A POROUS MEDIA APPROACH TOWARDS A DYNAMIC MECHANISTIC MODEL OF DRUG ELIMINATION BY THE LIVER. A Thesis Submitted to the

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1 A POROUS MEDIA APPROAH TOWARDS A DYNAMI MEHANISTI MODEL OF DRUG ELIMINATION BY THE LIVER A Thess Submtted to the ollege of Graduate Studes and Research n Partal Fulfllment of the Requrements for the Degree of Master of Scence n the Dvson of Bomedcal Engneerng Unversty of Saskatchewan Saskatoon, Saskatchewan By MOHAMMAD IZADIFAR opyrght Mohammad Izadfar, August, 0. All rghts reserved.

2 PERMISSION TO USE In presentng ths thess n partal fulfllment of the requrements for a Postgraduate degree from the Unversty of Saskatchewan, I agree that the Lbrares of ths Unversty may make t freely avalable for nspecton. I further agree that permsson for copyng of ths thess n any manner, n whole or n part, for scholarly purposes may be granted by the professor or professors who supervsed my thess work or, n ther absence, by the Head of the Department or the Dean of the ollege n whch my thess work was done. It s understood that any copyng or publcaton or use of ths thess or parts thereof for fnancal gan shall not be allowed wthout my wrtten permsson. It s also understood that due recognton shall be gven to me and to the Unversty of Saskatchewan n any scholarly use whch may be made of any materal n my thess. Requests for permsson to copy or to make other uses of materals n ths thess n whole or part should be addressed to: Head of the Dvson of Bomedcal Engneerng Unversty of Saskatchewan Saskatoon, Saskatchewan S7N 5A9 anada

3 ABSTRAT Hepatc drug elmnaton s a maor PK process contrbutng to loss of drug concentraton n the body. The predcton of hepatc clearance (and hence drug concentratons n the body) requres an understandng of the physology and mechansms of the hepatc elmnaton process and ther complaton nto a mechanstc model. Several physologcal models ncludng wellstrred model, parallel tube model and dsperson model have been developed to descrbe the hepatc elmnaton process and to determne how physologcal varables such as blood flow, unbound fracton and enzyme actvty may nfluence the hepatc clearance. However, each model has dstngushng advantages and lmtatons, whch lead sometmes to very dsparate predcton outcomes. Although hepatc drug elmnaton has been mathematcally descrbed by dfferent physologcal models, the mass transfer phenomena n the lver has not been descrbed from a porous meda vewpont usng local volume averagng method. The nherently porous structure of the lver allows us to descrbe the hepatc drug elmnaton process based on a porous meda approach such that structural propertes of the lver tssue, physco-chemcal propertes of the drug as well as transport propertes assocated wth the hepatc blood perfuson are ncluded n the model. Applyng local volume averagng method and local equlbrum to the lver as a porous medum, a governng partal dfferental equaton whch takes nto account lver porosty, tortuosty, permeablty, unbound drug fracton and hepatc tssue partton coeffcent, drugplasma dffusvty, aal/radal dsperson and hepatocellular metabolsm parameters was developed. The governng equaton was numercally solved usng mplct fnte dfference and Gauss-Sedel teratve method n order to descrbe changes n dug concentraton wth tme and poston across the lver followng an ntravenous drug admnstraton. The model was used to

4 predct hepatc clearance and boavalablty, whch were then compared to reported observatons. The predcted values of hepatc clearance and boavalablty had good agreement wth the reported observatons for hgh and low clearance drugs. As well, the model was able to successfully predct an unsteady state of hepatc drug elmnaton wth concentraton dependent ntrnsc clearance. When statstcally compared to the well-strred, parallel tube and dsperson models the proposed model suggested a smaller mean squared predcton error and very good agreement to reported observatons for eght drugs. A senstvty analyss revealed that an ncrease n lver porosty results n a slght decrease n the drug concentraton gradent across the lver whle hgher tssue partton coeffcent values ncrease the concentraton gradent. The model also suggested that the boavalablty was senstve to the nteracton between unbound fracton and ntrnsc clearance. Ths study ndcates that the lver and hepatc drug elmnaton can be successfully eplored from a porous meda vewpont and may provde better mechanstc predctons of drug elmnaton processes by the lver.

5 AKNOWLEDGMENTS Frst of all, I would lke to thank God, the Almghty, for havng made everythng possble by gvng me strength and courage to do ths work. I would lke to gratefully acknowledge Dr. Oon-Doo Bak, my supervsor, for unflnchng supports n varous ways durng my study. As well, I would lke to record my deepest grattude to Dr. Jane Alcorn, my advsor from the dvson of Pharmacy, for gudance, encouragement and valuable comments. I apprecate comments and the tme spent on revewng the drafts of ths thess by Dr. Danel hen, my GA char. I am also thankful to Dr. Davd M. Janz for servng as the eternal eamner and for hs constructve comments on the draft of ths thess. Fnally, sncere and nfnte thanks go to my dearests, my father and mother for ther patence and encouragement durng my study abroad; the patence and encouragement that I really apprecate. v

6 TABLE OF ONTENTS PERMISSION TO USE... ABSTRAT... AKNOWLEDGMENTS... v TABLE OF ONTENTS... v LIST OF FIGURES... v LIST OF TABLES... NOMENLATURE... HAPTER : BAKGROUND AND LITERATURE REVIEW.... Introducton.... The lver s central role n ADME processes Systemc crculaton Dstrbuton concepts Volume of dstrbuton Blood proten bndng and unbound fracton Equlbrum and tssue partton coeffcent Systemc clearance concepts Elmnaton rate and etracton rato Admnstraton routes and pre-systemc elmnaton Lver anatomy and physology Hepatc clearance Intrnsc clearance Hepatc perfuson rate Drug unbound fracton Transport medated uptake Boavalablty Physologcal model for hepatc drug elmnaton Well-strred model Parallel tube model Dstrbuted snusodal perfuson model Dsperson model Interconnected-tubes model Tanks-n-Seres model Recent model orentatons Porous meda concepts and applcatons n bomedcal engneerng Representatve elementary volume (REV) Local volume averagng method (LVA) Length scales n porous meda and LVA valdty condton Applcaton of LVA method to bologcal systems Research obectve HAPTER : A POROUS MEDIA APPROAH FOR MEHANISTI MODELING OF DRUG ELIMINATION BY THE LIVER Introducton Theory Local Volume Averagng (LVA) method v

7 .. Mathematcal Modelng Numercal Soluton of the model Pharmacoknetc and structural parameters for the smulaton Results and dscusson oncluson HAPTER 3: GENERAL DISUSSION Porous meda physologcal model Valdatng the proposed model PM model predctablty compared to WS, PT and DP models Smulaton and senstvty analyses The effect of aal dsperson on the drug dstrbuton profle The nfluence of the lver porosty on the lver drug dstrbuton profle The effect of tssue partton coeffcent on the lver drug dstrbuton profle The senstvty of lver drug dstrbuton profle to ntrnsc clearance The senstvty of lver drug dstrbuton profle to unbound fracton The senstvty of hepatc clearance to perfuson rate and ntrnsc clearance The senstvty of boavalablty to perfuson rate and ntrnsc clearance The nfluence of unbound fracton and ntrnsc clearance on boavalablty Smulaton of hepatc drug elmnaton wth nonlnear ntrnsc clearance General conclusons PM model lmtatons and future studes APPENDIX: v

