Computer-aided analysis of micro-morphological structure of porous membranes
- Juliusz L. Kulikowski^{1}Email author,
- Malgorzata Przytulska^{1} and
- Andrzej Chwojnowski^{1}
Received: 29 December 2017
Accepted: 18 April 2018
Published: 30 May 2018
Abstract
Background
The paper presents an approach and computer-aided method of numerical evaluation of the quality of porous membranes used as scaffolds for cultivation of chondrocytes in regeneration of biological tissues.
Materials
The scanning electron microscope (SEM) images of 300× and 1000× magnification presenting the sections of artificial polyvinylpyrrolidone membranes obtained in two alternative production processes are examined.
Theory and methods
There is presented a combined morphological and statistical method of the assessment of artificial membranes’ porosity, based on computer-aided segmentation and analysis of the size and shape of pores. Theoretical backgrounds of description pores as irregular objects in discrete 3-dimensional space are presented. The parameters characterizing the quality of pores: pores irregularity coefficient and pores density are defined. The quality of the examined specimens of materials is characterized by the size (mean 2-dimensional section areas) of pores. The main concept presented in the paper is the extraction of lacking information concerning the third dimension of pores from the 2-dimensional SEM images of their sections. Two approaches to evaluation of the parameters characterizing pores on the basis of computer-aided analysis of their cross-sections are proposed: (1) based on statistical extension of geometrical data and (2) based on analysis of brightness profiles. The corresponding methods are based on the assumption of isotropy of the examined porous materials. The results of automatic measurements of the areas of pores, lengths of their chords and recording the brightness profiles along fixed lines crossing the analyzed images are illustrated by examples.
Conclusions
Practical usefulness of the proposed methods to evaluation of the quality of porous membranes consists in their ability to be used in case if alternative methods for some reasons cannot be used.
Keywords
Background
Numerous physiological processes taking place in living organisms: absorption of nutritive products and gases, excretion of waste products etc. are based on the phenomenon of diffusion of various types of material particles through porous membranes. Similar processes play a substantial role in numerous areas of medical therapy based, in particular, on artificial organs: artificial lungs-heart, artificial kidney, encapsulated drugs or living cells, cultivation of artificial tissues etc. [1–5]. Porous membranes application to water or air purification indirectly influences also human health [6–8]. In all the above-mentioned cases porous membranes play the role of filters separating some biochemical media or of scaffolds for seeding and cultivation of tissues. The mechanisms of separation or of settling particles (cells) depend on biophysical, mechanical and micro-structural properties of the porous materials. Examples of biophysical properties are electric- and heat-conductivity, dielectric constant, pH level, etc. Mechanical properties denote specific gravity, Young module, break- or fracture-resistance, etc. The notion of micro-structural properties concerns spatial distribution, density, size and shape of the components the investigated material consists of, in particular—of pores and/or of the walls that separate them.
In most cases the pores are of different size and highly irregular shape. However, the notion of irregularity needs some explanation. We call regular curves (contours) or 3D shapes which can be precisely described by simple analytical functions or geometrical objects. Weakly irregular are contours or 3D shapes which can be described as parameterized compositions of finite number of regular curves (contours) or 3D shapes. Curves (contours) or 3D shapes are strongly irregular if they are neither regular nor weakly irregular. The class of strongly irregular curves (contours) and 3D shapes can be divided into the sub-classes of random and chaotic objects. Random curves (contours) or 3D shapes are assumed to be able to be described by probabilistic models [14, 15]. Chaotic objects can be described by fractals [16] or any other, more sophisticated, theoretical tools. Below, it is assumed that pores with sufficient accuracy can be described as random-shape objects. Therefore, no geometrical model is suitable for their accurate description. The statistical geometry or morphology based methods [17, 18] seem to be more suitable for numerical analysis of this type of structures. However, considering images of pores as instances of specific 2D random fields seems to be more rational. On the other hand, porosity is a 3D rather than a 2D property of a rigid body. Therefore, it arises a non-trivial problem of obtaining information about the 3D structure of a porous material by analysis of images presenting its 2D cross-sections. The problem is relatively simple if the samples of the investigated material can be cut into regularly distanced parallel slices, as proposed in [12, 19] Unfortunately, this cannot be done, e.g. in the case of high fragility of the samples. In such case, we are faced with a more sophisticated statistical decision making problem to be solved by using advanced computer-based methods. In this paper some concepts concerning overcoming the difficulty of 3D morphological structures description not based on analysis of stacks, but of single 2D images, are presented. The missing information about the 3rd dimension of the examined porous materials’ samples can (at least partially) be extracted from the brightness level distribution in the cross-section images or should be deduced under the assumption of 3-dimensional morphological isotropy of the samples. Below, both methods will be shortly described. However, statistical porosity characteristics describe only some aspects of the investigated materials’ utility in their biomedical applications. The relationships between measurable morphological parameters and large-scale properties of porous materials are still an open problem requiring additional observations and experiments and it is not considered here.
