Advertisement

Heat and Mass Transfer

, 42:939 | Cite as

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

  • K. Khanafer
  • K. VafaiEmail author
Special Issue

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.

Keywords

Porous Medium Drug Release Apparent Diffusion Coefficient Diffusion Equation External Fluid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols

a

Empirical constant

aE

Einestein radius

ADC

Apparent diffusion coefficient

b

Empirical constant

BSat

Saturation concentration of the drug in the polymer phase

Bs

Undissolved drug concentration in the polymer

C

Volume average of concentration

CL

Drug concentration in the liquid phase

Co

Initial drug concentration

Csat

Saturation concentration of the drug

CSe

Drug concentration in the effective solid phase

Cs

Undissolved drug concentration in the pores

dp

Pore diameter

D*

Effective diffusion coefficient

DB

Polymer diffusion coefficient

ECS

Extracellular space

fn

Viscosity function

F

Geometric function

F1, F2

Correction factors

F(C)

Uptake term

hm

Mass transfer coefficient

k

Permeability

kdis

Dissolution rate constant

kero

Surface erosion constant

KB

Forward rate constant

KC

Backward rate constant

KDB

Dissolution rate constant in polymer

KDC

Dissolution rate constant in pore

KHero

Hyperbolic erosion rate constant for bulk erosion

KLero

Linear erosion rate constant for bulk erosion

Km

Michele-menten constant

KSero

‘S’ erosion rate constant for bulk erosion

M

Cumulative amount of drug released at time infinity

Mt

Cumulative amount of drug released at time t

MRI

Magnetic resonance imaging

P

Fluid pressure

ro

Pore radius

Rs

Radius of microparticles

s

Mass source density

Sh

Sherwood number

t

Time

v

Velocity vector

V

Representative elementary volume

V1

Effective volume of the microsphere

Vmax

Rate constant

Vp

Pore volume

Greek symbols

ρf

Fluid density

ɛ

Porosity

λg

Geometrical tortuosity

λx, λ y, λ z

Tortuosity components

μf

Dynamic viscosity of the pure fluid

σ

Surface area

References

  1. 1.
    Vafai K, Tien CL (1981) Boundary and inertia effects on flow and heat transfer in porous media. Int J Heat Mass Transfer 24:195–203zbMATHCrossRefGoogle Scholar
  2. 2.
    Vafai K, Tien CL (1982) Boundary and inertia effects on convective mass transfer in porous media. Int J Heat Mass Transfer 25:1183–1190CrossRefGoogle Scholar
  3. 3.
    Nield DA, Bejan A (1995) Convection in porous media, 2nd edn. Springer, Berlin Heidelberg New YorkGoogle Scholar
  4. 4.
    Vafai K (2000) Handbook of porous media, 1st edn. Marcel Dekker, Inc., New YorkzbMATHGoogle Scholar
  5. 5.
    Vafai K (2005) Handbook of porous media, 2nd edn. Taylor and Francis Group, New YorkGoogle Scholar
  6. 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, NYGoogle Scholar
  7. 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, NYGoogle Scholar
  8. 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)Google Scholar
  9. 9.
    Ai L, Vafai K (2006) A coupling model for macromolecule transport in a stenosed arterial wall. Int J Heat Mass Transfer (in press)Google Scholar
  10. 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 94:103–115CrossRefGoogle Scholar
  11. 11.
    Yang M, Zhang X, Vafai K, Ozkan C (2003) High sensitivity piezoresistive cantilever design and optimization for analyte-receptor binding. J Micromech Microeng 13:864–872CrossRefGoogle Scholar
  12. 12.
    Khanafer K, Khaled ARA, Vafai K (2004) Spatial optimization of an array of aligned microcantilever biosensors. J Micromech Microeng 14:1328–1336CrossRefGoogle Scholar
  13. 13.
    Khaled ARA, Vafai K (2004) Optimization modeling of analyte adhesion over an inclined microcantilever-based biosensor. J Micromech Microeng 14:1220–1229CrossRefGoogle Scholar
  14. 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 3:721–729CrossRefGoogle Scholar
  15. 15.
    Kuffler SW, Potter DD (1964), Glia in the leech central nervous system: physiological properties and neuron-glia relationship. J Neurophysiol 27:290–320Google Scholar
  16. 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 80:542–548Google Scholar
  17. 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 279–294Google Scholar
  18. 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 457–476Google Scholar
  19. 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 217–336Google Scholar
  20. 20.
    Barrie PJ (1995) NMR applications to porous solids. Ann Rep NMR Spectrosc 30:37–95CrossRefGoogle Scholar
  21. 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 19:28–37Google Scholar
  22. 22.
    Stejskal EO, Tanner JE (1965) Use of spin-echo pulsed magnetic field gradient to study anisotropic restricted diffusion and flow. J Chem Phys 43:3579–3603CrossRefGoogle Scholar
  23. 23.
    Taylor DG, Bushell MC (1985) The spatial mapping of translational diffusion by the NMR imaging technique. Phys Med Biol 30:345–349CrossRefGoogle Scholar
