Team:iBowu-China/Model
Modeling
1.SUMMARY
We built our model using Python, and you can find all the source code on Github:
https://github.com/XZJ-AKM/Modeling-Code-iBowu-China
One of the most serious problems of cancer treatment is off-target. Most of the cancer drugs cannot be delivered accurately toward the designated location and kill the cancer cell.
This year, we developed a novel cancer treatment that combines targeted therapy with hyperthermia by using ligand GE11 as the targeting agent and magnetic protein crystals as the heating agent. In order to demonstrate the differences in targeting accuracy when an external alternating magnetic field is applied, we constructed mathematical models that explored the targeting process of our project.
2. BACKGROUND
2.1.Assumptions
- 1.The targeted region is located in a straight blood vessel.
- 2.Magnetic protein crystal-containing liposomes undergoes linear magnetization in an external alternating magnetic field.
- 3.The interaction between liposomes and fluid can be neglected so that the liposomes can be assumed to move in a stationary fluid flow
- 4.For simplicity, only two-dimensional fluid flow is considered.
- 5.Collisions between liposomes and upper or lower vessel wall are imperfectly elastic.
- 6.Periodic conditions are adopted at the left and right boundaries.
- 7.External alternating magnetic field is generated by a straight charged wire close to the targeted area.
- 8.Drugs are effective when liposomes are within certain range of the targeted area (red region in the simulation).
- 9.Parameters used in the simulation are in reasonable range.
2.2.Theory and Principles
Blood flow is simulated by a two-dimensional stationary fluid flow in a rectangular region of a blood vessel. The fluid velocity is nearly zero at the blood vessel walls and reaches its maximum in the central line of the vessel in a quadratic way. Liposomes moving in the vessel are subjected to two kinds of forces [1].
One is the viscous force F_v of the fluid flow, which can be characterized by [1]:
where u(y)=v_c-4(v_c-v_w )/H^2 y^2 is the fluid velocity along the vertical (y) axis. Apparently the fluid velocity reaches its maximum v_c at the central line and decreases to v_w at both the upper and lower vessel walls. In the above formula, v is the velocity vector of a given liposome, if the horizontal velocity of the liposome is slower than the fluid velocity, the fluid will drag it forward; similarly, if the horizontal velocity is faster than the fluid velocity, the fluid will hold it until the horizontal velocity of the liposome is identical to the local fluid velocity. The strength of this effect is controlled by the coefficient γ.
The other force is the magnetic force F_m exerted on liposomes by external alternating magnetic field. Since we only consider linear magnetization region of the liposomes under the alternating magnetic field generated by a straight charged wire, F_m can take the following form [1]:
where r is the distance between the wire and a certain liposome. The strength of this effect is characterized by coefficient χ, which is eventually proportional to the current of the charged wire.
Given these two forces, the motion of a liposome can be fully described by Newton’s laws of motion. Only four variables are needed, i.e. the horizontal and vertical positions of a liposome and its corresponding velocities in these two directions. Additionally, when liposomes approach vessel walls, they are bounced back in an imperfectly elastic manner. In this way, the trajectory of a liposome in a two-dimensional stationary fluid flow can be simulated by solving ordinal differential equations (ODEs) with four dimensions as followed:
3.PARAMETERS
To better describe our model, here is a framework of the main construction along with several key parameters attached to it. Below in the chart are the parameters and their values used in the simulation.
4.MODELS
4.1.Basic Model
As aforementioned, the ODEs are solved using ‘odeint’ of scipy package of Python with a time-step of dt=0.1. At t=0, all liposomes are released from a static uniform distribution across the vessel. We count the liposomes that flow inside the capture region at each steps to see the dynamics of effective liposomes along the time course.
4.2.Results
Simulation results of both scenarios, with and without external alternating magnetic field, are presented below:
Background colormap shows the fluid velocity corresponding to u(y), with its maximum at central line and minimums at both vessel walls. Colormap of liposomes shows the current velocity of a given liposome, with red moving fast and blue moving slow. We can see from the simulated videos, when an external magnetic field is added, liposomes are attracted to the targeted area which is not seen when the alternating magnetic field is absent.
By counting the number of liposomes passing through targeted area and filtering short-range fluctuations by moving average technique, we plot graphs to show the dynamic trend of the number of effective liposomes with and without alternating magnetic field.
Obviously seen from the picture, adding an external alternating magnetic field can largely increase the number of effective liposomes, thus contributing to cancer treatment efficacy by increasing the accuracy of drug delivery.
4.3.Conclusion
By adding an external alternating magnetic field outside the fluid field, the number of magnetic protein crystals loaded liposomes that are eventually captured in the targeted area has increased significantly. So, for our project, we can see that an external alternating magnetic field can increase the accuracy of drug delivery and kills cancer cells precisely.
5.REFERENCES
[1] Kelly, M., Yeoh, G. H., & Timchenko, V. (2015). On Computational Fluid Dynamics Study of Magnetic Drug Targeting. The Journal of Computational Multiphase Flows, 7(1), 43-56.