A spatial classification applied to convectional reaction-diffusion boundary problems basing on a geometrical polymorphism of biological objects

Convectional reaction-diffusion is a common physical process which forms a background for existence of all biological objects. Along with the fluid flows it provides a supplying/removing of chemical compounds in a living system. In mathematical modelling PDE border problems are usually used for decryption of such kind phenomena. In the present study the classification of mathematical problems is developed on the base of the general geometrical properties of the considering living objects. Moreover, the uniform/non-uniform and the homogenic/non-homogenic structure of the boundaries and the internal area are taken into consideration. The type of classified models is determined by the structure of both an outer boundary and the inner space of the contemplated objects (i.e. their homogeneity/heterogeneity, H/H Classification). Furthermore, several dimensionless metrics are also introduced to estimate geometrical features influence the spatial-time distributions of chemical compound concentrations. The proposed classification can be applied to any kind of biological systems to consider a spatial polymorphism and its influence on the metabolic gradients.


Introduction
Considering the main properties of biological objects, one obviously highlights that there are several essential features which are inextricably linked to them.First of all, the area of consideration is always 3D.Despite possible symmetry and the difference in the size, it means that the processes must be considered under essential spatial dependence of variables, and, as a result, they are spatially distributed.Furthermore, the distribution could be non-homogeneous and spatially localized in several compartments [1,2].The second aspect of the problem is the mutual combination of different physical and chemical processes [3].The most of them occur simultaneously (figure 1).Under such conditions the complexity of the considered problems seems to be unsolvable, or the ways of simplifications cause distortion of biological formulation background.Another essential aspect of reaction-diffusion with convection is a boundary condition.It is inextricably linked to geometry of the analyzed system and the algorithm of a digital phantom creation.If the distribution of chemical compounds is a subject of the analysis, then biochemical reactions and their spatial non-homogeneity must be taken into account [4].Thus, the final presentation of the biological claims forming a physical model to be modelled is a combination of known important processes.No one of them can be really excluded without losing of essential properties of the considered object.

Figure 1.
A schematic example of a living object with indication of physical processes and structures.A diffusion area is the medium where the chemical compounds (metabolites) can move forming a nonuniform concentration distribution.Convection appears as a result of a solvent movement inside the medium due to a difference of pressure.The area of convection may be an interstitial fluid among cells or especially organized space (not shown) like a blood stream, lymphatic channels, etc.The boundaries are forming the border under where the variables are determined.The medium is filled up with many internal compartments where the biochemical reactions occur.The figure is created with BioRender.com.
Having evaluated different mathematical formulation, one can conclude that the problems of reactiondiffusion with convection are able to be classified by geometrical aspects of the system.The main advantage of such a classification is to get a systemic approach to forming a mathematical formulation of the problem meeting to biological demands and features of the considered object.Indeed, the same mathematical models may correspond to the similar biological systems, but the aim of the modelling causes the final choice.In the present study the classification is represented in terms of the properties of spatial areas and localization of the physical processes.This type of systemic ordering for mathematical problems achieves the maximum connection to biological demands.

Mathematical description of convection reaction-diffusion in a biological system
As per the usual approach, the mathematical formulation of metabolites N diffusion in a limited area Ω can be depicted as follows:  and u correspond to metabolites concentration, system parameters and convectional velocity field respectively.The reactions in the medium provided by enzymes activities in the metabolic pathways are described by the term . It also includes the influence of some external parameters such as electric potential and so on.One should note that some indications in equation ( 1) are common but very essential.The shape and the property of a diffusion area (  ) are accompanied with the structure and the type of the boundary conditions for the object border (  ).Usually, both real object geometric parameters mentioned above are represented using the simple primitives.For symmetrical system this way seems to be acceptable.However, it is not always possible because of complexity and asymmetric distribution of geometrical shape of the object.Nevertheless, the geometric classification classes that will be introduced are extremely easy to understand because they are based on the analysis of elementary interactions in set theory.
It is essential to highlight that the introduces classification is quite different from other ones.For instance, the parameter space was earlier fully classified in terms of the types and stability of the uniform steady state for the reaction-diffusion systems of two chemical species on stationary rectangular domains [5].Furthermore, anomalous diffusion in animal movement data can be classified using power spectral analysis [6].Various statistical testing approaches may apply to fractional anomalous diffusion classification [7].Nevertheless, the creation of a new classification is influenced by the requirement to create a phantom basing on real biological geometry.

The general classes of geometrical types of considered problems
As already mentioned above the complexity of a biological system creates formidable difficulties to construct 3D phantom and to set up the boundary conditions in equation (1).Indeed, the most important problem is to choose a type of condition and furthermore evaluate the appropriate value.Nevertheless, all cases can be assigned to two general classes only.Class I represents a ranged area with continual external boundary.The most part of models will get into this class.Indeed, the processes are usually included into the area of consideration as a part of continual properties of the medium.This claim is true for both convection and reaction-diffusion.However, under some conditions one needs to exclude a part of the object area out of scrutiny.Under these circumstances the second Class of the models appears.Its especial feature is a forming of an internal boundary which is the part of the object border, and the boundary conditions are defined there.The schematic illustration of the mentioned classes is represented in figure 2.

