discriminant values that come from observations – and the target model is often based on creating an approximation function. In such cases, if the field of consideration / observation and the obtained results were a set of positive values – therefore, in the case of calculations, we take into account the range of positive x values coming from the modeled function. Negative values of x – this is only the result of the approximated function – we do not take them into account in this case

1. Adapted from P. Wilkinson et al., “Pharmacokinetics of Ethanol after Oral Administration in the Fasting State,” Journal of Pharmacokinetics and Biopharmaceutics 5 (1977): 207–24.

2. Sources:


March 28th, 2021