Difference between revisions of "Team:ETH Zurich/Model/In Vivo"

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         <p class="emphasize">The 3D model presented here is used to model the behavior in conditions as close as possible to real-life scenario of tumor colonization by including the partial differential equations for diffusion of AHL and Azuin, which also helps validate the assumptions and simplifications used in the <a href="https://2017.igem.org/Team:ETH_Zurich/Model/Environment_Sensing/model"><em>in-vitro</em> model</a>. Since in experiments, there is no diffusion, our 3D model helps us model scenarios that CATE will encounter, as close as possible to reality. We are also able to test the behavior of our <a href="https://2017.igem.org/Team:ETH_Zurich/Experiments/Tumor_Sensor">tumor sensor</a>.</p>
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         <p><em>The 3D model presented here is used to model the behavior in conditions as close as possible to real-life scenario of tumor colonization by including the partial differential equations for diffusion of AHL and Azuin, which also helps validate the assumptions and simplifications used in the <a href="https://2017.igem.org/Team:ETH_Zurich/Model/Environment_Sensing/model"><em>in-vitro</em> model</a>. Since in experiments, there is no diffusion, our 3D model helps us model scenarios that CATE will encounter, as close as possible to reality. We are also able to test the behavior of our <a href="https://2017.igem.org/Team:ETH_Zurich/Experiments/Tumor_Sensor">tumor sensor</a>.</em></p>
 
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Revision as of 20:53, 1 November 2017

Modelling the Behavior of CATE inside Tumor

FIXME

We developed a model to gauge the behavior of our circuit in the real life conditions of solid tumor colonization.

This page gives details about the 3D model we developed to simulate the behavior of CATE inside the tumor and healthy tissue with different colonization patterns. This section presents details about the following:

  • The Geometry used in the COMSOL model.

  • The domain-wise Partial Differential Equations used for modelling the Growth, AND-gate Tumor sensing Switch and the Lysis & Diffusion effects.

  • Values of the Parameters used

  • Simulation Results for the AND-Gate tumor sensing switch in the different environmental conditions to validate its functioning in context of the intended application.

The 3D model presented here is used to model the behavior in conditions as close as possible to real-life scenario of tumor colonization by including the partial differential equations for diffusion of AHL and Azuin, which also helps validate the assumptions and simplifications used in the in-vitro model. Since in experiments, there is no diffusion, our 3D model helps us model scenarios that CATE will encounter, as close as possible to reality. We are also able to test the behavior of our tumor sensor.

Geometry

Our model structure consists of 2 domains - Tumor and Layer.

As mentioned in system specifications, the tumour has been chosen as a solid sphere of radius 20 mm and the bactierial colonization pattern as a homogenous distribution in a spherical shell-shaped 0.5 mm thick layer in the tumour at a distance of 10 mm from the centre of the tumor, as shown in Figure 1.

Geometry of tumor and bacterial colony
Figure 1: Geometry of the tumor and bacteria colony (green area: colonized by E. coli Nissle)
Details about the geometry

Due to the spherical symmetry of the system, a 2D axisymmetric COMSOL model was used as shown in Figure 2 - a semicircle of radius 20 mm represents the tumor and the 0.5 mm thick layer at a distance of 10 mm from the center of the tumor represents the bacteria colonization pattern as explained in the system specifications. COMSOL then sweeps the semi-circle between 0° and 360 ° to simulate the entire 3D problem. The symmetry helps decrease the computational time and space requirements, without having to apply simplifications based on assumptions.

FIXME
Figure 2: Geometry of the COMSOL Model - 2D Axisymmetric; Coordinate system: (r,z); axes in mm; Tumor is a solid sphere of radius 20 mm located at (0,0), Bacterial colonization is in a spherical layer of thickness 0.5 mm at a distance of 10 mm from the center of the tumor.

Equations

Transport of Diluted Species physics was used in COMSOL to integrate diffusion into our model. The partial differential equation for diffusion of a species C with reaction source rate RC is \[\frac{\partial \text{[C]}}{\partial t} + \nabla \cdot (-D_{\text{C}} \nabla \text{[C]})= R_{\text{C}}\]. The reaction rates of the species depends on the domain – tumor (no production and only extracellular degradation) or bacterial layer (production and intracellular degradation). Read here about the details of the domain-wise reaction rates for each species (AHL, LuxI and Azurin) and equations used.

