BEE 2600 Fall 2016 Case Study #1

Diffusion and Pharmacokinetics of Anti-VEGF Drugs Written by Sachiye Koide

Due Date: 09/27/2016 Macular degeneration is the leading cause of age-related vision loss and affects more than 10 million Americans today. Considered an incurable disease, macular degeneration is caused by deterioration in the macula, the central portion of the retina, and is the consequence of overexpression of vascular endothelial growth factor (VEGF). VEGF is a signal protein that stimulates vasculogenesis and angiogenesis, and helps restore oxygen supply to tissues as part of the cardiovascular system when circulation is inadequate. When functioning normally, VEGF creates new blood vessels, such as during embryonic development, after injury, and when bypassing blocked vessels. However, overexpression of VEGF can have harmful consequences such as macular degeneration and cancer metastasis.

Anti-VEGF drugs currently being used to treat macular degeneration include Bevacizumab and Ranibizumab, which can be administered straight to the eyes. However, direct injection of the anti-VEGF drugs into the eyes may cause an uneven drug concentration profile and require frequent eye injections for long-term treatment. Recently, coatings of hydrophilic gels (commonly referred to as hydrogels) have been used as drug delivery vehicles and have the advantage of time-controlled drug release [4]. Hydrogels are extensive polymer networks whose hydrophilic structure of chemical and physical crosslinks (entanglements, crystallites, and hydrogen-bonded structures) allows them to absorb copious amounts of biological fluid ten to twenty times their molecular weight without dissolving [1]. Hydrogels have proved to be extremely useful in biomedical and pharmaceutical applications due to their high water content, similarity to natural tissue, and biocompatibility [10]. Drugs can be concentrated within the polymer and released through a diffusion mechanism that allows for reduced dosing frequency. The physical properties of the hydrogel, drug-polymer interactions, drug concentration and drug solubility determine the diffusion kinetics, duration, and rate of drug release from the hydrogel [15]. There are several types of controlled-delivery hydrogel systems, including: diffusion-controlled, swelling-controlled, chemically-controlled, and environmentally-controlled [15]. In part 1 of this case study, we will focus on diffusion-based drug delivery hydrogels in the distribution of Bevacizumab and Ranibizumab to patients with macular degeneration. Part 1:

In treating macular degeneration, the hydrogel containing the drugs is injected into the vitreous humor of the eye where the anti-VEGF drugs diffuse into the surrounding eye tissue. The hydrogel has a spherical shape with the drugs concentrated in the center of the gel (Fig. 2). The outer radius of the hydrogel sphere is 0.72 mm. The drug concentration at the center is 12.351 !!

!". After a certain time, the hydrogel reaches a steady state where the concentration of the

drug at the interface of the hydrogel and eye tissue is zero (the drug is immediately absorbed by the tissue), while the concentration of the drug in the center of the gel continues to be constant. In

Fig.  1:Physiology  of  macular  degeneration  in  the  eye.  

this study, we will model the drug diffusion through the hydrogel by only considering a small section (the rectangle section illustrated in Figure 2). We will assume there is no curve in this small section, and model it as a slab of drug and hydrogel. Assume that the hydrogel is in direct contact with the tissue throughout this section.

Bevacizumab has a diffusion coefficient of 4.1×10−7  cm2/s  and  a  degradation  rate  of  7.943   x   10-­‐7   µg/mm3/s.     Ranibizumab   has   a   diffusion   coefficient   of   6.7×10−7   cm2/s   and   a  degradation  rate  of  α  +  βz  where  α  =  3.972  x  10-­‐4  µg/mm3/s  and  β  =  -­‐5.516  x  10-­‐4  µg/mm4/s. For this exercise, assume the anti-VEGF drug diffuses directly from the center of the hydrogel into the eye tissue in one dimension only, and the porosity of the hydrogel is 1. We are only concerned with diffusion from the center of the hydrogel into the eye tissue. Ignore any diffusion into the vitreous liquid.

Figure 2: Structure of the hydrogel containing the drug in the center.

1. Perform a literature search on the two drugs to determine the underlying mechanism of

using anti-VEGF drugs in treating macular degeneration (this will be included in your introduction)

2. Draw a schematic diagram modeling the diffusion of Bevacizumab and Ranibizumab through the hydrogel. Be sure to include all boundary conditions and list all variables (including definition and units for each variable) and assumptions.

