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PLoS One. 2010; 5(12): e14350.
Published online 2010 December 31. doi:  10.1371/journal.pone.0014350
PMCID: PMC3013093

Multiscale Modeling of Light Absorption in Tissues: Limitations of Classical Homogenization Approach

Giuseppe Chirico, Editor

Abstract

In biophotonics, the light absorption in a tissue is usually modeled by the Helmholtz equation with two constant parameters, the scattering coefficient and the absorption coefficient. This classic approximation of “haemoglobin diluted everywhere” (constant absorption coefficient) corresponds to the classical homogenization approach. The paper discusses the limitations of this approach. The scattering coefficient is supposed to be constant (equal to one) while the absorption coefficient is equal to zero everywhere except for a periodic set of thin parallel strips simulating the blood vessels, where it is a large parameter An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e001.jpg The problem contains two other parameters which are small: An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e002.jpg, the ratio of the distance between the axes of vessels to the characteristic macroscopic size, and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e003.jpg, the ratio of the thickness of thin vessels and the period. We construct asymptotic expansion in two cases: An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e004.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e005.jpg and prove that in the first case the classical homogenization (averaging) of the differential equation is true while in the second case it is wrong. This result may be applied in the biomedical optics, for instance, in the modeling of the skin and cosmetics.

Introduction

Physical background of the problem

In the present paper we consider the Helmholtz equation with rapidly oscillating large potential with a periodic support having a small measure of its intersection with a period. This absorption coefficient An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e006.jpg of the potential (more precisely, it is the ratio between the absorption coefficient and the diffusion coefficient) depends on three small parameters: An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e007.jpg is a standard homogenization parameter that is the ratio of the period of the potential and the characteristic macroscopic size; An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e008.jpg is the ratio between the measure of the intersection of the support of An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e009.jpg and the period; An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e010.jpg is a small parameter standing for the inverse of the ratio of the maximal value of the coefficient An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e011.jpg and the diffusion coefficient multiplied by the square of the characteristic macroscopic size of the problem (we will consider the case when An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e012.jpg takes only two values: An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e013.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e014.jpg).

The Helmholtz equation

equation image
(1)

is considered below as a model of the light absorption in tissues under hypothesis that this absorption takes place only in the set of parallel thin blood vessels (where An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e016.jpg) and this absorption is ignored outside of these vessels.