Equations

Growth Model

An exponential growth model is used, as shown below.

\[\begin{aligned} \frac{\mathrm{d} d_{\text{cell}}}{\mathrm{d} t} &= \frac{1}{\tau} d_{\text{cell}} \left ( 1-\frac{d_{\text{cell}}}{d_{\text{cell,ss}}} \right ) \\ \Rightarrow d_{\text{cell}} &= \frac{ d_{\text{cell,0}} \, e^{\frac{t}{\tau}}}{1 - \frac{d_{\text{cell,0}}}{d_{\text{cell,ss}}} + \frac{d_{\text{cell,0}}}{d_{\text{cell,ss}}}e^{\frac{t}{\tau}}} \end{aligned}\]
Diffusion Model

Transport of Diluted Species physics of COMSOL was used to model the diffusion of AHL and Azurin. The equations below represent the diffusion models in both the Tumor and Layer domains. For simulating the effect of lysis, the diffusion of Azurin is triggered when temperature reaches 42°C, while AHL diffuses all the time. The value of the diffusion coefficients of AHL, D and Azurin, DAzu are given in the parameter list below.

\[\begin{aligned} \frac{\partial \text{[AHL]}}{\partial t} + \nabla \cdot (-D_{\text{AHL}} \nabla\text{[AHL]}) &= R_{\text{AHL}} \\ \frac{\partial \text{[Azu]}}{\partial t} + \nabla \cdot (-D_{\text{Azu}} \nabla\text{[Azu]}) &= R_{\text{Azu}} \end{aligned}\]
Reaction Rates, RC = d[C]/dt

Inside the Layer domain: AHL, LuxI and Azurin are both produced and degraded. This is modelled as:

\[\begin{aligned} R_{\text{AHL}} &= d_{\text{cell}}(t)\, a_{\text{AHL}} \text{[LuxI]} - d_{\text{AHL,out}}\text{[AHL]} \\ R_{\text{LuxI}} &= a_{\text{LuxI}} \, [k_{\text{LuxI}}+(1-k_{\text{LuxI}})P_{\text{Lux-}\text{Lac}}] - d_{\text{LuxI}}\text{[LuxI]} \\ R_{\text{Azu}} &= d_{\text{cell}}(t)\, k_{\text{Azu-LuxI}}\,(R_{\text{LuxI}}+d_{\text{LuxI}}\text{[LuxI]}-d_{\text{Azu}}\text{[Azu]} ) \\ \end{aligned}\]

where PLux-Lac is given by:

\[\begin{aligned} P_{\text{Lux-}\text{Lac}} &= P_{\text{Lux}}P_{\text{Lac}} \\ \text{where } P_{\text{Lux}} &= \frac{\left(\frac{[\text{LuxR-AHL}]}{K_{\text{LuxR}}} \right)^{n_{\text{LuxR}}}}{1 + \left(\frac{[\text{LuxR-AHL}]}{K_{\text{LuxR}}} \right)^{n_{\text{LuxR}}}} \\ \text{and } P_{\text{Lac}} &= \frac{\left(\frac{\text{[Lac]}}{K_{\text{Lac}}} \right)^{n_{\text{Lac}}}}{1 + \left(\frac{[\text{Lac}]}{K_{\text{Lac}}} \right)^{n_{\text{Lac}}}} \end{aligned}\]

where [Lac] is 1 mM in a healthy tissue and 5 mM in a tumor, as already mentioned in the system specifications.