3. Separately derive the steady-state concentration for Bevacizumab and Ranibizumab through the hydrogel using Fick’s Second Law of Diffusion in terms of the variables only.(You need to solve for k1 and k2 constants)

4. Graph both drug concentration profiles with respect to the depth of the hydrogel on the same plot.

5. Find the expressions for the flux of Bevacizumab and Ranibizumab at the hydrogel-eye tissue interface, first in terms of variables, and then substitute the values and units into the expressions. Leave final answers in units of [µg/mm2/s].

6. How long does it take for a small molecule of each drug to diffuse from the hydrogel into the eye tissue [hours]?

7. Which drug is more suitable for the time-controlled released delivery for macular degeneration from a transport point of view (considering the flux and time of diffusion)? Remember that your aim is to minimize the need for frequent injections by choosing a method that releases drug slowly over time.

Part II:

Besides being a treatment for macular degeneration, Bevacizumab is the first anti-angiogenic antibody approved by the FDA for metastatic cancers [4]. However, the doses are much higher than the amount given to treat macular degeneration, at around 100 mg per dose via intravenous injection [8]. The higher dosage of Bevacizumab in healthy organs would result in a number of side effects. In Part II of the assignment, we will evaluate the toxicity of Bevacizumab and its pharmacokinetics.

Typically, cancer patients receive an injection of Bevacizumab every three weeks, or when Bevacizumab levels in the blood go down to lower than 0.008 mg/mL. For an adult patient, assume the volume of blood in the entire body is 5.5L. The elimination half-life of bevacizumab is 20 days.

In the treatment of metastatic cancers with Bevacizumab, consider an IV to be the reservoir for the drug and can hold 1000 mL of fluid. The elution rate of the fluid into the blood is 7.72 mL/min [3]. Bevacizumab subsequently moves into the tissue at a rate of Kt=0.082 ℎ𝑟!! and the kidney at a rate of Kk= 0.041 ℎ𝑟!!. From the kidney, the drug is excreted at a rate of Ke= 0.056 ℎ𝑟!!. From the tissue, some of the drug moves back into the bloodstream at a rate Kb= 0.00025 mg/hr and some gets metabolized at a rate Km= 0.02 ℎ𝑟!!. Assume that all the Bevacizumab is eventually either metabolized by the tissue or excreted from the body by the kidney.

1. Draw a pharmacokinetic model for the process of transport and excretion of

Bevacizumab in the body using Word or PowerPoint. Include the IV reservoir, blood, tissue, kidney, and all the appropriate rate constants. List all variables and assumptions.

2. Derive the rate equations for the mass of the Bevacizumab in the IV reservoir, blood, tissue, and kidney at time t. Start with a word equation and mass balance for each. Write all expressions in terms of variables and do not solve the equations.

3. Use the rate equation for blood to find an expression in terms of variables for the mass of Bevacizumab in the blood with respect to time.

4. If the patient is given a 100 mg dose through the IV, calculate the time it takes to deliver all of the contents of the IV bag containing the Bevacizumab. What is the concentration of drug in the blood once the entire contents of the IV has been delivered (considering absorption and elimination processes of the drug)?

5. How long until the patient can receive another dose? Hint: use the elimination half-life of Bevacizumab for your calculation.

References:

1. Ahmed, EM. 2015. Hydrogel: preparation, characterization, and applications: a review. Journal of Advanced Research. 6(2):105-121. doi:10.1016/j.jare.2013.07.006

2. Bhattarai N, Gunn J, Zhang M. 2010. Chitosan-based hydrogels for controlled, localized drug delivery. Advanced Drug Delivery Reviews. 62(1): 83-99. doi:10.1016/j.addr.2009.07.019

3. Fournier RL. 1998. Basic transport phenomena in biomedical engineering. CRC Press. 4. Li SK, Liddell MR, Wen H. 2011. Effective electrophoretic mobilities and charges of

anti-VEGF proteins determined by capillary zone electrophoresis, J of Pharmaceutical and Biomedical Analysis. 55(3):603-607. doi:10.1016/j.jpba.2010.12.027

5. Lin C, Metters AT. 2006. Hydrogels in controlled release formulations: network design and mathematical modeling. Advanced Drug Delivery Reviews. 58: 1379-1408. doi:10.1016/j.addr.2006.09.004

6. Marcarelli R. 2015. Injectable ‘self-healing’ hydrogel could target cancer cells, treat macular degeneration. HNGN [Internet]. [cited 2016 March]. Available from: http://www.hngn.com/articles/71174/20150220/

