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A MATHEMATICAL MODEL FOR ANTIBIOTIC EFFECTIVENESS IN BACTERIAL COLONIES
Piseth Koy
References:
- 1.0 TEXTBOOK: STEWART, James., DAY, Troy. (2019). Biocalculus: Calculus Probability and Statistics for Life Sciences. NEW YORK: CENGAGE LEARNING.
Textbook References:
- S. Imre et al., “Validation of an HPLC Method for the Determination of Ciprofloxacin in Human Plasma,” Journal of Pharmaceutical and Biomedical Analysis 33 (2003): 125-30.
- W. Bär et al., “Rapid Method for Detection of Minimal Bactericidal Concentration of Antibiotics,” Journal of Microbiological Methods 77 (2009): 85-89, Figure 1.
- A. Firsov et al., “Parameters of Bacterial Killing and Regrowth Kinetics and Antimicrobial Effect Examined in Terms of Area under the Concentration-Time Curve Relationships: Action of Ciprofloxacin against Escherichia coli in an In Vitro Dynamic Model.” Antimicrobial Agents and Chemotherapy 41 (1997): 1281-87.
Study Summary
Description
The growth of bacteria population in a human host treated with an antibiotic depends on both: (a) the strength or the initial concentration of the antibiotic used for the treatment and (b) the profile in time of the antibiotic concentration.
Therefore both, antibiotic initial concentration, and the long-term effectiveness of the antibiotic, as its concentration decays overtime, will impact the profile of the bacteria population in the human host.
In this study we are developing a model to show the relationship between the concentration of the antibiotic Ciprofloxacin, its long-term effectiveness, and the E. coli bacteria population in a human host.
Name of antibiotic: Ciprofloxacin
Name of Bacterium: E. coli
Per Reference [1], the model used to represent the antibiotic concentration over time is the following exponential decay function: (SEE PDF)
where:
C(t) is the antibiotic concentration as a function of time C0 is the initial antibiotic concentration
k empirical constant
Figure 1: (SEE PDF)
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Figure 1 [1] shows that the antibiotic concentration increases extremely quickly followed by slow decay.
Figure 2: (SEE PDF)
- Figure 2: Show the relationship between the concentration of ciprofloxacin and bacteria growth rate.
- Figure 2 represents the rate of change of the concentration function, C(t), or derivative of function C(t) = C0 e-kt (shown in Figure 1). From one concentration to another, the rate of change of antibiotic is different which then also affects the population of the bacteria as well. As time the concentration of ciprofloxacin is decreasing in time, the bacteria begin to repopulate in the host.
- At t=0 the antibiotic concentration is C0. The lowest concentration of an antimicrobial that will inhibit the visible growth of a microorganism after a certain period, is called the Minimum Inhibitory Concentration (MIC) and is given as 0.013 ug/ml [1]. The initial given dose surges immediately to its maximum concentration (Figure 1) after which time it begins to decay eventually reaching a value of MIC= 0.013 ug/ml. At this point the growth rate of the bacteria population suddenly increases, becoming positive. Figure 2 shows the rate of population change (dP(c)/dc) becoming positive, therefore indicating an increase in the bacteria population.
Purpose
The purpose of this study is to explore the relationship between the bacteria population growth in a human host and the effectiveness and concentration of the antibiotic ciprofloxacin.
The study develops a mathematical model of the bacteria population as a function of:
a. _____Time
b. _____Concentration of ciprofloxacin
c. _____Magnitude of antibiotic treatment quantified by the parameters (α, ρ, τ)
d. _____Antibiotic effectiveness quantified by the parameters ( T, Δ)
Acronyms & Units
- (SEE PDF)
Assumptions and Definitions
The following values are extracted from Reference [1]
P (0) = 6 CFU/mL – The population of bacteria at the beginning, t = t0
k = 0.175 [1/hours] – Positive constant for exponential decay function
a = 5.7 ln (77 C0) – time when population is rebounds
b = 6.6 ln (77 C0) – time when population stay at 12 CFU/mL
A = (77 C0) ^ (-2.2) – empirical correlation
Calculations
Bacteria Population Models
Two models are proposed for the bacteria population growth [1], shown below as model (2a) and model (2b).
- Population Growth Function – P(t) – for Antibiotic Concentration Lower than MIC
- (SEE PDF)
- When the time in hours is less than 2.08 (t < 2.08), the bacteria population is P(t) = 6et/3
- When the time in hours is greater or equal to 2.08 (t > 2.08), the bacteria population is P(t) = 12 CFU/ml
- For (2a) model, the population of bacteria is defined based on time and if co < 0.013
- Population Growth Function – P(t) – for Antibiotic Concentration Greater than MIC
- (SEE PDF)
- If co > 0.013 and t < 5.7 ln(77 co), the bacteria population is 6e-t/20.
- If co > 0.013 and 5.7 ln(77 co) < t < 6.6 ln(77 co), the population is 6 Aet/3
- If co > 0.013 and t > 6.6 ln(77 co), the population is 12 CFU/mL.
For the (2b) model, the population of bacteria is defined for co > 0.013 and it depends on time, which has its own restriction that associate with the values of “a” and “b”
Calculations of Bacteria Population P(t) and Development of “Kill Curves”
- (SEE PDF)
Description: When the antibiotic concentration is 0 μg/mL, the population is increasing exponentially for approximately 2 hours until the population is constant at 12 CFU/mL. This function describes the bacteria growth in the absence of the antibiotic.
