Skip to main content

Advertisement

SpringerLink
Log in

The role of porous media in biomedical engineering as related to magnetic resonance imaging and drug delivery

  • Special Issue
  • Published:
Heat and Mass Transfer Aims and scope Submit manuscript

Abstract

Pertinent works associated with magnetic resonance imaging (MRI) and drug delivery are reviewed in this work to demonstrate the role of transport theory in porous media in advancing the progress in biomedical applications. Diffusion process is considered significant in many therapies such as delivering drugs to the brain. Progress in development of the diffusion equation using local volume-averaging technique and evaluation of the applications associated with the diffusion equation are analyzed. Tortuosity and porosity have a significant effect on the diffusion transport. Different relevant models of tortuosity are presented and mathematical modeling of drug release from biodegradable delivery systems are analyzed in this investigation. New models for the kinetics of drug release from porous biodegradable polymeric microspheres under bulk erosion and surface erosion of the polymer matrix are presented in this study. Diffusion of the dissolved drug, dissolution of the drug from the solid phase, and erosion of the polymer matrix are found to play a central role in controlling the overall drug release process. This study paves the road for the researchers in the area of MRI and drug delivery to develop comprehensive models based on porous media theory utilizing fewer assumptions as compared to other approaches.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Abbreviations

a :

Empirical constant

a E :

Einestein radius

ADC:

Apparent diffusion coefficient

b :

Empirical constant

B Sat :

Saturation concentration of the drug in the polymer phase

B s :

Undissolved drug concentration in the polymer

〈C〉:

Volume average of concentration

C L :

Drug concentration in the liquid phase

C o :

Initial drug concentration

C sat :

Saturation concentration of the drug

C Se :

Drug concentration in the effective solid phase

C s :

Undissolved drug concentration in the pores

d p :

Pore diameter

D * :

Effective diffusion coefficient

D B :

Polymer diffusion coefficient

ECS:

Extracellular space

f n :

Viscosity function

F :

Geometric function

F 1, F 2 :

Correction factors

F(C):

Uptake term

h m :

Mass transfer coefficient

k :

Permeability

k dis :

Dissolution rate constant

k ero :

Surface erosion constant

K B :

Forward rate constant

K C :

Backward rate constant

K DB :

Dissolution rate constant in polymer

K DC :

Dissolution rate constant in pore

K Hero :

Hyperbolic erosion rate constant for bulk erosion

K Lero :

Linear erosion rate constant for bulk erosion

K m :

Michele-menten constant

K Sero :

‘S’ erosion rate constant for bulk erosion

M ∞ :

Cumulative amount of drug released at time infinity

M t :

Cumulative amount of drug released at time t

MRI:

Magnetic resonance imaging

P :

Fluid pressure

r o :

Pore radius

R s :

Radius of microparticles

〈s〉:

Mass source density

Sh:

Sherwood number

t :

Time

〈v〉:

Velocity vector

V :

Representative elementary volume

V 1 :

Effective volume of the microsphere

V max :

Rate constant

V p :

Pore volume

ρf :

Fluid density

ɛ:

Porosity

λg :

Geometrical tortuosity

λ x , λ y , λ z :

Tortuosity components

μf :

Dynamic viscosity of the pure fluid

σ:

Surface area

References

  1. Vafai K, Tien CL (1981) Boundary and inertia effects on flow and heat transfer in porous media. Int J Heat Mass Transfer 24:195–203

    Article  MATH  Google Scholar 

  2. Vafai K, Tien CL (1982) Boundary and inertia effects on convective mass transfer in porous media. Int J Heat Mass Transfer 25:1183–1190

    Article  Google Scholar 

  3. Nield DA, Bejan A (1995) Convection in porous media, 2nd edn. Springer, Berlin Heidelberg New York

    Google Scholar 

  4. Vafai K (2000) Handbook of porous media, 1st edn. Marcel Dekker, Inc., New York

    MATH  Google Scholar 

  5. Vafai K (2005) Handbook of porous media, 2nd edn. Taylor and Francis Group, New York

    Google Scholar 

  6. Hadim H, Vafai K (2000) Overview of current computational studies of heat transfer in porous media and their applications-forced convection and multiphase transport. Adv Numer Heat Transfer 2:291–330. Taylor and Francis, NY

