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Molecular Concentration Prediction Model

2022 WHU-China

Molecular Concentration Prediction Model

     Overview

     After colonizing the intestine, engineering bacteria cells will secrete drug（SAMe）. SAMe firstly appears in the intestine and then enters the blood to produce medicinal effect. As we all know, the effectiveness of drug is closely related to the concentration in the blood. Therefore, it is crucial and necessary to determine the concentration by setting up a model, which aims to clarify the persistence of drug effect and offer guidance to the treatment of depression.
    
     In the second model, we set various parameters (such as the number of bacteria in a capsule, the growth rate of bacteria, the degradation rate of the drug, etc.) and established differential equations to find the relationship between the drug concentration and time.
    
     In order to provide a more complete discussion and analysis, we divided the modelinto two parts. The first part is a single-dose model that focuses on the variation of drug concentration over time under the condition that the patient takes drug just once. To make the results of our model more vivid, we have designed an interactive window , where you can enter parameters and subsequently observe the changes in concentration under various conditions. The second part of the model is a multiple dosing model, which is closer to a real-life situation. The patient takes the medicine based on recommended intervals from doctors. With our model, we simulate the concentration of the drug at different moments and thus are able to know if the drug works at all times.
    

     Assumption

  •     The effect of individual differences on the model is not considered.
  •     The concentration of the drug is zero until SAMe enters the bloodstream for the first time.
  •     The model only focuses on the change of SAMe concentration in blood.
  •     The degradation rate of SAMe is proportional to its concentration in the blood.
  •     Within a reasonable range of blood drug concentrations, SAMe has no side effects on humans.
    

     Symbol Description

    symbol     meaning
N 0 N_0     The number of engineering bacteria in the single dose mode
N t N_t     The number of bacteria which colonizes the human intestine（at time moment t)
α \alpha     Proportion of bacteria entering the intestine after taking the drug
β \beta     The difference between the absolute values of apoptosis rate and growth rate
γ \gamma     A scale factor which depicts the linear relationship between the rate of SAMe production and the number of bacteria in the intestine
ψ \psi     Proportional constants between the rate of SAMe entry into the bloodstream and its production rate
m m     The total amount of drug in the blood
v t v_t     The rate of SAMe entry into the bloodstream
k k     SAMe concentration degradation rate constant
c c     Concentration of uric acid oxidase in blood
V V     Total volume of human blood
r r     Iteration factor
T T     The dosing cycle
t m a x t_{max}     A variable which represent the time it takes for the concentration in the blood to reach its maximum
c m i n c_{min}     The lowest concentration at which the drug works
c m a x c_{max}     The highest concentration in the blood
    

    Single-dose model

    Colonize the human intestine

     At first, we assume that the number of engineering bacteria in the single dose mode is N 0 N_0 and the number of bacteria which colonizes the human intestine is N N （when t = 0 t=0 ，we use N t = 0 N_{t=0} to represent the initial value）. Besides, we harbor the idea that the colonized number is proportional to the number of ingested bacteria, while the proportionality factor is α \alpha . The following equation can depict the relationship:
N t = 0 = α N 0 ( 1 ) N_{t=0}=\alpha N_0 \quad \quad (1)

    Secretion of drugs in the intestinal tract

     In the intestine, the engineering bacteria undergo a normal physiological cycle, including growth, reproduction, and death. However, based on the intestinal environment, there will be more deaths than proliferations. We have this differential equation to describe the physiological cycle, where β \beta is the difference between the absolute values of apoptosis rate and growth rate, N N is the number of bacteria which colonizes the human intestine.
d N t d t = − β N t ( 2 ) \frac{dN_t}{dt}=-\beta N_t \quad \quad (2)
     Obviously,the two equations( (1) and (2) ) satisfy the initial value problem of the differential equation. We are able to find the number of bacteria at any time moment t.
N t = α N 0 e − β t ( 3 ) N_t=\alpha N_{0} e^{-\beta t} \quad \quad (3)
     Figure 1 Schematic diagram of colonization process and drug secretion process
     After exploring the changes in the population of bacteria, we focus on the secretion process. Assuming that the rate of SAMe production is linearly related to the number of bacteria in the intestine with a scale factor of γ γ . The rate of SAMe entry into the bloodstream is proportional to its production rate with a factor of ψ \psi . As a result, the rate of SAMe entry into the bloodstream v t v_t is proportional to the number of bacteria in the intestine N t N_t .
     It can be expressed as the following equation:
v t = γ ψ N t ( 4 ) v_t=\gamma \psi N_t \quad \quad (4)