Case 2:
- (SEE PDF)
Description: When the antibiotic concentration is 0.038 μg/mL, the population of bacteria is decreasing exponentially for the first 6 hours. Then, the population increases exponentially for approximately 3 hours until the population is constant at 12 CFU/mL after 9 hours.
Case 3:
- (SEE PDF)
Description: When the antibiotic concentration is 0.075 μg/mL, the population of bacteria is decreasing exponentially for the first 10 hours. Then, the population increases exponentially for approximately 4 hours until the population is constant at 12 CFU/mL after 14 hours.
Case 4:
- (SEE PDF)
Description: When the antibiotic concentration is 0.15 μg/mL, the population of bacteria is decreasing exponentially for the first 14 hours. Then, the population increases exponentially for approximately 4 hours until the population is constant at 12 CFU/mL after 18 hours.
Case 5:
- (SEE PDF)
Description: When the antibiotic concentration is 0.3 μg/mL, the population of bacteria is decreasing exponentially for the first 18 hours. Then, the population increases exponentially for approximately 5 hours until the population is constant at 12 CFU/mL after 23 hours.
Case 6:
- (SEE PDF)
Description: When the antibiotic concentration is 0.6 μg/mL, the population of bacteria is decreasing exponentially for the first 22 hours. Then, the population increases exponentially for approximately 5 hours until the population is constant at 12 CFU/mL after 27 hours.
Case 7:
- (SEE PDF)
Description: When the antibiotic concentration is 1.2 μg/mL, the population of bacteria is decreasing exponentially for the first 26 hours. Then, the population increases exponentially for approximately 6 hours until the population is constant at 12 CFU/mL after 32 hours.
Graphs of the bacteria population for each of the seven cases described above are calculated in Appendix A (EXCEL Workbook) and are shown below in Figure 3.
The resulting curves P(t) = f (t, c0) are called “kill curves”.
Figure 3: (SEE PDF)
Description of Population Kill Curves:
- P(t) = f (t, C0) – the population depends on time and concentration.
- The higher the concentration, the longer it takes the bacteria to start repopulating again as concentration of antibiotic is decreasing over time.
Therefore, we now need to define the metrics that describe the “effectiveness” of each antibiotic concentration. These parameters are:
- Δ − the difference between the minimum bacteria population and initial population. Δ is dependent on initial concentration C0, Δ = f (C0).
- T – time that it takes population to drop to 90% of initial population.
Antibiotic Effectiveness Metrics
We start by defining the following parameters:
- (SEE PDF)
Figure 4: (SEE PDF)
- As initial concentration increases, it takes a short amount of time for concentration to reach its maximum, Cmax.
- When the concentration is very low approaching zero and approaching MIC, it takes quite amount of time and the effectiveness of dose decreases as time goes by.
- (SEE PDF)
Figure 5: (SEE PDF)
- Δ − the difference between the minimum population and initial population. Δ is dependent on initial concentration C0, Δ = f (C0).
- T – time that it takes the population to drop to 90% of initial value
Find the expression for Δ = f(c0) and for T = f(c0)
- Δ = f(c0) – The difference of population from initial to minimum it gets in term of initial concentration.
- T = f(c0) – time it takes population to goes down 90% when it is based on initial concentration.
Develop expressions for both Δ and T
(SEE PDF)
Table 2: (SEE PDF)
Figure 6: (SEE PDF)
Based on Figure 6: as α increase, Δ also increase as well. These also make sense in terms of biological and algebraically. We know that Δ is the difference between the minimum population and initial population and it varies based on each concentration. For this equation we know that 𝛼 = 1 𝐶0 which also 𝑘 𝑀𝐼𝐶 depends on concentration. Both Δ and α show a relation with C0. As C0 increase, both Δ and α also increase as well.
Table 3: Calculation of T = -20 ln (0.9) (SEE PDF)
Figure 7: The Relationship between α and T (SEE PDF)
Based on the expression for T = -20 ln (0.9) = 2.107 sec, there is no relation to initial concentration which makes it constant no matter how’s high or low antibiotic concentrations are, but α shows a relation with C0.
DISCUSSION OF RESULTS AND CONCLUSIONS
Population of bacteria in a human host depends, on the initial antibiotic concentration and the duration its strength in time as shown in Figure 3.0. At the same time each concentration has its own unique effectiveness, and therefore unique impact on the bacteria population (implicitly shown in Figure 6.0)
Antibiotic effectiveness parameters are defined as population rebound (Δ) and time of population drop to 90% (T); while the rebound parameter (Δ) is shown to depend on the initial concentration (C0) the time of drop to 90% (T) is shown to be constant for all concentrations.
In the bacteria population model P(t) in equation (2b), the population decreases at the same rate for all initial concentrations governed by P(t) = 6e-t/20, when t < a. Thus, the time to reach 90% of the initial population size (T) is the same for initial concentration (C0) as seen in Figure 7.
The duration of the population decline phase increases as the initial concentration increases because the concentration remains above the minimum inhibitory concentration (MIC) for a longer period of time. Therefore, the larger initial concentrations lead to longer time periods in the population decline phase which, in turn, lead to larger drops in the population size before rebound (Δ) as seen in Figure 5.
Appendix A: Excel Workbook Calculations (SEE PDF)