    Google Scholar 

  7. Vafai K, Hadim H (2000) Overview of current computational studies of heat transfer in porous media and their applications- natural convection and mixed convection. Adv Numer Heat Transfer 2:331–371. Taylor and Francis, NY

    Google Scholar 

  8. Yang N, Vafai K (2006) Modeling of low-density lipoprotein (LDL) transport in the artery- effects of hypertension. Int J Heat Mass Transfer (in press)

  9. Ai L, Vafai K (2006) A coupling model for macromolecule transport in a stenosed arterial wall. Int J Heat Mass Transfer (in press)

  10. Khaled ARA, Vafai K, Yang M, Zhang X, Ozkan CS (2003) Analysis, control and augmentation of microcantilever deflections in bio-sensing systems. J Sens Actuators B 94:103–115

    Article  Google Scholar 

  11. Yang M, Zhang X, Vafai K, Ozkan C (2003) High sensitivity piezoresistive cantilever design and optimization for analyte-receptor binding. J Micromech Microeng 13:864–872

    Article  Google Scholar 

  12. Khanafer K, Khaled ARA, Vafai K (2004) Spatial optimization of an array of aligned microcantilever biosensors. J Micromech Microeng 14:1328–1336

    Article  Google Scholar 

  13. Khaled ARA, Vafai K (2004) Optimization modeling of analyte adhesion over an inclined microcantilever-based biosensor. J Micromech Microeng 14:1220–1229

    Article  Google Scholar 

  14. Khaled ARA, Vafai K (2004) Analysis of oscillatory flow disturbances and thermal characteristics inside fluidic cells due to fluid leakage and wall slip conditions. J Biomech 3:721–729

    Article  Google Scholar 

  15. Kuffler SW, Potter DD (1964), Glia in the leech central nervous system: physiological properties and neuron-glia relationship. J Neurophysiol 27:290–320

    Google Scholar 

  16. Prokopova-Kubinova S, Vargova L, Tao L, Ulbrich K, Subr V, Sykova E, Nicholson C (2001) Poly [N-hydroxypropyl) methacrylamide] polymers diffuse in brain extracellular space with same tortuosity as small molecules. Biophys J 80:542–548

    Google Scholar 

  17. Nicholson C, Rice ME (1991) Diffusion of ions and transmitters in the brain cell microenvironment. In: Flux K, Agnati LF (eds) Volume transmission in the brain, novel mechanisms for neural transmission. Raven Press, New York, pp 279–294

    Google Scholar 

  18. Nicholson C (1979) Brain cell microenvironment as a communication channel. In: Schmidt FO, Worden FG (eds) The neurosciences fourth study program. MIT Press, Cambridge, pp 457–476

    Google Scholar 

  19. Sykova E (1991) Activity-related ionic and volume changes in neuronal microenvironment. In: Flux K, Agnati LF (eds) Volume transmission in the brain: novel mechanisms for neural transmission. Raven Press, New York, pp 217–336

    Google Scholar 

  20. Barrie PJ (1995) NMR applications to porous solids. Ann Rep NMR Spectrosc 30:37–95

    Article  Google Scholar 

  21. Bose B, Jones SC, Lorig R, Friel HT, Weinstein M, Little JR (1988) Evolving focal cerebral ischemia in cats: spatial correlation of nuclear magnetic resonance imaging, cerebral blood flow, tetrazolium staining, and histopathology. Stroke 19:28–37

    Google Scholar 

  22. Stejskal EO, Tanner JE (1965) Use of spin-echo pulsed magnetic field gradient to study anisotropic restricted diffusion and flow. J Chem Phys 43:3579–3603

    Article  Google Scholar 

  23. Taylor DG, Bushell MC (1985) The spatial mapping of translational diffusion by the NMR imaging technique. Phys Med Biol 30:345–349

