L8. Drug Dispersion and Diffusion in Biological Systems (2) April 26, 2018
2. Diffusion in Water (3) Diffusion of small molecule in water drug molecules/solvents are identical to solvent Assume molecules are aligned on a cubic lattice in the liquid - each molecule is in contact with six identical neighbors - center-to-center distance is 2a (a ~ molecule radius) a = 1 2 ( v ) 1/3 N Av D AA = k BT 4πμa = k BT 2πμ (N Av v: molar volume of the molecules; N Av : Avogadro s number, 6.02 x 10 23 v )1/3 e.g. H 2 O: v = 18 ml = 18x10-6 m 3 D AA = 2.07 x10-5 cm 2 /s AA: double subscript indicates that this is the self-diffusion coefficient of solvent molecules. drug molecules are similar to solvent but not identical, empirical equation: D A = 7.4 10 8 T μv ψ solvent M W,solvent ψ solvent : association constant ψ solvent = 2.6 2a
3. Diffusion in Polymer Solution and Gels Drug molecules often must diffuse through some complex fluid instead of water in order to reach their site of action Complex fluid: membranes; polymer solutions; and polymer gels
3. Diffusion in Polymer Solution and Gels (1) Diffusion through membranes Renkin equation: D A,membrane a: drug molecule radius r p : pore radius (average) Define: λ = a r p = radius of the drug radius of the pore D A, = (1 λ) 2 [1 2.1044λ + 2.089λ 3 0.948λ 5 ] D A,membrane : diffusion coefficient of drug A in the membrane D A, : diffusion coefficient of drug A in water
3. Diffusion in Polymer Solution and Gels (2) Diffusion through polymer solution Stretched exponential: D A,polymer D A, = exp( α c υ M γ I β a δ ) A few parameters will affect the D of the particle: Polymer concentration: c Polymer size (molecular weight): M w Ionic strength for polyelectrolyte: I Particle radius: a Experimentally: α: constant value ᶹ = 0.5~1 β = -0.8 (polyelectrolyte) or 0 (non-polyelectrolyte) δ = 0.3~0.5 (polyelectrolyte) or 0 (non-polyelectrolyte) γ = 0.8
3. Diffusion in Polymer Solution and Gels (3) Diffusion through polymer gel Exponential function D A,gel D A, = exp( a 3πλ f ln( L r f ) Effective medium approach: Polymer gel: continuous network within the liquid phase The network is maintained by chemical or physical interactions between polymer chains, including covalent cross-links, hydrogen bonds, physical entanglement of molecules The extracellular space of tissue is an aqueous gel of proteins and polysaccharides, which provide additional resistance to the diffusion of drugs and particles due to volume exclusion and hydrodynamic interactions. 1 a: particle/drug radius 2 ) r f : fiber diameter (µm) λ f : length density of the fibers (µm/µm 3 ) L: fiber length (µm) Steric effect: due to the volume excluded by the fibers in the gel, which is inaccessible to the diffusing particles. Hydrostatic effect: due to increased hydrodynamic drag on the diffusing particles caused by the presence of the fibers.
4. Diffusion in Extracellular Space D in tissue is significantly slower than for the same subject in water D A, tissue << D A, As the size of the diffusing subject increases, this difference becomes more pronounced.
