Team:Sheffield/Model
Modelling
Introduction
When investigating the use of our device within the application for a possible diagnostic device for antimicrobial resistance one of the most important questions to answer is: How quickly will the device be able to produce a result? For example, if the bacteria take a long time to grow, they will have to be left in the device in the presence of the antibiotic for a longer amount of time to confirm that the antibiotics are being effective. Whilst running the device we will be growing a bacteria in plain media without antibiotics as a control against the ones that are growing in the presence of antibiotics so that growth can be clearly seen in the control wells. The control wells will allow us to be confident that it is the antibiotic that is preventing bacteria from growing. Estimating exactly how long it takes to receive a result, however, will help inform the physician and therefore allow them to treat the patient quicker.
Factors that might affect how long the device takes to reach a reliable conclusion include:
Specific Growth Rate of Bacteria: The smaller the specific growth rate of a bacterium the longer it will take to grow. It will, therefore, take a longer time to see whether the antibiotics are effective in preventing the bacteria to grow, and the time that we have to leave the device running will be longer to ensure that a reliable conclusion about the effectiveness of the antibiotic can be made.
Growth Conditions: Nutrient supply, pH and temperature can also greatly affect the pace of bacterial growth. These environmental factors can easily be controlled when growing the bacteria in a well plate using a nutrient-rich medium and keeping the bacteria incubated at an optimum temperature in a warm room can ensure that the growth is as fast as can be.
Initial Inoculum Size: The rate of bacterial growth in a well is highly dependant on the initial bacterial concentration. Very small initial concentrations of bacteria will take a longer time to grow and therefore take longer to be detected by the device. In an infection, the concentration of the bacteria (the colony forming units per ml, CFU/ml) that are present can vary significantly and thus can affect how long it will take to receive a result. For this reason, we wanted to build a model to estimate exactly how the initial inoculum size affects the time it takes to start seeing bacterial growth. This would, therefore, allow us to estimate the effects of low inoculum concentration on the time it takes to get a result.
Assumptions
Due to the health and safety restrictions in the lab where our team was based over the summer, we were only able to do experiments on E. coli. Therefore, an example model was developed for demonstrating how we could estimate the amount of time it would take to get a result from testing a urinary tract infection (UTI) from a patient that was caused by E. coli, which is the primary cause of UTIs in many cases. This would show the model’s helpfulness in determining a rough time to wait before the results of the device could be received. This model would then be able to be applied to calculate the time required for other bacteria, providing that experimental data for them could be fed into it.
One consideration that was taken into account was whether the initial bacterial concentration of the solution would have any effect on the specific growth rate of that sample of bacteria and therefore would this also have an effect on how long it takes to receive a reliable result from the device. However, there was a comparison between the different specific growth rates calculated from different starting concentrations of bacteria, and it was shown that there was no observable difference between the specific growth rates of the samples from different starting concentrations. This is shown in the graph below.
Figure 1. This graph shows the specific growth rate of E. coli grown in a 96 well plate in a 37 degrees Celsius room. As you can see, the initial concentration of the bacteria in the solution has no significant effect on their resulting specific growth rates.
Therefore, for this reason, we can assume the specific growth rate does not depend on the initial number of bacteria in the sample, and we can discount it in our model. Instead, we can focus on how the detection time varies depending on the initial inoculum of the bacteria. Detection time is defined as the time it takes for the device to detect the bacterial growth. The time before this can be known as the “apparent" lag time, when there is no noticeable increase in the absorbance, and therefore it may appear as though the bacteria are not growing. However, the detection time can be dependant on the sensitivity of the device, for example the bacteria could be growing at an exponential rate, however, because the device is not sensitive enough, it is unable to detect the very small number of bacteria and unable to detect the very small changes in the size of the bacterial population.
Figure 2. The graph above shows how the different aspects involved in the model relate to a usual bacterial growth curve as discussed previously.
Experiments
To perform data gathering experiments for the model, Victor plate reader was used. Initial samples of bacteria were grown overnight and then different dilutions of the bacteria were produced from this overnight culture. The first dilution was produced by diluting 1 ml of bacterial stock solution into 9 ml of plain media (LB broth). The next solution was then created by diluting 1 ml of the previous dilution with 9 ml of LB broth (a serial dilution). This continued until we had 8 different dilutions. These bacteria were dispersed into a 96 well plate, with a different dilution in each row, this is shown below:
Figure 3: A diagram demonstrating how the different dilutions of bacterial solution were distributed across the 96 well plate. Each row had a different dilution.