    Article  Google Scholar 

  24. Gelderen PV, DeVleeschouwer MH, DesPres D, Pekar J, VanZijl PCM, Moonen CT (1994) Water diffusion and acute stroke. Magn Reson Med 31:154–163

    Article  Google Scholar 

  25. Norris DG, Niendor T, Leibfritz D (1993) A theory of diffusion contrast in healthy and infracted tissue. In: Proceedings of SMRM, 12th Annual Meeting. SMRM, New York, 579

  26. Latour LL, Svoboda K, Mitra P, Sotak CH (1991) Time dependent diffusion of water in a biological model system. Proc Natl Acad Sci USA 91:1229–1233

    Article  Google Scholar 

  27. Moseley ME, Cohen Y, Mintorovitch J, Chileuitt L, Shimizu H, Kucharczyk J, Wendland MF, Weinstein PR (1990) Early detection of cerebral ischemia in cats: comparison of diffusion and T2-weighted MRI and spectroscopy (1990). Magn Reson Med 16:330–346

    Article  Google Scholar 

  28. Mintorovitch J, Moseley ME, Chileuitt L, Shimizu H, Cohen Y, Weinstein PR (1991) Comparison of diffusion and T 2-weighted MRI for the early detection of cerebral ischemia and reperfusion in rats. Magn Reson Med 18:39–50

    Article  Google Scholar 

  29. Benveniste H, Hedlund LW, Johnson GA (1992) Mechanism of detection of acute cerebral ischemia in rats by diffusion weighted magnetic resonance microscopy. Stroke 23:746–754

    Google Scholar 

  30. Helpern JA, Ordidge RJ, Knight RA (1992) The effect of cell membrane water permeability on the apparent diffusion coefficient of water. In: Proceedings of of SMRM, 11th Annual Meeting, SMRM, Berlin, 1201

  31. Nicholson C (2001) Diffusion and related transport mechanisms in brain tissue. Rep Prog Phys 64:815–884

    Article  Google Scholar 

  32. Nicholson C, Phillips JM (1981) Ion diffusion modified by tortuosity and volume fraction in the extracellular microenvironment of rat cerebellum. J Physiol 321:225–257

    Google Scholar 

  33. Mota M, Teixeira JA, Keating JB, Yelshin A (2004) Changes in diffusion through the brain extracellular space. Biotechnol Appl Biochem 39:223–232

    Article  Google Scholar 

  34. Dai L, Miura R (1999) A lattice cellular automated model for ion diffusion in the brain-cell microenvironment and determination of tortuosity and volume fraction. SIAM J Appl Math 59:2247–2273

    Article  MATH  MathSciNet  Google Scholar 

  35. Szafer A, Zhong JH, Gore JC (1995) Theoretical model for water diffusion in tissues. Magn Reson Med 33:697–712

    Article  Google Scholar 

  36. Amiri A, Vafai K (1994) Analysis of dispersion effects and nonthermal equilibrium, non-Darcian, variable porosity incompressible flow through porous media. Int J Heat Mass Transfer 37:939–954

    Article  Google Scholar 

  37. Carslaw HS, Jaeger JC (1959) Conduction of heat in solids, 2nd edn. Clarendon, Oxford

    Google Scholar 

  38. Lubarsky DA, Smith LR, Sladen RN, Mault JR, Reed RL (1995) Defining the relationship of oxygen delivery and consumption-Use of biological system models. J Surg Res 58:508–803

    Article  Google Scholar 

  39. Horn AS (1979) Characteristics of dopamine uptake. In: Horn AS et al (eds) The neurobiology of dopamine. Academic, London, pp 217–35

    Google Scholar 

  40. Lehner FK (1979) On the validity of Fick’s law for transient diffusion through a porous medium. Chem Eng Sci 34:821–825

    Article  Google Scholar 

  41. Mota M, Teixeira JA, Yelshin A (2001) Biotechnol Appl Biochem 17:860–865

    Google Scholar 

  42. El-Kerah AW, Braunstein SL, Secomb TW (1993) Effect of cell arrangement and interstitial volume fraction on the diffusivity on monoclonal antibodies in tissue. Biophys J 64:1638–1646