The linear dimensions of the vessels are much smaller than the linear dimensions of the body as a whole. Typically the distance between two neighboring large micro-vessels (of about 15 An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e017.jpgm in the diameter) is around 180 An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e018.jpgm in primate cerebral cortex [1]. This distance is also used in 3D simulation of tumor growth and angiogenesis [2]. Assume at the first approximation that the tissue is a nearly periodic structure. Moreover, consider the two-dimensional idealization of this periodic structure that is, the periodic set of the parallel strait narrow identic vessels separated by the homogeneous tissue. Let L be the macroscopic characteristic size and let An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e019.jpg be the distance between two neighboring vessels (strips). It means that the parameter An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e020.jpg stands here for the ratio of the distance between two neighboring vessels and the characteristic macroscopic size An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e021.jpg. It is assumed throughout that this ratio is a small parameter. Indeed, if the characteristic macroscopic size (L) is equal to 10 mm (it is a typical value of the diffuse optical tomography [3]) and the distance between the vessels is equal to 0.18 mm, then An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e022.jpg. The second parameter of the model is the ratio of the thickness of the vessels and the distance between neighboring vessels denoted An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e023.jpg. This An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e024.jpg as well is supposed to be a small parameter. Thus, in the discussed above structure we have An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e025.jpg. In the spectral window 500 nm–700 nm, the oxyhaemoglobin extinction coefficient shows a wide dynamic from 275 (mole/L)An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e026.jpg cmAn external file that holds a picture, illustration, etc.
Object name is pone.0014350.e027.jpg at 690 nm (the lowest) and 55 500 (mole/L)An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e028.jpg cmAn external file that holds a picture, illustration, etc.
Object name is pone.0014350.e029.jpg at 575 nm [4]. In order to quantify the maximum variation we compare arterial blood (90 percent saturated) with pure water at 690 nm and 575 nm then the ratio of the absorption coefficient is respectively 300 and 400000. At 700 nm the absorption of tissue (without blood) is about ten times less than that of the pure water [5]. In the visible window the maximum of this ratio can be more than 10000. That is why in the idealized model we consider the case when the absorption coefficient is equal to zero out of vessels. As we have mentioned above the light absorption process is described here by the Helmholtz equation (1), where An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e030.jpg is just non-dimensionalized absorption coefficient, equal to zero out of vessels and equal to the great (dimensionless) parameter An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e031.jpg within the vessels. Let us remind that the physical sense of this great parameter is the ratio of the absorption and diffusion effects, more exactly, the ratio of the absorption coefficient and the diffusion coefficient multiplied (i.e. the ratio is multiplied) by the square of the characteristic macroscopic size An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e032.jpg. Let the haemoglobin concentration be 150 g/liter. Then in order to convert the molar extinction coefficient An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e033.jpg to the absorption coefficient (in mmAn external file that holds a picture, illustration, etc.
Object name is pone.0014350.e034.jpg), one has to multiply it by 0.00054. At the wavelength 575 nm for the scattering coefficient close to 2.5 mmAn external file that holds a picture, illustration, etc.
Object name is pone.0014350.e035.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e036.jpg is about 23000. Then An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e037.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e038.jpg In the case of penetrating vessels with the average diameter close to 65 An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e039.jpgm, at the same wavelength, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e040.jpg. Then the case when An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e041.jpg is greater than one can be found in a spectral window below 600 nm and for vessels of large diameter. In neurophotonics, the highly vascularized pie-matter correspond to the maximal values of An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e042.jpg. The highly vascularized tumors could be also a special case when An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e043.jpg is very high.

Models of this “composite medium” are widely considered in biophysics. Let us give a short review of these results. The partial differential equation (PDE) diffusion with absorption is one of the most important in life science, and in particular, in tissue optics [6]. The researchers are interested in two types of results:

  1. when the optical properties are unknown, and they want to find them using observed measurements, it is the harder well-known inverse problem.
  2. when the optical properties are known, then they can use many strategies to calculate various quantities of interest (reflectance, transmission, fluence). This is often referred to as the forward problem.

But in all these approaches, the community of research in biophotonics uses the volumetric averaging of the absorption coefficient. This problem is crucial in the domain of in vivo diffuse optical tomography [3], [7], and is always present in in vivo optical neuromethods [8].

There are as well some experimental papers on the problem of averaging of blood absorption of light in tissue but without the homogenization theory [9][14].

Our goal in the present paper is to show that the classical homogenization (averaging) approach to the solution of equation (1) leading to the approximation

equation image
(2)

where An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e045.jpg is often approximated to the volumic mean value of An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e046.jpg has some limitations. Indeed, we will show that it is right for some combination of magnitudes of parameters An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e047.jpg: it can be proved that An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e048.jpg but it is inapplicable for some other combinations. In this paper we do not construct the expansions for all possible combinations; we only prove the classical homogenization result in the case

equation image

and we show that the homogeneous model (2) is inapplicable in the case

equation image

It means that the “diluting” of vessels in multiscale modeling of light absorption of tissues may be applied only in justified cases, in particular, in case A. In fact, the situations when the coefficients of equations depend on two or more small parameters and it leads to a fail of classical homogenization are not too surprising now: first results of this type could be found in [15][17]. For some cases the non-classical asymptotic expansions were constructed. However, to our knowledge, such examples have not been constructed for rapidly oscillating large potentials with narrow support. The present paper tries to contribute in this important case of biophysical applications.

In order to simplify technical details of the analysis we consider the boundary value problem for the Helmholtz equation set in a layer An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e051.jpg with the Dirichlet conditions on the boundary of the layer; we assume that the right-hand sides of the equation and of the boundary condition do not depend on An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e052.jpg. In this case we may seek a solution independent of An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e053.jpg as well. The Helmholtz equation takes the following form:

equation image
(3)

where An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e055.jpg stands for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e056.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e057.jpg is a small positive parameter, such that, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e058.jpg is an integer.