8 LIST OF FIGURES Fgure.. Schematc dagram of the blood crculaton and three maor admnstraton stes... Fgure.. Schematc dagrams of the lver heagonal unts (a) and the mcrostructure of an acnus Fgure.3. Schematc dagram of representatve elementary volume and the varaton of the medum property wth the sze of representatve elementary volume Fgure.4. Schematc dagram of a porous medum and the assocated length scales Fgure.. Schematc dagram of the lver mcrostructure wth the assocated length scales Fgure.. Schematc dagram of the smplfed geometry (a) and the porous dfferental element (b) of the lver Fgure.3. Senstvty analyss for determnaton of optmum mesh sze for the numercal soluton Fgure.4. Predcted values of hepatc clearance from Porous Meda based model versus reported observatons (Shbata et al., 00) for eght drugs Fgure.5. Predcted values of boavalablty from Porous Meda based model versus reported observatons (Shbata et al., 00) for eght drugs Fgure.6. Plasma unbound drug concentraton gradent across the lver at dfferent tmes n the absence (a) and presence (b) of aal dsperson (Dn) for ldocane Fgure.7. Varaton of plasma unbound drug concentraton at the hepatc ven versus tme and aal dsperson number for ldocane... 6 Fgure.8. The nfluence of porosty on plasma unbound drug concentraton gradent across the lver at dfferent tmes for ldocane for aal dsperson number (D n ) of 0.7:, =0.06;, =0.;, = Fgure.9. Senstvty of plasma unbound drug concentraton gradent across the lver to the partton coeffcent for ldocane for aal dsperson number (D n ) of 0.7:, K * =0.70;, K * =0.6;, K * = Fgure.0. Influence of ntrnsc clearance (a) and unbound fracton (b) on plasma unbound drug concentraton gradent across the lver for ldocane at an aal dsperson number of Fgure.. Varaton of hepatc clearance wth ntrnsc clearance and hepatc perfuson rate for ldocane at a snusodal porosty of 0. and perfuson rate of 500 mlmn v

9 Fgure.. Varaton of boavalablty wth ntrnsc clearance and hepatc perfuson rate for ldocane at a snusodal porosty of 0. and perfuson rate of 500 mlmn Fgure.3. Varaton of boavalablty wth ntrnsc clearance and unbound fracton for ldocane at a snusodal porosty of 0. and perfuson rate of 500 mlmn Fgure.4. Varaton of ntrnsc clearance (a) and plasma unbound drug concentraton of - hydroymdazolam across the lver at dfferent tmes v

10 LIST OF TABLES Table.. Pharmacoknetc parameters of drugs used n the smulaton of hepatc clearance Table.. omparson of model predctons wth observed values of the hepatc clearance assocated wth seven drugs Table.3. omparson of model predctons wth observed values of boavalablty assocated wth seven drugs Table.4. Mean squared predcton error (MSE) and coeffcent of determnaton (R) values of hepatc clearance and boavalablty for well-strred, parallel tube, dsperson and porous meda models... 59

11 NOMENLATURE A ross sectonal area (m ) a Parameters n Eqs. (.4), (.9) and (.5) A b Arteral blood drug concentraton (mg/ml) Total blood drug concentraton (mg/ml) Drug concentraton (mg/ml) P Local volume average plasma drug concentraton (mg/ml) p ss u V l H l h_dp l h_lva l h_pt l h_ws l n_nvvo l n-overall l nt l organ l s D AB Plasma drug concentraton (mg/ml) Steady state plasma drug concentraton (mg/ml) Unbound drug concentraton (mg/ml) Venous blood drug concentraton (mg/ml) Hepatc clearance (ml/s) Hepatc clearance from dsperson model (ml/s) Local volume averaged hepatc clearance (ml/s) Hepatc clearance from parallel tube model (ml/s) Hepatc clearance from well strred model (ml/s) Invvo ntrnsc clearance (/s) Overall hepatc ntrnsc clearance (ml/s) Intrnsc clearance (ml/s) Organ clearance (ml/s) Systemc clearance (ml/s) Drug-plasma molecular dffuson coeffcent (m /s) d D Aal dsperson coeffcent (dmensonless) d D d p D n dv ER F F H Radal dsperson number (dmensonless) Pore sze (m) Dsperson number (dmensonless) Dfferental volume (ml) Etracton rato (dmensoless) Boavalablty (dmensonless) Hepatc boavalablty (dmensonless)

12 f u(b) Unbound drug concentraton (dmensonless) H Hematocrt (dmensonless) Node number nde (dmensonless) Tme step nde (dmensonless) K Tssue hydraulc conductvty (m ) K * K e K M Tssue partton coeffcent (dmensonless) Dagonal matr of elmnaton rate constant (/s) Affnty term (mg/ml) k phys Snusodal based permeablty (m ) L Length (m) l Representatve elementary volume length scale (m) M The matr coeffcent of the mass echange (/s) MRT Mean resdence tme of the drug n the body (s) mˆ Local volume averaged hepatocellular metabolsm rate (mg/ml.s) met N Pe PS u,efflu PS u,nflu P Q Q h t t u u B V V Number of nodes or compartments (dmensonless) Peclet number (dmensonless) Efflu membrane permeablty of unbound drug (ml/s) Influ membrane permeablty of unbound drug (ml/s) Pressure drop (Pas.) Blood perfuson rate (ml/s) Hepatc perfuson rate (ml/s) Tme (s) Tme step sze (s) Lnear velocty (m/s) Blood Darcy velocty (m/s) Dagonal matr of blood velocty (m/s) Volume (ml) V d Volume of dstrbuton (ml) V ma Mamum enzyme actvty capacty (mg/ml s) V ss Steady state volume of dstrbuton (ml)

13 V t W b X IV X 0 Z z Tssue volume (ml) Body weght (kg) Poston (m) Intravascular dose of the drug (mg) Dose (mg) Mesh sze (m) Dmensonless poston along the lver (dmensonless) Poston (m) Greek letters Parameter n Eqs. (.0) to (.3) Parameter n Eqs. (.0) to (.3) Porosty (dmensonless) <> Local volume average property Parameter n Eqs. (.0) to (.3) Vscosty (kg/m s) Parameter n Eqs. (.0) to (.3) Tortuosty (dmensonless) Hepatocellular metabolsm rate (mg/s) Elmnaton rate (mg/s) Heterogenety coeffcent (dmensonless)

14 HAPTER BAKGROUND AND LITERATURE REVIEW. Introducton Physologcal modelng contrbutes substantally to drug dscovery and development through ts ablty to provde better nsght nto mechansms and processes attrbuted to pharmacoknetcs (PK) and ntegrate these wth pharmacodynamc (PD) processes. A mechanstc model whch can adequately descrbe n-vvo physologcal phenomena based on lmted avalable n-vtro data can reduce the costs of anmal eperments and loss n tme due to poor selecton of possble lead canddates. Such a model can perform senstvty analyses to estmate epected PK profles of new drugs n human. However, a reducton n predcton uncertantes only follow from the avodance of oversmplfcatons or assumptons that often eclude crtcal determnants of the PK propertes of a compound. Today, pharmaceutcal companes focus efforts on developng more predctve models to descrbe the physologcal processes more precsely so that the anmal eperments can be reduced, refned and replaced by mathematcal models. In addton, developng predctve physologcal/mechanstc models are requred for better understandng and analyzng pathophysologcal events n the body. Hepatc drug elmnaton s a maor PK process contrbutng to loss of drug concentraton n the body. The predcton of hepatc clearance (and hence drug concentratons n the body) requres an understandng of the physology and mechansms of the hepatc elmnaton process and ther complaton nto a mechanstc model. Several physologcal models, namely the well-

15 strred (WS) model, parallel tube (PT) model and dsperson (DP) model, have been developed to descrbe the hepatc elmnaton process and to determne how physologcal varables such as blood flow, unbound fracton and enzyme actvty may nfluence the hepatc clearance. However, each model has dstngushng advantages and lmtatons, whch lead sometmes to very dsparate predcton outcomes. Smplfcatons and assumptons whch are made to develop and solve models are the maor sources of uncertantes of predctons. To mprove the predctablty of a mechanstc model for hepatc drug elmnaton, the mathematcal formulaton must be consstent wth the physologcal and physcal phenomena takng place n the lver. In the WS model, drug s assumed to be nstantaneously and homogeneously med wth the blood n the lver resultng n very unform drug concentraton across the lver. Ths dealzed oversmplfcaton adds mportant assocated uncertantes n predctons by the WS model. Furthermore, the WS model gnores the concentraton gradent across the lver whch s contradctory to realstc phenomenon n the lver. Although the PT and DP models assume an eponental drug concentraton gradent across the lver, these models do not take nto account the nfluence of lver structural characterstcs such as tssue porosty and tortuosty and tssue partton coeffcents. The DP model has an advantage over PT model by takng nto account the dsperson mechansm of the drug transport n the lver; however, DP model assumes that the hepatocellular metabolsm rate s constant such that the drug concentraton n hepatocytes s ndependent of tme, and physco-chemcal propertes reman unchanged wth the poston across the lver. In realty, enzyme actvty changes wth the drug concentraton across the lver and the drug concentraton n hepatocytes as well as physco-chemcal propertes can change wth tme.

