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UCSD BENG 221 - DRUG DIFFUSION THROUGH A COATED STENT

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BENG%221%%%%DRUG%DIFFUSION%THROUGH%A%COATED%STENT%%%%%%%%%!!12TH%NOV E MBER,%20 1 0 %%%%1. INTRODUCT IO N : %In! this! repor t,! a! bioengineering!application! of! the! one3dimensional! diffusion! equation,!solved! in! both!rectangular! and! cylindrical! coordinate s ,! is! conside re d .! An! example! of! this!is!the!diffusion!of!a!drug!through!an!a rterial!w all!via!a!coa ted !sten t.!!A!stent!is!a!metallic! prosthesis!implanted!into!the!arterial! w all!and!coated!with!a!layer!of!a!therapeutic!drug.! It! is! used! to! treat! heart! diseases! such! as! atherosclerosis,! which! can! be! considered! a! form ! of!chronic! inflammation.! When! it!affects!coronary!arteries,!symptoms! such! as! angina! pectoris! and! h eart!attack! can! occur[1].! So! as! to ! revascular ize ! coronary ! arteries,! a! stent! can! be! used .! ! A! stent! is ! an!expandable! metal! or! polymeric! tubular! mesh.! A! drug! diffusing! stent! is! a! normal! m etal! stent! that! has!been!coated!with! a! pharmacologic!agent!(drug)!that! is!known!to!interfere!with!the!process!of! restenosis!(reblocking).!!In!this!p ro b le m,!a!drug3diffusing!stent!coated!w ith!heparin!is!considered.!Once!the!stent!is!inserted!into!an! artery,! it! starts! diffusing! through! the! arterial! wall.! The! solution! indicates! the! concentration! of! the!drug! at! different! distances! in! the! wall,! at! different! time! intervals.! This! is! important! as! it! helps! to!determine!the!rate! at! w hich! the! drug! is! diffusing!through! the! artery.!Depending! on!the!time! at!which!all!drug!has!diffused!through!the!wall,!we!can!determine!if!modifications! in!the!system! are!required! for!sustained!drug!delivery.!2. SET9UP:%!!CASE!II! ! ! ! ! ! ! CASE!I!Figure%2.1!Cross!sect io n !o f!a r te ry !a.) CASE%I:!Consider!o n e !p o rt io n !o f!t h e!a rt e ria l!w a ll!!We!can!make!the!following!assumptions!in!this!condition:!• As!the! vacant! space! in! the! artery! is! much!larger!than!the!thickness!of!either!the! stent! or! the!wall,! we!can!consider!only! one!part!of!the!blood!vessel,!and!assume!the!stent!and!wall!to!be!two!parts!of!a !slab.!Considering !this!assumption ,!the!problem!can!be!solved!in!rectangular!coordinates.!• The!diffusivities!of!both!the!coated!stent!and!the!wall!are!constant.!• The!stent!is!impervious!to!flow!of!blood!(that!is,!the!drug!will!only!diffuse!in!one!direction).!• The!outer!edge!of!the!wall!is!impermeable!to!the!drug.!• Differential!equation:! !• Initial!con d it io n :!C (x ,0) = 0;!in it ia lly,!th e !dru g!co nce ntra tion !in!the !wa ll!is!zero.!• Boundary!condition:!!!1. C(0,t)=Cmax=1!mol/cm3;!the! amo u n t! of!drug!o n ! the! surface ! of!th e ! stent! is!m u c h ! highe r!than!at!any!other!poin t!in!the!wall.!So,!the!stent!is!a!constan t!sourc e!of!the!dru g!ove r!time.!!2. Dc/dt(L,t)=0;!at!the!end!of!the!arterial!wall,!the!flux!is!zero!due!to!impermeability.!This!problem!is!first!solved!analytically!considering!the!entire!setup!to!be!made!up!of!one!slab.!Then,!the!problem !is!solved !nu m erica lly!cons iderin g!tw o!laye rs.!b.) CASE%II:!Consider! the!artery!to!be!a!cylinder.!A!stent!is!an!expandable!metal!or!polymeric!tubular!mesh.!It!is!difficult!to!solve!this!problem!with!the!stent!as!a!mesh,!as!it!is!then!not!a!uniform!source!of!the!drug.!Certain!assumptions!are!therefore!m ade!to!m od ify!our!problem!into!a!more!convenient!form,!mathematically.!The!assumptions!made!in!this!case!are:!