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journa l homepage: www. int l .e lsev ierhea l th .com/ journa ls /cmpb

tracer metric numerical model for predicting tortuosityactors in three-dimensional porous tissue scaffolds

. Starlya, E. Yildirimb, W. Sunb,∗

School of Industrial Engineering, University of Oklahoma, Norman, OK 73019, USADepartment of Mechanical Engineering and Mechanics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA

r t i c l e i n f o

rticle history:

eceived 3 December 2006

eceived in revised form

5 April 2007

ccepted 16 April 2007

eywords:

hree-dimensional tortuosity

orous media

a b s t r a c t

One of the critical functions of a tissue-engineered construct is to be able to provide ade-

quate nutrient and oxygen supply into the interior of the construct. An insufficient supply

will lead to slower cellular proliferation rates and eventual apoptosis. The supply of the

nutrients is largely governed by the transport properties of the construct which in turn is

dependent on the porosity, tortuosity and surface chemistry of the tissue construct. The

design and fabrication of scaffolds with tailored properties is thus a crucial step in the

growth of tissue within their host environment. This paper discusses the development of

a numerical characterization technique to measure the three-dimensional tortuosity fac-

tors for any given interconnected porous design. Tortuosity factors are obtained in the three

racer analysis

issue engineering

orthogonal principal directions for several candidate unit cell architectures. The proposed

numerical technique has been validated with models of known tortuosity. The developed

technique will provide a basis for the study of transport properties of the designed scaffold

and its effect on cellular function and response through the development of dynamic culture

bioreactors.

tion. The consumption of nutrients by the cells must be locally

. Introduction and background

issue-engineered scaffolds can potentially provide clinicalherapies in wound healing and tissue regeneration appli-ations [1,2]. These scaffolds provide a three-dimensionalramework upon which transplanted and/or encapsulatedells can adhere, grow and differentiate to provide thentended tissue function. The classical strategy for in vitroissue engineering involves either of the two following

ethods—the direct seeding of cells on the scaffolds [3,4] or byhe encapsulation of cells within hydrogels [5,6]. The scaffoldsontaining living cells are subjected to and maintained under

onditions that support appropriate tissue growth throughhe use of dynamic culture vessels and bioreactors. To facil-tate the growth of tissue into the interior of the scaffold, a

∗ Corresponding author. Tel.: +1 215 895 5810.E-mail address: sunwei@drexel.edu (W. Sun).

169-2607/$ – see front matter © 2007 Elsevier Ireland Ltd. All rights resoi:10.1016/j.cmpb.2007.04.003

© 2007 Elsevier Ireland Ltd. All rights reserved.

highly porous and a well interconnected network is built intothe scaffold model. The porous architecture allows for themovement of nutrients and essential signaling molecules bymaintaining flow through the scaffold.

A manmade tissue substitute must not only serve as astructural support for cells but also provide a nutritional path-way for the transfer of nutrients and molecules to regionswhere the new formation of tissue is desired [7]. Cells requireaccess to nutrient molecules such as oxygen, glucose andamino acids combined with the need for the efficient removalof metabolite byproducts for proper cell survival and prolifera-

balanced by the delivery of fresh nutrients into the scaffoldfrom the medium. This rate of consumption of nutrients islargely governed by the cell type, nutrient requirements and

erved.

m s i

22 c o m p u t e r m e t h o d s a n d p r o g r a

cell seeding density, while the delivery of nutrients into thescaffold is governed by scaffold porosity, permeability and flowconditions. Often these variables are conflicting, for exam-ple, scaffolds with larger porosity and lower tortuosity factorsensure the fastest delivery of nutrients to the cells, but willhave poor mechanical properties. Therefore, scaffold designstrategies must seek to optimize the design parameters thatmodulate its mechanical and transport properties.

Permeability, diffusion, porosity and tortuosity are criticalparameters to be measured when fluid flow within scaffoldsis being studied [8]. Malda et al. in their landmark study haveshown a direct correlation of oxygen gradients and type ofscaffold architecture. Scaffolds with controlled architectureand well interconnected pores as opposed to random architec-ture have shown to significantly enhance the differentiationand growth of human chondrocytes for cartilage tissue regen-eration. Larger interconnected pores provide a more efficientpathway for nutrient access and waste removal [9,10]. Thesestudies and others suggest that the scaffold architecture playsa significant role in the tissue regeneration process. A com-plete understanding of the effect of transport properties oncellular response cannot be undertaken without a proper mea-surement of the porosity and tortuosity factors of the givenscaffold. These fundamental parameters play a major role inthe determination of the permeability and diffusion charac-teristics of nutrients through porous media.