16 An ablty to make robust predctons of drug elmnaton by the lver requres a mechanstc model that smulates real physologcal events n the lver. Ths study focuses on the development and valdaton of a mechanstc model that represents the lver s structural propertes based on the concepts of porous meda and consders unsteady state propertes n the lver and concentraton dependent metabolc ntrnsc clearance (the nherent ablty of lver cells to elmnate the drug) and transport mechansms of a drug.. The lver s central role n ADME processes Absorpton, dstrbuton, metabolsm and ecreton (ADME) of a drug are the maor processes whch are quanttatvely dscussed n pharmacoknetcs (PK). Together wth physologcal and pharmacodynamcs propertes, ADME processes determne the therapeutc profle of a drug. For eample, f a drug compound s poorly absorbed n the ntestne and s hghly metabolzed n the lver or rapdly ecreted by the kdneys, the drug wll be unable to provde ts optmum therapeutc effects so that hgher dose levels of the drug wll be requred to acheve suffcent drug concentraton n the target organ for full therapeutc effect of the drug (Yann and Thakker, 007). Also, f the drug s poorly lpophlc, drug dstrbuton n a target tssue.e. bran wll be so poor that the drug effcacy wll be low despte ts hgh plasma drug concentraton. Therefore, t s very mportant to understand and quantfy ADME processes. Among ADME processes, hepatc drug metabolsm s consdered as an mportant process for drug elmnaton from the body and can be an mportant determnant of plasma drug concentratons. The lver s the maor ste of drug metabolsm n the body and s responsble for blood detofcaton whereby toc substances n the blood are metabolzed by the lver wth the same prncples as drug elmnaton processes. The lver s also a prncpal organ n the 3

17 mantenance of homeostass. In order to understand the role of the lver n drug PK then, requres a knowledge of PK concepts and parameters, drug physco-chemcal propertes and lver anatomy, archtecture and physology... Systemc crculaton The systemc crculaton supples blood to all tssues throughout the body ecept for the pulmonary crculaton whch has ts nclusve blood crculaton system. The contracton of heart s left ventrcle forcefully pushes the fully oygenated blood through the aorta whch branches nto many small arteres and further nto arteroles and termnatng nto capllary beds throughout organs and tssues of the body. Arteres, arteroles and capllares are responsble for oygen and nutrent delvery to the tssues. Oygen, nutrents, drug compounds as well as any possble toc substances are transferred from capllares to the tssues whle wastes are transferred back nto the capllares whch conduct the low oygenated blood to the heart through venules whch collect nto larger vessels, the vens. Then the pulmonary crculaton system takes over oygenatng the blood n lungs. In the systemc crculaton, the heart supples the oygenated blood to the whole body va the arteral blood supply. The oygenated blood s dstrbuted n tssues and organs where the oygen, nutrents and/or drug compounds are echanged wthn tssue capllary beds. The varous organs and tssues of the body receve a fracton of the cardac output n parallel. Also, the blood passes through the portal crculaton around the gastrontestnal tract from whch nutrents as well as drug compounds are absorbed from the ntestne nto the portal blood. The seral nature of the blood supply from gastrontestnal tract through the portal ven to the lver and hepatc ven dstngushes the portal ven blood supply over the parallel nature of blood dstrbuton n maor organs n the body. Portal ven drects the nutrent enrched blood to the lver where the blood s med wth the hepatc arteral blood n snusods and eventually s 4

18 draned nto the hepatc ven. The hepatc ven ons the vens from the rest of the body towards the heart where deoygenated blood s pumped to the pulmonary crculaton... Dstrbuton concepts The dstrbuton process s defned as the reversble movement of drug between the systemc crculaton and tssues of the body. The reversble drug dstrbuton between the blood and tssues can be characterzed by the rate of dstrbuton and the etent of dstrbuton. Although the rate of drug dstrbuton s mportant, the etent of drug dstrbuton s consdered as a key PK process. Tssue perfuson rate, the permeablty of tssue membranes, the drug bndng ablty wth the blood protens and tssue, and the lpophlctyof the drug can nfluence the etent and the rate of drug dstrbuton.... Volume of dstrbuton Blood volume n human body s varable but roughly t can be appromated as much as 8% of the body weght. An average adult has about 5 L blood n the body. Unlke the defnton of volume n chemcal reacton engneerng where a compound s dstrbuted throughout a fed volume, n pharmacoknetcs the avalable space for a drug compound dstrbuton n the body can change due to the dsease state, physologcal condton and physco-chemcal propertes of the drug. Therefore, a dfferent measure s requred to well defne the space n whch the drug s dstrbuted n the body (Benet, 00). Volume of dstrbuton s the measure that descrbes the apparent volume nto whch a drug dstrbutes n the body at equlbrum condton. Ths mportant PK has no physologcal realty, rather represents a fcttous volume to dentfy the etent of drug dstrbuton n tssues. Theoretcally the dstrbuton of a drug cannot eceed the total body water (vascular flud, etracellular and ntracellular flud); however, dependng on the physco-chemcal propertes of the drug (.e. proten bndng), dsease state and physologcal condtons, the volume of dstrbuton can be smaller or much larger than the total body water 5

19 volume. For eample, f the drug s poorly plasma proten bound, t s hghly dstrbuted n the body tssues rather than the plasma; therefore, the apparent volume of dstrbuton wll be larger than the blood volume n the body. The apparent volume of dstrbuton s a proportonalty rato calculated by defnton as: X IV Vd (.) p where V d s the apparent volume of dstrbuton (L), X IV s the ntravascular (IV) dose (mg) and p s the plasma concentraton at dstrbuton equlbrum (mg/l). The volume of dstrbuton s ndependent of drug elmnaton. Snce systemc clearance (see secton..3), whch s the measure of elmnaton effcency, and volume of dstrbuton are bascally ndependent of each other, the volume of dstrbuton at steady state can be defned based on the mean resdence tme (MRT) of the drug n the body as (Benet, 00): V ss l MRT (.) s where V ss s the volume of dstrbuton at steady state (L), l s s systemc clearance (L/h) and MRT s the mean resdence of the drug n the body (h). onceptually, the volume of dstrbuton depends on several physologcal determnants ncludng blood volume, blood flow, partton coeffcent (see secton...3) and proten bndng. The greater the blood volume and a larger blood flow n a tssue allows a larger amount of drug to be avalable to the tssue and a rapd drug presentaton to the tssue. These factors enhance the etent and the rate of the drug dstrbuton n the tssue. Also, ncreased drug lpophlcty results n a larger drug parttonng n the tssue. Increased lpophlcty can lead to more etensve dstrbuton of the drug n the tssue n the presence of suffcently hgh blood flow and volume n the tssue. When the equlbrum between the plasma and the tssue s 6

