Efficient Methods to Calculate Partial Sphere Surface Areas for a Higher Resolution Finite Volume Method for Diffusion-Reaction Systems in Biological Modeling

Round 1

Reviewer 1 Report

The authors present an interesting approach to calculate areas on a sphere intersecting with rectangular finite volumes. The Monte Carlo method, through icosahedron-based technique, is found to provide the best performance in terms of computing efficiency. However, the  application and validation of the model to actual  biological systems modelling is missing. Additionally, the authors should consider the use of quasi-Monte Carlo method in their discussion.  

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

This manuscript presents an intriguing computational geometry problem: Compute the spherical surface areas created in each of eight octants when a sphere is cut by
three orthogonal planes. For the most general problem presented here, the intersection of the three planes is assumed to be inside the sphere, although not necessarily at the center of the sphere. The authors were unable to find analytic formulas for the partial surfaces (and it isn't obvious such formulas exist) and so develop a numerical for computing these surface areas.

This computational problem is well motivated by an interest in modeling reaction diffusion on the surface of biological cells (the spheres in the present model) and the interaction of the reactants with a surrounding fluid. A finite volume scheme on a uniform Cartesian mesh is used to model the fluid. To account for the effects of reactions on the surface of the cells, each finite volume mesh cell must know how much of any one biological cell is contained within the volume.

The only minor criticism is that the authors have not made clear how much accuracy they need for their finite volume scheme.   If the dimensions of the sphere are on the order of a finite volume mesh cell, why would they ever need a high generation icosahedral mesh?  A bit more discussion on the actual processes on the sphere might be a good start.

The paper is well written, although the authors should consider the comments below.

Figure 1. This figure would be clearer if only the slicing planes were shown. Can the authors comment on why the linear system is underdetermined? From the linear algebra, it is obvious that more constraints are needed on the system. The authors choose to compute several of the areas explicitly to constrain the solution. But more generally, why is the original system not sufficiently constrained? What exactly is missing? Are there other ways to constrain it, other than just supplying some of the areas directly? What are the practical accuracy requirements for their surface calculations? If each sphere fits roughly inside of a finite volume mesh cell, it seems that the resolution of the surface diffusion is much higher than the solution in the bulk fluid. Can the authors get away with much coarser approximations to areas of surface domains? This would certainly make the code more efficient.  On the other hand, does it make sense to use integral formulations to get accurate area computations?  Or would this be overkill?  "We couldn't find any analytical way to calculate ..." should probably be rephrased to something like "Analytic formulas are not readily available ...". How are the errors being computed? Do the authors have an exact solution to which to compare their numerical results? Figure 7. These would provide more useful information using a log scale for each axis. Or better, it might make sense to put all error calculations on one graph, and calculation times on a second graph. This would make it easier to compare results of the three methods. The paper is generally well written, but the authors should read it over carefully to catch awkward phrasing. For example,

-- "The rest three values ..." should read "the remaining three values ...".
-- "The set of randomly generated one hundred cells ... " should read "The set of 100 randomly generated cells ...".
-- "Stochasticity of the sample points appeared to ruin the line-like behavior ..." is awkward. Maybe something like "Stochasticity of the sample points negatively impacts the convergence rate"
-- " ... the number of sample points from a uniform distribtuion can be any number." should read " can be arbitrary".

Author Response

Please see the attachment.

Author Response File: Author Response.pdf