Problem Set 3 solution BE.462J/3.962J Issued: Day 6 Spring 2003 Due: Day 8 (20 pts total) A recent study of controlled release of a model small-molecule drug from poly(lactide-co-glycolide) microspheres prepared by the single-emulsion method found that the diffusion constant of the drug through the polymer was best related to the polymer’s molecular weight according to: f D(t) = D 0 + M(t) In this equation, f and D 0 are constants, and M(t) is the molecular weight of the matrix polymer. From data obtained on PLGA microspheres, the constants were determined to be: f = 2.1x10 -11 cm 2 (kg/mole)/s D 0 = 4.9x10 -12 cm 2 /s We can use this expression for D(t) in the Charlier controlled release model to obtain modified expressions for h(t) and Q(t) (we’ll call this model B, and the expression derived in class model A). Assume that the molecular weight M(t) = M 0 e -kt , where M 0 is the initial molecular weight and k is the degradation rate constant for PLGA hydrolysis. A reasonable estimate for k is: Degradation rate constant for PLGA hydrolysis: k = 9.8E-03 hr -1 1. (5 pts) Quantitatively, will the diffusion constant in model B given above differ significantly from that obtained from model A derived in class over experimentally-relevant timescales? The difference in diffusion constants depends significantly on the initial value of the molecular weight, M 0 . For release from a high molecular weight matrix with M 0 = 80,000 g/mole, we have: BE.462 PS 3 1 of 4 BE.462 PS 3 2 of 4 0.00E+00 5.00E-12 1.00E-11 1.50E-11 2.00E-11 2.50E-11 3.00E-11 3.50E-11 4.00E-11 0 2 4 6 8 10 time (days) D (cm^2/s) Faisant MW-D relationship Charlier MW-D relationship, discrete integral …where the diffusion constant starts similar in both models and becomes greatly disparate after several days of hydrolysis. 0 = 5000 g/mole), the diffusion constant begins quite disparate and becomes similar in each model after several days: 0.00E+00 5.00E-12 1.00E-11 1.50E-11 2.00E-11 2.50E-11 3.00E-11 3.50E-11 4.00E-11 0 2 4 6 8 10 time (days) D (cm^2/s) Faisant MW-D relationship Charlier MW-D relationship, discrete integral This analysis indicates that for a high molecular weight matrix, the two models will at least initially predict similar release profiles, which will become different as time goes on (after only 24-48 hours). molecular weight matrix, the difference in release profiles will be apparent immediately. 2. (5 pts) Using the model B formula above for the diffusion constant, derive a new expression for the thickness of the diffusion field h(t) in the Charlier model. decay with time as derived in class. In contrast, if a low molecular weight matrix is used (e.g., the plot below is for M For a low Assume M(t) has an exponential BE.462 PS 3 3 of 4 0.00E+00 1.00E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00 7.00E+00 8.00E+00 9.00E+00 0 2 4 6 8 10 time (hr) h (μm) Faisant MW-D relationship Charlier MW-D relationship 3. measure Q(t) t) would need to be carried out to distinguish which of the two models for the diffusion constant (D = D 0 e kt as derived in class, or the expression given above) best represents release of HGH from a PLGA matrix in the framework of the Charlier model. (Hint: plot Q(t) for each of the two models; solve for Q(t) in model B by numerically integrating an expression dQ = (…)dt.) Solubility of HGH in PLGA matrix: C s = 6.12E-04 g/cm 3 Concentration of HGH encapsulated in the matrix: C 0 = 0.02 g/cm 3 Surface area of release matrix: A = 1.67 cm 2 Initial molecular weight of the matrix: M 0 = 78,000 g/mole (10 pts) Using the data above and that given below, determine how long release experiments that (total amount of drug released at time BE.462 PS 3 4 of 4 Using the derived expressions, we can compare release predicted by model A and model B: 0.00E+00 5.00E-03 1.00E-02 1.50E-02 2.00E-02 2.50E-02 3.00E-02 0 2 4 6 8 10 time (days) Q (mg drug released) Charlier MW-D relationship Faisant MW-D relationship Charlier MW-D relationship, discrete integral Thus, for the given parameters, release experiments carried out for at least 2 days would be necessary for the 2 models to deviate from one another significantly. should allow an unequivocal determination of which model better fits the experimental system. Release experiments carried out for 10 days