The analysis of irregular structures (in particular, of pores) is not a new problem. Its origins date the Buffon’s needle problem which initiated the development of stochastic geometry [20]. Roughly speaking, this area of investigations concerns the properties of random compositions of regular geometrical objects. This is not quite the problem of highly irregular shapes investigation arising in biomedical engineering, material engineering, geophysics, geomorphology, etc. [11, 14, 17, 21]. However, each of the disciplines mentioned here uses its specific experimental data acquisition methods and this causes that the corresponding data processing methods are not directly applicable in other application areas. For example, echo-sounding methods of geophysical data acquisition and SEM imaging of pores provide formally different types of experimental data, despite the fact that in both cases they concern less or more irregular morphological structures.
The paper is organized as follows. The “Materials” section shortly presents the materials used as a source of image data subjected to computer analysis. The “Theory and methods” section consists of four subsections. The “Preliminary image processing” subsection presents initial operations aimed at enhancement the quality of SEM images before their analysis. The “Basic assumptions” subsection describes some discrete geometry objects used in computer analysis of pores. The “Characteristics of irregular object” subsection contains definitions of basic parameters used to describe irregular shapes of pores. The “Rough evaluation of 3D objects’ porosity” subsection is devoted to presentation of proposed two approaches to evaluation of the volumes of pores on the basis of analysis of their 2D sections: 1st based on statistical extension of geometrical data and 2nd based on analysis of brightness profiles. “Conclusions” contain remarks summarizing the work.
Materials
The materials used in this study have been provided by the Laboratory of Semi-permeable Membranes and Bioreactors of the Nalecz Institute of Biocybernetics and Biomedical Engineering, Polish Academy of Sciences. They had the form of series of images presenting the cross-sections of samples of porous membranes produced in the Laboratory. The membranes were produced by an inverse-phase method using the Poli-l-lactide (PLLA), and polyvinylpyrrolidone (PVP) as basic input components [22]. Several variants of technological processes were used in order to select the most effective ones considering the highest membrane’s quality and the highest production process stability. The samples of investigated materials were cut and covered by a 7‒10 nm coat of gold and in this form they have been used as specimens for analysis in a scanning electron microscope (SEM) type Hitachi TM1000 using a 15 kV acceleration voltage. The cross-section images in 300× and/or 1000× magnifications in electronic form of (approximately) 1100 × 1280 pix TIFF standard visual objects were then provided for computer analysis.
Theory and methods
Preliminary image processing
All calculated morphological characteristics of pores are based on the distances expressed in pixels (pix), areas expressed in pix^{2} etc., recalculated to metric units. In order to make the parameters evaluated in various samples of material mutually comparable it is necessary to present the images in standardized size, magnification and scale (in pix/μm). Moreover, in order to outline the pores more exactly the contrast of the original SEM images should be enhanced. Typical image enhancement methods are described in [23, 24].
Basic assumptions
An open discrete line-segment can be defined as a minimal linearly ordered set of points Λ = [p_{ 0 }, p_{ 1 }…, p_{ κ },…, p_{ K }] such that p_{ 0 } and p_{ K } are connective to exactly one, while all the others to exactly two other points of the set. A closed discrete line segment (a contour) C is defined similarly excepting that it does not contain any point connective to less than two other points of the segment (p_{ K } is connective both to p_{ K−1 } and p_{ 0 }).