Once dilutions from the original broth had been made up, a small amount of the highest dilution was plated out on an agar plate. Each bacterium on the plate divides and gives rise to a single colony. This value is known as a colony forming unit (CFU) and can be used to calculate the number of bacteria in the original sample. This then allowed us to investigate the effect of initial concentration of bacteria in the sample to the time it took the device to detect any growth (ie: detection time).
The lowest absorbance readings (600 nm) that the Victor plate reader was able to make was in the range of 0.04-0.49 with an average of 0.0446, which was estimated after running the plate reader on a 96 well plate containing plain LB medium with no bacteria present. Therefore for this experiment, the detection time is defined as the time it takes for the readings to reach an absorbance of 0.5, which is above the smallest values recorded for a plain plate without any bacteria.
Absorbance data across the plate was recorded every hour for 6 hours, and was then recorded again at 24 hours. Below is an example of the absorbance data collected for each inoculum:
Figure 4: This graph shows the bacterial growth resulting from different initial inoculum sizes. As you can see in the graph the smaller initial inoculum sizes have longer “apparent” lag phases, however, these may be due to the fact that the device is not sensitive enough to detect small amounts of bacteria or changes in the population that are very small.
This experiment was completed twice with 16 different dilutions of bacteria, each with 12 different wells, resulting in information for 192 different bacterial growth curves.
Data Analysis
The detection times measured were then plotted against the different initial bacterial concentrations, resulting in this scatter plot below:
Figure 5: This graph shows the different detection times that result from different initial inoculum sizes. This graph shows an exponential decrease between the detection time and the initial inoculum size.
Using GraphPad Prism 7 a non-linear regression analysis could be performed on the data and an exponential decay model was applied to the data as shown below:
Figure 6: This graph shows the non-linear regression analysis on the data. This gives us a “line of best fit” with an equation that we can use to estimate what detection time we would roughly expect depending on the initial concentration of bacteria in our sample.
This model produced the following equation to relate inoculum size to detection time:
$$T_{Det} = (T_{det-0}-Plateau) \times e^{(-K\times Conc)}+Plateau$$
Where:
\(T_{Det}\) : The detection Time
\(T_{det-0}\) : The theoretical detection time if the bacterial concentration in the sample is 0.
\(Plateau\) : The theoretical detection time as the initial concentration in the sample keeps increasing to a very high number.
\(K\) : A constant which helps shape the curve produced.
\(Conc\) : The initial bacterial concentration in the solution.
After inputting the data into a software programme called “GraphPad Prism 7” the following values were calculated for each variable:
$$T_{Det} = (404.1-129.9) \times e^{(-4.763\times 10^{-6}\times Conc)}+129.9$$
Which can be simplified to:
$$T_{Det} = 274.2e^{(-4.763\times 10^{-6}\times Conc)}+129.9$$
Example
Now we have constructed a simple model to determine: How long will it take to see if a UTI infection is resistant to certain antibiotics using a Victor plate reader depending on how many bacteria are present in the sample.
This model is just an example with the Victor plate reader using E.coli bacteria for analysis of a UTI infection. After a calibration between the device and the Victor plate reader has been completed, this model could then be applied to the individual device that each user creates. As the sensor response can vary between each component, a quick calibration will be required between every different device that is created from the open-source instructions. If information could be collected about different bacteria, this model could then be applied to different infections.
However, using this simple model example, we can now estimate how long it will take the Victor plate reader to determine that the E. coli bacteria from a UTI infection are resistant or susceptible. For some UTI infections, a diagnosis is confirmed with around 100 CFU per ml of bacteria in the urine, and some infections would expect a concentration of 10,000 CFU per ml in the urine.
Using these values as an example for a concentration of 100 CFU per ml of bacteria would need 403.969 minutes (6.73 hours) before any growth could be detected and a concentration of 10,000 CFU per ml would need 391.346 minutes (6.522 hours) before any growth could be initially detected.
This demonstrates that use of this device to determine whether a UTI infection is resistant to a certain type of bacteria will take around 7-8 hours, which is a reasonable time for a diagnostic device.