D A,eff D A,pore = h(ζ) 4. Diffusion in Extracellular Space D A,eff : effective diffusion coefficient D A,pore : diffusion coefficient of A in unbound pore space fluid ζ: complete geometric description of the material/extracellular matrix For diffusion in a porous material: h ζ = 1 F τ F: shape factor of the pore (pore structure constricted, creating local sites with decreased permeability, F>1). τ: tortuosity of the pore (diffusional path length is increased because of windiness in the diffusional path, τ > 1). Empirically, Tortuosity = F τ (no way to separate these two factors) D A,eff = 1 D A,pore F τ From previous section: D A,pore D A, D A,eff = D A, gel D A, 1< τ <3 1< F τ <10 = f(solute size, volume fraction) D A, = actual tortuosity (additional tortuorsity) F τ f(ϕ, a)
4. Diffusion in Extracellular Space Diffusion with binding in tissue Drug/nanoparticle diffusing through a tissue often interact with elements of the tissue, non-specific and/or specific (ligand-receptor binding), which therefore influences the movement of the drug/nanoparticle through a tissue. e.g. IgG antibody in the interstitial space of tumors (in living animal) D 0 (IgG) = 3.9x10-7 cm 2 /s D tumor (IgG) = 1.3x10-7 cm 2 /s D tumor (IgG)/D 0 (IgG) 0.3 Specific binding: At low antibody concentration: D tumor(specific antibody) 0 At high antibody concentration: free antibody increase close to nonspecific antibody Bound fraction [IgG] nmol Specific IgG Nonspecific IgG
4. Diffusion in Extracellular Space Model for ligand binding to cell surface receptor Cell With receptors on surface Ligand, L 0 Ligand molecules must diffuse to cell surface before binding to a receptor
4. Diffusion in Extracellular Space Model for ligand binding to cell surface receptor L+R k on k off L-R Ligand Receptor Cell surface Cell surface Overall reaction, k f k r c L : ligand density c R : receptor density Rate of L-R formation = k on c L c R Rate of L-R dissociation = k off c L-R At equilibrium: k on c L c R = k off c L-R Cell surface K d = 1 K a = c Lc R c L R = k off k on k on : M -1 min -1 k off : min -1
On steady state: 4. Diffusion in Extracellular Space [Rate of diffusion at cell surface] + [Rate of ligand disappearance due to binding] = 0 4πR 2 j x Fick s first law 4πR 2 D dc L dr Fick s second law c x t = D 2 c x x 2 D 1 d r 2 dr x=r + k on c L c R = 0 r2 dc L dr r=r + k on c L c R = 0 4πR 2 D dc L dr (Rectangular coordinate) r=r = k on c L c R = 0 (Spherical coordinate) (c L = c L,0 @ r = ) Solution: C L r = k onc R Rc L,0 1 4πDR + k on c R r + c L0 (j x : net rate of particle movement per unit area) R r
4. Diffusion in Extracellular Space [Overall rate of disappearance at cell surface] = 4πR 2 D dc L dr r=r = [Overall rate of binding at cell surface] = k f c L,0 k f = [Rate] c L,0 k onc R Rc L,0 4πR2 D 4πDR + k on c R R 2 = (4πDR)k onc R c L,0 4πDR + k on c R c L0 is constant k f depends on k +, k on, c R = (4πDR)c Rk on 4πDR + c R k on Rate constant for diffusion-limited reaction: k f = k +c R k on k + + c R k on If k + >> k on, k f,max = c R k on k f k f,max = k + k + + c R k on k + = 4πDR K f,max,: Maximal rate of the overall reaction
5. Diffusion within Cells If a drug acts within a cell, it must move from the point of entry to the site of action. Diffusion is an essential mode of molecular transport within individual cells. Intracellular diffusion has not been extensively studied. Small molecule Rapidly spread in the cell Protein Much slower diffusion (binding, reaction)
5. Diffusion within Cells Diffusion in cytoplasm depends on molecular size The functional dependence of diffusion coefficient on molecular size is similar to that observed for diffusion of proteins in condensed polymer solutions or gels. e.g. serum albumin protein: D cyto /D water 1/5 Characteristic times for diffusion in cells 10-9 cm 2 /s 10-7 cm 2 /s 10-5 cm 2 /s 1 µm 10 s 0.1 s 0.001 s 10 µm 1000 s 10 s 0.1 s 100 µm 100000 s 1000 s 10 s Diffusion is an efficient method for distributing molecules throughout a small cell (1 µm). Large cells (100 µm) can not rely on diffusion to distribute substrates or newly produced proteins.
Drug-loaded nanoparticles: Entry: endocytosis Drug release: 5. Diffusion within Cells Small molecules: diffuse out of endosome Large hydrophilic molecules (protein, gene): - cannot actively diffuse out of endosome - less useful Endosome escape represents one of the most challenging tasks for nanoparticle drug delivery