    Article  Google Scholar 

  43. Limbach KW, Wei J (1990) Restricted diffusion through granular materials. AIChE J 36:242–248

    Article  Google Scholar 

  44. Blanch HW, Clark DS (1996) Biochemical engineering. Marcel Dekker, New York

    Google Scholar 

  45. Deen WM (1987) Hindered transport of large molecules in liquid-filled pores. AIChE J 33:1409–1425

    Article  Google Scholar 

  46. Netrabukkana R, Lourvanij K, Rorrer GL (1996) The Diffusion of glucose and glucitol in microporous and mesoporous silicate catalysts. Ind Eng Chem Res 35:458–464

    Article  Google Scholar 

  47. Pfeuffer J, Dreher W, Sykova E, Leibfritz D (1998) Water signal attenuation in diffusion-weighted 1H NMR experiments during cerebral ischemia: influence of intracellular restrictions, extracellular tortuosity, and exchange. Magn Reson Imaging 16:1023–1032

    Article  Google Scholar 

  48. Archie GE (1942) The electrical resistivity log as an aid in determining some reservoir characteristics. Trans Am Inst Min Metall Petrol Eng Inc 146:54–62

    Google Scholar 

  49. Woerly S, Petrov P, Sykova E, Roitbak T, Simonova Z, Harvey AR (1999) Neural tissue formation within porous hydrogels implanted in brain and spinal cord lesions: ultrastructural, immunohistochemical, and diffusion studies. Tissue Eng 5:467–488

    Article  Google Scholar 

  50. Koegler WS, Patrick C, Cima MJ, Griffith LG (2002) Carbon dioxide extraction of residual chloroform from biodegradable polymers. J Biomed Mater Res Part B 63:567–576

    Article  Google Scholar 

  51. Abbott NJ, Bundgaard M, Cserr HF (1985) Tightness of the blood brain barrier and evidence for brain interstitial fluid flow in the cuttlefish, Sepia officinalis. J Physiol (London) 368:213–226

    Google Scholar 

  52. Rosenberg GA, Kyner WT, Estrada E (1980) Bulk flow of brain interstitial fluid under normal and hyperosmolar conditions. Am J Physiol 238:F42–F49

    Google Scholar 

  53. Rosenberg GA, Kyner WT (1980) Gray and white matter brain-blood transfer constants by steady-state tissue clearance in cat. Brain Res 193:59–66

    Article  Google Scholar 

  54. Khaled A-RA, Vafai K (2003) The role of porous media on modeling flow and heat transfer in biological tissues. Int J Heat Mass Transfer 46:4989–5003

    Article  MATH  Google Scholar 

  55. Khanafer K, Vafai K, Kangarlu A (2003) Computational modeling of cerebral diffusion-application to stroke imaging. Magn Reson Imaging 21:651–661

    Article  Google Scholar 

  56. Khanafer K, Vafai K, Kangarlu K (2003) Water diffusion in biomedical systems as related to magnetic resonance imaging. Magn Reson Imaging 21:17–31

    Article  Google Scholar 

  57. Vafai K (1984) Convective flow and heat transfer in variable-porosity media. J Fluid Mech 147:233–259

    Article  MATH  Google Scholar 

  58. Vafai K (1986) Analysis of the channeling effect in variable porosity media. ASME J Energy Resour Technol 108:131–139

    Article  Google Scholar 

  59. Amiri A, Vafai K, Kuzay TM (1995) Effects of boundary conditions on non Darcian heat transfer through porous media and experimental comparisons. Numer Heat Transfer Part A 27:651–664

    Article  Google Scholar 

  60. Chien YW (1992) Novel drug delivery systems, 2nd edn. Marcel Dekker, New York

    Google Scholar 

  61. Fan LT, Singh SK (1989) Controlled release: a quantitative treatment. Springer, Berlin Heidelberg New York

    Google Scholar 

  62. Jalil R, Nixon JR (1990) Biodegradable poly (lactic acid) and poly(lactide-co-glycolide) microcapsules: problems associated with preparative techniques and release properties. J Microencapsul 7:297–325