20 establshed, the unbound fracton of the drug n the tssue can nfluence the volume of dstrbuton. Larger unbound fracton of the drug n the tssue leads to the drug transfer from the tssue to the plasma and t reduces the etent of drug dstrbuton n the tssue.... Blood proten bndng and unbound fracton A drug can undergo bndng to dfferent blood consttutes ncludng red blood cells, whte blood cells and plasma protens. Therefore, dfferent knds of drug concentraton can be defned as blood drug concentraton, plasma drug concentraton and unbound drug concentraton. In terms of PK processes the only fracton of drug avalable for crossng bologcal membranes s the free or unbound drug n blood. Therefore, ncreases n the unbound drug fracton (rato of unbound drug concentraton to total drug concentraton) results n greater dstrbuton of drug nto tssues, whch ncreases the volume of dstrbuton. Due to the hghly sgnfcant varablty of plasma proten bndng, a large varaton n the volume of dstrbuton can be observed for a sngle drug. In addton, dseases and drug-drug nteractons can nfluence the plasma proten bndng of a drug whch consequently alters the volume of dstrbuton of the drug. The clncal term of drug unbound fracton s defned as the rato of the unbound drug concentraton to the total drug concentraton. If the unbound fracton of a drug s smaller than 0., the drug wll be consdered hghly proten bound and senstve to changes n proten bndng; however, f the unbound fracton s larger than 0.8, clncally any changes n proten bndng can lead to nsgnfcant nfluences on dsposton and elmnaton processes (see secton..5..3). Another measure whch descrbes the etent of bndng of a drug n the blood cells s bloodto-plasma concentraton rato whch s defned as the rato of the drug concentraton n blood cells to that n unbound plasma. Snce hematocrt s defned as the volumetrc fracton of blood 7

21 cells wth respect to the blood volume, the blood-to-plasma concentraton rato can be formulated by defnaton as: b H p (.3) f H u( B) where H s hematocrt, b s the total blood drug concentraton, p s the plasma drug concentraton, f u(b) s the unbound fracton and s the blood-to-plasma concentraton rato....3 Equlbrum and tssue partton coeffcent Drugs can be hydrophlc (soluble n plasma water and etra-ntracellular fluds) or lpophlc (plasma proten bound) n nature. Most lpophlc drug compounds ehbt reversble proten bndng nteractons nvolvng weaker chemcal bonds wth protens. Wthn the plasma or a tssue compartment, lpophlc drugs undergo reversble bndng wth the actve stes of the protens to reach an equlbrum between bound and unbound drug molecules. The equlbrum condtons determne the plasma unbound drug concentraton whch s the only form of drug molecules to traverse the membrane and dffuse nto tssues. The mgraton of plasma unbound drug molecules to the tssue depends on the degree of lpophlcty of molecules and the rate of presentaton of drugs to the tssue by the blood flow. For hgh lpophlc drugs, the lmtng factor s the blood flow whle for low lpophlc wth low solublty n the membrane the plasma-tssue echanged s a dffuson lmted process (Levtt, 00). The etent of the drug parttonng between the plasma and the tssue can be represented by the physologcal measure of partton coeffcent. Partton coeffcent s defned as the rato of drug concentraton n the tssue to the unbound drug concentraton n plasma when the tssue and the plasma are n equlbrum. Provdng suffcently hgh blood flow rate, the drug partton coeffcent can be used for determnaton of the etent of drug dstrbuton n a tssue; however, 8

22 partton coeffcent s unable to evaluate how quckly the drug compound s dstrbuted n the tssue. The rate of drug dstrbuton n a tssue reles on the perfuson rate and the dffusvty of the drug across the membrane...3 Systemc clearance concepts After a drug compound reaches the systemc crculaton, t s dstrbuted n tssues and organs some of whch elmnate the drug compound from the body. Drugs are mostly elmnated from the body ether n unchanged form (parent drug) or changed form (metabolzed drug). Dependng on the water solublty or lpophlcty, the drug can be ecreted by the kdneys or be metabolzed by the lver. The kdney s the most mportant organ for ecreton of polar drugs and ther metaboltes; however ecreton may also occur va the lungs, skn, mlk and ble. Lver s the maor elmnaton organ responsble for the metabolsm of lpophlc drugs and convertng them nto water soluble metaboltes whch are ecreted through the kdneys. Blary ecreton of parent drug or metaboltes may occur wth a potental for ther reabsorpton from the ntestne. In order to descrbe the effcency of the elmnaton process by the maor elmnaton organs, lver and kdney, a physologcal measure of clearance s defned. The concept of the clearance was orgnally ntroduced by Rowland et al. (973). Total body clearance or systemc clearance s defned as the volume of the blood cleared of drug per unt tme n the body. Hepatc and renal clearance are defned as the volume of the blood cleared from the drug per unt tme by the lver and kdneys, respectvely...3. Elmnaton rate and etracton rato The systemc clearance at steady state can be calculated based on the steady state drug plasma concentraton ( ss ), avalablty of the drug n the blood (F), admnstraton nterval () and the dose (X 0 ) as follows (Rowland et al., 973): 9

23 l s FX 0 (.4) ss Rowland et al. (973) and Wlknson and Shand (975) ntroduced the concept of organ elmnaton rate defned as the product of the blood flow rate by the drug concentraton dfference between the arteral and venous blood of the organ as: Q A V (.5) s the elmnaton rate, Q s the perfuson rate to the elmnaton organ, A and V are the drug concentratons n arteral and venous blood, respectvely. Dvdng Eq. (.5) by the arteral drug concentraton leads to the followng equaton descrbng the organ clearance as: l organ Q A A V (.6) where ( A - V )/ A s called the etracton rato (ER) and defned as the fracton of arteral drug elmnated by the organ. The ER s bascally a functon of blood perfuson rate of the organ, the ntrnsc ablty of the organ to elmnate the drug, and the unbound fracton of the drug n the blood. Borrowng the concept of well strred reactor model, Wlknson and Shand (975) assumed the lver as a homogenous reactor vessel n order to ncorporate the above physologcal factors nto the conceptual defnton of hepatc clearance. A physologcally defned hepatc clearance was then ntroduced as: l H Qf u(b) lnt (.7) Q f l u(b) nt where l H s the hepatc clearance and l nt s the ntrnsc clearance representng the mamum enzymatc metabolsm capacty of the lver. Hepatc clearance has the central role n elmnatng 0

24 lpophlc drugs. As t was dscussed, the cell membranes are lpd n nature and lpd soluble drugs can readly traverse the membranes of tssue cells. In the same way, lpophlc drugs can easly enter the lver cells and undergo hepatc metabolsm the effcency of whch s gven by the hepatc clearance...3. Admnstraton routes and pre-systemc elmnaton Drug admnstraton can be performed through dfferent etravascular routes, such as oral, rectal, subcutaneous, ntramuscular, or ntravascular admnstratons. Dependng on the ste of admnstraton, the fracton of the drug that reaches the systemc crculaton (.e. boavalablty) can vary. Fgure. llustrates a schematc dagram of the blood crculaton system and the maor stes of the drug admnstraton. Admnstraton stes and represent IV admnstraton stes, and ste 3 ndcates the gastrontestnal absorpton of a drug admnstered orally. As t can be seen, after the blood s crculated through the pulmonary crculaton, t s pumped nto the systemc crculaton whle the oygenated blood s drected to the gastrontestnal tract where drug and nutrent absorpton takes place. Then the blood perfuses the lver where drug metabolsm occurs. The blood leavng the lver through the hepatc ven ons the venous blood comng from the general crculaton n body and s conducted towards the heart. If the drug s admnstered at Ste, provdng that the lung has no contrbuton to the drug elmnaton, samplng from the ven assocated wth the general crculaton can lead to a good estmaton of systemc clearance. Lkewse, f the drug s admnstered at Ste representng ntra-arteral admnstraton, t wll be dstrbuted n the body durng ts frst passage wthout undergong any elmnaton process. However, oral admnstraton of a drug s followed by drug absorpton n gastrontestnal tract, Ste 3. If the admnstraton occurs at Ste 3, the drug s absorbed nto the portal ven that drects the blood to the lver. The portal blood contanng the drug compound