Porous structures are generally characterized by theirporosity, pore distribution and tortuousity factors whichin turn determine the effective mechanical properties andtransport characteristics of the structure. Traditionally, poros-ity measurements have been undertaken using techniquessuch as mercury intrusion porosimetry [11–13], scanningelectron microscopy [13], mechanical testing [14,15], opticalmicroscopy, etc. for scaffolds that are produced using chemicalmethods of fabrication. These techniques are often destruc-tive and hence do not provide direct accurate estimates ofthe structural properties, but nevertheless they are capable ofestimating structural properties at a high resolution. With thedevelopment of solid freeform fabrication (SFF) technologiesas one among the favored scaffold fabrication methods, thevirtual CAD model of the scaffold can now be used for struc-tural characterization using analytical or numerical methods[16–18]. Such characterization tools can be devised due to theunit cell nature of SFF-based tissue scaffolds. Since unit cellsare sufficiently repeated throughout the structure, the globalproperties of the scaffold are “embedded” within the unit cellstructure and hence the unit cells by themselves can be usedfor virtual characterization studies through the use of homog-enization techniques [19].

A survey of the literature has shown two primary meth-ods that are used to measure tortuosity: (1) Monte Carlosimulations and (2) image-based analysis. The Monte Carlo(MC)-based simulations are performed to determine theeffective diffusion coefficient from which the tortuosity ofthe model is estimated. Tao et al. have performed three-dimensional diffusion studies using MC simulations over a set

time scale to predict the tortuosity values for the brain extra-cellular space [20]. On the other hand, image-based techniquesare independent of time and make use of imaging modalitiessuch as CT/MRI/X-ray images of the model. Image process-

n b i o m e d i c i n e 8 7 ( 2 0 0 7 ) 21–27

ing techniques are analyzed to obtain the distance traveledby “tracers” from a given entry point to the exit point. Forinstance, Ramaswamy et al. have reported using digital imageanalysis techniques to measure the tortuous characteristics ofpaper fabrics [21]. CT image sections of the paper fabric wereobtained to measure the tortuosity factors in the two prin-cipal orthogonal directions. The tortuousity factors obtainedusing image analysis techniques were compared with thoseobtained from lattice Boltzmann simulations and have shownreasonable agreement. Eze et al. using sine wave simulationshave derived three-dimensional values for the tortuosity fac-tors of the artery using serial sections of the CT image dataset. Their clinically adaptive technique measures the succes-sive differences in the arterial midline coordinates to calculatethe tortuosity coefficients [22]. To the author’s knowledge, noreported work has been published to provide a technique toestimate the three-dimensional tortuosity factors for scaffolddesigns used in tissue engineering.

The current technique for measuring scaffold tortuosityincludes a simple one-dimensional estimation of the tortu-osity factor of the scaffold through the use of simple graphingtechniques. While porosity of a scaffold has been predomi-nantly been used as a scaffold design parameter, the tortuosityof a scaffold plays an equally important role in the transportbehavior of the scaffold. Limited effort has been undertakenin developing an analytical or numerical technique to quan-tify the tortuosity and relate it to the transport properties of agiven scaffold design. The objective of this paper is to presenta numerical technique to predict the tortuosity factors in thethree principal orthogonal directions of a given unit cell archi-tecture.

The numerical model is validated by comparing the resultsobtained from a model with known tortuosity. The algorithmpresented here is valid for any porous structure provided aCAD model of the structure exists. Section 2 of the paperdescribes the several definitions of tortuosity that exist inliterature. Section 3 details the numerical technique that esti-mates the tortuosity factors in the three principal orthogonaldirections. Section 4 describes the results obtained using thistechnique followed by conclusions and discussions.

2. Definition of tortuosity

Besides the pore characteristics and its effect on mechani-cal properties, tortuosity of the scaffolds plays a major rolein tissue growth and proliferation. The transport propertiesof the designed scaffolds are based on the degree of tortuos-ity which determines the resistance to the transport of fluidmolecules (nutrients and oxygen) through the scaffold. Thesimplest definition of tortuosity is defined as the ratio of theshortest distance taken by the fluid particles (L0) to the short-est straight line (L) that connects the entry point to the exitpoint as given by Eq. (1) and shown in Fig. 1.

