Brian&Bober&Kyle&Buchholz&Kevin&Young&Jessica&We n&ALCOHOL&DIFFUSION&IN&THE&STOMACH&LINING&&Introdu ctio n&Whenðyl& alcohol&is& consumed,& the& liquid& first&enters& the& stomach.& Alcohol& in&the&stomach&can&be&absorbed& through& the& stomach& lining& into& the& bloodstream,& or& it& can& continue& on& to& the& small&intestine ,&where&it&can&again&be&absorbed&into&the&bloodstream.&Intoxication&occurs&after&absorption&into&the&b lo o d st re am,&due&to&th e &p r e s e n c e &o f&a l cohol&in&the&blood.&&&Although& alcohol& can& really& be& absorbed& into& the& bloodstream& at& any& point& along& the&gastrointestinal&digestive&tract&from&the&mouth&during&ingestion&to&the&small&intestine,&the&majority&of& the& alcohol& enters& the& bloodstream& via& either& the& stomach&lining&or&the&small&intestine&walls&(Chemcases ,&20 03).&Approximately& 20%&ofðyl&alcohol&consumed& is&absorbed&in&the& stomach&an d&80%&is &absorb ed &in& th e&small&intestine &(Chemcases,&200 3),&while&a&neglig ible&amount&is&a b s o r b e d&in&the& remainder& of& the& digestive& tract.& The& difference& between& absorption& in& the& stomach& and& the&intestine &is &d u e &t o&th e&differe nce &in&surface&areas&that&limits&diffusion&rates&(.045 &m2&is&the&surface&area& of& the& stomach,& 600& m2&is& the& surface& are a& of& the & sm al l& intes tin e ).& Because& more& alcohol& is&absorbed& in& the& small& intestine& than& in& the& stomach,& most& alcohol& absorption& studies& have&traditionally& studied,& simulated,& or& experimentally& measured& the& changing & rate& of& alcoho l&concentration&in& the&blood&or&the&breath&and&compared&that&to&the&rate&of&gastric&emptying&into&the&small&intestine&(Watkins&and&Adler,&1993).&The&rate&of&alcohol&absorption&overall&is&dependent&on&the&ra te&of&gastric&emptying&since &alcohol&absorption&occurs&primarily&from&the&small&bowel&(Holt&et&al.,&1980).&Thus,&most&studies& testing&breathalyzers&typically&use&this&relationship&to&determine&the&accuracy&of&the&breathalyzer,&a&dev ic e &that&measures &breath&alcohol&conten t&(BrAC),&wh i ch&is&in &turn&a&fraction&of&blood&alcohol&content&(BAC),&which&prov id e s &a &q uantificatio n &o f&i n toxication .&&&Because&the&typical&profile&of&alcohol&in&the&blood&following&a&short&period&of&drinking&then&stopping&resembles&a&bell&curve,& certain& driving&under& the& influe n c e & (DUI)& cases& may& be & fou g h t& ag a in s t & by&criminal& defense&attorneys& who& can&pro ve& that& although&BAC & leve ls& were & abov e& the& legal& limit& at&time& of&breathalyzer & testing&by& the&po lice& officer,&B AC&leve l s & were& not& about& the& legal& limit& at&the &time& that& the&drive r& was&actually&driving&the&vehicle& (CalDUI,&2 010 ).& It& is&d ifficult& to& pinpoint& the&exact&BAC&back&in&time,&and&to&determine&exactly&where&the&person&was&on&the&bell&curve&at&the&time&of& driving.& However,& this& process& would& involve& considering& exactly& what& drink& was& consumed,&exactly& what& time& drinking& occurred,& and& quantifying& alcohol& absorption& into& the& bloodstream&through&both &the&stomach&and&the&small&intestine .&&&In&this&problem ,&w e &intend&to &model&the& absorption& of&alcohol&through& the&stomach&lining&in&order&to&determine &ho w &m uc h&o f&bloo d&a lcoh ol&co nten t&is&du e&to &absorption&in&the&stomach&alone.