    Article  Google Scholar 

  63. Hanes J, Chiba M, Langer R (1998) Degredation of porous poly (anhydride-co-imide) microspheres and implications for controlled macromolecule delivery. Biomaterials 19:163–172

    Article  Google Scholar 

  64. Feng SS, Chien S (2003) Chemotherapeutic engineering: application and further development of chemical engineering principles for chemotherapy cancer and other diseases. Chem Eng Sci 58:4087–4114

    Article  Google Scholar 

  65. Crank J (1975) The mathematics of diffusion, 2nd edn. Clarendon Press, Oxford UK

    Google Scholar 

  66. Harland RS, Dubernet C, Benoit J-P, Peppas NA (1988) A model of dissolution-controlled, diffusion drug release from non-swellable polymeric microspheres. J Control Release 7:207–215

    Article  Google Scholar 

  67. Hopfenberg HB (1976) Controlled release from erodible slabs, cylinders, and spheres. In: Controlled release polymeric formulations. ACS Symposium Series 33:26

  68. Higuchi T (1961) Rate of release of medicaments from ointment bases containing drugs in suspension. J Pharm Sci 50:874–875

    Article  Google Scholar 

  69. Higuchi T (1963) Mechanism of sustained-action medication: theoretical analysis of rate of release of solid drugs dispersed in solid matrices. J Pharm Sci 52:1145–1149

    Article  Google Scholar 

  70. Peppas NA (1985) Analysis of Fickian and non-Fickian drug release from polymers. Acta Helv 60:110–111

    Google Scholar 

  71. Kim J, Finn NC (1996) Shape modeling of dissolution profiles by non-integer kinetic orders. Int J Pharm 143:223–232

    Article  Google Scholar 

  72. Sezer AD, Akbuga J (1995) Controlled release of piroxicam from chitosan beads. Int J Pharm 121:113–116

    Article  Google Scholar 

  73. Kosmidis K, Argyrakis P, Macheras P (2003) A reappraisal of drug release laws using Monte Carlo simulations: the relevance of the Weibull function. Pharm Res 20(7):988–995

    Article  Google Scholar 

  74. Weibull W (1951) A statistical distribution of wide applicability. J Appl Mech 18:293–297

    MATH  Google Scholar 

  75. Siepmann J, Lecomte F, Bodmeier R (1999) Diffusion-controlled drug delivery systems: calculation of the required composition to achieve desired release profiles. J Control Release 60:379–389

    Article  Google Scholar 

  76. Flynn GL, Yalkowsky SH, Roseman TJ (1974) Mass transport phenomena and models: theoretical concepts. J Pharm Sci 63:479–510

    Article  Google Scholar 

  77. Charlier A, Leclerc B, Couarraze G (2000) Release of mifepristone from biodegradable matrices: experimental and theoretical evaluations. Int J Pharm 200:115–120

    Article  Google Scholar 

  78. Bezemer JM, Radersma R, Grijpma DW, Dijkstra PJ, Feijen J, Blitterswijk CA (2000) Zero-order release of lysozyme from poly(ethylene glycol)/poly(butylenes terephthalate) matrices. J Control Release 64:179–192

    Article  Google Scholar 

  79. Heller J, Baker RW (1980) Theory and practice from controlled drug delivery from bioerodible polymers. In: Baker RW (ed) controlled release of bioactive materials. Academic Press, New York, pp 1–18

    Google Scholar 

  80. Lee PI (1980) Diffusional release of a solute from a polymeric matrix-approximate analytical solutions. J Membr Sci 7:255–275

    Article  Google Scholar 

  81. Joshi A, Himmelstein KJ (1991) Dynamics of controlled release from bioerodible matrices. J Control Release 15:95–104

    Article  Google Scholar 

  82. Batycky RP, Hanes J, Langer R, Edwards DA (1997) A theoretical model of erosion and macromolecular drug release from biodegrading microspheres. J Pharm Sci 86:1464–1477

    Article  Google Scholar 

  83. Lemaire V, Belair J, Hildgen P (2003) Structural modeling of drug release from biodegradable porous matrices based on a combined diffusion/erosion process. Int J Pharm 258:95–107