� = L0 (1)

L

Several authors in their studies of porous structures haveexpressed this parameter based on the diffusion rates ofmolecules within the structure [20,23–25] as given by the fol-

c o m p u t e r m e t h o d s a n d p r o g r a m s i n

l

�

wfceat

atvfttnga

(

(

Fig. 1 – Tortuosity definition.

owing equation:

=√

D

De(2)

here D = diffusivity of molecules in bulk, De = effective dif-usivity. The factor calculated from this expression, however,annot be used to fully describe the nature of the tortuousffects within the structure and is applicable to symmetricrchitectures since the obtained value gives us an estimate ofhe tortuousity within the whole model.

From the generally accepted definition, it is clear that forheterogeneous structure, one factor alone cannot be used

o describe the nature of tortuosity. To obtain a more realisticalue for the degree of tortuosity in a three-dimensional scaf-old structure, an algorithm is developed to calculate threeortuosity factors to describe the level of resistance in thehree principal axes of the model based on a Cartesian coordi-ate system. These factors are determined purely based on theeometric nature of the model and have made the following

ssumptions:

a) The algorithm does not take into account the type of dif-fusing molecule;

Fig. 2 – Image extra

b i o m e d i c i n e 8 7 ( 2 0 0 7 ) 21–27 23

b) The surface characteristics of the scaffolding material areassumed hard;

(c) The scaffolds have little or no dead pore spaces within thescaffold.

In this paper, we have defined the three tortuosity factors,namely �x, �y, �z, from which a single three-dimensional valuecan be obtained from their geometric means, i.e.

�3D = (�2x + �2

y + �2z )

1/2(3)

Each tortuosity factor coefficient is defined as the ratio of thedistance traveled by a particle from entry plane to exit plane ina given preferred direction to the shortest distance that con-nects the two planes. If the particle meets an obstacle alongits path of movement, a side step is taken until an open spaceis detected.

3. Three-dimensional tracer metricnumerical model to estimatethree-dimensional tortuosity

3.1. Image extraction

The first step involves the selection of the unit cell architecturethat is sufficiently repeated within the scaffold structure. Ifsuch a representative structure cannot be selected, the entirescaffold structure is used for analysis. The three-dimensionalvolumetric unit cell assembly, V, is sliced at regular intervalsinto layers to obtain sliced contours. Each slice contour is con-verted to a binary image filled with 0s and 1s, with each porebeing assigned a value 0 and the solid structure is assigned thevalue 1. The process is shown in Fig. 2. Each binary image slicelevel is stacked upon each other to obtain the complete binaryimage data set for the entire unit cell assembly. This sequenceis defined by the slice thickness parameter, h which is used todecompose the three-dimensional volume, V. A finite discon-tinuous voxel set C is introduced to indicate the discrete imagelayers into which V is decomposed:

C = {ck; a ≤ c1 < c2 < · · · < ck−1 < ck < ck+1 < · · · < cn < cn+1

≤ b, k = 1, 2, . . . , n} (4)

where the subscript k represents the number of the kth decom-position in a total of n sliced layers, and constants a and b

represent the values of lowermost and uppermost sequencerespectively for a particular object V. The entire image dataset is stored in a three-dimensional matrix that contains thevoxel value.

ction process.

m s i n b i o m e d i c i n e 8 7 ( 2 0 0 7 ) 21–27

24 c o m p u t e r m e t h o d s a n d p r o g r a

3.2. Tracer metric analysis

Once the three-dimensional matrix of voxel values that definethe unit cell structure has been assembled, the tortuosity inthe three orthogonal directions can be calculated by measur-ing the distance traveled by a particle traversing through themodel. The steps given below describe the process by whichthe distance traveled by “tracers” between two given points iscalculated. Appendix A presents the flowchart of the process.

(1) Tracers are defined on an initial slice plane that representsone of the exterior faces of the model and is selected tobe the entry face. The tracers traverse through the poresof the unit cell structure until it meets the exit face. Ifthe tracers encounter a solid structure, it cycles throughits neighborhood until a pore voxel is found. For exam-ple, voxels at the plane z = 0 are initialized as input tracersand allowed to propagate through the model in a preferreddirection, in this case the positive z-direction. A tracer, �

continues to move along the z direction as long as it trav-els along a pore voxel (0). If it encounters a voxel withvalue 1, a side step is taken, defined by an eight-level voxelneighborhood connectivity criterion as follows:

N8(�) = {(x + i, y), (x − i, y), (x, y − i), (x, y − i),

(x + i, y + i), (x + i, y − i), (x − i, y + i), (x − i, y − i)} (5)

where (i = 1, 2, 3,. . .). Each alternate step from the currentposition �(x, y) is traversed in the order given in Eq. (2) andchecked to see if a pore voxel is met. If it cannot find a porevoxel, after cycling through all of its nearest neighbors(i = 1), the value of i is incremented to increase the searchneighborhood. The value of i is allowed to increment untilthe search neighborhood has met the outer bounds of themodel or if a pore voxel is reached. If it cannot find a porevoxel within its search neighborhood, it is classified as adead pore and the search is abandoned (Fig. 3).

