Overview
Our project aims to develop a safe and effective targeted drug delivery system. However, it is not enough to create vesicles with builtin cellcell fusion protein to treat patients effectively. That’s why we came up with the idea to find out the best way to deliver our modified vesicles to the human body by means of computer modelling.
Any mechanical contact to the tumour can cause active hallmarks of cancer manifestation, for example, angiogenesis [5]. It can be difficult to deliver a drug directly into the tumour. In addition, we want to destroy the metastases, which spread through the bloodstream. Thus, we must inject our drug up into the blood. To explore this approach, we develop a model to simulate the vesicle behaviour in the blood flow. The purpose of our model is to evaluate the efficiency of vesicle collision with diseased endothelium lining the surface of a vessel. The obtained data will make it possible to understand how many vesicles are required for effective drug delivery.
Model development
During the modelling, we had to revise the model parameters to make it more realistic.
Firstly, we decided to take the V. N. Kaneva et al. article [1] as a foundation. The authors of the article developed a novel twodimensional particlebased computational model of microvessel thrombosis. We have taken the basic principles of haemodynamics and model assumptions from this paper. The platelets adhesion mechanism is similar to vesicle adhesion. However, they do not form a clot. Therefore, we may omit rotation equations and complex particleparticle interactions. Following the article, we consider a blood vessel as a 2D model due to the computational complexity of 3D modelling. Vesicles are moving under the following forces: Stokes force and Van der Waalspotentialbased forces [2]. We expect that all the vesicles hitting the diseased endothelium, which lines the vessel, will get to the tumour cells. So, it is enough for us to determine the number of the stuck vesicles. We have to note that this assumption does not reflect the real situation properly, and the actual number of the stuck vesicles is smaller than predicted. Thus, our model provides only an upper limit for this quantity. The blood flow in the vessel obeys the HagenPoiseuille law, and the blood parameters are physiological. However, initial calculations showed that with the blood flow, extracellular vesicles are too light and are carried away by the bloodstream without adhering to surface cells.
Then, we revised our idea. From the model of the continuous medium, we moved to the mesoscopic scale model. It is based on G. Fullstone et al. article [3]. The authors have demonstrated that red blood cells are highly important for effective nanoparticle distribution within capillaries. In addition, our vesicles have nanoscales, and their ability to target and enter tissues from the blood should be highly dependent on their behaviour under blood flow too. In the new version, we assumed elastic collisions between two particles types: vesicles and red blood cells. We do not take into account other types of cells in connection with a small number of them. The interaction occurs according to the Van der Waals potential and flow forces. Besides, we had to take into account the technical moments: our particles should spawn randomly but do not overlap, vesicles should stick to the vessel wall, we should be able to count them. If you are interested in details, you can write to rogacheva.av18@physics.msu.ru (Anastasia Rogacheva) or ask us for the source code.
Assumptions
The main assumptions are listed here:
 Mesoscopic scale model: vesicles and red blood cells in the flow.
 HagenPoiseuille laminar plasma flow in the vessel. Parabolic velocity profile [7]:
 Stokes 3D force for sphere particles [7]:
 Van der Waalspotentialmediated motion [7], [2]:
 Elastic collisions between all the particles and red blood cells and vessel wall.
 Velocity verlet integration [4].
Physiological parameters are:
Parameter  Value 

vessel diameter  30 µm 
plasma dynamic viscosity  1.2 mPa*s 
red blood cell mass  27 pg 
red blood cell density  5*10^{6} cells per µl 
red blood cell diameter  7 µm 
vesicle mass  27*10^{3} pg 
vesicle diameter  90 nm 
pressure difference  1600 Pa 
Hamaker coefficient  10^{19} J 
Results

Vesicle behaviour in tumour vessel.
As we expected, vesicles and red blood cells are quickly carried away by the flow. However, a portion of vesicles adheres to the vessel wall and drops out of the flow. It confirms that our delivery concept is theoretically possible.
the model with 30 vesicles:
the model with 50 vesicles:
the model with 110 vesicles:

The capture index.
To measure the effect of vesicles number on therapeutic effect quality for tumour tissue, we measured the ratio of vesicles delivered to tumours against an initial number of vesicles and called it a capture index.
Number of vesicles Capture index 10 0.60 20 0.45 30 0.37 40 0.48 50 0.36 60 0.42 70 0.36 80 0.61 90 0.49 100 0.5 110 0.39 120 0.38 
Accumulated vesicle distribution among the vessel.
We have calculated adsorbed vesicle distribution among the X (green graph) and Y (blue graph) axis.
The number of vesicles stuck to the vessel. The initial number of vesicles is 50.
The number of vesicles stuck to the top and bottom walls. The initial number of vesicles is 50.
The number of vesicles stuck to the vessel. The initial number of vesicles is 100.
The number of vesicles stuck to the top and bottom walls. The initial number of vesicles is 100.
The inner diameter of many cancer cell lines is greater than 10 µm, in contrast to normal cells [6]. We can see that vesicles get in almost every malignant cell. The distribution of vesicles is near uniform between the bottom and top vessel walls.

The change in capture ratio with an increasing number of vesicles. Further estimations require high computer performance.
The number of vesicles stuck to the top and bottom walls. The initial number of vesicles is 100.
However, we can expect two ways of system behaviour. First: efficiency will oscillate near the asymptote. Second: the efficiency will decrease dramatically since the large group of vesicles will be carried out by the flow. This trend can be seen from simulations with a high number of vesicles. All this leads to the conclusion that the capture index maximum is near 0.6. To expect the therapeutic effect, we must take it into account.
We have estimated the required number of RNA molecules per cancer cell. There are average 3.42 transcripts per gene [8]. So, to provide effective treatment, we have to deliver about ten vesicles to every cell. Further calculations depend on tumour size and the average diameter of the cancer cell. These parameters can be determined for each tumour individually.
Probably, the injection with a high number of vesicles at once should be avoided and separated into a few injections. Thereby we will reach the optimum between capture index and the number of vesicles. To calculate the last parameter, we should analyze the fourth formula for every specific case.
The correction factor (CF) is necessary because we have modelled a 2D vessel. To count CF, we considered the natural size of the tumour vessel. We assume that the size of each computational layer crosssection has a red blood cell size. So, we should use the fifth formula to count the required number of vesicles.
Conclusions
 Our delivery concept is theoretically possible.
 Vesicles get in almost every malignant cell.
 The distribution of vesicles is near uniform between the bottom and top vessel walls.
 Сapture index maximum is near 0.6 for the vessel with 100 µm length and 30 µm diameter.
 We should use the fifth formula to count the required number of vesicles.
References
 V. N. Kaneva et al: “Modeling Thrombus Shell: Linking Adhesion Receptor Properties and Macroscopic Dynamics”, 4 Biophysical Journal 120, 334–351, January 19, 2021.
 J. N. Israelachvili: “The Calculation of Van Der Waals Dispersion Forces between Macroscopic Bodies”, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. Vol. 331, No. 1584 (Nov. 21, 1972), pp. 3955 (17 pages).
 G. Fullstone et al: “Modelling the Transport of Nanoparticles under Blood Flow using an Agentbased Approach”, Sci Rep 5, 10649 (2015).
 Verlet, L. 1967. Computer ‘‘experiments’’ on classical fluids. I. Thermodynamical properties of LennardJones molecules. Phys. Rev. 159:98–103.
 Charlotte F J M Peeters et al: “Vascular density in colorectal liver metastases increases after removal of the primary tumour in human cancer patients”. Int J Cancer 2004 Nov 20;112(4):5549.
 Babita Shashni et al: “SizeBased Differentiation of Cancer and Normal Cells by a Particle Size Analyzer Assisted by a CellRecognition PC Software”, Biol Pharm Bull. 2018 Apr 1;41(4):487503.
 Thomas Andrew Waigh: “Applied Biophysics: A Molecular Approach for Physical Scientists”, 2007.
 KuoFeng Tung et al: “Topranked expressed gene transcripts of human proteincoding genes investigated with GTEx dataset”, Sci Rep 10, 16245 (2020).