25 perfuses the lver where a fracton of the drug s metabolzed before t reaches the systemc crculaton. Elmnaton durng the frst passage of the drug through the lver s called the frst pass effect whch causes an error n the estmaton of systemc clearance. In addton, the drug may be subected to metabolsm n gastrontestnal tract lumen, gastrontestnal tract mucosa, the ntestnal and the portal ven further reducng the fracton of parent drug reachng the systemc crculaton. Therefore, dependng on the admnstraton and samplng stes, a seres of dfferent organs may contrbute to the frst pass effect. Fgure.. Schematc dagram of the blood crculaton and three maor admnstraton stes. Obvously the lver has a central role n ADME processes as t s quanttatvely and qualtatvely a very crucal ste of drug metabolsm. The frst pass effect caused by the lver can sgnfcantly nfluence the plasma concentraton and the effcacy of the drug. onsequently, t s requred to have an nsght nto the lver anatomy and physology as well as the mechansm of hepatc elmnaton process.

26 ..4 Lver anatomy and physology Lver s the largest sold organ n the body and serves a crtcal functon n blood detofcaton, drug metabolsm, ble ecreton for fat breakdown, blood sugar level regulaton and cholesterol metabolsm regulaton. Lver s mostly stuated n the rght sde of the abdomnal cavty ust below the daphragm. For adults t weghs about 600 g n average wth an average volume of 60 and 34 ml, for males and females, respectvely (Anderson et al., 000). Under normal condtons n human, a quarter of the cardac output (between 00 and 500 ml/mn) flows through the lver every mnute. The lver blood s suppled by two man sources - portal ven and hepatc artery. Appromately 80% of the hepatc blood flow s suppled by the portal ven contanng partally deoygenated but nutrent-enrched blood mostly orgnatng from the gastrontestnal tract (Garcea and Maddern, 009). The rest of the hepatc blood flow s suppled by the hepatc artery contanng fully oygenated blood comng from the celac trunk and descendng aorta. The regulaton of the hepatc blood flow s performed by controllng the hepatc arteral flow (Bonfglo et al., 00). The blood supply from portal ven and hepatc artery are med n the lver and are eventually draned nto the hepatc ven leavng the lver to travel to the heart. Fgure. llustrates a schematc dagram of the cross secton of the lver tssue. A mcroscopc vew nto the lver tssue reveals that the lver s composed of heagonal crosssectonal unts called acn (Fg..a). Each heagonal unt, acnus, s mostly made hepatocytes radally arranged n thn layers from nsde to the outsde (Fg..b). The blood from the branches of hepatc artery and portal ven are med n capllares called snusods through whch the blood flows to the central ven. Snusods are vascular channels made of a thn layer of endothelal cells whch are separated from the underlyng hepatocytes by space of dsse. As the blood flows n snusods, the blood s fltered by the endothelal cells so that oygen, drug 3

27 compounds and/or toc substances dffuse nto the space of dsse from whch mass transfer takes place nto the hepatocytes (Bonfglo et al., 00; Teutsch, 005). Metabolsm takes place n hepatocytes by Phase I and Phase II enzyme-medated processes. Ble, the secreted product of hepatocytes, s ecreted nto a network of ble canalcul that conduct the ble to the ble ducts from whch the ble s eventually draned nto the gall bladder. Eventually, ble enters the promal duodenum upon stmulaton by the consumpton of a meal. Fgure.. Schematc dagrams of the lver heagonal unts (a) and the mcrostructure of an acnus...5 Hepatc clearance The hepatc clearance s defned as the volume of the blood that perfuses to the lver and s cleared of drug compound per unt tme. Hepatc drug elmnaton results from the drug metabolsm and/or blary ecreton of drug n the lver. Drug metabolsm s a process by whch a drug s chemcally changed to a metabolte wth concomtant loss n pharmacologcal actvty. Blary ecreton of a drug occurs due to the concentraton gradent of unbound drug across the hepatocytes such that a hgher plasma unbound drug concentraton can enhance the secretary 4

28 transport of the drug n the lver. For lpophlc drugs whch are not suffcently ecreted by kdneys, hepatc metabolsm alters them nto more water soluble compounds to be elmnated n urne and/or ble (Nebert and Russell, 00). For many drugs lver s the most mportant elmnatng organ such that t urges us to dentfy the physologcal determnants attrbuted to the hepatc drug uptake and removal governng the hepatc clearance. These physologcal determnants are hepatc blood flow, proten bndng of the drug to the blood, nherent ablty of the lver for elmnatng the drug (ntrnsc clearance), and hepatc transport medated uptake (Pang and Gllette, 978). Based on how the physologcal determnants nfluence the hepatc clearance, the drug can be hgh or low n hepatc ER (secton..3.) whch s defned as the fracton of the drug at the lver nlet that s elmnated due to the hepatc clearance. A hgh or low ER drug s a drug whch s hghly or poorly elmnated from the blood, respectvely, as t s passng though the lver. The nterrelatonshp of the physologcal determnants of hepatc clearance determnes low or hgh hepatc ER. For eample, f the nherent ablty of the lver for elmnatng a drug s poor, the hepatc ER of the drug remans low even f the drug presentaton to the lver s hgh due to a hgh hepatc blood flow. Lkewse, the hepatc ER of the drug can be stll low f the drug s poorly delvered to the lver despte a hgh nherent ablty of drug metabolsm by the lver. Below the rate lmtng factors nfluencng the hepatc clearance wll be elaborated...5. Intrnsc clearance Intrnsc clearance s an ndcaton of nherent enzyme actvty n hepatocytes. Intrnsc clearance represents the mamal ablty of hepatocytes to rreversbly elmnate unbound drug molecules from lver water assumng blood flow, proten bndng and cell membrane 5

29 permeablty are not rate lmtng. Therefore, unbound ntrnsc clearance can eceed the hepatc clearance n most cases. Generally the hepatc metabolsm of drugs s medated by hepatc Phase I and Phase II enzymatc metabolsm n hepatocytes. The enzymatc metabolsm s performed through odaton/reducton, hydrolyss and nactvaton of functonal group of the parent drug. Phase I enzymes are responsble for odaton/reducton and hydrolyss whle n Phase II pathway an endogenous molecule s conugated to the functonal group of the parent drug molecule beng functonally nactvated. Another mportant functon of Phase II enzymes s the transformaton of reactve molecules that may be produced by Phase I drug metabolsm (Park et al., 005). The metabolzng P450 enzymes and some Phase II enzymes are located n the smooth endoplasmc retculum wthn the hepatocytes (Ortz de Montellano, 995). Hepatocellular metabolsm can be nfluenced by drug-drug nteractons and genetc polymorphsms that cause nterndvdual varablty n hepatc drug metabolsm (Trona et al., 003; Martnez-Jmenez et al., 005). In many cases, Mchaels-Menten knetcs descrbes the metabolsm of a drug and ntrnsc clearance, then, can be quanttatvely determned as: l nt ma (.8) K V M u where V ma s the mamum enzyme actvty capacty, K M s the nverse functon of affnty between the drug molecules and enzyme, and u s the unbound plasma drug concentraton. V ma s a functon of the enzyme concentraton at the metabolzng ste whle the affnty term (K M ) s defned as the unbound drug concentraton that leads to half of the mamum enzymatc metabolsm rate. In other words, the affnty term represents the unbound drug concentraton causng half of the actve stes of metabolc enzymes to be saturated. Snce t s mpossble to 6