(2) The process is repeated for all tracers on the plane z = 0.The distance traveled by all the tracers given by the dis-tance metric is calculated and divided by the shortestlength from the tracer entry plane to the exit plane. The

average tortuosity, �z is then easily determined for the zdirection.

(3) Similarly, tracers are allowed to propagate from the planes(x = 0 and y = 0) and Steps 1–2 are repeated to obtain �x, and�y, respectively for any given unit cell assembly model.

Fig. 3 – Tortuosity measurement usin

Fig. 4 – Test model with theoretical tortuosity of 1.3, whilethe numerical algorithm predicts a value of 1.376.

The above procedure is implemented using the MicrosoftVisual Studio 6.0 Developer Studio. Appendix A provides aflowchart for a detailed description of the algorithm. The algo-rithm is tested on a Pentium IV 1 GB RAM system which takesabout 120 s for processing, given a 256 × 256 pixel image and256 image plane slices.

4. Results

The described technique has been validated by a model ofknown tortuosity. The model shown in Fig. 4 has a theoret-ical tortuosity of 1.3 while the algorithm predicts a tortuosityof 1.376. The slightly higher prediction can be attributed tothe fact that the total distance traveled by multiple tracersentering the tube inlet is included in the calculation. In con-trast, the theoretical tortuosity equation (given by Eq. (1))

takes into account only the sum total of the mid line dis-tance (5 + 3 + 5/10 = 1.3) traveled by a single tracer. Tortuosityprediction has been performed for different unit cell patternsas shown in Fig. 5. The tortuosity values for Models A and B are

g tracer distance metric analysis.

c o m p u t e r m e t h o d s a n d p r o g r a m s i n b i o m e d i c i n e 8 7 ( 2 0 0 7 ) 21–27 25

Fig. 5 – Two models with different porosity but same recta

Fig. 6 – Two models with constant porosity of 48% butdifferent tortuosity factors due to the difference information factors or internal architecture variations. (a)Model C tortuosity: �x = 1.21, �y = 1.35, �z = 1.11. (b) Model Dtortuosity: �x = 1.091, �y = 1.035, �z = 1.206.

Fig. 7 – Four types of scaffolds with varyin

ngular pore shape having different tortuosity factors.

close to unity, though a dependence on porosity is observed. Itmust be noted that if Models A and B were infinitely repeated,the tortuosity of both models will tend to unity irrespective ofthe differences in structural porosity.

Two unit cell structures of the same porosity can have dif-ferent values of tortuosity as shown in Fig. 6. This indicates thedependence of tortuosity on the geometric characteristics ofthe model which is symbolically represented as the formationfactor, F [26]. Hence the tortuosity factors are clearly depen-dent on the manner in which material is distributed withinthe structure. The tortuosity factors in the x, y and z direc-tions can be designed to obtain different transport properties,while keeping porosity constant. In Fig. 6, it is clear that forModel C, the transport of nutrients would take place faster inthe z-axis than the x- and y-axis due to the lower z-tortuosityfactor. However, in the case of Model D, the tortuosity in z-axisis much higher than x and y axes indicating that the nutrientswill take a larger tortuous flow path. This indirectly suggeststhat for the scaffold Model D, cellular proliferation rates inthe z-axis will be limited. By being able to selectively designthe tortuosity factors in the three principal axes, directed cell

growth and proliferation can be attained since the faster per-meation of nutrients in a preferred direction contributes tohigher cellular proliferation rates. Selecting unit cell architec-tures in a scaffold that have varying degrees of tortuosity can

g tortuosity particularly in the z-axis.

m s i

26 c o m p u t e r m e t h o d s a n d p r o g r a

also help us to understand the role played by the transport ofnutrients on cellular activity as shown in Fig. 7.