We consider a boundary condition corresponding to a constant solution in the case of the constant absorption coefficient An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e059.jpg and of a constant right-hand side An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e060.jpg. Then An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e061.jpg. So, these boundary conditions are:

equation image
(4)

Similar problems were addressed by many authors [18][22]; however the potentials considered in these works are typically An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e063.jpg or An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e064.jpg. In this work, there are three parameters and we study all relevant asymptotic regimes. We follow the ideas and methods of [15][17], [23], [24] in the sense that we construct asymptotic expansions for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e065.jpg and analyze all the important asymptotic regimes of the parameters An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e066.jpg.

Mention that the Helmholtz equation (1) (or (3)) is not a perfect model for the light absorption process: it is not more than an approximation of a more adequate model based on the radiation transfer equation [25]. Indeed, one can introduce the average (over all directions An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e067.jpg) for the diffused radiation intensity An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e068.jpg

equation image

In the case of the described above plane geometry the diffusion approximation of the radiation transfer has a form [25]

equation image

where An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e071.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e072.jpg are the coefficients corresponding to the light absorption and dispersion respectively, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e073.jpg is the blood heat radiation intensity and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e074.jpg is the external radiation intensity, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e075.jpg is the thickness of the irradiated skin strip, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e076.jpg is the dimension factor. Introducing the new variable corresponding to the relative optical depth of the radiation process

equation image

we get for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e078.jpg (here An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e079.jpg):

equation image

where An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e081.jpg and functions An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e082.jpg depend on An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e083.jpg, solution of equation

equation image

Making the change of unknown function

equation image

we get finally equation (3):

equation image

with An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e087.jpg

The choice of the boundary conditions (4) is not too important for our analysis: the asymptotic approach developed below can be applied in the case of other conditions, for instance,

equation image

corresponding to a constant solution for equation (3) with An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e089.jpg replaced by its volumic mean An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e090.jpg or

equation image

or the periodicity conditions (mention that the diffusion approximation of the radiation transfer may loose its precision near the boundary).

Mathematical statement of the problem

Consider interval An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e092.jpg and let An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e093.jpg Let An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e094.jpg be a 1-periodic function defined on the basic period An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e095.jpg by

equation image
(5)

Let An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e097.jpg be smooth enough (this assertion will be formulated more precisely later). We are interested in studying the asymptotic behaviour as An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e098.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e099.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e100.jpg of An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e101.jpg solving the following boundary value problem:

equation image
(6)
equation image
(7)

Note that we have the following a priori estimate for solution of equation (6) with boundary condition (7) replaced by the homogeneous one

equation image

Proposition 0.1. There exists a constant An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e105.jpg independent of small parameters such that

equation image

Here An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e107.jpg may be taken equal to An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e108.jpg. A well-known imbedding theorem gives:

Corollary 0.1. There exists a constant An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e109.jpg independent of small parameters such that

equation image

Applying equation, we get:

Corollary 0.2. There exists a constant An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e111.jpg independent of small parameters such that

equation image

Returning now to equation (6) with non-homogeneous boundary condition (7) we can present the solution as a sum of the solution of equation (6) with homogeneous boundary condition and the solution of the homogeneous equation (6) with non-homogeneous boundary condition (7). Applying the maximum principle argument, we get:

Proposition 0.2. There exists a constant An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e113.jpg independent of small parameters such that solution of problem (6)-(7) satisfies:

equation image

Remark 0.1. If boundary condition (7) is replaced by another one:

equation image

with some numbers An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e116.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e117.jpg then the estimate of Proposition 0.2 takes form:

equation image

Let us mention one more useful inequality for functions An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e119.jpg of An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e120.jpg

equation image
(8)

Proof: For any An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e122.jpg An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e123.jpg Choose An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e124.jpg such that An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e125.jpg We get:

equation image

If An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e127.jpg, then the estimate is proved. If An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e128.jpg, then the sign of An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e129.jpg is constant on the whole interval An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e130.jpg; consider the case An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e131.jpg Then An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e132.jpg is increasing and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e133.jpg and so,

equation image

and we get (8). In the same way we consider the opposite case when An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e135.jpg

