Correction function for accuracy improvement of the Composite Smeared Finite Element for diffusive transport in biological tissue systems
Introduction
In investigating supply of nutrients or drugs to target cells, it is essential to achieve efficient transport through capillary walls as the biological barrier. In order to help in that goal it is desirable to have adequate simple, but accurate, computational models. Transport within capillaries and tissue has been the subject of study over past decades, especially molecular/particulate convective and diffusive transport within blood vessels and through the vessel walls, since it is the fundamental process in drug delivery. Capillaries are filled with plasma and cells, while tissue can be considered as a porous medium with very complex microstructure composed of extracellular fibrous matrix and cells. Interaction between particles/molecules and the capillary wall components can dominate the transport through the wall. This interaction on a molecular level may be incorporated into a continuum transport model [[1], [2]], and appropriate multiscale model has been developed and applied to various bioengineering problems [[3], [4]].
We here consider transport of matter by diffusion within biological systems, with a specific focus on solid tumors. Morphology of tumor vasculature has been intensively investigated, with experimental evaluation of the role of capillary density within tissue, capillary lengths and diameters [[5], [6], [7], [8], [9]]. In most of the tissue, and especially in tumors, geometry of blood vessels is highly heterogeneous with irregular blood vessel branching and variability of diameters and lengths. Since the number of vessels is extremely large (total number of capillaries in the body billions [10], modeling of each vessel within an organ or tumor would not at present be feasible. Also, tumor tissue is very heterogeneous and involves zones of significantly different permeability for fluid and particulate transport. Additionally, capillary networks are different for different tumor types; they vary from patient to patient, and are changing over the period of tumor growth.
Frequent mass exchange takes place in a heterogeneous environment among several phases, where mass partitioning (hydrophobicity) may occur at the interface of phases. Hydrophobicity is usually present at the interface between solid and fluid phase [11], which has to be considered in computational models. Our numerical results [11], based on the analysis of diffusion kinetics in the parameter space of partitioning and diffusivity, showed that partitioning is an extremely important parameter which may control equilibrated mass distribution.
Flow through capillary branching is three-dimensional, which would require detailed 3D continuum models. However, 3D models of branching, in general, lead to significant effort in the 3D FE mesh generation and are computationally very demanding. Detailed models, therefore, are not suitable for large pipe networks, as in case of capillary systems. Consequently, it is desirable to have a simpler, efficient model which can be used as a good approximation of the flow within large pipe networks. One of the options is to use 1D pipe capillary network for capillary vessels which are immersed in real tissue domain. In [12] a transport model was introduced which can be applied to large vascular systems [13]. The model is efficient since it uses 1D finite elements for larger vessels and equivalent continuum FEs for capillary beds. The introduced computational model relies on a number of approximations, summarized in these references, with respect to the real biological conditions within capillaries and tissue.
Furthermore, even with using 1D finite elements, there is still a problem of reconstructing complex geometries, so that FE meshes are with large number of nodes and large number of equations to be solved. All these facts have been a motivation to explore possibility of formulation of a smeared computational model. Generally, smeared procedures have been applied for crack distribution within materials in fracture mechanics studies, or for material properties of complex materials (e.g. [[14], [15]]). Our recently introduced smeared modeling concept [16] provides models for large domains which can effectively be employed for mass transport in the capillary system and tissue [[12], [13]]. This formulation is easy for implementation since it does not require numerical homogenization or 1D capillary modeling, and it can be used for modeling mass transport within entire body organs or tumors.
The basic requirement of the smeared concept is that the transport characteristics of the system should appropriately be preserved. This means that smeared model has to be as accurate as a true 3D model and needs to have approximately the same predictions regarding drug and nutrient transport to and from tissue, which is crucial in prediction for cancer and tumor treatment. Despite the appropriate theoretical considerations presented in [16], we found that there are still differences in the overall mass transport when comparing a true 3D model and smeared model. Following these findings, the main goal of this study is to introduce a numerical technique for improving accuracy of smeared model, especially in case of common drugs which are often hydrophobic.
The main idea of improving smeared model accuracy is to introduce correction function by which the diffusion coefficients of the wall are to be multiplied, in order that evolution of concentration in the tissue domain, obtained by the smeared model, is approximately the same as when using a true 3D model.
Here, we give the basic relations used in the formulation of the Composite Smeared Finite Element (CSFE) according to [16]. Formulation of the composite finite element has been present in the FE literature, as, for example in [[17], [18]] where beam and continuum 3D are coupled. First, the diffusive transport in tissue can be described by a differential equation, based on Fick’s law and mass balance equation [19]: where are diffusion tensor coefficients for the coordinate directions, c is concentration and q is a source term. Then, the corresponding FE balance equation, for the equilibrium iteration i is [[12], [19]] where C is the nodal concentration vector (C corresponds to start of time step ), is the external flux; and “mass” matrix M, diffusion matrix K and the source vector (evaluated at end of time step), are
The CSFE contains two domains, capillary and tissue domain, Fig. 1. Diffusive transport within capillary domain consists of (approximated) 1D diffusion in the capillary net, which can be transformed into a continuum form by formulating a diffusion tensor,
where is the capillary diffusion coefficient, is the capillary internal cross-sectional area, is the total cross-sectional area as sum of , and are the directional cosines of the capillary axis with respect to the coordinate axes i and j. Then, diffusion within elementary capillary domain volume dV occurs according to Eq. (1), where is the capillary volumetric fraction, or capillary density, and dV is the total elementary volume of continuum.