30 measure the unbound drug concentraton at the enzyme, the estmaton of K M s made based on the unbound plasma concentraton, u. Accordng to Eq. (.8) for low unbound drug concentraton, the ntrnsc clearance can be assumed as a constant value of the rato of V ma /K M. However, f the affnty of a drug to the metabolc enzymes n hepatocytes s hgh or the concentraton of drug s hgh, the value of K M wll be neglgble compared to the unbound drug concentraton resultng n a concentraton dependent ntrnsc clearance defned as the rato of V ma / u. In ths case, ntrnsc clearance decreases as the unbound drug concentraton ncreases. Ths s because the actve stes of the metabolzng enzymes approach saturaton when unbound concentraton ( u ) eceeds K M. Snce dfferent enzymes can be responsble for metabolzng a partcular drug, the overall metabolsm process can be defned as: l nt N K V ma, n n M, n u, V (.9) where n represents the n th ndvdual metabolc enzyme n hepatocytes and u,v s the unbound drug concentraton n the hepatc ven. Usng the defnton of ntrnsc clearance smply gven n Eq. (.8), the hepatc metabolsm rate can be defned as: V ( ma l nt f u B) p f u( B) p (.0) K M u where s the rate of metabolsm by the hepatocytes and p s the total drug concentraton n plasma. In case of good drug delvery to the lver tssue (due to suffcent blood flow) and suffcently hgh unbound fracton of the drug n the blood, f the drug transfer to hepatocytes s not lmted by the hepatc tssue membrane permeablty, the ntrnsc clearance wll be the rate lmtng factor of the hepatc clearance. In ths case, f the ntrnsc clearance s low, the metabolsm of 7

31 the drug remans low despte the presence of the drug compounds. Therefore, hgh or low ntrnsc clearance determne hgh or low hepatc ER of the drug...5. Hepatc perfuson rate Provdng that the drug s poorly bound wth blood cellular components and plasma protens (hgh unbound fracton), that passve dffuson or transporters adequately medate drug transport across cell membranes, and that the hepatocellular enzyme actvty s nherently hgh, the drug presentaton to the hepatocytes, governed by the blood flow rate, wll be the determnstc factor of the hepatc clearance. Thus, the faster the drug s suppled to the lver the hgher rate of drug elmnaton by the lver. Therefore, the perfuson rate governs the hepatc elmnaton rate and consequently wll be the lmtng factor of the hepatc clearance. In ths case, hgher blood flow results n hgher hepatc ER (secton..3.); however, f the ntrnsc clearance s low, the hepatc ER wll be low despte a hgh blood flow. In general, f other rate lmtng processes ests, the blood perfuson rate wll no longer be the lmtng factor of the hepatc clearance Drug unbound fracton For drugs wth low hepatc ER (secton..5) whch can be assocated wth low ntrnsc clearance and poor transport medated drug uptake, the unbound drug fracton becomes a rate lmtng factor of hepatc clearance because wth a poor transport medated drug uptake, only unbound drugs can traverse the membranes and be metabolzed. Equaton (.0) also ndcates that an ncrease n unbound fracton has a more sgnfcant role n enhancng the metabolsm rate when the ntrnsc clearance s low. In contrast, f the hepatc ER of a drug s nherently hgh, t ndcates that a sgnfcant fracton of the drug at the lver nlet s well elmnated from the blood. Therefore, even f the proten bndng of the drug to the blood s hgh, t wll not be a ratelmtng factor. As the free drug s quckly metabolsed n hepatocyes, unbound drug compound 8

32 s transferred from the plasma to the tssue. It results n a departure from equlbrum n the plasma where a fracton of bound drug dssocates nto unbound form. The unbound drug wll be ready to traverse the membrane to be avalable to the enzymes. onsequently, for hgh ER drugs, changes n unbound fracton of a drug have no nfluence on the hepatc clearance. For the cases where metabolc ER s ntermedate, a regon of the lver may ehbt hgh metabolc ER whle another regon poses low metabolc ER due to non unformty of the dstrbuton of enzymes and ther actvty. In ths case, hepatc metabolsm rate can be senstve to the unbound fracton n the regons wth low ER whle nsenstve to the unbound fracton n the regons wth hgh ER Transport medated uptake The hepatc metabolsm rate can be lmted by hepatc uptake transporters. A hepatc ntrnsc clearance whch ncludes the concepts of ntrnsc clearance, nflu and efflu membrane permeablty can be descrbed as (Pang et al., 978; Yamazak et al., 996): l n _ overall l nt PSu,nflu (.) lnt PSu, efflu where PS u,nflu and PS u,efflu are membrane permeablty surface area of unbound drugs across the snusodal membrane for the nflu and efflu processes, respectvely. If the efflu s neglgble compared to the ntrnsc clearance, the overall ntrnsc clearance s gven as: l PS n _ overall u,nflu (.) In ths case, the hepatc uptake transporters wll be rate lmtng factor and subsequently determne the net ntrnsc clearance and hepatc metabolsm rate. It mples that even f the metabolc enzyme actvty n hepatocytes s very hgh, the hepatc metabolsm rate can be low f uptake transporters poorly perform the transport process. In contrast, f the efflu s consderably larger than the ntrnsc clearance, the overall ntrnsc clearance can be gven as: 9

33 l n _ overall PS l nt u,nflu (.3) PSu, efflu In ths case, both nflu and efflu processes affect the overall ntrnsc clearance. In case of the equalty of nflu and efflu processes wth rapd permeaton of the drug across the snusodal membrane, the hepatc transporters have no rate lmtng effect on the hepatc clearance where the net ntrnsc clearance wll be equal to the ntrnsc clearance...6 Boavalablty It was dscussed n Secton..6, the oral admnstraton s followed by the drug absorpton from the ntestnal tract nto the portal ven that transports the drugs to the lver. A fracton of the drug can be potentally metabolzed by the enzymes n the ntestnal wall cells as well as n the blood of the portal ven. However, the maorty of the drug s subect to the hepatc metabolsm through the frst passage across the lver. Followng ntestnal absorpton the concentraton of the drug n the portal ven s consderably hgh and the fracton of the drug whch s subected to hepatc elmnaton on the frst pass through the lver can be sgnfcant. The fracton of the absorbed drug whch reaches the systemc crculaton s called hepatc boavalablty (F H ) and defned as: F H A V ER (.4) A Equaton (.4) s based on the assumpton of full perfuson of the portal ven n the lver. Equaton (.4) mples that f a drug s effcently etracted by the lver, a small fracton of the drug may reach the systemc crculaton f t s admnstered orally. For ths type of drug, hepatc dsease or drug-nduced alteratons, whch reduce the hepatc elmnaton effcency, can sgnfcantly ncrease the systemc boavalablty. 0

34 Oral boavalablty s defned as the fracton of the dose that reaches the systemc crculaton. Oral boavalablty takes nto account the unabsorbed fracton of the dose n the ntestne as well as the drug loss due to the metabolsm occurrng n the ntestne wall, portal ven and lver untl the drug reaches the systemc crculaton..3 Physologcal model for hepatc drug elmnaton Physologcal models are mathematcal equatons whch can descrbe the mechansms of the processes n the body based on degrees of physologcal and physcal smplfcatons and assumptons of the process. Unlke emprcal models whch are manly based on the measured data, physologcal/mechanstc models are developed based on the physcs and the bology behnd the phenomena. Physologcal, chemcal and physcal propertes and parameters are ncorporated n a mechanstc model so that the model provdes us a better nsght nto a process occurrng n the body compared to knetc and emprcal models. The lver possesses a complcated physology and structure and has been an attractve organ for those researchers who were tryng to mathematcally descrbe hepatc functonalty for predcton of hepatc drug elmnaton. A physologcally predctve model can allow us to analyze the nfluence of physologcal, pharmacologcal and pathophysologcal events on the effcency of the hepatc drug elmnaton. In addton, such a model can be used for estmatng PK parameters of a new drug durng the drug dscovery process. Several physologcal models have been proposed to descrbe the hepatc clearance. The four well known models are wellstrred (WS), parallel tube (PT), dstrbuted snusodal perfuson (DSP), and dsperson (DP) models whle some other conceptual models have been proposed recently based on dfferent sets