Methods

Asymptotic expansion in the case An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e136.jpg

Consider the An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e137.jpg-th level approximation An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e138.jpg of An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e139.jpg given by an ansatz from [15]:

equation image
(9)

where An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e141.jpg are 1-periodic functions, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e142.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e143.jpg is a smooth function which will be sought in a form

equation image
(10)

Substituting the expression for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e145.jpg from (9) in (6) we obtain

equation image
(11)

where

equation image
(12)

and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e148.jpg is given by

equation image
(13)

In writing expressions (11), (12) we followed the convention that An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e150.jpg if at least one of the two indices An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e151.jpg or An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e152.jpg, is negative.

We choose An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e153.jpg being a constant such that the equation (12) has a solution, i.e.

equation image
(14)

where An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e155.jpg Also, periodic solutions An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e156.jpg of equation

equation image
(15)

are chosen to be of average zero for all An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e158.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e159.jpg such that, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e160.jpg; An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e161.jpg is chosen equal to An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e162.jpg. We get: An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e163.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e164.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e165.jpg for all An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e166.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e167.jpg is a 1-periodic solution of equation

equation image
equation image

We solve the family of equations (15) by determining for each fixed An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e170.jpg the solutions An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e171.jpg for all An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e172.jpg. Due to the form of the equation (15) the behavior of An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e173.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e174.jpg will be similar.

With this background, we now state the principal result concerning the 1-periodic functions An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e175.jpg.

Lemma 0.1. Let An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e176.jpg. For each An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e177.jpg (of the form An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e178.jpg or An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e179.jpg) and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e180.jpg,

  1. Functions An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e181.jpg are piecewise polynomials on An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e182.jpg with respect to the partition An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e183.jpg.
  2. There exists a constant An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e184.jpg independent of the small parameters such that, for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e185.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e186.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e187.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e188.jpg.

Proof. The proof is by induction on An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e189.jpg.

Step (i). Let An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e190.jpg. That is, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e191.jpg. We get: An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e192.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e193.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e194.jpg satisfy equations (15) with the last two terms equal to zero; so An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e195.jpg, for all An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e196.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e197.jpg for all An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e198.jpg and the assertion of lemma for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e199.jpg is evident.

Step (ii). Let An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e200.jpg and let all An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e201.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e202.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e203.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e204.jpg be bounded by An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e205.jpg, where An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e206.jpg stands for the integer part of An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e207.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e208.jpg is a constant independent of the small parameters. Consider equation (15) for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e209.jpg. For An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e210.jpg it takes form

equation image

with the piecewise polynomial right hand side; moreover, the primitive (with vanishing mean value) of this right hand side is also bounded by the same order. That is why, integrating this equation twice and keeping every time the vanishing mean for the primitive, we prove the assertion of lemma for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e212.jpg. In the same way we prove it for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e213.jpg.

Now we apply again (15), express An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e214.jpg from this equation and prove by induction on An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e215.jpg that all An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e216.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e217.jpg, the right hand sides and so, finally, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e218.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e219.jpg are bounded by An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e220.jpg. The lemma is proved.

Substituting the expression for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e221.jpg from (9) in (7) we obtain

equation image
(16)
equation image

and applying the An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e224.jpgperiodicity of functions An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e225.jpg and the divisibility of An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e226.jpg by An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e227.jpg we get:

equation image
(17)

Substituting the expression for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e229.jpg (10) in (11),(16),(17), we get:

equation image
(18)

and

equation image
(19)
equation image
(20)

Here

equation image
(21)
equation image
(22)

and

equation image
(23)

Define now functions An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e236.jpg from boundary value problems:

equation image
(24)

and

equation image
(25)
equation image
(26)

Here An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e240.jpg is supposed to be greater than An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e241.jpg. These equations have a form

equation image

where An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e243.jpg are some real numbers and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e244.jpg is a smooth function. In the same way as in the first section we prove that

equation image
equation image

and so,

equation image

where the constants An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e248.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e249.jpg are independent of small parameters.