[LuxR-AHL] is obtained by soliving the following 2 equations:

\[\begin{aligned} \text{[LuxR-AHL]} &= K_{\text{LuxR-AHL}} [\text{LuxR}]^2 [\text{AHL}]^2 &\text{(rapid binding equilibrium)} \\ \text{[LuxR]} &= [\text{LuxR}]_0 - 2 [\text{LuxR-AHL}] &\text{(mass conservation)} \end{aligned} \]

Inside the Tumor domain: AHL, LuxI and Azurin are not produced and only degraded. This is modelled as:

\[\begin{aligned} R_{\text{AHL}} &= - \, d_{\text{AHL,out}}\text{[AHL]} \\ R_{\text{LuxI}} &= 0 \\ R_{\text{Azu}} &= - \, d_{\text{Azu}}\text{[Azu]} ) \\ \end{aligned}\]

Disclaimer

The concentration of LuxI, LuxR and LuxR-AHL modelled above represents the intracellular concentration i.e. the concentration at a point had a bacteria been there.

However, since AHL diffuses everywhere rapidly and freely through the cell-memberane, a rapid equilibrium between the intercellular and extracellular concentration of AHL is reached, and thus we multiply the LuxI controlled AHL production rate by dcell, since the layer has all the bacteria colonizing the tumor.

Similarly, since we are interested in the Azurin concentration after lysis, we have a dcell multiplication factor in its reaction rate, as after lysis Azurin from all the cells diffused out of the layer. This is also then used to find the

Parameters

The parameters that were used in the COMSOL model were obtained partly from literature, partly from characterizations of previous iGEM teams and finally the most important ones were estimated by fitting our experimental data and tuning the fitted-results in the context of the intended applciation, as explained in detail by the Functional Parameter Search. Check out details about the model to read more about the different parameter values used.

For more details about the model go to the detailed description and Functional Parameter Search.

Results

We could simulate for a geometry of the system closer to the real-life tumor conditions

Since it was not practically feasible to conduct experiments of bacterial colonization inside tumors, we simulated the bacterial colonization in a thin spherical layer inside a solid tumor considering the simplifications and assumptions as mentioned in the system specifications. This helped us to test our tumour sensing AND-Gate switch functionality in all the possible real-life scenarios that CATE might encounter in context of the intended application.

Exact diffusion physics of AHL was included witout any simplifications

Our MATLAB model uses a simplified AHL diffusion model with the assumption of negligible degradation inside the layer and and not taking into consideration the diffusion of AHL far from the source. Extending the diffusion physics ordinary differential equations into partial differential equations using the COMSOL model helped us gauge and verify the behavior of our tumor-sensing circuit in more real-life conditions pertaining to the intended application context of a solid spherical tumor. Using the results obtained from our simulations, we could check the behavior of the AND Gate Switching in different conditions of dcell and lactate.

Diffusion physics of Azurin was included to simulate the effect of lysis

To simulate the effect of lysis, our COMSOL model stops the production of Azurin and starts its diffusion when temperature reaches 42°C. This simulates the effect of increase in temperature with FUS to cause cell lysis. Using data obtained from such a simulation, we could also find the temporal-maximum concentrations of Azurin at each point in the tumor, effectively helping us to estimate the killing area and the time-scale of the treatment.

Simulation of different colonization patterns

Using our model, we also tried a few other colonization patterns to show our system works as expected inside a tumor while stays dormant in healthy tissue. We simulated the following patterns:

  • Homogeneous distribution in a Single spherical-shell-shaped layer in Tumor

  • Heterogeneous distribution in a Single spherical-shell-shaped layer in Tumor

  • Heterogeneous distribution in Double spherical-shell-shaped layer in Tumor

  • Homogeneous distribution in Healthy tissue

Final Conclusions

Our model finally helped us implement a comprehensive in-silico test to prove that our system already exhibits an excellent performance for the clinical application it has been designed for. This helped us to take into account the fitting of the parameters using our experimental data and integrate our own hybrid promoter tuned in reference to the intended application context, to verify the functioning of our environment sensing function.

Limitations

Our model has some limitations. We do not model protein E production and cell lysis caused by it. Instead lysis is just simulated in effect as the end of production of AHL and Azurin and start of diffusion of Azurin. Moreover, a step signal is used as a trigger for the lysis. As mentioned in the parameters description, Azurin production is taken to be 10 times proportional to LuxI production. Also, killing mechanism of Azurin has not been modelled since that was not necessary to demonstrate the working of our project CATE in the scope of iGEM.

Tools used