Team:UGent2 Belgium/Model

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Model

Intro

In this exceptional year of lockdowns and public restrictions, we could barely access the lab for our iGEM project. Besides our genetic engineering experiments, we also wanted to investigate the diffusion of naringenin (our active component) from pearl to the small intestine. Because we lacked time and could barely access the lab due to the COVID-19 pandemic, we could not proceed with these so called, dissolution experiments. Therefore, it was the perfect opportunity to build a functional diffusion model to replace one of our planned experiments. This model gives us first insights into the diffusion of naringenin (considered the active component) from pearl to the small intestine. 
We consider three different approaches:

1. Diffusion within a solid sphere: The naringenin molecules migrate to the pearl's surface and diffuse in the medium. 

2. Pearl as a solid sphere: The naringenin concentration is constant in the pearl. This model is valuable when the pearl is small and/or the diffusion is very rapid. 

3. Pearl as a dissolving sphere: The pearl dissolves; this means that diffusion is negligible since naringenin is released when the pearl dissolves. 


All code for the different models is written in Julia. This is a relatively new coding language with high-speed performance and intuitive syntax. Dr. Ir. Michiel Stock was very eager to help us out with the project's modeling part. Finally, we wish to remark that these models do not consider several factors like pH and intestinal motility. Our models are a first approximation of reality: tThey form the basis for further incorporating different factors as aforementioned. Besides that, we need to conduct experiments to determine essential constants as diffusion constants. We based ourselves on literature data and self-computed values for the models. In the future, we need to determine the real values of these constants using appropriate experiments. For instance, the diffusion constant of naringenin in the model is only approximating the diffusion constant of naringenin in the pearl. We also considered the diffusion in the pearl as from pearl to environment as the same. To compute more reliable diffusion constants, we should conduct real diffusion experiments and measure the diffusion flux from pearl into the environment. All these improvements are all for later. For when we can access the lab.

Part 2: Diffusion Within a Sphere

Diffusion within sphere

By dividing a sphere in an infinite number of shells with infinitly small thickness, we can derive the heat equation (for diffusion, cf. Fick's law) on a sphere:


$ 4\pi r^2 \dfrac{\partial c(r,t)}{\partial t} = - D8\pi r \dfrac{\partial^2 c(r,t)}{\partial r^2} +\mu 8 \pi r\, $

With:

  • $c(r,t) $: concentration at radius $r$ [m] and time $t$ [s];
  • $D$: diffusion constrant within the sphere [$\dfrac{m^2}{s}$];
  • $ \mu $: production/decay of molecules per unit mass per time [$ \dfrac{kg}{m^2 s} $]. We consider $ \mu = 0 $.

Finally, we need $k$ [$\dfrac{m^2}{s}$], which respesents the diffusion from the sphere into the medium. Here, we consider $ D = k $.

Since we cannot solve this directly (or won't), let us use the finite element method to approximate the system in $n$ shells of finite thickness. Then, we will construct a system of ODE for each shell and solve it using eigenvalue decomposition.

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This figure indicates the naringenin concentration profile per shell of the pearl. It shows that the nargingenin cargo releases fast in the pearl's outer region. After 10000 s (~ 167 min), the remaining naringenin cargo is negligable. This is still under the mean intestinal retention time (Sivius & Itani, 2019).

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This figure indicates the concentration profile in the pearl when the third shell (outer shell = first shell) is directly in contact with the environment.

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This animation gives a clear 2D view on the diffusion of naringenin in the pearl and from pearl to environment.

References

Echeverría, J., Opazo, J., Mendoza, L., Urzúa, A., & Wilkens, M. (2017). Structure-activity and lipophilicity relationships of selected antibacterial natural flavones and flavanones of Chilean flora. Molecules, 22(4). https://doi.org/10.3390/molecules22040608

Rebello, C. J., Beyl, R. A., Lertora, J. J. L., Greenway, F. L., Ravussin, E., Ribnicky, D. M., Poulev, A., Kennedy, B. J., Castro, H. F., Campagna, S. R., Coulter, A. A., & Redman, L. M. (2020). Safety and pharmacokinetics of naringenin: A randomized, controlled, single-ascending-dose clinical trial. Diabetes, Obesity and Metabolism, 22(1), 91–98. https://doi.org/10.1111/dom.13868

Svihus, B., & Itani, K. (2019). Intestinal Passage and Its Relation to Digestive Processes. Journal of Applied Poultry Research, 28(3), 546–555. https://doi.org/10.3382/japr/pfy027


Part 3: Solid Sphere With Homogeneous Concentration

Diffusion Within Sphere

In this model, we consider the pearl as a solid sphere. The initial naringenin concentration is homogeneously distrubuted in the pearl.