Applying the induction on An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e250.jpg, we get:

equation image

where the constants An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e252.jpg are independent of small parameters. Applying now (8), we get: for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e253.jpg

equation image

and so,

equation image

Remark 0.2. Functions An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e256.jpg depend on small parameters but their asymptotic expansion can be constructed by classical boundary layer technique (see [23]). For instance, if An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e257.jpg, then An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e258.jpg has a form:

equation image

Moreover, if An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e260.jpg, then An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e261.jpg is An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e262.jpg times differentiable and satisfies the estimate (see the above a priori estimate for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e263.jpg):

equation image

In the same way we construct by induction the expansions of functions An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e265.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e266.jpg. Assume that An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e267.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e268.jpg. Consider the right hand side An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e269.jpg having the form of a linear combination with some bounded coefficients An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e270.jpg of functions An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e271.jpg having a form:

equation image

where An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e273.jpg are An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e274.jpg times differentiable functions independent of small parameters An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e275.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e276.jpg, such that, there exist constants An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e277.jpg independent of small parameters satisfying for any real An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e278.jpg inequalities

equation image

An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e280.jpg are An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e281.jpg times differentiable on An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e282.jpg,

equation image

The right hand sides of the boundary conditions as well have a similar form: they are some linear combinations with bounded coefficients An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e284.jpg of constants An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e285.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e286.jpg having a form:

equation image
equation image

where An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e289.jpg are independent of small parameters.

The expansion of An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e290.jpg has a similar form of a linear combination with bounded coefficients of functions

equation image

where An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e292.jpg satisfy equations (functions with the negative indices are equal to zero)

equation image

and exponentially decaying at infinity functions An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e294.jpg satisfy equations

equation image

and boundary conditions

equation image

exponentially decaying at infinity functions An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e297.jpg satisfy equations

equation image

and boundary conditions

equation image

and of the exponents

equation image
equation image

Here

equation image

Applying the induction on An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e303.jpg, the explicit expressions for the right hand sides of the problems (24),(25),(26) for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e304.jpg and the estimates of Lemma 0.1 we prove that there exist constants An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e305.jpg independent of small parameters, such that,

equation image
(27)

This estimate and Lemma 0.1 are crucial for evaluation of the discrepancies An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e307.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e308.jpg:

equation image
(28)

Now, applying the a priori estimate of the Remark after Proposition 0.2 we get the estimate

equation image

Assume that there exists a positive real An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e311.jpg such that, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e312.jpg is bounded by a constant independent of small parameters. Then the last bound yields:

equation image

Taking An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e314.jpg, we get:

equation image

On the other hand, (27) and Lemma 0.1 give:

equation image

So, from the triangle inequality we get

equation image

Theorem 0.1. Assume that there exists a positive real An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e318.jpg such that, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e319.jpg is bounded by a constant independent of small parameters. Let An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e320.jpg belong to An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e321.jpg. Then there exists a constant An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e322.jpg independent of small parameters such that

equation image

Consider now the very first term of the expansion An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e324.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e325.jpg we get:

Corollary 0.3. There exists a constant An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e326.jpg independent of small parameters such that

equation image

Mention that the error of order An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e328.jpg is much smaller than An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e329.jpg that is of order An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e330.jpg, and so the relative error of the approximation of the exact solution An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e331.jpg by An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e332.jpg is small.

Remark 0.3. Here we have considered the case when An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e333.jpg stands for a large parameter tending to the infinity. The case when An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e334.jpg is a positive finite constant may be considered by a classical technique [15] and the estimate of Corollary 0.3 becomes

equation image

Asymptotic analysis: case An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e336.jpg

Here we will construct an example where the behavior of the solution is completely different from the behavior described in the previous section. In particular, the solution of the problem is different from the behavior of the solution An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e337.jpg of the homogenized equation. For the sake of simplicity consider the right hand side An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e338.jpg. Consider an auxiliary function An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e339.jpg defined on the interval An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e340.jpg as a function from An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e341.jpg independent of small parameters such that it equals to zero on the interval An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e342.jpg and it equals to one on An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e343.jpg. Let us keep the same notation An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e344.jpg for the 2-periodic extension of this function on An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e345.jpg. Consider an approximation for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e346.jpg having a form:

equation image
(29)

where An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e348.jpg and extend it An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e349.jpgperiodically to An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e350.jpg. By a simple calculation we find that An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e351.jpg satisfies the boundary conditions exactly and the equation with a discrepancy of order An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e352.jpg, i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e353.jpg. Assume that there exists a positive real An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e354.jpg such that, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e355.jpg. Then for any positive An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e356.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e357.jpg.