7. Medscape: drugs and diseases [Internet]. c1994-2016. WebMD LLC: [cited 2016 March]. Available from: http://reference.medscape.com/drug/avastin-bevacizumab-342257 [H]

8. Michels S. 2006. Is intravitreal bevacizumab (Avastin) safe? BMJ. 90(11):1333-1334 9. Peppas NA, Colombo P. 1997. Analysis of drug release behavior from swellable polymer

carriers using the dimensionality index. Journal of Controlled Release. 45(1): 35-40. doi:10.1016/S0168-3659(96)01542-8

10. Peppas, NA. 1997. Hydrogels and drug delivery. Current Opinion in Colloid & Interface Science. 2(5):531-537. doi:10.1016/j.addr.2009.07.019

11. Pike DB, Cai S, Pomraning KR, Firpo MA, Fisher RJ, Shu XZ, Prestwich GD, Peattie RA. 2006. Heparin-regulated release of growth factors in vitro and angiogenic response in vivo to implanted hyaluronan hydrogels containing VEGF and bFGF. Biomaterials. 27(30):5242-5251

12. Porter TL, Stewart R, Reed J, Morton K. 2007. Models of hydrogel swelling with applications to hydration sensing. PMC. 7(9):1980-1991

13. RxList: the internet drug index [Internet]. c2016. RxList Inc.: [cited 2016 March]. Available from: http://www.rxlist.com/lucentis-drug.htm

14. Salter JT, Miller KD. 2006. Targeting VEGF for breast cancer: safety and toxicity data with bevacizumab. Medscape CME and Education [Internet]. C1994-2016. WebMD LLC:[cited 2016 March].

15. Wei C, Kim C, Kim H, Limsakul P. 2012. Hydrogel drug delivery: diffusion models. [cited 2016 March].

SOLUTIONS  1.)  Literature  Search    2.)  Variables    

 Given:  L  =  0.72  mm  Concentration  of  Bevacizumab/Ranibizumab  inside  the  hydrogel  =  12.351 !!

!"  

ε = porosity = 1    

Variables:  CB(z)= concentration of Bevacizumab at a given depth of z  CB,hydrogel = CB(0) = concentration of Bevacizumab at center of hydrogel (z=0)     = 12.351 µg/uL  

CB,vitreous  =  CB(L) = concentration of Bevacizumab at vitreous-hydrogel boundary  DS,B= Bevacizumab diffusion coefficient = 4.1×10−7 cm2/s = 4.1×10−5 mm2/s  RB = Bevacizumab degradation rate = 7.943 x 10-7 µg/mm3/s    

CR(z)= concentration of Ranibizumab at a given depth of z  CR,hydrogel = CR(0) = concentration of Ranibizumab at center of hydrogel (z=0)     = 12.351 ug/uL  

CR,tissue  =  CR(L) = concentration of Ranibizumab at vitreous-hydrogel boundary  DS,R = Ranibizumab diffusion coefficient = 6.7×10−7 cm2/s = 6.7 x 10-5 mm2/s  RR = Ranibizumab degradation rate = α + βz  

α = 3.972 x 10-4 µg/mm3/s

β = -5.516 x 10-4 µg/mm4/s  

Assumptions: • Drug diffuses directly from the center of the hydrogel into the eye tissue in one direction. • Only consider drug diffusion in one small section and model it as a slab of the hydrogel with uniform thickness. • Porosity of the hydrogel is 1 • Concentration of Bevacizumab /  Ranibizumab in the hydrogel is the same as at z = 0 (the center of the gel), and is assumed to be constant throughout the diffusion process

 

Eye  Tissue  

z  =  0,  𝐶!,!!"#$%&' = 12.351 !!!"  

𝐶!,!!"#$%&' = 12.351µμ𝑔𝑢𝐿  

 z  =  L,  CB,  tissue  =    C(L)  =  0  !!!"

 

 

Hydrogel  

Bevacizumab/  ranibizumab  diffusion  

L  

• The eye tissue absorbs the drug instantly, so that the concentration at the interface of hydrogel and eye tissue is zero. • Bevacizumab is degraded at a rate of R in the hydrogel wall and Ranibizumab  is  degraded  at  a  rate  R  =  α  +  βz  that  decreases  with  increasing  depth

• The  mass  of  Bevacizumab/  Ranibizumab  is  conserved   • Diffusion  rate,  porosity,  and  thickness  are  constant  throughout  the  hydrogel. • The  concentrations  of  Bevacizumab/  Ranibizumab  are  homogeneous  throughout  the  center  of  the  hydrogel  at  time  =  0  

• System  is  in  steady  state

3.) Derivation of Concentration Bevacizumab

𝜀𝛿𝐶!𝛿𝑡 =  𝐷!,!