5. Conclusion

A tracer metric numerical model to predict the tortuosity fac-tors in three principal orthogonal directions of a given unit cellscaffold is presented. Knowing these tortuosity factors, thetransport properties of the scaffolds can be correlated usingavailable models published in the literature. The developedtechnique can be used for generic porous structure providedan initial virtual model of the sample exists. The results ofthe study show that the tortuosity factors are not entirelydependent on the porosity of the model but on the geometricarrangement of the material inside the porous structure.

The tortuosity factors derived here are dimensionless andits accuracy is dependent upon the resolution of the binaryimages. For accurate prediction of the tortuosity factors, theimage slice thickness must be thin and voxel size must be assmall as possible. Larger slice thickness can result in the loss ofsmall features in the model and are typically attributed to theerrors in slicing. This can lead to an erroneous prediction of thetortuosity factor. This is analogous to picking the mesh sizein a classical finite element analysis technique. The smallerthe slice thickness and voxel size, better the accuracy of thetortuosity values.

Future cell culture experiments would entail the study ofthe effect of the three-dimensional tortuosity factors on cellu-lar response and behavioral characteristics in an in vitro studykeeping all other variables constant (biomaterial surface, celltype). The authors hypothesize that the tortuosity may also be

one of the important factors contributing to cell proliferation,among other scaffold design and biological parameters. To val-idate this hypothesis, the authors plan to conduct in vitro

n b i o m e d i c i n e 8 7 ( 2 0 0 7 ) 21–27

cell culture experiments to study the effect of the designedthree-dimensional tortuosity factors on subsequent cellularbehavior, particularly cell growth kinetics.

The algorithm presented in this paper can be used for pre-dicting the tortuosity values of any porous structure including,scaffolds prepared from particulate leaching/gas foamingtechniques. The only requirement is that a virtual voxel orCAD model of the structure must be prepared through eithermicroCT (�CT) scanning or other compatible reverse engi-neering schemes. Porous structures with very random interiorarchitectures have a higher probability of encountering thepresence of dead pores within the model. The eight voxelneighborhood search criteria presented in Eq. (5) will attemptat enabling tracers to find open pores. The resolution of the�CT will be a limiting factor on the accuracy of the predictedtortuosity factors. The factors produced by the algorithm willbe only as good as the resolution of the �CT scan images. Thehigher the resolution of the images, the better is the accu-racy of the predicted tortuosity. Future research efforts willbe undertaken to study the dependence of image resolutionon the accuracy of prediction. This can be performed by fab-ricating scaffolds with known tortuosity preferably throughSFF techniques and then �CT scanning them at various res-olutions. These scanned images can now be entered into thealgorithm for testing and eventual evaluation.

Acknowledgement

The authors would like to acknowledge the partial fundingsupport from NSF-0427216 grant for this research.

Appendix A. Tracer metric flowchart to predicttortuosity values

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e f e r e n c e s

[1] R. Langer, Tissue engineering: a new field and its challenges,Sci. Am. 280 (1999) 86–89.

[2] R. Lanza, J. Vacanti, Principles of Tissue Engineering,Academic Press, 2000.

[3] E. Sachlos, J.T. Czernuszka, Making tissue engineeringscaffolds work. Review: the application of solid freeformfabrication technology to the production of tissueengineering scaffolds, Eur. Cell Mater. 5 (2003) 29–39(discussion, 39-40).

[4] B.M. Wu, S.W. Borland, R.A. Giordano, L.G. Cima, E.M. Sachs,M.J. Cima, Solid free-form fabrication of drug deliverydevices, J. Control. Rel. 40 (1996) 77–87.

[5] A.D. Augst, H.J. Kong, D.J. Mooney, Alginate hydrogels asbiomaterials, Macromol. Biosci. 6 (8) (2006) 623–663.

[6] D.R. Albrecht, V.L. Tsang, R.L. Sah, S.N. Bhatia, Photo- andelectropatterning of hydrogel-encapsulated living cellarrays, Lab Chip. 5 (1) (2005) 111–118, Jan.

[7] T.S. Karande, J.L. Ong, C.M. Agrawal, Diffusion inmusculoskeletal tissue engineering scaffolds: design issuesrelated to porosity, permeability, architecture, and nutrientmixing, Ann. Biomed. Eng. 32 (12) (2004) 1728–1743.

[8] E.A. Botchwey, M.A. Dupree, S.R. Pollack, E.M. Levine, C.T.Laurencin, Tissue engineered bone: measurement ofnutrient transport in three-dimensional matrices, J. Biomed.Mater. Res. A 67 (1) (2003) 357–367.