    Blood Environment

     Furthermore, we consider the situation in the blood: Suppose that SAMe is uniformly distributed in the blood and is constantly degraded, then let the total amount of drug in the blood be represented as m m . Moreover, since the rate of drug degradation from the blood satisfies first-order reaction kinetics, we decide that the rate of degradation is proportional to the total amount of drug in the blood with a scale factor of k k . The rate of SAMe entry into the blood is v t v_t . Based on the above assumptions, the variation of the total amount of drug over time is shown as follows:
d m d t = v t − k m ( 5 ) \frac{d m}{d t}=v_t-k m \quad \quad (5)
     Assuming that the blood volume is V V ， we divide both sides of the equation by V V . A model for the blood drug concentration at time t is shown below:
d c d t = v t V − k c ( 6 ) \frac{d c}{d t}=\frac{v_t}{V}-kc \quad \quad (6)
     Substituting the expression for v t v_t and N t N_t into the above equation (6), we get the final model
d c d t = γ ψ α N 0 e − β t V − k c ( 7 ) \frac{d c}{d t}=\frac{\gamma \psi \alpha N_{0} e^{-\beta t}}{V}-kc \quad \quad (7)
     To solve the equation, we let the initial value of concentration be 0, ie. c t = 0 = 0 c_{t=0} =0 , and we get the following parsing solution:
c t = γ ψ α N 0 V ( β − k ) ( e − k t − e − β t ) ( 8 ) c_t=\frac{\gamma \psi \alpha N_{0}}{V(\beta-k)}\left(e^{-k t}-e^{-\beta t}\right) \quad \quad (8)

    Visualization of results

     After reviewing relevant information and discussing with the wet lab group, we decided to set the parameters like this（the molecular weight of SAMe is 398.44g/mol)
  • N 0 N_0 : 15000 pieces（per pill）
  • α α ：0.9
  • ψ ψ ：0.8
  • β β ：0.07
  • k k ：0.1
  • V V ：4500ml
  • γ γ ：0.001μmol（per piece & per time unit）
    As a result，we get the c-t image below:
     Figure 2 The c-t image of single-dose model
     From the graph, it shows that with the change of time, the concentration increases sharply and then decreases slowly, and finally decreases to 0.
     Moreover, we have designed an interactive window to show the c-t image of single-dose model under different conditions. Everyone who is interested in this model can choose different combinations of parameters to observe changes of concentration over time.
     We offer the following choices for each parameter:
    symbol     meaning     different value choices
α \alpha     Proportion of bacteria entering the intestine after taking the drug     0.8 0.9
N 0 N_0     The number of engineering bacteria in the single dose mode     15000 17000 20000
ψ \psi     Proportional constants between the rate of SAMe entry into the bloodstream and its production rate     0.7 0.8
     Here is the interactive window. Click on each parameter and you can design your own model and get your own results. Just try it!
α
value = 0.8
value = 0.9
N0
value = 15000
value = 17000
value = 20000
ɸ
value = 0.7
value = 0.8
     After changing these parameters, it is obvious that α \alpha has a greater impact on single-dose process. Therefore, we should pay more attention to it during our wet experiment and try our best to improve its value.