𝛿!𝐶!𝛿𝑧! − 𝑅!  

 𝜀 !!!

!"=  0   𝑠𝑡𝑒𝑎𝑑𝑦  𝑠𝑡𝑎𝑡𝑒  𝑎𝑛𝑑  𝜀 = 1  

 

𝐷!,!𝛿!𝐶!𝛿𝑧! =  𝑅!  

 𝛿!𝐶!𝛿𝑧! 𝑑𝑧 =  

𝑅!𝐷!,!

𝑑𝑧  

 𝛿𝐶!𝛿𝑧 𝑑𝑧 =   (

𝑅!𝐷!,!

𝑧 + 𝑘!)𝑑𝑧  

   

𝐶! 𝑧 =  𝑅!2𝐷!,!

𝑧! + 𝑘!𝑧 + 𝑘!  

 At  z  =  0,  CB  (0)  =  CB,hydrogel  

𝑅!2𝐷!,!

(0)! + 𝑘!(0)+ 𝑘! =  𝐶!,!!"#$%&'  

 𝑘! =  𝐶!,!!"#$%&'  

 At  z  =  L,  CB(L)  =  CB,tissue  

𝑅!2𝐷!,!

(𝐿)! + 𝑘!𝐿 +  𝐶!,!!"#$%&' =  0  

 

𝑘! =  −𝐶!,!!"#$%&' −  

𝑅!2𝐷!,!

(𝐿)!

𝐿    

𝑘! =  −𝐶!,!!"#$%&'

𝐿 −  𝑅!2𝐷!,!

𝐿  

   

 𝑪𝑩 𝒛 =𝑹𝑩𝟐𝑫𝒔,𝑩

𝒛𝟐 −𝑪𝑩,𝒉𝒚𝒅𝒓𝒐𝒈𝒆𝒍

𝑳 +  𝑹𝑩𝟐𝑫𝒔,𝑩

𝑳 𝒛+ 𝑪𝑩,𝒉𝒚𝒅𝒓𝒐𝒈𝒆𝒍    

 

Derivation  for  Ranibizumab:    

𝜀𝛿𝐶!𝛿𝑡 =  𝐷!,!

𝛿!𝐶!𝛿𝑧! − 𝑅!  

 𝛿𝐶!𝛿𝑡 =  0   𝑠𝑡𝑒𝑎𝑑𝑦  𝑠𝑡𝑎𝑡𝑒  𝑎𝑛𝑑  𝜀 = 1  

   

𝐷!,!𝛿!𝐶!𝛿𝑧! =  𝛼 +  𝛽𝑧  

 𝛿!𝐶!𝛿𝑧! 𝑑𝑧 =  

𝛼 +  𝛽𝑧𝐷!,!

𝑑𝑧  

 𝛿𝐶!𝛿𝑧 𝑑𝑧 =  

𝛼𝑧𝐷!,!

+𝛽𝑧!

2𝐷!,!+  𝑘!   𝑑𝑧  

   

𝐶! 𝑧 =  𝛼𝑧!

2𝐷!,!+  

𝛽𝑧!

6𝐷!,!+  𝑘!𝑧 +  𝑘!  

Determining  constants  at  the  boundary  conditions:  At  z  =  0,  CR  (0)  =  CR,hydrogel    

𝐶!,!!"#$%&' =  𝛼(0)!

2𝐷!,!+  𝛽(0)!

6𝐷!,!+  𝑘!(0)+  𝑘!  

 𝑘! =  𝐶!,!!"#$%&'  

 At  z  =  L,  CR  (L)  =  CR,tissue  =  0    

0 =  𝛼𝐿!

2𝐷!,!+  

𝛽𝐿!

6𝐷!,!+  𝑘!𝐿 +  𝐶!,!!"#$%&'  

 

−𝑘!𝐿 =  𝛼𝐿!

2𝐷!,!+  

𝛽𝐿!

6𝐷!,!+  𝐶!,!!"#$%&'  

 

𝑘! =  −𝛼𝐿2𝐷!,!

−  𝛽𝐿!