    Article  Google Scholar 

  84. Lee AJ, King JR, Hibberd S (1998) Mathematical modeling of the release of drug from porous, nonswelling transdermal drug-delivery devices. IMA J Math Appl Med Biol 15:135–163

    Article  MATH  Google Scholar 

  85. Cohen DS, Erneux T (1988) Free boundary problems in controlled release pharmaceuticals. II: Swelling controlled release. SIAM J Appl Math 48:1466–1474

    Article  MATH  MathSciNet  Google Scholar 

  86. Fujita J (1961) Diffusion in polymer-diluent systems. Fortschr Hochpolym Fortschr 3:1–47

    Article  Google Scholar 

  87. Siepmann J, Peppas N (2001) Modeling of drug release from delivery systems based on hydroxypropyl methylcellulose (HPMC). Add Drug Deliv Rev 48:139–157

    Article  Google Scholar 

  88. Siepmann J, Gopferich A (2001) Mathematical modeling of bioerodible, polymeric drug delivery systems. Adv Drug Deliv Rev 48:229–247

    Article  Google Scholar 

  89. Siepmann J, Faisant N, Benoti J-P (2002) A new mathematical model quantifying drug release from bioerodible microparticles using Monte Carlo simulations. Pharm Res 19:1885–1893

    Article  Google Scholar 

  90. Zhang M, Yang Z, Chow LL, Wang CH (2003) Simulation of drug release from biodegradable polymeric microspheres with bulk and surface erosions. J Pharm Sci 92(10):2040–2056

    Article  Google Scholar 

  91. James MA, Matthew SS (1997) Biodegradation and biocompatibility of PLA and PLGA microspheres. Adv Drug Deliv Rev 28:5–24

    Article  Google Scholar 

  92. Breitenbach A, Pistel KF, Kissel T (2000) Biodegradable comb polyesters. Part II. Erosion and release properties of poly(vinyl alcohol)-g-poly-(lactic-co-glycolic acid). Polymer 41:4781–4792

    Article  Google Scholar 

  93. Wong HM, Wang JJ, Wang CH (2001) In vitro release of human immunoglobulin G from biodegradable microspheres. Ind Eng Chem Res 40:933–948

    Article  Google Scholar 

  94. Beate B, Christian W, Karsten M, Thomas K (1999) Degradation and protein release properties of microspheres prepared from biodegradable poly-(lactide-co-glycolide) and ABA triblock copolymers: influence of buffer media on polymer erosion and bovine serum albumin release. J Control Release 60:297–309

    Article  Google Scholar 

  95. Chia HH, Yang YY, Chung TS, Steve NG, Heller J (2001) Auto-catalyzed poly(ortho ester) microspheres: a study of their erosion and drug release mechanism. J Control Release 75:11–25

    Article  Google Scholar 

  96. Wang JP, Yang YY, Chung TS, Tan D, Steve NG, Heller J (2001) POE-PEG-POE triblock copolymeric microspheres containing protein. II. Polymer erosion and protein release mechanism. J Control Release 75:129–141

    Article  Google Scholar 

  97. Bear J (1969) Hydrodynamic dispersion. In: de Weist RJM (ed) Flow through porous media. Academic Press, New York, pp 109–199

    Google Scholar 

  98. Breitenbach A, Pistel KF, Kissel T (2000) Biodegradable comb polyesters. Part II. Erosion and release properties of poly(vinyl alcohol)-g-poly- (lactic-co-glycolic acid). Polymer 41:4781–4792

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Vascular Mechanics Lab, Biomedical Engineering Department, University of Michigan, Ann Arbor, MI, 48109, USA

    K. Khanafer

  2. Mechanical Engineering Department, University of California, Riverside, CA, 92505, USA

    K. Vafai

Authors
  1. K. Khanafer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Vafai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. Vafai.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Khanafer, K., Vafai, K. The role of porous media in biomedical engineering as related to magnetic resonance imaging and drug delivery. Heat Mass Transfer 42, 939–953 (2006). https://doi.org/10.1007/s00231-006-0142-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00231-006-0142-6

Keywords

Search

Navigation