35 of assumptons and theores (Nestorov, 007; Pang et al., 007; Sun and pang, 00). The prncples of the models wll be revewed n ths secton..3. Well-strred model WS model, whch was nspred from the concept of well-med reactor for petroleum crackng n chemcal reacton engneerng, was frst proposed by Gllette (97) and then establshed by Rowland et al. (973) and Wlknson and Shand (975). WS model assumes that the lver s equvalent to a perfectly med reacton vessel whch s contnuously strred so that the fresh blood enterng the lver nstantaneously mes wth the blood n the lver. Accordng to WS model the drug s homogenously dstrbuted n the lver causng no concentraton gradent across the lver. onsequently, mass balance over the lver at steady state results n (Rdgway et al., 003): V H A (.5) Q H Q f l u(b) nt where V and A are the drug concentraton at the lver nlet and outlet, respectvely, and Q H s the perfuson rate. Applyng Eq. (.5) to the defnton of the hepatc clearance and boavalablty results n boavalablty and hepatc clearance mathematcal epressons as: F H f l u(b) Q H nt (.6) l H QH fu( B) lnt (.7) Q f l H u( B) nt Although the WS model s based on an dealzed stuaton whch makes the model oversmplfed, t s easy to use and understand and can be easly appled for rough estmaton of rate lmtng factors. For eample, f the ntrnsc clearance s much smaller than the blood flow rate, the denomnator of Eq. (.7) reduces to Q H and Eq. (.7) s smplfed to hepatc

36 clearance as a functon of ntrnsc clearance and unbound fracton. In ths case for a reasonably hgh unbound fracton, the enzyme actvty wll be rate lmtng factor of the hepatc clearance. On the other hand, WS model s only vald for a steady state condton and unable to take nto account the comple network of the vascular anatomy of the lver and some mportant physochemcal propertes such as tssue partton coeffcent..3. Parallel tube model The concept of parallel tube (PT) model was nspred from the plug flow reactors n chemcal reacton engneerng. PT model became a better representaton of the lver compared to WS model by assumng the snusods as parallel tubes wth equal blood velocty and ntrnsc clearance (Iwatsubo et al., 996). PT model assumes that the drug compounds n the blood at the lver nlet enter the lver at the same tme and then travel through the lver wth a constant and equal velocty n parallel cylndrcal tubes (Roberts and Rowland, 986). Based on PT model, mass balance of the drug at steady state results n a nonlnear drug concentraton gradent functon as (Nro et al., 003): d Q dz fu ( B) lnt (.8) L Accordng to the analytcal soluton of Eq. (.8), boavalablty and hepatc clearance are defned as: F H l H fu( B) lnt ep (.9) QH Q H fu( B) lnt ep (.0) QH Smlar to WS model, the PT model can be easly used as a useful tool to assess the relatonshp between the hepatc clearance and unbound fracton, ntrnsc clearance and the 3

37 hepatc blood flow (Wlknson and Shand, 975). Although PT model takes nto consderaton the non-unform drug dstrbuton n the lver and also offers a pcture of the blood flow n the lver, t lacks the mportant transport mechansms of dffuson, convecton and dsperson for drug transport across hepatocellular membranes. In addton, both WS and PT model eclude the nfluence of the comple vascular network estng n the lver..3.3 Dstrbuted snusodal perfuson model DSP model s the physologcally etended verson of PT model. Unlke the PT model, the DSP model represents snusods as non-dentcal parallel tubes allowng the blood flow to pass through the lver whle each tube contans a fracton of the hepatc blood flow (Iwatsubo et al., 996; Bass et al., 978). Accordng to DSP model, the hepatc boavalablty derved for the PT model must be epressed by each ndvdual tube wth the assocated fracton of the hepatc blood flow rate. Snce the tubes are not dentcal, each tube suggests ts own ntrnsc clearance as the assocated dsperson and flow mng effects all of whch offer the lver heterogenety whch s defned as (Bass et al., 978): N l l Q Q nt, nt H, H (.) where l nt, and Q H, are ntrnsc clearance and blood flow rate assocated wth each ndvdual tube, s the tube nde and N s the number of tubes. Subsequently, the flow-weghted boavalablty s defned as (Bass et al., 978): F H lnt l nt ep (.) QH QH 4

38 Although the DSP model ncorporates a more realstc pcture of the lver blood flow compared to PT and WS models, t s unable to characterze the dstrbuton of resdence tme of the drug n the lver. In addton, smlar to WS and PT models, the DSP model s vald at steady state..3.4 Dsperson model Incorporatng the aal dsperson and convecton of blood flow n cylndrcal tubes that represent the snusods, the DP model has been wdely accepted as a physologcal based model of the lver (Roberts and Rowland, 987). The DP model s based on the resdence tme dstrbuton of the drug n the lver (Roberts and Rowland, 986). Assumng a very small volume much smaller than the organ volume but much larger than the mean free path of molecules, the DP model was developed accordng to the varaton of the amount of tme spent on respectve volumes n the organ (Roberts and Rowland, 986). The resdence tme s nfluenced by the convecton and dsperson effects of the blood flow. Aal dsperson s a maor mass transport mechansm n the lver where the snusodal blood velocty s suffcently hgh to cause a nonunform snusodal blood velocty and subsequently non-unform resdence tme dstrbuton n the lver. Péclet number s a dmensonless number representng the magntude of convecton effect over the dsperson effect of a flud flow. The nverse of Péclet number s called dsperson number (D n ) as an ndcaton of the degree of mportance of the dsperson n the drug transport n the lver. Dsperson number s defned as: D n Pe D d ul (.3) Where Pe s Péclet number, d D s the aal dsperson coeffcent, and u s the lnear velocty along the length of L of the cylndrcal tubes. When the dsperson number goes to nfnty, the mng frequency of the blood becomes so hgh that the system behaves smlar to WS model. 5

39 On the other hand, f the dsperson number goes to zero, the system ehbts the behavour of PT model (Nro et al., 003). Assumng that the ntrnsc clearance s ndependent of the drug concentraton, Rowland and Roberts (986) presented the varaton of resdence tme dstrbuton of the drug n the lver at steady state as (Roberts and Rowland, 986): d d QH Dn Q lnt 0 H (.4) dz dz where s the plasma drug concentraton normalzed to the rato of the drug dose to the volume of dstrbuton, and Z s the dmensonless poston along the lver. For the analytcal soluton of Eq. (.4) the Dankwerts boundary condton was defned as (Dankwerts, 95): D d dz n 0 d dz at n Z at Z 0 (.5) The analytcal soluton of Eq. (.4) led to the hepatc boavalablty epresson as (Dankwerts, 95): F H ( a) 4a ( a ) ep ( a) Dn ( a) ep Dn (.6) where a s defned as: / 4DN fu( B) lnt a (.7) QH An etenson to DP model has been made by ncorporatng a second vascular compartment to the nterconnectng snusods (Roberts and Anssmov, 999). Although DP model takes nto account the physologcal characterstcs of snusodal blood dstrbuton and the assocated resdence tme of the drug n the lver, t requres a general assumpton whereby no parameter 6