6𝐷!,!−  𝐶!,!!"#$%&'

𝐿  

   

𝑪𝑹 𝒛 =  𝜶𝒛𝟐

𝟐𝑫𝒔,𝑹+  

𝜷𝒛𝟑

𝟔𝑫𝒔,𝑹−  

𝜶𝑳𝟐𝑫𝒔,𝑹

+  𝜷𝑳𝟐

𝟔𝑫𝒔,𝑹+  𝑪!,!!"#$%&'

𝑳 𝒛+  𝑪!,!!"#$%&'  

   4.)  Matlab  Code  and  Graph    Figure 1: Concentration profiles with respect to the depth of the hydrogel wall for Bevacizumab and Ranibizumab  

   5.  Bevacizumab  Flux:  

𝐹𝑙𝑢𝑥 =  −𝐷!,!𝛿𝐶!𝛿𝑧

𝛿𝐶!𝛿𝑧 =  

𝑅!𝐷!,!

𝑧 +  −𝐶!,!!"#$%&'

𝐿 −  𝑅!2𝐷!,!

𝐿  

 

𝐹𝑙𝑢𝑥 =  −𝐷!,!𝑅!𝐷!,!

𝑧 +  −𝐶!,!!"#$%&'

𝐿 −  𝑅!2𝐷!,!

𝐿

At z = L, (this part can be substitute later)

𝐹𝑙𝑢𝑥 = −𝑅!𝐿 +  𝐷!,! ∗ 𝐶!,!!"#$%&'

𝐿 +  𝑅!𝐿2

𝑭𝒍𝒖𝒙 =   −𝑹𝑩𝑳𝟐 +  

𝑫𝒔,𝑩 ∗ 𝑪!,!!"#$%&'𝑳  

 

𝐹𝑙𝑢𝑥 =  −(7.943  x  10−7   µμg

𝑚𝑚!𝑠)(0.72  mm)

2 +  4.1×10!!mm

!

s ∗ 12.351  µμ𝑔/𝑚𝑚!

0.72  𝑚𝑚  

 

𝐹𝑙𝑢𝑥 = 𝟕.𝟎𝟑  𝐱𝟏𝟎!𝟒µμ𝐠

𝒎𝒎𝟐 ⋅ 𝒔        Ranibizumab  Flux    

𝐹𝑙𝑢𝑥 =  −𝐷!,!𝛿𝐶!𝛿𝑧

𝛿𝐶!𝛿𝑧 =  

𝛼𝑧𝐷!,!

+𝛽𝑧!

2𝐷!,!  −

𝛼𝐿2𝐷!,!

−  𝛽𝐿!

6𝐷!,!−  𝐶!,!!"#$%&'

𝐿

𝐹𝑙𝑢𝑥 =  −𝐷!,!𝛼𝑧𝐷!,!

+𝛽𝑧!

2𝐷!,!  −

𝛼𝐿2𝐷!,!

−  𝛽𝐿!

6𝐷!,!−  𝐶!,!!"#$%&'

𝐿

At z = L,

𝐹𝑙𝑢𝑥 =  −𝛼𝐿 −𝛽𝐿!

2 +𝛼𝐿2 +  

𝛽𝐿!

6 +  𝐷!,! ∗ 𝐶!,!!"#$%&'

𝐿

𝑭𝒍𝒖𝒙 =  −𝜷𝑳𝟐

𝟑 −𝜶𝑳𝟐 +  

𝑫𝒔,𝑹 ∗ 𝑪𝑹,𝒉𝒚𝒅𝒓𝒐𝒈𝒆𝒍𝑳

𝐹𝑙𝑢𝑥 =

 −− 5.516  x  10−4 µμg

𝑚𝑚4∗𝑠!.!"!! 2

!−

3.972  x  10−4 µμg𝑚𝑚3∗𝑠

!.!"  !!

!+  

(6.7  x  10−5mm2

s )(!".!"#   !"!!!)

!.!"  !!

𝑭𝒍𝒖𝒙 =  𝟏.𝟏𝟎𝟐  𝒙  𝟏𝟎!𝟑  µμ𝒈/𝒎𝒎𝟐/𝒔

6.    Time  of  Diffusion                

𝑥! = 2𝑛𝐷!𝑡    

t  =   !!

!!!!  

 For  Bevacizumab:  𝑡 = (!.!"#  !")!

!∙!∙(!.!"!"!!  !"!/!)  =  6321  s  =  1.76  hours  

 For  Ranibizumab: 𝑡 = (!.!"#  !")!

!∙!∙(!.!"!"!!!!!/!) = 3867 s = 1.07 hours

7.