A finite discrete 2D or 3D object can be defined as a finite set Ω = {p_{ κ }}, κ = 1,2,…,K, such that each point p_{ k } is connective to at least one other point of Ω. The surface of Ω is defined as a maximal subset S_{ Ω }⊆ Ω such that each its point p_{ κ } is connective to at least one point not belonging to Ω. An object Ω is called thin if S_{ Ω }≡ Ω. Evidently, contours and surfaces of objects are thin objects themselves. The above-defined discrete objects have also a property of compactness, i.e. any pair of their points can be linked by a line totally belonging to the object.
A discrete length of a line segment Λ or of a contour C, measured in pixels [pix], is defined as the number of points constituting them. Similarly, a discrete area of a surface S_{ Ω }, measured in [pix^{2}] and a discrete volume of an object, measured in [pix^{3}], are defined as numbers of points that constitute them.
The higher-connective models suit, in general, better to approximation of continuous shapes then the low-connective models. However, evaluation of distance, area or volume measures in discrete models of geometrical objects by counting the numbers of pixels belonging to the objects may lead to substantial differences concerning the respective Euclidean measures of the objects. This should be taken into account in morphological analysis of pores.
Characteristics of irregular objects
Regularity coefficients η calculated in various ways
Type of connectivity/radius | 4-connectivity | 8-connectivity | Continuous (Euclidean) | ||||||
---|---|---|---|---|---|---|---|---|---|
L | S | η | L | S | η | L | S | η | |
1 | 8 | 9 | 1.767 | 4 | 5 | 3.927 | 6.283 | 3.142 | 1 |
3 | 24 | 45 | 0.982 | 16 | 37 | 1.816 | 18.850 | 28.274 | 1 |
5 | 40 | 97 | 0.762 | 28 | 96 | 1.539 | 31.416 | 78.540 | 1 |
10 | 80 | 349 | 0.685 | 56 | 348 | 1.394 | 62.832 | 314.159 | 1 |
It follows from the Table that assumption of 8-connectivity of the discrete space suits to describe irregular contours better than 4-connectivity. Moreover, in both cases the regularity coefficient η is better suited for characterization of larger than of small objects. In any case, the type of connectivity should be a priori strongly established in order to make the results of porosity evaluation mutually comparable. However, it still remains the problem, how to evaluate the values of S and V of 3D objects observable only in 2D space without a possibility to measure them directly in all three dimensions.
Rough evaluation of 3D objects’ porosity
- (a)
Mean pores’ volume m_{ V },
- (b)
Standard deviation of pores’ volume ∆_{ V },
- (c)
Total pores’ density:
$$\rho = \frac{{V_{\sum } }}{V}$$(7)
In order to calculate the above-listed parameters the volumes V_{ i } of pores in the observation area should be evaluated. In [26] a method of statistical evaluation of V_{ i } based on the assumption of isotropy of the morphological structure of pores and of their quasi-spherical shapes was described. Below, two other approaches to solution of the problem are presented.
Approach based on statistical extension of geometrical data
Another situation arises in the case of isotropic type of porosity. Evaluation of porosity can then be based on the following assumptions.
Assumption 1
The morphological structure of the examined porous material is isotropic in 3D; therefore, the probability distributions of the lengths of parallel chords crossing the pores do not depend on the direction of the chords.
Assumption 2
The cross-sections of the pores in (y, z) plane are of elliptic form with the major and minor axes ordered in parallel to the y and z coordinate axes.
Assumption 3
The pore’s cross-sections in the (x, y) plane are of compact irregular form.
Assumption 3 seems to suit better to characterization of irregular shapes of pores than this based on the concept of effective pore diameter proposed in [27–29].
Thus, it is assumed that for any given (i-th) horizontal chord a conditional probability distribution (cpd) of the length of the assigned to it vertical chords w(h|l_{ i }) can be taken into consideration and the same cpd is in effect also for the length z of horizontal chords crossing the i-th chord and directed perpendicularly to the visualization plane.