So, the discrepancy of the equation is An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e358.jpg Applying now Corollary 1.1, we get an error estimate in An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e359.jpgnorm of the same order:

equation image

Mention that for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e361.jpg the right hand side of this bound is much smaller than the values of approximation An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e362.jpg, and so the relative error of the approximation of the exact solution An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e363.jpg by An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e364.jpg is small.

Results and Discussion

The main observations on the asymptotic analysis

We record some observations in the following remark.

Remark 0.4. Let us compare the asymptotic behavior of solution for An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e365.jpg in the cases

equation image

and

equation image

In the case (A), the leading term is equal to the solution An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e368.jpg of the homogenized equation (2), that is the constant An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e369.jpg plus two exponents rapidly decaying from the boundary: their contribution can be neglected at the distance of order of An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e370.jpg. In the case (B) the approximate solution is completely different: it is An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e371.jpgperiodic piecewise quadratic function of order An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e372.jpg except for the small intervals where the potential is large. That is why the homogeneous model (2) is inapplicable in this case.

Thus, the homogenized model (2) may be applied only in the case An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e373.jpg, when it is justified theoretically. The homogenized model may be inapplicable in the case if An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e374.jpg is not small.

On the effective absorption coefficient, the volumetric mean and the total absorption coefficient

Biological tissues are highly heterogeneous media at the microscopic scale. For years, the patterns of the mammalian cortex micro-angioarchitecture have attracted the interest of many research groups [1], [26][28]. At different scales the blood vessels have very different physical characteristics. Moreover the absorption coefficient is great in the vessels and small out of the vessels in UV and visible parts of the spectrum. At the end they induce multiscale complex photon propagation and absorption. On the other hand the experimental data are obtained mainly at the macroscopic scale, and so one of the main questions is how to make a passage from the micro-scale to the macro-scale. The main mathematical tool of such passage from one scale to another (the up-scaling) is the homogenization theory, see [15], [19], [23], [24] and the references there. Normally, the microscopic description of the heterogeneous medium can be replaced by an equation with constant coefficients. This equation is called the homogenized equation and the constant coefficients are effective coefficients. This homogeneous approximation for the heterogeneous medium is justified if the solutions of both models are close in some norms. In some cases a heterogeneous medium cannot be approximated by a homogeneous one [16], [17], [23], [24]. Then the notion of an effective coefficient cannot be introduced in the above sense.

In particular, in the present paper we prove that if product An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e375.jpg is small then the homogenized model (2) is justified and the effective absorption coefficient is equal to the volumetric mean value An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e376.jpg of function An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e377.jpg. The smallness of the value of this product An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e378.jpg means that the relative error of the approximation of the heterogeneous medium by the homogeneous one is of order of this product. The effective absorption coefficient is an important quantity because the detailed knowledge of the tissue macroscopic optical properties is essential for an optimization of optical methods i.e. for modeling the color of skin, of port-wine stains and for tumor detection; it helps to adapt an appropriate photodynamic therapy, in particular, some laser treatment.

On the other hand the present paper shows the limitations of the homogenized models for the absorption problems. Indeed, if product An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e379.jpg is not small then the homogenized model (2) as well as other homogenized models (with other possible constant values of the effective absorption coefficient) do not approximate the initial microscale model, because the solution of problem (2) does not oscillate for any choice of the effective absorption coefficient, while the solution of equation (6) with boundary condition (7) rapidly oscillates (see (29)).