Compared to Ranibizumab, Bevacizumab takes longer time to diffuse and has a smaller flux at the interface of the hydrogel and the vitreous fluid. Therefore, Bevacizumab is a better drug for this macular degeneration treatment.

 

Part  2.  

 

1.) Pharmacokinetic  model      

IV  

U  K  

T  

kb  

 RV  

kk  

kt  

ke  

L = mass of Bevacizumab in IV [mg] = 100 mg B = mass of Bevacizumab in blood [mg] T = mass of Bevacizumab in tissue [mg] K = mass of Bevacizumab in kidney [mg] U = mass of Bevacizumab excreted [mg] D* = mass of Bevacizumab metabolized [mg] r = elution rate = 7.72 mL/min v = volume of IV bag = 1000 mL c = concentration of drug in IV = L/v = 100 mg/1000 mL = 0.1 mg/mL RV = rc =7.72 mL/min x 0.1 mg/mLx 60min/1h = 46.32[mg/h] kb = rate of movement of drug from tissue back into bloodstream = 0.00025 [mg/s] kt = rate of movement of drug into tissue = 0.082 [1/hr] kk = rate of movement of drug into kidney = 0.041 [1/hr] ke = rate of excretion of drug from kidney = 0.056 [1/hr] km = rate of metabolism of drug in tissue = 0.02 [1/hr] t = time since administration of dose [hr] VB = volume of blood = 5.5 L

thalf=drug elimination half life= 20 days Bthreshold= maximum mass of drug in blood for additional dose = 0.008mg/mL

Assumptions: • The concentration of Bevacizumab in the IV reservoir and in the blood is constant • No new dose is needed if concentration of drug is above 0.008 mg/mL in the

blood. • There are 5.5 L of blood in the patient’s bloodstream. • The IV reservoir has a 1000 mL volume. • Flow rates are constant and unaffected by other bodily functions. • Bevacizumab only enters through the IV. • Bevacizumab diffuses completely into the bloodstream. • Diffusion rate and porosity are constant across the bloodstream. • The presence of Bevacizumab does not affect metabolism of Bevacizumab. • Temperature and pH of the system are uniform and thus do not affect rates.

D*  km  

B  

• There is no drug in the blood initially (at t = 0).      IV:  𝐶ℎ𝑎𝑛𝑔𝑒  𝑜𝑓  𝑚𝑎𝑠𝑠  𝑜𝑓

 Bevacizumab𝑖𝑛  𝐼𝑉

=   𝑀𝑎𝑠𝑠  𝑜𝑓  Bevacizumab𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔  𝐼𝑉 −   𝑀𝑎𝑠𝑠  𝑜𝑓  𝐵𝑒𝑣𝑎𝑐𝑖𝑧𝑢𝑚𝑎𝑏𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔  𝑏𝑙𝑜𝑜𝑑𝑠𝑡𝑟𝑒𝑎𝑚  

 ∆𝐿 = −𝑚!  !"  !∆𝑡       →    

∆!∆!= −𝑅!      

 𝒅𝑳𝒅𝒕 =  −𝑹𝑽  

 Blood:  𝐶ℎ𝑎𝑛𝑔𝑒  𝑖𝑛  𝑚𝑎𝑠𝑠  𝑜𝑓  

Bevacizumab  𝑖𝑛  𝑏𝑙𝑜𝑜𝑑

=

  𝑀𝑎𝑠𝑠  𝑜𝑓  Bevacizumab𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔  𝑏𝑙𝑜𝑜𝑑𝑠𝑡𝑟𝑒𝑎𝑚 −   𝑀𝑎𝑠𝑠  𝑜𝑓  Bevacizumab𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔  𝑡𝑖𝑠𝑠𝑢𝑒 −

  𝑀𝑎𝑠𝑠  𝑜𝑓  Bevacizumab𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔  𝑘𝑖𝑑𝑛𝑒𝑦 +   𝑀𝑎𝑠𝑠  𝑜𝑓  Bevacizumab𝑟𝑒 − 𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔  𝑏𝑙𝑜𝑜𝑑𝑠𝑡𝑟𝑒𝑎𝑚    

 

∆𝐵 = 𝑚!  !"  !∆𝑡 −𝑚!  !"  !∆𝑡 −𝑚!  !"  !∆𝑡 +𝑚!  !"  !∆𝑡               →                ∆𝐵∆𝑡

= 𝑅! − 𝑘!𝐵  −  𝑘!𝐵 + 𝑘!