     Multi-dose model

    Model modifications

     Based on the reference[1], we add an iteration factor r r to our multi-dose model, which is the following equation:
r = 1 − f n 1 − f ( 9 ) \mathrm{r}=\frac{1-f^{n}}{1-f} \quad \quad (9)
     In the multi-dose model, we decide to multiply each indicator by the iteration factor r to reflect cyclicality. Then we get a more complex differential equation, where the dosing cycle is set to be T T .
c n t = γ ψ α N 0 V ( β − k ) ( 1 − e − k n T 1 − e − k T e − k t − 1 − e − β n T 1 − e − β T e − β t ) ( 10 ) c_{nt}=\frac{\gamma \psi \alpha N_{0}}{V(\beta-k)}\left(\frac{1-e^{-k n T}}{1-e^{-k T}} e^{-k t}-\frac{1-e^{-\beta n T}}{1-e^{-\beta T}} e^{-\beta t}\right) \quad \quad (10)
     If n tends to positive infinity (i.e. many many doses are taken), the equation for the change in concentration over time is as follows :
c t = γ ψ α N 0 V ( β − k ) ( 1 1 − e − k T e − k t − 1 1 − e − β T e − β t ) ( 11 ) c_t=\frac{\gamma \psi \alpha N_{0}}{V(\beta-k)}\left(\frac{1}{1-e^{-k T}} e^{-k t}-\frac{1}{1-e^{-\beta T}} e^{-\beta t}\right) \quad \quad (11)
     If we take the derivative of above equation and make the derivative equal to 0, then we can obtain the t m a x t_{max} ,which represents the time it takes for the concentration in the blood to reach its maximum after taking drug many times and then taking the next drug.
t max ⁡ = 1 β − k ln ⁡ β ( 1 − e − k T ) k ( 1 − e − β T ) ( 12 ) t_{\max }=\frac{1}{\beta-k} \ln \frac{\beta \left(1-e^{-k T}\right)}{k\left(1-e^{-\beta T}\right)} \quad \quad (12)
     After substituting the variable t m a x t_{max} into the original equation we get the highest drug concentration value.
c max ⁡ = γ ψ α N 0 V ( β − k ) ( e − k t max ⁡ 1 − e − k T − e − β t max ⁡ 1 − e − β T ) ( 13 ) c_{\max }=\frac{\gamma \psi \alpha N_{0}}{V(\beta -k)}\left(\frac{e^{-k t_{\max }}}{1-e^{-k T}}-\frac{e^{-\beta t_{\max }}}{1-e^{-\beta T}}\right) \quad \quad (13)

    Results

     If we set the dosing cycle T T to 8 hours, the following graphical line (concentration curve for the first four cycles of dosing) can be made:
     Figure 3 The c-t image of multiple-dose model
     Based on the FDA standards, the minimum concentration for SAMe to exert its effect is 1.25μg/ml，which is depicted as c m i n c_{min} in the c-t image above.
     From the result image, we know that according to the previous parameter settings, the minimum concentration in the blood is just sufficient for the drug to work, i.e. 1.25μg/ml.

    Discussion

     If we change the parameters, then we will get more situations and examples.
     1)change α \alpha from 0.9 to 0.6
     Figure 4 The c-t image of multiple-dose model（after changing α \alpha )
     It is obvious that under this condition，the minimum concentration of the drug does not meet the requirements to work. Therefore, it reminds us that we should improve α \alpha by using a few methods, such as making it easier for our engineering bacteria to colonize the intestine. This process can guide the wet group to improve their experiments.
    
     2) change N 0 N_0 from 15000 to 17000
     Figure 5 The c-t image of multiple-dose model（after changing N 0 N_0 )
     Under this condition，the minimum concentration of the drug is able to meet the requirements.
     Therefore, we can slightly improve the number of engineering bacteria in the single dose mode to get better treatment results, which is also a guidance for the wet group.
    

     Conclusion

     Firstly, our single-dose model provides insight into the trend of drug molecule concentrations in the blood over time: a sharp increase followed by a slow decrease. Secondly, our multiple-dose model is closer to the real situation and can be used to guide the wet lab group to adjust their experiment. For instance, we try to prevent leakage of the suicide module to ensure sufficient numbers of engineering bacteria in the capsule. Moreover, it is a good idea for us to retrofit engineering bacteria so that they can colonize the intestine more easily and quickly.
     In summary，our model is close to daily life and offers guidance to the wet lab group.
    

     Reference

    [1]Senthilkumari S, Lalitha P, Prajna N V, et al. Single and multidose ocular kinetics and stability analysis of extemporaneous formulation of topical voriconazole in humans[J]. Current eye research, 2010, 35(11): 953-960.