I am trying to mathematically model the dynamics of a particular cell in circulation and its binding to another nanoparticle that is intravenously injected. The concentration of this cell is very low (about 5 cells per mL of blood). I want to see how many nanoparticles are needed to bind to as many of these rare cells as possible.

So first, I am trying to model the cell dynamics in the blood flow. I am not sure how to start. I wanted to model the cell movement through the entire cardiovascular system, but I would have to simplify it. I am thinking of modeling the blood flow through some of the major organ systems, and estimating the absorption through the capillaries. I am thinking something like the physiologically-based pharmacokinetic model (PBPK). However, there would be some parameters that I wouldn't know about that I would need for the model (at least that is what I think, I am new to pharmacokinetics) Also for the actual fluid dynamics, I'm not sure how to model this, because the blood isn't just a fluid, and has many other cells, so I don't know if there are some other effects.

So, my question is, in what direction should I go to model the dynamics of this cell in circulation? I want to model the circulation in the entire cardiovascular system, because I need to also model the other nanoparticle dynamics, and how likely it is to find and bind to some of the receptors on the cell. Is there an example of a similar model that has been done?

  • $\begingroup$ I'm not terribly knowledgeable in this field, but I've come across it once or twice in my work doing pharmacokinetics/pharmacodynamics (PK/PD) studies on both large and small molecules. To my knowledge, there should be existing published studies on modeling the circulation of cells in blood. While you are correct that blood itself is a complex non-Newtonian fluid, its physics are pretty well established by now. Nanoparticles are also becoming more and more common, so hopefully you should be able to find multiple published reports on modeling them as well, then put the two together. $\endgroup$ – MattDMo Oct 16 '16 at 15:37
  • $\begingroup$ So, you should be fairly accurately able to determine the dispersion of your cell type of interest as well as the nanoparticles, then it's just a matter of calculating the probability of their meeting and the avidity needed for them to bind - say, for example, that the nanoparticle has a receptor-specific scFv coated on its surface, and the correlating receptor density on your target cell is such-and-such, and the Pk of binding is so many nM or pM or whatever. Good luck, I'm not a math guy (I just take care of the biology)! $\endgroup$ – MattDMo Oct 16 '16 at 15:43
  • $\begingroup$ This question is a bit opinion-based and would most likely be put on hold for that. However, this is my suggestion. Start with the simplest model and add complexities if the model is insufficient. Don't start with modelling the blood flow: you'll get into PDEs which are more complicated than ODEs. Start with an assumption that the blood stream is a well mixed vessel. You can consider different organs as compartments with different volumes (based on how much vascularised they are). $\endgroup$ – WYSIWYG Oct 17 '16 at 9:23
  • $\begingroup$ @WYSIWYG in what sense is it opinion-based? There are only 2-3 possible models that have been previously done. So, it might be a little opinion-based, but is it really bad that it would be put on hold? $\endgroup$ – TanMath Oct 18 '16 at 3:45
  • $\begingroup$ I am not saying that it will definitely be put on hold. But different people would have different suggestions on how to model a certain system. Technically, there are infinite ways to model something. If your question is explicitly about reference request (previous models etc) then it is fine; you can add the relevant tag and clearly mention that you are looking for references. $\endgroup$ – WYSIWYG Oct 18 '16 at 4:53

To go the PBPK route:

I would recommend using the organism parameters and structure from the whole body platform model by Shah and Betts.

  • Shah, D.K. & Betts, A.M. J Pharmacokinet Pharmacodyn (2012) 39: 67. doi:10.1007/s10928-011-9232-2

They have assigned a plasma space volume, plasma flow, blood cell space volume and blood cell flow for each organ and the total body. The base model structure for circulation is complete if you use the equations for the blood cell space in this paper as the foundation (Eq 2 and Eq 5).

Once the model structure is complete, step 2 is to model all processes that can affect the cell in each compartment. If the cell is adhering to a nanoparticle, a compartment equation could then be:

d[cell]/dt = (flow in - flow out)/volume + Kon x [nano] x [cell] - Koff x [cell-nano complex]

To model the interaction of the cell with the nanoparticle you will need to model both within the same structure. A review of PBPK modelling for nanoparticles can be found here:

  • Physiologically Based Pharmacokinetic Modeling of Nanoparticles Mingguang Li, Khuloud T. Al-Jamal, Kostas Kostarelos, and Joshua Reineke ACS Nano 2010 4 (11), 6303-6317 doi: 10.1021/nn1018818

A pharmacokinetic (top-down, classical) model for nanoparticles in the circulatory system may also give insight to how to model the processes that affect nanoparticles in vivo. This model accounts for fluid dynamics, pore size and specific nanoparticle properties.

  • Kirtane, A. R., Siegel, R. A. and Panyam, J. (2015), A Pharmacokinetic Model for Quantifying the Effect of Vascular Permeability on the Choice of Drug Carrier: A Framework for Personalized Nanomedicine. J. Pharm. Sci., 104: 1174–1186. doi:10.1002/jps.24302
  • $\begingroup$ I am not understanding the compartment equation. Wouldn't it be one for each organ, modeling the concentration and flux of the cell into and out of the organ? I plan to keep the model simple and assume there are no major effects on the cell, in terms of growth, and elimination. However, maybe the elimination of the nanoparticle by immune response might be important. $\endgroup$ – TanMath Oct 18 '16 at 3:41
  • $\begingroup$ You're right. The words "compartment" and "organ" are interchangeable. Every organ is modeled as a compartment. $\endgroup$ – Polymania Oct 19 '16 at 15:34
  • $\begingroup$ Can you fully explain the differential equation you wrote down? Also what parameters do I need to know about the nanoparticle and the cell? Where can I find a good introduction to PBPK modelling? $\endgroup$ – TanMath Oct 21 '16 at 1:36
  • $\begingroup$ What does the Kon and Koff model? $\endgroup$ – TanMath Nov 3 '16 at 22:15
  • $\begingroup$ Kon is the rate constant used for calculating the rate that the nanoparticle and the cell adhere together, proportional to the concentration of each. The term [Kon x nano x cell] gives you a rate in mol/time of nano-cell complexes that are formed. Koff is the reverse rate constant. en.wikipedia.org/wiki/Binding_constant $\endgroup$ – Polymania Nov 4 '16 at 15:23

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