 𝒅𝑩𝒅𝒕 =  𝑹𝑽 − 𝒌𝒕𝑩−  𝒌𝒌𝑩+ 𝒌𝒃  

     

Tissue:  𝐶ℎ𝑎𝑛𝑔𝑒  𝑖𝑛  𝑚𝑎𝑠𝑠  𝑜𝑓    Bevacizumab𝑖𝑛  𝑡𝑖𝑠𝑠𝑢𝑒

=   𝑀𝑎𝑠𝑠  𝑜𝑓  Bevacizumab𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔  𝑡𝑖𝑠𝑠𝑢𝑒 −   𝑀𝑎𝑠𝑠  𝑜𝑓  Bevacizumab𝑟𝑒 − 𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔  𝑏𝑙𝑜𝑜𝑑𝑠𝑡𝑟𝑒𝑎𝑚

−   𝑀𝑎𝑠𝑠  𝑜𝑓  Bevacizumab𝑚𝑒𝑡𝑎𝑏𝑜𝑙𝑖𝑧𝑒𝑑  𝑖𝑛  𝑡𝑖𝑠𝑠𝑢𝑒

   

∆𝑇 = 𝑚!  !"  !∆𝑡 −𝑚!  !"  !∆𝑡 −𝑚!"#$%&'()"*∆𝑡       →      ∆𝑇∆𝑡 = 𝑘!𝐵 − 𝑘! − 𝑘!𝑇  

 

𝒅𝑻𝒅𝒕 =  𝒌𝒕𝑩−  𝒌𝒃 −  𝒌𝒎𝑻  

   Kidney:  

𝐶ℎ𝑎𝑛𝑔𝑒  𝑖𝑛  𝑚𝑎𝑠𝑠  𝑜𝑓  Bevacizumab𝑖𝑛  𝑖𝑛  𝑘𝑖𝑑𝑛𝑒𝑦

=   𝑀𝑎𝑠𝑠  𝑜𝑓  Bevacizumab𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔  𝑘𝑖𝑑𝑛𝑒𝑦 −   𝑀𝑎𝑠𝑠  𝑜𝑓  Bevacizumab𝑒𝑥𝑐𝑟𝑒𝑡𝑖𝑜𝑛  

 

∆𝐾 = 𝑚!  !"  !∆𝑡 −𝑚!"#$!%!&∆𝑡       →      ∆𝐾∆𝑡 = 𝑘!𝐵 − 𝑘!𝐾

 𝒅𝑲𝒅𝒕 =  𝒌𝒌𝑩− 𝒌𝒆𝑲  

 3.  Mass  of  Bevacizumab  in  the  blood  

   

𝑑𝐵𝑑𝑡 =  𝑅! − 𝑘!𝐵 −  𝑘!𝐵 + 𝑘!  

 𝑑𝐵

𝑅! − 𝑘!𝐵 −  𝑘!𝐵 + 𝑘!=  𝑑𝑡  

 𝑑𝐵

𝑅! − 𝑘!𝐵 −  𝑘!𝐵 + 𝑘!=   𝑑𝑡  

   

(−1

𝑘! +  𝑘!) ln 𝑅! − 𝑘!𝐵 −  𝑘!𝐵 + 𝑘! =  𝑡 + 𝑐  

 𝑙𝑛 𝑅! − 𝐵 𝑘! + 𝑘! + 𝑘! = −(𝑘! + 𝑘!)(𝑡 + 𝑐)  

 𝑒!(!!!!!)(!!!) =  𝑅! − 𝐵 𝑘! + 𝑘! + 𝑘!  

 *simplify  left  side  

𝑒!(!!!!!)(!!!) =  𝑒!!(!!!!!)  𝑒!!(!!!!!)      

𝑒!!(!!!!!) = 𝑐!    

𝑒!(!!!!!)(!!!) =   𝑐!𝑒!!(!!!!!)    

𝑐!𝑒!!(!!!!!) =  𝑅! − 𝐵 𝑘! + 𝑘! + 𝑘!  

 

𝐵 =  𝑅!  +  𝑘! − 𝑐!𝑒!! !!!!!

𝑘! + 𝑘!

*Apply boundary conditions: when t=0, B=0

0 =  𝑅!  +  𝑘! − 𝑐!𝑒!

𝑘! + 𝑘!

0 =  𝑅!  +  𝑘! − 𝑐!

𝑘! + 𝑘!

𝑐! = 𝑅! +  𝑘!

*Plug c1 back in

𝐵 =  𝑅!  +  𝑘! − 𝑅!  +  𝑘! 𝑒!! !!!!!