[9] S. Miot, T.B.F. Woodfield, A.U. Daniels, R. Suetterlin, I.Peterschmitt, M. Heberer, C.A. van Blitterswijk, J. Riesle, I.Martin, Effects of scaffold composition and architecture onhuman nasal chondrocyte redifferentiation andcartilaginous matrix deposition, Biomaterials 26 (15) (2005)2479–2489.

[10] J. Malda, T.B.F. Woodfield, F. van der Vloodt, F.K. Kooy, D.E.Martens, J. Tramper, C.A. van Blitterswijk, J. Riesle, The effectof PEGT/PBT scaffold architecture on oxygen gradients intissue engineered cartilaginous constructs, Biomaterials 25(26) (2004) 5773–5780.

[11] L. Safinia, A. Mantalaris, A. Bismarck, Nondestructivetechnique for the characterization of the pore sizedistribution of soft porous constructs for tissue engineering,Langmuir 22 (7) (2006) 3235–3242.

[12] S.H. Lee, B.S. Kim, S.H. Kim, S.W. Kang, Y.H. Kim, Thermallyproduced biodegradable scaffolds for cartilage tissueengineering, Macromol. Biosci. 4 (8) (2004) 802–810.

[13] W.J. Li, J.A. Cooper Jr., R.L. Mauck, R.S. Tuan, Fabrication andcharacterization of six electrospun poly(alpha-hydroxy

b i o m e d i c i n e 8 7 ( 2 0 0 7 ) 21–27 27

ester)-based fibrous scaffolds for tissue engineeringapplications, Acta Biomater. 2 (4) (2006) 377–385.

[14] S. Gong, H. Wang, Q. Sun, S.T. Xue, J.Y. Wang, Mechanicalproperties and in vitro biocompatibility of porous scaffolds,Biomaterials 27 (20) (2006) 3793–3799.

[15] D.W. Hutmacher, T. Schantz, I. Zein, K.W. Ng, S.H. Teoh, K.C.Tan, Mechanical properties and cell cultural response ofpolycaprolactone scaffolds designed and fabricated viafused deposition modeling, J. Biomed. Mater. Res. 55 (2)(2001) 203–216.

[16] W. Sun, B. Starly, A. Darling, C. Gomez, Computer-aidedtissue engineering: application to biomimetic modelling anddesign of tissue scaffolds, Biotechnol. Appl. Biochem. 39(2004) 49–58.

[17] S.J. Hollister, R.D. Maddox, J.M. Taboas, Optimal design andfabrication of scaffolds to mimic tissue properties andsatisfy biological constraints, Biomaterials 23 (20) (2002)4095–4103.

[18] S. Hollister, Porous scaffold design for tissue engineering,Nat. Mater. 4 (2005) 518–524.

[19] Z. Fang, C. Yan, W. Sun, A. Shokoufandeh, W. Regli,Homogenization of heterogeneous tissue scaffold: acomparison of mechanics, asymptotic homogenization, andfinite element approaches, J. Bionic Appl. Biomech. 2 (2005)17–29.

[20] A. Tao, L. Tao, C. Nicholson, Cell cavities increase tortuosityin brain extracellular space, J. Theor. Biol. 234 (4) (2005)525–536.

[21] S. Ramaswamy, M. Gupta, A. Goel, U. Aaltosalmi, M. Kataja,The 3D structure of fabric and its relationship to liquid andvapor transport, Colloids Surf. AL Physiochem. Eng. 241(2004) 323–333.

[22] C.U. Eze, R. Gupta, D.L. Newman, A comparison ofquantitative measures of arterial tortuosity using sine wavesimulations and 3D wire models, Phys. Med. Biol. 45 (2000)2593–2599.

[23] N. Wakao, J.M. Smith, Diffusion in catalyst pellets, Chem.Eng. Sci. 17911 (1962) 825–834.

[24] M. Zalc, S. Reyes, E. Iglesia, The effects of diffusionmechanism and void structure on transport rates andtortuosity factors in complex porous structures, Chem. Eng.Sci. 59 (2004) 2947–2960.

[25] J. Hrabe, S. Hrabetova, K. Segeth, A model of effectivediffusion and tortuosity in the extracellular space of the

brain, Biophys. J. 87 (2004) 1606–1617.

[26] Y. Nakashima, T. Yamaguchi, DMAP.m: a Mathematica®

program for three-dimensional mapping of tortuosity andporosity of porous media, Bull. Geol. Survey Jpn. 55 (3/4)(2004) 93–103.