This just means that $ k = D $, so the partial differential equation becomes a simple ODE:

$ V\dfrac{\text{d}c(t)}{\text{d}t} = -k \cdot A \cdot c(t) $

which can be solved as:


$c(t) = c_0 \exp(\dfrac{-k \cdot A \cdot t}{V})$

Consider: $ k = 1000 \cdot D $

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In this plot, we see that the naringenin is released exponentially into the environment. The model indicates that in 20 minutes, all naringenin cargo is released. This is much lower than the mean intestinal retention time (Svihu & Itani, 2019).

References

Echeverría, J., Opazo, J., Mendoza, L., Urzúa, A., & Wilkens, M. (2017). Structure-activity and lipophilicity relationships of selected antibacterial natural flavones and flavanones of Chilean flora. Molecules, 22(4). https://doi.org/10.3390/molecules22040608

Rebello, C. J., Beyl, R. A., Lertora, J. J. L., Greenway, F. L., Ravussin, E., Ribnicky, D. M., Poulev, A., Kennedy, B. J., Castro, H. F., Campagna, S. R., Coulter, A. A., & Redman, L. M. (2020). Safety and pharmacokinetics of naringenin: A randomized, controlled, single-ascending-dose clinical trial. Diabetes, Obesity and Metabolism, 22(1), 91–98. https://doi.org/10.1111/dom.13868Svihus, B., & Itani, K. (2019). Intestinal Passage and Its Relation to Digestive Processes. Journal of Applied Poultry Research, 28(3), 546–555. https://doi.org/10.3382/japr/pfy027

Svihus, B., & Itani, K. (2019). Intestinal Passage and Its Relation to Digestive Processes. Journal of Applied Poultry Research, 28(3), 546–555. https://doi.org/10.3382/japr/pfy027


Part 4: Dissolving Sphere

Sphere That Dissolves

Finally, when the sphere dissolves with a rate $a$, proportional to the surface area:

$ \dfrac{\text{d}V(t)}{\text{d}t} = - \dfrac{\text{d}V(t)}{\text{d}t} = - \dfrac{\text{d}(4/3\pi r(t)^3)}{\text{d}t} = - 4\pi r(t)^2\dfrac{\text{d}(3/4\pi r(t))}{\text{d}t}= - a 4 \pi r^2 $

So, decay proportional with $a$, indicating the mass flow per unit of surface and time: $ \dfrac{kg}{m^2 \cdot s} $.


We estimate a value for $a$ by first calculating the density of naringinine in the outer shell of the pearl. Considering $ t = 1200 s $ as a good approximation for the time that it takes to release the naringenine cargo.

$ a = \dfrac{m}{A*t} $

where:

  • $ m $ is the mass of naringenin in the outer shell [kg]
  • $ A $ is the outer surface of the pearl [$m^2$]
  • $ t $ the end point of the considered time frame ($ 0 $, t) [s]


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These two figures indicate that the pearl dissolves during its time in the small intestin. In this model, it takes approximately one minute to dissolve the pearl completely. This is far under the mean intestinal retention time (Svihu & Itani, 2019).

This figure indicates that the release of naringenine is completed after one minute. This is far under the mean intestinal retention time (Svihu & Itani, 2019).

References

Rebello, C. J., Beyl, R. A., Lertora, J. J. L., Greenway, F. L., Ravussin, E., Ribnicky, D. M., Poulev, A., Kennedy, B. J., Castro, H. F., Campagna, S. R., Coulter, A. A., & Redman, L. M. (2020). Safety and pharmacokinetics of naringenin: A randomized, controlled, single-ascending-dose clinical trial. Diabetes, Obesity and Metabolism, 22(1), 91–98. https://doi.org/10.1111/dom.13868Svihus, B., & Itani, K. (2019). Intestinal Passage and Its Relation to Digestive Processes. Journal of Applied Poultry Research, 28(3), 546–555. https://doi.org/10.3382/japr/pfy027

Svihus, B., & Itani, K. (2019). Intestinal Passage and Its Relation to Digestive Processes. Journal of Applied Poultry Research, 28(3), 546–555. https://doi.org/10.3382/japr/pfy027