This theoretical argument is confirmed by some physical reasons and by experimental observations. Many authors [10], [12], [14], [29] show that the assumption of a homogeneous distribution of blood in the tissue may strongly overestimate the total blood absorption when absorption is high and/or the vessels have sufficient diameter. For large vessels less of light reaches the center of the vessel, and the absorbers in the center of a vessel contribute less and less to the total attenuation of the light. However, the total blood absorption is an important qualitative characteristic of the absorption process and so it should be calculated with a great precision. Let us apply the above asymptotic analysis provided for equation (6) with boundary condition (7) and calculate the total blood absorption in cases A and B. Let us define the total absorption coefficient An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e380.jpg as a ratio of the integrals

equation image

For the particular case of a constant coefficient An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e382.jpg we get evidently An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e383.jpg. In the case A substituting the leading term of an asymptotic expansion we get that the asymptotic behavior of An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e384.jpg is given by an approximate formula An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e385.jpg, i.e. it is close to the volumetric mean of An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e386.jpg that is, An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e387.jpg. In the case B the value of this total absorption coefficient An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e388.jpg is much less than the volumetric mean (that was observed in the discussed above experiments): the direct computations for the approximation (29) show that An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e389.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e390.jpg

The authors of papers [10], [12], [14], [29] characterize the fall of the total absorption coefficient by the correction factor An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e391.jpg and discuss the situations when this factor is different from 1. In our asymptotic analysis we see that in the case A An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e392.jpg but in the case B An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e393.jpg.

We hope that our result will help to analyze the link between the total absorption coefficient An external file that holds a picture, illustration, etc.
Object name is pone.0014350.e394.jpg and other parameters which may be applied in optical tomography [3], [7].

Acknowledgments

The authors are grateful to Andrey Amosov for fruitful discussions.

Footnotes

Competing Interests: The authors have declared that no competing interests exist.

Funding: The results were supported by the Program Emergence of the Region Rhone-Alpes. The theoretical part was supported by the CNRS postdoctoral grant and by the grant of the Russian Federal Agency on Research and Innovations Contract No 02.740.11.5091 “Multiscale Models in Physics, Biology and Technologies: Asymptotic and Numerical Analysis”. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