&&This&approach&further& allows&us&to&model&the& changing&concentration&of& alcohol&absorbed&in&stomach&as&a&person’s&drinking&behavior&changes&over&time.&&Problem&Statement&Students& A ,&B,&and&C &decide&to&go&out&for&a&night&of&drinking&in&Pacific&Beach&after&they&turn&in&their&last&m a th &assignment&of&the&quarter.&Student&A&goe s &to&the &bar s&and&downs&drink&after&drink&until&his&stomach&is&full.&He&then&realizes&he&needs&to&drive,&and&stops&drinking&immediately.&Student&B&decides&to&go&all&out,&and&can&be&found&constantly&slowly&drinking&beer&from&a&CamelBak&on&his&back&all& night.& Student& C’s& drinking& pattern& is& consistent& throughout& the& night.&He&drinks&beer,&then&drinks&water,&drinks&beer,&then&drinks&water,&drinks&beer,&and&so&forth,&throughout&the&night.&&&Brian&Bober&Kyle&Buchholz&Kevin&Young&Jessica&We n&&Model& the& concentration& of& alcohol& at& the& outer& edge& of& the& stomach& over& time&for&each& of& the&students.&Determine&the&concentration&profile&over&time&at&the¢er&of&the&stomach,&at&the&inner&lining&of&th e &stom ac h ,&and&the&oute r&lining&of&the&stom ac h.&How&lon g&mu s t&Stud en t&A&wait&until&he&can&safely&drive&the&others&home?&You&may&define&y our&own&criteria&for&being&able&to&safely&drive.&&Problem&Set&Up&Diffusion&in&the&stomach& walls&can&be&most&simplest&modeled& by&using&Fick’s&2nd&Law&of& Diffusion& in&one& dimension& in& Cartesian& coordinates& where& C&is&concentration,&t&is&time,&x&is&position,&D% is&diffusivity,&and&Q&denotes&a &d r iv in g &fo r c e&(E qu ation&1 ).&&&&(1)&&&However,& because&the&stomach& can&be&assumed& to&be&roughly&spherical,&Fick’s&2nd&Law&of&Diffusion&in&three&d im e n sio n s &in&s p he ric a l&co o rd in at es &w o u ld &b e &m o re &a cc u ra te &(E q u at ion &2 ).&&&(2)&&&In& ord er& to& simplify& the& pro blem,& we & will& assume & constant& mixing& in& the& stomach&in&order&to&approximate&this&as&a&one&dimensional&problem&(Equation&3).&&&(3)&&&For&simplicity,&we&assume&that&the&stomach&is&shap ed&as&a&sphere&and&can&be&modeled&as&a&sphere.&The&stomach&normally&holds&a&volume&of&about&1&liter&(Hypertextbook).&A&sphere&with&a&radius&of&about&6&cm&w ould& h ave& this& volum e.& The&stomach&wall& h as& a&thickness&of& 0.51& ±&0.11&cm&(Rapaccini,&et&al.&1988).&Th e r e fo r e ,& we&ap proximat e d & the&sto mach&as & a&sph e re & with& an&oute r & radius& of&6.5& cm&and&an&inner&radius&of&6.0&cm&to&a cc ou n t &for &a&0 .5&cm&thick&wall.&&&Another&difficulty&in&settin g&u p &the&pro b le m &is&keep ing&the&conce ntration&the&sam e&throu ghou t&the&stomach.& Note& here& that& we& are& only& considering& diffusion& through& the& stomach& wall& itself.&However,&there&is&volume/space&inside&the&stomach&itself!&In&order&to &accomp lish &this,&we&assume&that&the&diffusivity&in&the&volume&of&the&stomach,&or&in&the&stomach&chamber&from&the¢er&of&the&stomach& to&the& inner&e dge& of&the& wall& is&extremely& high:&1015&cm2/s.&This& forces&the&inside&of&the&stomach& to& be& well& mixed& an d& does& not& allow& the& con centration& to& vary& w ith& position& inside&
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