In this paper, we present a simplified electrical model for tissue culture. We derive a mathematical structure for overall electrical properties of the culture and study their dependence on the frequency of the current. We introduce a method for recovering the microscopic properties of the cell culture from the spectral measurements of the effective conductivity. Numerical examples are provided to illustrate the performance of our approach.
Cell culture production processes, such as those from stem cell therapy, must be monitored and controlled to meet strict functional requirements. For example, a cell culture of cartilage, designed to replace that in the knee, must be organized in a specific way.
Hyaline cartilage is located on the joint surface and play an important role in body movement. In normal articular cartilage, there is a depth-dependent stratified structure known as zonal organization. As a simplified model, cartilage comprises three different layers : a superficial zone in outer , a middle zone that is of the height, and a deep zone consisting in the inner . At the microscopic level, cartilage tissue is composed of cells, collagen fibers, and glycosaminoglycans (GAGs). The concentration and organization of each microstructure differs among the three layers. In the superficial zone, cells are anisotropic and horizontally aligned, collagen orientation is also horizontal and GAGs have a lower concentration than in the other layers. In the middle zone, there are fewer cells and they are isotropic, collagen is randomly oriented and there is a medium concentration of GAGs. In the deep zone, cells are isotropic, cell density is higher than in the middle zone, collagen is vertically aligned and there is a high GAG density. As these parameters all contribute to the function of collagen in the knee, and must be replicated in the cell culture.
It is important that the method for monitoring cell cultures is non-destructive. Destructive methods require hundreds of samples to be cultured for a single functional tissue, and for the samples to be monitored multiple times during maturation. Here, we propose a microscopic electrical impedance tomography (micro-EIT) method for monitoring cell cultures that exploits the distinctive dielectric properties of cells and other microstructures. In this method, electrodes inject a current into the medium at different frequencies and the corresponding dielectric potentials are recorded, thus enabling reconstruction of the microscopic parameters of the medium. The parameters of interest are cell density, collagen orientation, and GAG density, as well as the orientation and shape of cells.
EIT uses a low-frequency current (below 500 kHz) to visualize the internal impedance distribution of a conducting domain such as a tissue sample or the human body. Recent studies measured electrical conductivity values and anisotropy ratios of engineered cartilage to distinguish extracellular matrix samples containing differing amounts of collagen and GAGs. During chondrogenesis over a six-week period, these measurements could distinguish the stages of the process and provide information regarding the internal depth-dependent structure.
In this work, we provide a mathematical framework for determining the microscopic properties of the cell culture from spectral measurements of the effective conductivity. For simplicity, we consider a microstructure comprising two components in a background medium. One of the components has a frequency dependent on the material parameters arising from the cell membrane structure, while the other has constant conductivity and permittivity over the frequency range. First, we derive in Theorem 2 the overall electrical properties of the culture, which depend on the volume fraction of each component and associated membrane polarization tensors defined by (10) and (11). Then, we show that the spectral measurements of the overall electrical properties of the culture can be used to determine the volume fraction of each component and the anisotropy ratio of the first component. For doing so, we study the dependence of the membrane polarization tensors on the operating frequency and use the spectral theorem to recover in Proposition 9 from the measurement of the effective conductivity on a range of frequencies the coefficients of its expansion with respect to the frequency. Proposition 9 also provides the anisotropy ratio of the cell culture.
This paper is organized as follows. In Section 2, we present a simplified model of the tissue culture. In Section 3, we derive an equivalent effective conductivity for the solution at the macroscopic scale. In Section 4, we present a method based on spectral measurements, in which microscopic properties are measured from the effective conductivity. This process is known as inverse homogenization or dehomogenization [5,14]. Finally, we provide some numerical examples to illustrate our main findings.
2.The direct problem
In this section, we propose a simple electrical model for the tissue and derive an effective conductivity using periodic homogenization.
We consider the domain of interest – the cell culture – to be described by a domain . We assume that where denotes a floor of the culture medium; see Fig. 1. Following [11,17,18], we describe the conductivity of the medium by a scalar field
Now that we have an expression for the conductivity in the medium, as commonly accepted in EIT, we use the quasistatic approximation for the electrical potential. For an input current on the boundary , with , the real part of the corresponding time-harmonic potential, denoted by , satisfies the following problem approximately:
Let us briefly explain how the expression of in (1) is derived. We should note that the frequency dependent behaviors of in (2) are attributed to thin cell membranes. Imagine that we inject an oscillating current at the angular frequency ω into the cube . Then, the resulting time-harmonic potential in is governed by
2.2.Homogenization of the tissue
We are now interested in getting rid of the microscale oscillations of , since boundary measurements will only allow us to image macroscale variations of the conductivity. To this end, we proceed to the homogenization of equation (2). Assume that is bounded from below and from above:
3.Imaging the microstructure from effective conductivity measurements
In this section, we do not care about the space dependence of , and will therefore drop it. We will thus assume that is constant equal to some matrix in . We will show what kind of information on the microstructure we can recover from the knowledge of in a range of frequencies . First, in Section 3.1, we will obtain a simple representation of the effective conductivity in the dilute case, where the volume fraction of both cells and other inclusions is small compared to the volume of biological fluid. Then, in the following sections we will use this representation and will show how to recover information about the microstructure using the spectral measure.
