High-accuracy adaptive modeling of the energy distribution of a meniscus-shaped cell culture in a Petri dish
Introduction
A Petri dish is a cylindrical plate often used for cell cultures. To induce and control the growth of these cell cultures, they may be exposed to various electromagnetic (EM) fields. One common scenario is to place the Petri dish into a rectangular waveguide that is illuminated with a polarized wave radiating at a particular frequency, [1], [18], [17].
In order to ensure the proper growth of the cell cultures, it is necessary to guarantee the high and uniform distribution of the EM energy (typically measured in terms of SAR – Specific Absorption Rate–) within the Petri dish [1], [20]. Some authors state that in addition to control the SAR, one also needs to impose some additional conditions, e.g., polarization, on the distribution of the full electromagnetic fields to secure the proper evolution of the cell culture (see [22] and references therein). Furthermore, the meniscus shape developed at the interface of the liquid with the dish provides a complex shape, whose geometry is typically expressed as a mathematical formula involving exponential and hyperbolic functions.
Suitable numerical simulation methods for these scenarios need to have the following features. First, they should be able to handle three-dimensional geometries, including the Petri dish shape, and the meniscus shape. Second, they should be able to efficiently deal with the discontinuous material properties at the air-liquid interface. Third, they should be flexible enough to enable simulation of all possible scenarios, including various geometries, polarizations, and frequencies. Finally, and more importantly, since the observed electromagnetic fields produced by modifications on the design system are often small but nonetheless important, the simulation software should be highly accurate for all considered models. Moreover, it should provide an error estimation in order to minimize uncertainty and guarantee the correctness of the solution for each model.
Several numerical methods have been employed for Petri dish simulations exposed to EM fields in different configurations, e.g., [3], [20], [21], [16], [1], [22], [23]. Approaches based on differential formulations are mainly used because of their flexibility to deal with complex geometrical and material configurations. Among them, the most common numerical technique is finite differences (FD); typically, in time domain.
SAR data is obtained by averaging the energy distribution within a small cube shaped volume (known as voxel), which is the natural choice in FD grids. The presence of non-Cartesian geometries, boundary layers, and field singularities encountered on the resulting field solution together with internal resonances, requires the use of tiny voxels. Unfortunately, the use of small voxel sides of the order of one tenth (or even one hundredth) of a wavelength may not be enough in some cases to have confidence on the results. In those situations, the voxels close to the solid/liquid interface are skipped from the SAR distribution assessments, as it is reported in [22].
In here, we propose to employ a highly accurate method that works under all the above scenarios and provides an error estimation that guarantees the correctness of the solution. It is based on a Finite Element Method (FEM) that utilizes “adapted” meshes to both the geometry of the problem domain and its solution. A sequence of adapted meshes is generated in an automatic fashion by refining a given mesh in certain areas of the domain. Simultaneous h and p refinements, i.e., local variations of the element size h and the polynomial order of approximation p throughout the mesh are supported (the so called hp-adaptivity [5], [6]). Preliminary results of the 3D implementation of the hp-adaptivity proposed by the authors [9] applied to the Petri dish problem were presented in [10], where the cell cultures were modeled as a circular dielectric, i.e., the meniscus shape was not included in the geometry. In this paper, the meniscus shape is included in the computational model. The main focus of this work is to analyze: (a) the numerical effect of the geometrical refinements in the hp-adaptivity due to the presence of a meniscus-shaped object, and (b) the differences obtained in the physical results due to the presence of such complex geometrical object.
Section snippets
Mathematical modeling and formulation
We consider a Petri dish filled with a cell culture and placed inside a rectangular waveguide, as depicted in Fig. 1. Dimensions of the waveguide and dish are described Fig. 1. The meniscus shape is modeled as described in [20]. Its mathematical expression is displayed in Fig. 1 (we selected c = 2.01 mm). The waveguide is electromagnetically excited with its fundamental mode, known as TE10 (details can be found in any of the numerous books in the subject, e.g., [4]). Operating frequency is
hp-FEM and automatic hp-adaptivity in 3D
The 3D implementation of the hp-adaptivity proposed by the authors (see [9] and the references therein) is based on a self-adaptive strategy devised in [19] and further improved in [15], [6]. The hp-adaptive strategy supports anisotropic refinements on irregular meshes with hanging nodes, and isoparametric elements as well as exact-geometry elements. It supports geometrical refinements through the use of its own geometrical package to accurately model the problem geometry. Hexahedral H(curl)
Numerical results
We start by analyzing the effect of geometrical refinements that are needed to reproduce the geometry of the problem, namely, the Petri dish. Refinements forced to geometrically adapt to a complex geometry affect to the meshes delivered by the adaptivity and hence have impact on the numerical error of the finite element solution.
For that analysis, we consider the following four geometrical models: a Hexa (a waveguide without any object in its interior), a Cylinder (a waveguide containing a
Conclusions
We have simulated the electromagnetic problem of the irradiation of a meniscus shaped cell culture within a Petri dish using an hp-adaptive FEM. The proposed method delivers highly accurate solutions with error control. The effect of the meniscus-shaped geometry that develops in the air/liquid interface significantly difficults its modeling, while it heavily influences the growth of the cell culture. The effect of the geometrical refinements in the performance of the adaptivity in such a
Acknowledgment
The authors would like to thank Prof. González-García of University of Granada, Spain for providing the geometry of the waveguide with Petri dish. Also to Ph.D. student Adrián Amor for providing some numerical results used for code verification. The authors would like also to acknowledge the support of Ministerio de Educación y Ciencia of Spain under project TEC2010-18175/TCM. David Pardo has received funding from the European Union's Horizon 2020 research and innovation programme under the
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