References

1. Risser L, Plouraboue F, Cloetens P, Fonta C. A 3D-investigation shows that angiogenesis in primate cerebral cortex mainly occurs at capillary level. International Journal of Developmental Neuroscience. 2009;27:185–196. [PubMed]
2. Shirinifard A, Gens JS, Zaitlen BL, Poplawski NJ, Swat M, et al. 3D multi-cell simulation of tumor growth and angiogenesis. PLoS ONE. 2009;4(10):e7190. [PMC free article] [PubMed]
3. Vignal C, Boumans T, Montcel B, Ramstein S, Verhoye M, et al. Measuring brain hemodynamic changes in a songbird: responses to hypercapnia measured with functional MRI and near-infrared spectroscopy. Phys Med Biol. 2008;53:2457–2470. [PubMed]
5. Ramstein S, Vignal C, Mathevon N, Mottin S. In-vivo and non-invasive measurement of songbird head's optical properties. Applied Optics. 2005;4:6197–6204. [PubMed]
6. Arridge SR, Cope M, Delpy DT. The theoretical basis for the determination of optical pathlengths in tissue - temporal and frequency-analysis. Physics in Medicine and Biology. 1992;37:1531–1560. [PubMed]
7. Gibson AP, Austin T, Everdell NL, Schweiger M, Arridge SR, et al. Three-dimensional whole-head optical passive motor evoked responses in the tomography of neonate. Neuroimage. 2006;30:521–528. [PubMed]
8. Mottin S, Laporte P, Cespuglio R. The inhibition of brain NADH oxidation by chloramphenicol in the freely moving rat measured by picosecond time-resolved emission spectroscopy. J Neurochem. 2003;84:633–642. [PubMed]
9. Liu H, Chance B, Hielscher AH, Jacques SL, Tittel FK. Influence of blood-vessels on the measurement of hemoglobin oxygenation as determined by time-resolved reflectance spectroscopy. Medical Physics. 1995;22:1209–1217. [PubMed]
10. Svaasand L, Fiskerstrand EJ, Kopstad G, Norvang LT, Svaasand EK, et al. Therapeutic response during pulsed laser treatment of port-wine stains: dependence on vessel diameter and depth in dermis. Lasers in Medical Science. 1995;10:235–243.
11. Firbank M, Okada E, Delpy DT. Investigation of the effect of discrete absorbers upon the measurement of blood volume with near-infrared spectroscopy. Physics in Medicine and Biology. 1997;42:465–477. [PubMed]
12. Verkruysse W, Lucassen GW, de Boer JF, Smithies D, Nelson JS, et al. Modelling light distributions of homogeneous versus discrete absorbers in light irradiated turbid media. Physics in Medicine and Biology. 1997;42:51–65. [PubMed]
13. Bradu A, Sablong R, Julien C, Tropres I, Payen JF, et al. Papazoglou TG, Wagnieres GA, editors. In vivo absorption spectroscopy in brain using small optical fiber probes: effect of blood confinement. 2001. pp. 85–90. 2001 Jun 19-21; Munich, Germany Spie-Int Soc Optical Engineering.
14. van Veen RLP, Verkruysse W, Sterenborg H. Diffuse-reflectance spectroscopy from 500 to 1060 nm by correction for inhomogeneously distributed absorbers. Optics Letters. 2002;27:246–248. [PubMed]
15. Bakhvalov N, Panasenko G. Homogenisation: averaging processes in periodic media. Mathematical problems in the mechanics of composite materials. 1989 Mathematics and its Applications (Soviet Series): Kluwer, Dordrecht/London/Boston.
16. Panasenko G. Multicomponent averaging of processes in strongly inhomogeneous structures. (in Russian) Mat Sb. 1990;181:134–142; translation in Math. USSR-Sb. (1991) 69(1): 143–153.
17. Panasenko G. Multicomponent homogenization of the vibration problem for incompressible media with heavy and rigid inclusions. C R Acad Sci Paris Série I Math. 1995;321(8):1109–1114.
18. Allaire G, Capdeboscq Y, Piatnitski A, Siess V, Vanninathan M. Homogenization of periodic systems with large potentials. Archive for Rational Mechanics and Analysis. 2004;174:179–220.
19. Bensoussan A, Lions JL, Papanicolaou G.Asymptotic analysis for periodic structures. 1978. Studies in Mathematics and its Applications: North Holland, Amsterdam.
20. Campillo F, Kleptsyna M, Piatnitski A. Homogenization of random parabolic operator with large potential. Stochastic Processes and Their Applications. 2001;93:57–85.
21. Fleming WH, Sheu SJ. Asymptotics for the principal eigenvalue and eigenfunction of a nearly first-order operator with large potential. Annals of Probability. 1997;25:1953–1994.
22. Weinstein MI, Keller JB. Hills equation with a large potential. SIAM Journal on Applied Mathematics. 1985;45:200–214.
23. Panasenko G. Dordrecht: Springer; 2005. Multiscale Modelling for Structures and Composites.
24. Panasenko G. Homogenization for periodic media: from microscale to macroscale. Physics of Atomic Nuclei. 2008;4:681–694.
25. Case KM, Zweifel PF. Reading, MA: Addison-Wesley; 1967. Linear transport theory.
26. Bar T. The vascular system of the cerebral cortex. Adv Anat Embryol Cell Biol. 1980;59:1–65. [PubMed]
27. Duvernoy HM, Delon S, Vannson JL. Cortical blood-vessels of the human-brain. Brain Research Bulletin. 1981;7:519–579. [PubMed]
28. Cassot F, Lauwers F, Fouard C, Prohaska S, Lauwers-Cances V. A novel three-dimensional computer-assisted method for a quantitative study of microvascular networks of the human cerebral cortex. Microcirculation. 2006;13:1–18. [PubMed]
29. Talsma A, Chance B, Graaff R. Corrections for inhomogeneities in biological tissue caused by blood vessels. Journal of the Optical Society of America a-Optics Image Science and Vision. 2001;18:932–939. [PubMed]

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