3.1.Effective conductivity in the dilute case
Here, we consider some reference cell and some reference inclusion with there boundaries and . We assume that and for some reference and let , where and respectively indicate the locations of the cell and inclusion and and their characteristic sizes. We assume that the conductivity of the inclusion is given by
The effective conductivity is therefore expressed as
Let , and . Then we have the following expansion:
We begin be reviewing properties of periodic layer potentials. Let us define the periodic Green’s function
We have the following jump relations along the boundary :
We denote by the operator . We write on and . Using these jump relations, we have the following representation theorem for , .
We have the following representation for :
For any , the system
As shown in the Appendix, and are invertible for . Moreover, since
We can now proceed to prove Theorem 4.
Let be a solution of (14), and let
We now proceed to compute the representation of the effective conductivity.
We have the following representation for :
We recall the expression of in (5):
We have the following expansion for :
Using this expansion, we obtain by exactly the same arguments as those in [3, Chapter 8] the following expansion, which is uniform in ,
3.2.Spectral measure of the tissue
Expansion (9) yields
can be extended to a self-adjoint operator , whose image is a subset of .
Let and . Let . Then obviously extends and its image is a subset of . Let us show that it is self-adjoint. Let . Let and respectively denote the -scalar product and the duality pairing between and . We have
From this result, we can now proceed. From the spectral theorem, there exists a spectral measure E such that for any and for any ,
Let be defined by (18). Then the following expansion of F in a neighborhood of 0 holds:
Identity (19) holds using the analyticity of F in a neighborhood of 0. We also have
In the following, we write
4.1.Imaging of the anisotropy ratio
The anisotropy ratio (the ratio between the largest and the lowest eigenvalue of the effective conductivity tensor) depends on the frequency . Furthermore, in the general case, the anisotropy orientation (the direction of the effective conductivity tensor eigenvectors) depends also on the frequency. However, in the special case where we have an axis of symmetry of a single inclusion or a cell, the anisotropy orientation is independent of the frequency.
We denote by the set of rotational matrices. Here, the superscript T denotes the transpose. For convenience, we write for and . We will need the following covariance result :
Let and . Then
We have, for any ,
The following corollary holds immediately.
Let . Then,
Let us begin with the two-dimensional case.
Let , and be an orthonormal basis of ; see Fig. 3. Let ξ be the orthogonal symmetry of axis . If , then
We have a similar result in three dimensions. The following proposition holds.
Let , and be an orthonormal basis of . Let (resp. ) be the orthogonal symmetry of axis (resp. ). If , then
The proof is exactly the same as in the case and is therefore omitted. □
It is also true that the symmetry axes of correspond to the eigenvectors of the polarization tensor . Therefore, the anisotropy direction of the frequency-independent background can also be recovered as the principal directions of .
Even if each of inclusion and cell has an axis of symmetry, the direction of eigenvectors of the effective conductivity tensor can be frequency dependent. The following numerical test is conducted to show an example of frequency dependency. There are an ellipsoidal inclusion with major axis and minor axis and an ellipsoidal cell with major axis and minor axis in the unit square as shown in Fig. 4(a). For the square domain , each axis length of cell and inclusion is , and . The center of ellipsoidal cell and inclusion are and respectively. The ratio between membrane thickness and size of a cell is . The conductivity value of medium, membrane, inclusion are 0.5 S/m, S/m, and S/m respectively. We use (5) to compute the effective conductivity tensor. For the numerical computation, we take advantage of using satisfying in Ω with boundary condition for . Then, can be replaced with . Hence, the eigenvectors of the effective conductivity can be computed and the main direction of anisotropy changes in terms of the frequency as shown in Fig. 4(b).
4.2.Implementation of the inverse homogenization
Following , we use the following values:
The size of cells: 50 μm;
Ratio between membrane thickness and size of a cell: ;
Medium conductivity: 0.5 S/m;
Membrane conductivity: S/m;
Background inclusion conductivity: S/m;
Membrane permittivity: F/m;
Frequency band: Hz.
In this case, we have values of for in Fig. 5. We consider a sample medium as follows: the cells are elliptic in shape, with axes lengths and , with . The background is composed of elliptic inclusions, with axes lengths and , with . Their orientation is given by the angles and respectively.
At each frequency, in order to compute the true effective conductivity given by (5), we perform a finite element computation using freefem . Comparison between the true effective conductivity and the expansion from Theorem 9 can be seen in Figs 6 and 7, in the case and , and .