- (a)
Based on the mean value:
$$g_{i} = M_{i} = \frac{1}{J}\sum\limits_{j = 1}^{J} {h_{j} } ;$$(10) - (b)
Based on the highest conditional probability.
Otherwise, if more than one sub-interval satisfies the condition of maximization n_{ κ } the number K should be reduced by 1 and the procedure should be repeated. In a marginal case, when K = 1 is reached, all h_{ j } have approximately the same value h* and g_{ i } = h* as a solution can be accepted.
Approach based on analysis of brightness profiles
The visualized pores are not only of various sizes and shapes but also of various brightness caused by their different depths and inner profiles. The profiles recorded along several horizontal lines crossing the image, shown in Fig. 10b, 1–4, suggest a possibility of extraction of information concerning the depths of pores in the third dimension. Therefore, the below presented method will be based on the following:
Assumption 4
The brightness levels in 2D visualization of pores are linearly proportional to the metric depths of pores with negative coefficient of proportionality.
The brightness scale of 0–255 units has been divided into sub-intervals by 10 units enumerated from 0 to 26. Each bar in the histogram represents the number of pixels whose brightness levels belong to the given sub-interval.
In all histograms domination of the 26-th bars over the several preceding bars is remarkable. This means that in the given case ε = 10 in the formulae (15, 16) can be used.
At the next step, the profiles shown in Fig. 10b should be taken into consideration. On the level 255 − ε the lengths D_{ α } of sub-intervals crossing the areas where 0 ≤ β <255 − ε should be recorded.
Conclusions
Artificial porous membranes have a wide spectrum of applications in various areas of medicine. Porosity of membranes directly influences their utility as filters separating some types of particles from a fluid matter or as scaffolds for cultivation of biological tissues. However, the notion of porosity contains several properties directly connected with morphological microstructure of pores. The size, density and shape of pores can be mentioned as the basic porosity parameters. Evaluation of the parameters is thus an auxiliary but still important problem to be solved in order to improve the quality of membranes and to increase their utility. However, the problem is not trivial because of irregularity of the shapes of pores and lack of access to some characterizing them data. In this paper some concepts concerning the methods of artificial membranes’ porosity evaluation based on computer-aided analysis of microscope images of their cross-sections are presented. The proposed methods are the result of a long-term cooperation between two working groups in the Nalecz Institute of Biocybernetics and Biomedical Engineering PAS: first (headed by Ph.D. Przytulska), specialized in computer-aided image processing methods and second (headed by prof., D.Sc. Chwojnowski) specialized in artificial membranes design and applications. The methods are based on combined morphological and statistical approach and on some assumptions concerning the statistical isotropy of the materials under investigation. Unlike other methods of 3D pores analysis (e.g. the Focused Ion Beam or Serial Block Face methods) assuming easy access to deep sections of the specimens of examined materials, the proposed methods can be recommended in case of pores analysis based on single SEM images of 2D sections of the materials. It is shown that all the parameters needed to practically use the proposed methods can be obtained by relatively simple numerical analysis of images: measuring distances, areas, brightness profiles, calculation of histograms and their parameters, etc. The necessary procedures are available in various statistical or image processing libraries of programs [23, 24]. However, in order to make our research more effective an original specialized software MeMoExplorer™ has been designed and it is still under development [27].
Declarations
Authors’ contributions
JLK carried out the detailed theoretical derivation, drafted and revised the manuscript; MP collected the data and carried out the experimental analysis, checked the theoretical derivation and revised the manuscript; AC supervised the theoretical development and revised the manuscript. All authors read and approved the final manuscript.
Acknowledgements
We would like to express our thanks to Prof. R. Koprowski for encouraging us to prepare the manuscript, the Reviewers for their valuable critical remarks and recommendations as well as to MSc. P. Mazur and MSc. R. Maksymowicz for their help to improve English style of the paper.
Competing interests
The authors declare that they have no competing interests.
Ethics approval and consent to participate
Not applicable.
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