  24. 24.
    Gelderen PV, DeVleeschouwer MH, DesPres D, Pekar J, VanZijl PCM, Moonen CT (1994) Water diffusion and acute stroke. Magn Reson Med 31:154–163CrossRefGoogle Scholar
  25. 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, 579Google Scholar
  26. 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 91:1229–1233CrossRefGoogle Scholar
  27. 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 16:330–346CrossRefGoogle Scholar
  28. 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 18:39–50CrossRefGoogle Scholar
  29. 29.
    Benveniste H, Hedlund LW, Johnson GA (1992) Mechanism of detection of acute cerebral ischemia in rats by diffusion weighted magnetic resonance microscopy. Stroke 23:746–754Google Scholar
  30. 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, 1201Google Scholar
  31. 31.
    Nicholson C (2001) Diffusion and related transport mechanisms in brain tissue. Rep Prog Phys 64:815–884CrossRefGoogle Scholar
  32. 32.
    Nicholson C, Phillips JM (1981) Ion diffusion modified by tortuosity and volume fraction in the extracellular microenvironment of rat cerebellum. J Physiol 321:225–257Google Scholar
  33. 33.
    Mota M, Teixeira JA, Keating JB, Yelshin A (2004) Changes in diffusion through the brain extracellular space. Biotechnol Appl Biochem 39:223–232CrossRefGoogle Scholar
  34. 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 59:2247–2273zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Szafer A, Zhong JH, Gore JC (1995) Theoretical model for water diffusion in tissues. Magn Reson Med 33:697–712CrossRefGoogle Scholar
  36. 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 37:939–954CrossRefGoogle Scholar
  37. 37.
    Carslaw HS, Jaeger JC (1959) Conduction of heat in solids, 2nd edn. Clarendon, OxfordGoogle Scholar
  38. 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 58:508–803CrossRefGoogle Scholar
  39. 39.
    Horn AS (1979) Characteristics of dopamine uptake. In: Horn AS et al (eds) The neurobiology of dopamine. Academic, London, pp 217–35Google Scholar
  40. 40.
    Lehner FK (1979) On the validity of Fick’s law for transient diffusion through a porous medium. Chem Eng Sci 34:821–825CrossRefGoogle Scholar
  41. 41.
    Mota M, Teixeira JA, Yelshin A (2001) Biotechnol Appl Biochem 17:860–865Google Scholar
  42. 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 64:1638–1646CrossRefGoogle Scholar
  43. 43.
    Limbach KW, Wei J (1990) Restricted diffusion through granular materials. AIChE J 36:242–248CrossRefGoogle Scholar
  44. 44.
    Blanch HW, Clark DS (1996) Biochemical engineering. Marcel Dekker, New YorkGoogle Scholar
  45. 45.
    Deen WM (1987) Hindered transport of large molecules in liquid-filled pores. AIChE J 33:1409–1425CrossRefGoogle Scholar
  46. 46.
    Netrabukkana R, Lourvanij K, Rorrer GL (1996) The Diffusion of glucose and glucitol in microporous and mesoporous silicate catalysts. Ind Eng Chem Res 35:458–464CrossRefGoogle Scholar
  47. 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 16:1023–1032CrossRefGoogle Scholar
  48. 48.
    Archie GE (1942) The electrical resistivity log as an aid in determining some reservoir characteristics. Trans Am Inst Min Metall Petrol Eng Inc 146:54–62Google Scholar
  49. 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 5:467–488CrossRefGoogle Scholar
  50. 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 63:567–576CrossRefGoogle Scholar
  51. 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) 368:213–226Google Scholar
  52. 52.
    Rosenberg GA, Kyner WT, Estrada E (1980) Bulk flow of brain interstitial fluid under normal and hyperosmolar conditions. Am J Physiol 238:F42–F49Google Scholar
  53. 53.
    Rosenberg GA, Kyner WT (1980) Gray and white matter brain-blood transfer constants by steady-state tissue clearance in cat. Brain Res 193:59–66CrossRefGoogle Scholar
  54. 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 46:4989–5003zbMATHCrossRefGoogle Scholar
  55. 55.
    Khanafer K, Vafai K, Kangarlu A (2003) Computational modeling of cerebral diffusion-application to stroke imaging. Magn Reson Imaging 21:651–661CrossRefGoogle Scholar
  56. 56.
    Khanafer K, Vafai K, Kangarlu K (2003) Water diffusion in biomedical systems as related to magnetic resonance imaging. Magn Reson Imaging 21:17–31CrossRefGoogle Scholar
  57. 57.
    Vafai K (1984) Convective flow and heat transfer in variable-porosity media. J Fluid Mech 147:233–259zbMATHCrossRefGoogle Scholar
  58. 58.
    Vafai K (1986) Analysis of the channeling effect in variable porosity media. ASME J Energy Resour Technol 108:131–139CrossRefGoogle Scholar
  59. 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 27:651–664CrossRefGoogle Scholar
  60. 60.
    Chien YW (1992) Novel drug delivery systems, 2nd edn. Marcel Dekker, New YorkGoogle Scholar
  61. 61.
    Fan LT, Singh SK (1989) Controlled release: a quantitative treatment. Springer, Berlin Heidelberg New YorkGoogle Scholar