Diffusion within tissue domain is governed by Eq. (1), with the corresponding tissue diffusion tensor, and within the volume .
Diffusion through capillary wall can be considered as a 1D process. Graphical interpretation of capillary is shown in Fig. 2.
Elementary area of the internal surface of the wall can be related to the elementary volume ; further can be related to the elementary total volume , so finally we have where is the surface ratio (capillary wall area-to-volume ratio, for a single straight capillary ).
The mass concentration can be considered as linearly distributed through the wall thickness (acceptable for thin capillary walls). Then, the flux , corresponding to the elementary surface through the wall at point 2, can be expressed as where are the systemic (capillary) and tissue concentrations at the end and start of time step, respectively; both matrices and are specified in [16]. As a result, we have a tissue continuum within which capillaries are distributed and are producing the source of the mass according to (6). Therefore, the nodal fluxes of a continuum finite element are where terms within the parenthesis (…) follow from (6), and are the continuum interpolation functions of the element with the volume V. When evaluating the integral (7), concentration is the current concentration within tissue at an integration point.
For hydrophobic drugs it is important to take partitioning effect into account. In case when partitioning phenomenon is present at the wall surfaces, the elementary mass flux can be expressed as [16] where and are the partitioning coefficients at the internal and external capillary wall surfaces.
Instead of using source terms at FE integration points, fictitious connectivity elements can be assigned at each continuum node (Fig. 2). Then, the balance equation for the connectivity element at continuum node I can be written as where and and are concentrations at node I at end and start of time step, respectively. Also, and are partitioning coefficients as in (8); is the wall diffusion coefficient, is the wall thickness at node I; and is the wall surface area belonging to the node I, which is with , and being the volumetric ratio, the area coefficient and the volume of the continuum which belongs to the node, respectively. The volume can numerically be evaluated as where summation includes all elements containing the node I. The parameters of the model, assigned to each continuum node I include geometrical data (the volumetric ratio of capillaries , the surface ratio , the wall thickness (); and material data of capillaries consisting of wall diffusion coefficient and partition coefficients and at the capillary surfaces (additional details given in [16]).
We note here that partitioning is usually present at the (blood plasma)–(internal wall surface) interface, while at the interface between the external wall surface and tissue can be discarded; hence in further presentation we take that , and use parameter P for .
Section snippets
Accuracy of the smeared model
Before specifying our concept of improving accuracy of the smeared models, we present simple examples to gain insight into the concentration distribution in tissue around capillary and then consider mass accumulation within tissue according to detailed and smeared models.
Numerical determination of the correction function
According to the presented results (and others not shown here), there can be notable differences in mass release from capillaries to tissue when compared true and smeared models. We here introduce a numerical procedure in order to find a table of correction factors, or correction functions, by which we will multiply the capillary wall diffusion coefficient to obtain the same mass exchange between capillaries and tissue for true and smeared models. As a result, we generate a table (or
Effects of partitioning on correction functions
A phenomenon called hydrophobicity may be important in diffusion through biological system. Majority of drugs are hydrophobic substances, meaning that those chemical compounds prefer the organic phase over water [20]. The simplest way to express partitioning phenomena in molecular transport is to take that, within a time step, the ratio between numbers of molecules passing the boundary between two media [11], where and are the numbers at solid and fluid side, respectively. We
Applicability of correction function to general conditions and practical aspects of application to smeared computational models of diffusion
In the previous section we introduced and evaluated correction function in terms of three parameters () to be used for multiplying the wall diffusion coefficient according to the relation (14). Here, we present some practical aspects of use of the scaling functions and implement that concept to general cases, including complex branching, unsteady diffusion, tissue with a number of capillaries and cells; and, finally, a simplified model of pancreatic tumor.
Conclusions
The goal of this study was to investigate accuracy of our recently introduced smeared finite element modeling concept [16]. The capillary network is replaced by a smeared continuum representation which mainly cannot capture detailed non-uniform pressure and concentration field within tissue, in the vicinity of capillaries. Although the corresponding Darcy’s and diffusion tensors are correctly derived, due to the mentioned characteristic of the smeared model, there is a difference between the
Acknowledgments
Dr. Ferrari acknowledges the support from NCI U54 CA210181 and The Ernest Cockrell Jr. Presidential Distinguished Chair at Houston Methodist Research Institute.
The authors acknowledge support from Ministry of Education and Science of Serbia, grants OI 174028 and III 41007, and City of Kragujevac.
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