To recover the moments from the effective conductivity, we approximate as a rational function,
Numerically, this is done as a simple least square inversion: the coefficients of the polynomials and are computed to minimize the quantity
We now consider a toy example where C is an ellipse in . In this case, if and are the eigenvalues of defined by (20) for , the ratio is independent of the volume fraction and is given by
Since the right-hand side of (21) can be regarded as a function of , the anisotropy ratio can be easily obtained by solving (21) with the known value r. In Fig. 8 (resp. in Fig. 9), we illustrate the reconstruction of the ratio r using the Padé approximation of F as a function of the anisotropy ratio compared to its theoretical value given by the preceding formula in the case where there is no inclusion B (resp. with an inclusion B with ). As we can see, the reconstruction is almost perfect in the case where there is no inclusion, and there is a slight bias induced by the inclusion B.
After recovering the anisotropy ratio , we can recover the volume fraction from the product of of the eigenvalues of . Indeed, we have
To reconstruct the angle of the inclusions, we simply use the orientation of the eigenvalues of the moments of for B and for C. This is illustrated by results in Fig. 10 when both B and C are ellipses of anisotropy ratio 2 and with .
AppendixSpectrum of some periodic integral operators
Let be a -domain for some . It is known that the non periodic operator is invertible on for [4,6]. The positivity of [12, Section 3.3] also implies that is invertible for . We extend these results to the case of periodic Green’s function.
For any , the operator is invertible.
We first show that the operator is a Fredholm operator. Note that, where is an integral operator with a smooth kernel and is therefore compact. Moreover, since has a dimension 1 kernel and image, it is a Fredholm operator. Therefore, is Fredholm. Now we show that is positive semi-definite, and the result will follow from the Fredholm alternative. Since
For , the operator is invertible on .
Since is invertible, is a compact operator , is a Fredholm operator and it is enough to show that it is one-to-one. The proof goes exactly as in . Let us assume that is not one-to-one. Then there exists some such that
This work was supported by the ERC Advanced Grant Project MULTIMOD–267184.
G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (1992), 1482–1518. doi:10.1137/0523084.
H. Ammari, J. Garnier, L. Giovangigli, W. Jing and J.K. Seo, Spectroscopic imaging of a dilute cell suspension, J. Math. Pures Appl. 105 (2016), 603–661. doi:10.1016/j.matpur.2015.11.009.
H. Ammari and H. Kang, Polarization and Moment Tensors. With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, Vol. 162, Springer, New York, 2007.
T.K. Chang and K. Lee, Spectral properties of the layer potentials on Lipschitz domains, Illinois J. Math. 52 (2008), 463–472.
E. Cherkaev, Inverse homogenization for evaluation of effective properties of a mixture, Inverse Prob. 17 (2001), 1203–1218. doi:10.1088/0266-5611/17/4/341.
E.B. Fabes, M. Sand and J.K. Seo, The spectral radius of the classical layer potentials on convex domains, in: Partial Differential Equations with Minimal Smoothness and Applications, Chicago, IL, 1990, IMA Vol. Math. Appl., Vol. 42, Springer, New York, 1992, pp. 129–137. doi:10.1007/978-1-4612-2898-1_12.
F. Hecht, New development in freefem, J. Numer. Math. 20 (2012), 251–266.
A. Khelifi and H. Zribi, Asymptotic expansions for the voltage potentials with two-dimensional and three-dimensional thin interfaces, Math. Meth. Appl. Sci. 34 (2011), 2274–2290.
S. Kim, E.J. Lee, E.J. Woo and J.K. Seo, Asymptotic analysis of the membrane structure to sensitivity of frequency-difference electrical impedance tomography, Inverse Problems 28 (2012), 075004. doi:10.1088/0266-5611/28/7/075004.
J.M. Mansour, Biomechanics of cartilage, in: Kinesiology: The Mechanics and Pathomechanics of Human Movement, Wolters Kluwer, Philadelphia, 2003, pp. 66–79.
G.W. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2001.
J.-C. Nédélec, Acoustic and Electromagnetic Equations – Integral Representations for Harmonic Problems, Applied Mathematical Sciences, Vol. 144, Springer, Berlin, 2001.
G. Nguestseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989), 608–623. doi:10.1137/0520043.
C. Orum, E. Cherkaev and K.M. Golden, Recovery of inclusion separations in strongly heterogeneous composites from effective property measurements, Proc. Royal Soc. A 468 (2012), 784–809. doi:10.1098/rspa.2011.0527.
C. Poignard, Asymptotics for steady state voltage potentials in a bidimensional highly contrasted medium with thin layer, Math. Meth. Appl. Sci. 31 (2008), 443–479. doi:10.1002/mma.923.
C. Poignard, About the transmembrane voltage potential of a biological cell in time-harmonic regime, ESAIM:Proceedings 26 (2009), 162–179. doi:10.1051/proc/2009012.
J.K. Seo, T.K. Bera, H. Kwon and R. Sadleir, Effective admittivity of biological tissues as a coefficient of elliptic PDE, Comput. Math. Meth. Medicine 2013 (2013), 353849.
T. Zhang, T.K. Bera, E.J. Woo and J.K. Seo, Spectroscopic admittivity imaging of biological tissues: Challenges and future directions, J. KSIAM 18(2) (2014), 77–105.