𝑘! + 𝑘!

𝑩 = (𝑹𝑽 + 𝒌𝒃)𝟏− 𝒆!𝒕 𝒌𝒌!𝒌𝒕𝒌𝒌 + 𝒌𝒕

   4.    

   

t  =  v/r  =  1000  mL/7.72  mL/min  =  130  min  =  2.17  hr  100  mg  dose  gives  a  drug  mass  flow  rate  of:    

𝑅! = 46.32[mg/h]    

Since  𝐵 = (𝑅! + 𝑘!)!!!!! !!!!!

!!!!!

 

𝐵 = (46.32  𝑚𝑔/ℎ + 0.00025  𝑚𝑔/ℎ)1− 𝑒!!.!"#! 0.041    !!!!!    !.!"#!!!!

0.041  ℎ𝑟!! + 0.082  ℎ𝑟!!

   

𝑩 = 𝟖𝟖.𝟐  𝒎𝒈      

 

𝐵𝑒𝑣𝑎𝑐𝑖𝑧𝑢𝑚𝑎𝑏  𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛  𝑖𝑛  𝑏𝑙𝑜𝑜𝑑   =  88.2  𝑚𝑔5500  𝑚𝑙 = 𝟎.𝟎𝟏𝟔

𝒎𝒈𝒎𝒍  

 5.    

𝑡!!"# = 20  𝑑𝑎𝑦𝑠          𝐵!"#$ = 0.008𝑚𝑔𝑚𝑙 ∗ 5500  𝑚𝑙 = 44.0  𝑚𝑔  

 

𝑛𝑢𝑚𝑏𝑒𝑟  𝑜𝑓  ℎ𝑎𝑙𝑓  𝑙𝑖𝑓𝑒  𝑐𝑦𝑐𝑙𝑒𝑠 =  𝑡

𝑡!!"#  

 

𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛  𝑜𝑓  𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙  𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛  𝑟𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 =𝐵!"#$𝐵  

 Therefore:  

Bsafe  =  B !!

!  =  B !

!

!!!!"#  where  x  is  the  number  of  half  life  cycles  

     

𝐵 ∗12

!!!!"#

= 𝐵!"#$      

12

!!!!"#

=𝐵!"#$  𝐵  

 𝑡

𝑡!!"#ln

12 = ln

𝐵!"#$  𝐵  

 

𝑡 = ln𝐵!"#$  𝐵 ∗

𝑡!!"#

ln 12  

   

𝑡 = ln  44.0𝑚𝑔  88.2  𝑚𝑔 ∗

20  𝑑𝑎𝑦𝑠

ln 12

 

 𝒕 = 𝟐𝟎  𝒅𝒂𝒚𝒔  

 Appendix:  code for generating graphs for Part I problem 4  part 1 %graph  concentrations  of  becvacizumab  and  ranibizumab  vs  depth  of  hydrogel  wall    close  all    

figure    hold  on    %variables    L  =  .72;  %thickness  of  hydrogel  wall    Dsb  =  4.1e-­‐5;  %  bevacizumab  diffusion  constant    Rb  =  7.943e-­‐7;  %  bevacizumab  degradation  rate    Dsr  =  6.7e-­‐5;  %ranibizumab  diffusion  constant    alpha  =  3.972e-­‐4;  %constant  for  ranibizumab  degradation  rate    beta  =  -­‐5.516e-­‐4;  %constant  for  ranibizumab  degredation  rate    Cl  =  12.351;  %drug  concentration  in  hydrogel        %%%%%plotting    z  =  linspace(0,.72,100);    k1b  =  (-­‐Cl/L)-­‐((Rb*L)/(2*Dsb));    Cb  =  (Rb/(2*Dsb))*(z.^2)  +  k1b*z  +  Cl;    plot(z,Cb,'r')    k1r  =  ((-­‐alpha*L)/(2*Dsr))-­‐((beta*(L^2))/(6*Dsr))-­‐(Cl/L);    Cr  =  ((alpha/(2*Dsr))*(z.^2))+((beta/(6*Dsr))*(z.^3))  +  (k1r*z)  +  Cl;    plot(z,Cr,'b')    axis([0  0.72  0  12.3531])    title('Concentration  of  Bevacizumab  and  Ranibizumab  vs  depth  in  hydrogel')        xlabel('depth  into  hydrogel  wall  [mm]')    ylabel('drug  concentration  [ug/mm^3]')    legend('Bevacizumab','Ranibizumab')    grid  on    hold  off    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%