Class I Class II
Figure 2. A scheme of two geometrical classes applied to the boundary problem of reaction-diffusion with convection.A green area indicated the diffusion area (  ), and the blue line corresponds to the object border (  ).In both classes the final phantom is formed as a unit of  and  .
These types of geometry determine the style of the digital phantom creation, and they partially influence the choice of boundary conditions.For example, if a researcher prefers to exclude the part of convection appearing due to a blood flow the area of inner cavity of the blood vessels will be omitted from the phantom.In this case Class I converted into Class II with appropriate boundary conditions on the inner part of  .The classification process is further complicated by the properties of  and  .

The detailed classification of reaction-diffusion problem on the base of homogeneity/heterogeneity of the phantom elements
The properties of the diffusion area and the boundaries are the most essential to be formulated in the body of the digital phantom [8].Modern capabilities allow for the conditions to be considered to be varied.The simplest case is a homogeneous medium with a homogeneous boundary.Under such circumstances the phantom has the following properties: ,; The next variation is characterized by a spatial heterogeneity being present both inside and on its borders.The non-homogeneous conditions in the medium are determined by the conditions: The heterogeneity of the boundary conditions is reflected in possible combination of different type of conditions and\or spatial non-homogeneity of one type of condition. ,; It is essential to notice that classification relies on the combination of multiple specific properties of the diffusion area and its boundaries.For instance, sub-subclass 2 in any Class is caused by the consideration of various phases in the biological object, like the membrane and cytosol, which have different diffusion coefficients.Furthermore, the boundary conditions can be homogenic, but they are combination of the Dirichlet Conditions and the Neumann Conditions on different surfaces.This feature of the problem immediately leads to subclass 2 in H/H Classification.The combination of classes and subclasses is represented in table 1.
. Table 1.The classification of convectional reaction-diffusion problems.The classes and subclasses indicated by the numbers.The main aspect is the combination of homogeneity and heterogeneity for both the diffusion area and the boundaries (H/H Classification).a The classification is represented as the final combination of numbered conditions.For example, Class I with homogenic boundary and homogenic diffusion area will be represented as H/H I.1.1 and for heterogenic diffusion area -H/H I.1.2,etc. b Boundary conditions determined by equations ( 3) and ( 5).c Homogeneity/heterogeneity of the diffusion area according to equations (2) and (4).

The parameters which can be used as metrics to estimate geometrical fluctuation of digital phantoms
Using introduced H/H Classification one is able to formulate clearly the types of the phantom parts and to consider their appropriate properties.However, sometimes the property of the area can be averaged according to geometry and the evaluated values.In particular, each biological surface has roughness, but this may be negligible when a concentration gradient is formed near it.Indeed, this aspect needs to be considered case to case.Nevertheless, some common recommendation of metrics is reasonable to take into account.The considered phantom always has the standard relations between medium properties and the geometrical polymorph sizes.The simplest metric is a relation between a diffusion coefficient and the constant of a medium consumption.If the third term of equation (1) includes the summand like ( ) then the relation is expressed in the following form: This parameter in a modified dimensionless form was firstly introduced for modeling of glucose gradients near pial vessels using sphere sources diffusion fields (SSDF) method [9].However, equation ( 6) may be reasonably modified taking into account the geometric sizes of both the phantom and the roughness.The expression looks like as follows:   where roughness  is a part of the considered phantom  related to irregular geometrical features of the biological object.The value of parameter introduced by equation ( 7) may be a common criterion of geometrical irregularity inclusion/exclusion in creation of a digital phantom.

Discussion
Modelling of reaction-diffusion coupled with convection and the gradients of electric field is always based on precise relation between real shape of the biological object and geometry and parameters distribution in the mimicking digital phantom.Under omitting of some part of geometry or simplifying of physical processes the researcher is in danger of losing significant information about the object being investigated.Therefore, to avoid this risk, one needs to consider right processes with right localization.In particular, for H/H I.2.2 models of glucose distribution in a neurovascular unit it was shown that the geometry of inner flux dysconnectivity barrier essentially determines the reply of the system on changes of cerebral blood flow in the capillary [10,11].The same effect of geometrical polymorphism appears in other system corresponding to H/H I.2.1.It was shown that a spatial polymorphism of the post-synaptic density of glycine receptors is important to form a physiological response to a neuromediator release [12].Thus, the introduced classification can be a useful tool for design of schemes of a digital biological phantom creation and further localization of boundary conditions and internal parameter distributions.