• The!stent!is!assumed!to!be!a!film!of!uniform!thickness!coating!the!inner!wall!of!the!blood!vessel.!• The!artery!is!assumed!to!be!a!cylinder!of!uniform!thickness.!• The!diffusivities!of!both!the!coated!stent!and!the!wall!are!constant.!• The!stent!is!impervious!to!flow!of!blood!(that!is,!the!drug!will!only!diffuse!in!one!direction).!• The!outer!wall!of!the!blood!vessel!is!impermeable!to!the!drug.!• Differential!equation:! !• Initial!con d it io n :!C (x ,0) = 0!• Boundary!conditions:!1. !C(r,t)=!Cmax=!1!mol/cm3,!!2. dc/dx(r2,t)=0!!3. SOLUTION:%%3.1 ANALYTICAL%SOLUTION%FOR%CARTESIAN %COORDINATE:%Diffusion%Equation%! (3.1.1)!Initial%Con d itio n %! (3.1.2)!Boundary%Conditions%! (3.1.3)!! (3.1.4)!! !This!is!a! diffusion! problem!in! one3dimensional!slab!with!constant!physical!properties!and!no!sources.!To!solve! this! equation,! solution ! can! be! separated! into! hom oge neou s! and! p articular! solutions.! Particular!solution!is!t h e!s te a d y!s ta te !so lu tio n !w h e re !λ =0 .!!! (3.1.5)!! (3.1.6)!! (3.1.7)!! (3.1.8)!!! (3.1.9)!! (3.1.10)!!! (3.1.11)!!% (3.1.12)%!!! (3.1.13)!!!! (3.1.14)!!! (3.1.15)!!! (3.1.16)!!! (3.1.17)!!! (3.1.18)!!! (3.1.19)!! (3.1.20)!!! (3.1.21)!! (3.1.22)!! (3.1.23)!!! (3.1.24)!! (3.1.25)!!! (3.1.26)!The!physically!relevant!solution!is!found!for!λ "! (3.1.27)!The!form!of!the!general!solution!is!!! (3.1.28)!! (3.1.29)!!!! (3.1.30)!! (3.1.31)!! (3.1.32)!!!!!! (3.1.33)!! (3.1.34)!=0! (3.1.35)!Applying!this!boundary!condition!gives!! (3.1.36)!! !!!!!!!!!!!(3.1.37)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(3.1.38 )!!! (3.1.39)!! (3.1.40)!! (3.1.41)!!! (3.1.42)!! (3.1.43)!!! (3.1.44)!! (3.1.45)!! (3.1.46)!= ! (3.1 .47) !! (3.1.48)!! (3.1.49)!! (3.1.50)!! (3.1.51)!! (3.1.52)!! (3.1.53)!! (3.1.54)!! (3.1.55)!! (3.1.56)!! (3.1.57)!! (3.1.58)!!! (3.1.59)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(3.1.60 )!% (3.1.61) !!!!!!!!!!!!!!!!!!!3.2.%ANALYTICAL%SOLUTION%FOR%CYLINDRICAL%COORDINATE:!Diffusion%Equation%! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(3.2.1)%Initial%Con d itio n %!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! ! ! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(3.2.2)!Boundary%Conditions%! ! ! ! ! ! ! ! ! ! !!!!!!!!!!!!!!!!!(3.2.3)!!!!! ! ! ! ! ! ! ! ! !!!!!!!!!!!!!!!!!(3 .2.4) !This!is!a!diffusion! problem!in!r3 direction!with!constant!physical!properties!and!no!sources.! To!solve!this!equation,! solution! can! be! separated!into!homogeneous!and!particular!solutions.!Particular!solution!is!the!steady!state!so lution !w he re!λ= 0.!!! ! ! ! ! ! ! ! ! !!!!!!!!!!!!!!!!!(3.2.5)!!! ! ! ! ! ! ! ! ! ! !!!!!!!!!!!!!!!!!(3.2.6)!!!! ! ! ! ! ! ! ! ! !!!!!!!!!!!!!!!!!(3 .2.7) !!! ! ! ! ! ! ! ! ! !!!!!!!!!!!!!!!!!(3.2.8)!At!r=0,!Cp(x,t)!should!be!finite.!Sin ce !ln(0)!is!no t!a!finite !value : !A=0!! ! ! ! ! ! ! ! ! ! !


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