  62. 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 7:297–325CrossRefGoogle Scholar
  63. 63.
    Hanes J, Chiba M, Langer R (1998) Degredation of porous poly (anhydride-co-imide) microspheres and implications for controlled macromolecule delivery. Biomaterials 19:163–172CrossRefGoogle Scholar
  64. 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 58:4087–4114CrossRefGoogle Scholar
  65. 65.
    Crank J (1975) The mathematics of diffusion, 2nd edn. Clarendon Press, Oxford UKGoogle Scholar
  66. 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 7:207–215CrossRefGoogle Scholar
  67. 67.
    Hopfenberg HB (1976) Controlled release from erodible slabs, cylinders, and spheres. In: Controlled release polymeric formulations. ACS Symposium Series 33:26Google Scholar
  68. 68.
    Higuchi T (1961) Rate of release of medicaments from ointment bases containing drugs in suspension. J Pharm Sci 50:874–875CrossRefGoogle Scholar
  69. 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 52:1145–1149CrossRefGoogle Scholar
  70. 70.
    Peppas NA (1985) Analysis of Fickian and non-Fickian drug release from polymers. Acta Helv 60:110–111Google Scholar
  71. 71.
    Kim J, Finn NC (1996) Shape modeling of dissolution profiles by non-integer kinetic orders. Int J Pharm 143:223–232CrossRefGoogle Scholar
  72. 72.
    Sezer AD, Akbuga J (1995) Controlled release of piroxicam from chitosan beads. Int J Pharm 121:113–116CrossRefGoogle Scholar
  73. 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):988–995CrossRefGoogle Scholar
  74. 74.
    Weibull W (1951) A statistical distribution of wide applicability. J Appl Mech 18:293–297zbMATHGoogle Scholar
  75. 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 60:379–389CrossRefGoogle Scholar
  76. 76.
    Flynn GL, Yalkowsky SH, Roseman TJ (1974) Mass transport phenomena and models: theoretical concepts. J Pharm Sci 63:479–510CrossRefGoogle Scholar
  77. 77.
    Charlier A, Leclerc B, Couarraze G (2000) Release of mifepristone from biodegradable matrices: experimental and theoretical evaluations. Int J Pharm 200:115–120CrossRefGoogle Scholar
  78. 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 64:179–192CrossRefGoogle Scholar
  79. 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 1–18Google Scholar
  80. 80.
    Lee PI (1980) Diffusional release of a solute from a polymeric matrix-approximate analytical solutions. J Membr Sci 7:255–275CrossRefGoogle Scholar
  81. 81.
    Joshi A, Himmelstein KJ (1991) Dynamics of controlled release from bioerodible matrices. J Control Release 15:95–104CrossRefGoogle Scholar
  82. 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 86:1464–1477CrossRefGoogle Scholar
  83. 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 258:95–107CrossRefGoogle Scholar
  84. 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 15:135–163zbMATHCrossRefGoogle Scholar
  85. 85.
    Cohen DS, Erneux T (1988) Free boundary problems in controlled release pharmaceuticals. II: Swelling controlled release. SIAM J Appl Math 48:1466–1474zbMATHCrossRefMathSciNetGoogle Scholar
  86. 86.
    Fujita J (1961) Diffusion in polymer-diluent systems. Fortschr Hochpolym Fortschr 3:1–47CrossRefGoogle Scholar
  87. 87.
    Siepmann J, Peppas N (2001) Modeling of drug release from delivery systems based on hydroxypropyl methylcellulose (HPMC). Add Drug Deliv Rev 48:139–157CrossRefGoogle Scholar
  88. 88.
    Siepmann J, Gopferich A (2001) Mathematical modeling of bioerodible, polymeric drug delivery systems. Adv Drug Deliv Rev 48:229–247CrossRefGoogle Scholar
  89. 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 19:1885–1893CrossRefGoogle Scholar
  90. 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):2040–2056CrossRefGoogle Scholar
  91. 91.
    James MA, Matthew SS (1997) Biodegradation and biocompatibility of PLA and PLGA microspheres. Adv Drug Deliv Rev 28:5–24CrossRefGoogle Scholar
  92. 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 41:4781–4792CrossRefGoogle Scholar
  93. 93.
    Wong HM, Wang JJ, Wang CH (2001) In vitro release of human immunoglobulin G from biodegradable microspheres. Ind Eng Chem Res 40:933–948CrossRefGoogle Scholar
  94. 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 60:297–309CrossRefGoogle Scholar
  95. 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 75:11–25CrossRefGoogle Scholar
  96. 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 75:129–141CrossRefGoogle Scholar
  97. 97.
    Bear J (1969) Hydrodynamic dispersion. In: de Weist RJM (ed) Flow through porous media. Academic Press, New York, pp 109–199Google Scholar
  98. 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 41:4781–4792CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Vascular Mechanics Lab, Biomedical Engineering DepartmentUniversity of MichiganAnn ArborUSA
  2. 2.Mechanical Engineering DepartmentUniversity of CaliforniaRiversideUSA

Personalised recommendations