麻省理工大学：《生物材料的分子结构》教学讲义（英文版）Problem set 3 solution

Problem set 3 solution BE462J3.962J Issued ay 6 Spring 2003 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 polymers molecular weight according to D(t)=D M(t In this equation, and Do 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 p=2. 1x10 cm(kg/mole)/s D0=49×1012cm2/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)=Moe", where Mo 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 (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, Mo For release from a high molecular weight matrix with Mo=80, 000 g/mole, we have BE.462 PS 3

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) = D0 + M(t) In this equation, f and D0 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 cm2 (kg/mole)/s D0 = 4.9x10-12 cm2 /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) = M0e-kt, where M0 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, M0. For release from a high molecular weight matrix with M0 = 80,000 g/mole, we have: BE.462 PS 3 1 of 4

3.00E-11 霸2.50E-11 Faisant MW-D 50E-11 elationship, discrete 1.00E-11 5.00E-12 0.00E+00 here the diffusion constant starts similar in both models and becomes greatly disparate after several days of hydrolysis. In contrast, if a low molecular weight matrix is used(e.g, the plot below is for Mo 5000 g/mole), the diffusion constant begins quite disparate and becomes similar in each model after several days 4.00E-11 3.50E-11 2.50E-11 一一- Faisant Mv-D 2.00E-11 1.50E-11 relationship, discrete 1.00E-11 5.00E-12 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). For a low 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. Assume M(t) has an exponential decay with time as derived in class BE.462 PS 3 2 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

中 (f) FROM THE CHARLIER MMODEL 中1A t kt t Cmo kt Dot p 9.00E+00 8.00E+00 7.00E+00 5.00E+00 E4.00E+00 Charlier MW-D relationship 3.0oE+0 0.00E+00 3.(10 pts)Using the data above and that given below, determine how long release experiments that measure Q(t)(total amount of drug released at time t)would need to be carried out to distinguish which of the two models for the diffusion constant (D=Doe 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 Concentration of HGH encapsulated in the matrix: Co=0.02 g/cm Surface area of release matrix Initial molecular weight of the matrix Mo=78,000 g/mole BE.462 PS 3 3 of 4

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 = D0ekt 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: Cs = 6.12E-04 g/cm3 Concentration of HGH encapsulated in the matrix: C0 = 0.02 g/cm3 Surface area of release matrix: A = 1.67 cm2 Initial molecular weight of the matrix: M0 = 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

AGAIN USING THE CHARLIER WOD:L ⊥d:DC RIi. Gi A Φ ≌(Dt;中( P。k sivcs G)is Nur STRNIGN FoRwArD To IATLGRATE, WL CAN OBTAIN A REASOMAe UoMctICAL ESTIMAT. Fok QLt) USING ()INSTLAD dQ A Du)Cc (v) AQ: Ab(t)Cs At, WRE AP: IS THE AMOUNT OF DRUG RCLEASD IN A SMALL TiME INTERVAL Q\s OBTAINED 8f SUMMING Au Tit 4<: FRoM TiME 1t) SINCE WE HAv. ExFuCIT EXPR<SSONS FoR D()AND hct),(iv)AND (v) A、PM有所AD/的An/CAe NUMRICAL CE7Au(t)W的LB Using the derived expressions, we can compare release predicted by model a and mode/ B 3.00E-02 2.50E-02 Charlier MW-D 2.00E-02 relationsh Charlier MW-D 1.00E-02 relationship, discrete 5.00E-03 0.00E+00 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. Release experiments carried out for 10 days should allow an unequivocal determination of which model better fits the experimental system BE.462 PS 3

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