40 vares along the lver and also the ntrnsc clearance remans ndependent of drug concentraton. In addton, the soluton of DP model s vald for a steady state appromaton of unbound drug concentraton n hepatocytes..3.5 Interconnected-tubes model Anssmov et al. ntroduced the concept of nterconnected-tubes (IT) model by modelng the hepatc elmnaton process usng a large number of parallel and nterconnected tubes nterchangng the blood flow along the length of the lver (Anssmov et al., 997). The IT model takes nto account the ntermng of the blood flow between snusods by a contnuous and constant nterchange of the drug between a set of parallel tubes at a steady state of hepatc elmnaton process. The IT model ncludes a heterogenety epresson whch characterzes the combned effects of non-unform dstrbutons of enzymes and flow rates between dfferent snusods and also the ntermng effects of the blood flow between snusods (Anssmov et al., 999). The IT model at steady state s mathematcally formulated as: d V K e M (.8) d where V s the dagonal matr of blood velocty assocated wth parallel tubes, K e s the dagonal matr of the elmnaton rate constant, and M s the matr of coeffcents of echange between tubes. Accordng to IT model, when the parallel tubes are poorly nterconnected, the predcted drug concentraton at the outlet s smlar to that n DSP model and DP model for small dsperson number values. Although IT model conceptualzes the nterconnectvty of snusods and ntermng of blood flow n the lver, t lacks the physologcal concept and parameters nvolved n hepatc clearance (.e. membrane permeablty, ntrnsc clearance, tssue partton coeffcent) and 7

41 physo-chemcal propertes of the drug (.e. unbound fracton). In addton, the elmnaton coeffcent s assumed constant for all the tubes along the tube lengths..3.6 Tanks-n-Seres model Murray et al. (987) treated the lver as a seres of adustable number of compartments (tanks) that allow the clearance to be dependent on blood flow. The model performs as a brdge between WS and PT models. When the number of tanks approaches, the model results become dentcal to those of WS model. For large number of tanks the model ehbts the PT model (Gray and Tam, 987). The model at steady state leads to the lver outlet drug concentraton and the boavalablty, respectvely, as (Gray and Tam, 987): N l N nt out n NQ (.9) H F H l NQ H N nt (.30) where N s the number of tanks. For the non-lnear elmnaton knetcs, the model s reformed as: a a a 4 K M K M V Q H ma N / (.3) where K m and V ma are the affnty and mamum metabolsm rate capacty terms. Lke WS and PT model, the tank-n-seres model s unable to physologcally descrbe the effect of dsperson and convecton and ntermng effects of blood flow n the lver. However, the key advantage of such compartmental models s the smplcty of the equatons. Also, 8

42 compartmental models can be easly reformulated for more comple stuatons such as heterogeneous enzyme dstrbuton n the lver..3.7 Recent model orentatons In recent years, researchers have focused on etendng the models so that the heterogenety n blood flow, enzymes, and transporters are ncluded. Lu and Pang (006) adopted an ntegrated approach to predct hepatc drug clearance n such a way to nclude the heterogenety n enzymes and transporters. Fan et al. (00) utlzed physologcally based pharmacoknetc ntestnal and lver models to predct the contrbutons of enzymes and transporters on ntestnal avalablty and the hepatc avalablty of the drug. They ncluded the mpact of the nflu and efflu transport processes to evaluate the ntestne and the lver clearances (Fan et al., 00). The use of the physologcal models becomes dffcult when unsteady state and nonlnear hepatc pharmacoknetcs and enzyme heterogenety est. Hsaka and Sugyama (998) solved the fundamental equaton of DM at nonlnear and unsteady state hepatc elmnaton of substances usng eplct fnte dfference method whch s less accurate than the mplct method. They also ncorporated ther method nto a nonlnear least-squares fttng algorthm to estmate PK parameters. Ther numercal model was a resemblance of a seres of m compartments correspondng to the free or bound drug n the vascular space, blood cells, or Dsse space, or n the cells at varous radal dstances from the vasculature; however, the model was unable to nclude the equlbrum condton, represented by tssue partton coeffcent, between the tssue cells and the blood. Also the model was lackng the structural propertes of the lver tssue (.e. porosty, tortuosty). A promsng approach that has emerged recently to assst scentsts to analyze bologcal systems s the porous meda approach. Snce bologcal systems are made of a dspersed phase (.e. cells) n a contnuous phase of a flud (.e. blood or etracellular flud), they can be treated 9

43 as porous meda. In the same way, porous meda prncples can be appled to human tssues/organs where dspersed cells are separated by connectve vods that allow the blood/etracellular matr flow though the tssue. Porous meda approach has been successfully appled to some bosystems ncludng cartlage tssue engneered scaffold development (Izadfar et al., 0), rado frequency enhanced etracton of ant-cancer compounds from a bologcal system (Izadfar and Bak, 00), and thermotherapy and human thermoregulaton system (Sherar et al., 00; Sanyal and Ma, 00). The lver s a hghly perfused tssue whch can be approprately treated as a porous medum. harles et al. (989) ntroduced a three dmensonal fnte element model for the flud flow and mass transfer n the lver based on the prncples of porous meda. Assumng a Reynolds number smaller than, they adopted the prncples of porous meda where the blood flow n the lver was a creepng flow descrbed by Darcy s law. They developed the momentum equaton accordng to the hydraulc permeablty of the tssue. For the mass transfer equaton, they assumed the Peclet number to be greater than whereby the dsperson could be neglected due to the domnant convecton. The mass transfer governng equaton was unable to descrbe the hepatocellular drug metabolsm whch can be lnear or nonlnear. In addton, the model was unable to nclude the tssue-blood equlbrum of the drug speces but the drug transfer from the blood to the cells was descrbed based on the cell membrane permeablty. The lack of nformaton of tssue partton coeffcent, the aal dsperson and hepatocellular metabolsm n the model were sgnfcant downsdes of the proposed model. The followng secton brngs a bref overvew of some mportant porous meda concepts to the readers. It wll be followed by ntroducng the obectve of ths research whch s to apply porous meda concepts to mathematcally descrbe hepatc drug elmnaton process. 30

44 .4 Porous meda concepts and applcatons n bomedcal engneerng Porous medum s defned as a sold matr wth nterconnected vods flled wth a flud, lqud or gas or both. The sold matr can be partcles of the sol, the polymer matr of a tssue engneerng scaffold, a packed bed of fertlzer partcles, or a vascularsed tssue. A porous medum s generally characterzed by porosty, length scales, tortuosty and permeablty. Modelng of transport phenomena of porous meda made sgnfcant progress recently and the models have been used for dfferent applcatons n engneerng and scence. In ths secton the concepts and bologcal applcatons of porous meda are revewed and dscussed..4. Representatve elementary volume (REV) A porous medum s a sold matr consstng of a sold phase and spaces among sold partcles whch can be flled wth a flud, lqud or gas. The range of pore sze of porous meda can vary from molecular sze (nm) to centmetre. When a sold matr cannot be descrbed wthn a pore sze, a REV wth a characterstc length of l and volume of V l s defned to represent the structure of the sold matr. REV s defned as the smallest dfferental volume of a porous medum that results n statstcally meanngful average propertes of the porous medum (Darcy, 856). Fgure.3 schematcally and graphcally depcts the concept of a REV. As t can be seen, a very small volume located n the pore of the porous medum (Fg..3a) results n the flud volumetrc fracton of as ndcated n Fg..3b. As the sze of the volume ncreases, more sold fracton s ncluded n the volume causng the flud volumetrc fracton to decrease. As t s shown n Fg..3 b, the varaton of the flud volumetrc fracton fluctuates wth the sze of the volume; however, t becomes nsenstve to any ncrease n the sze of the volume when the volume of V l s suffcently large. The volume of V l, whch s called REV, s the smallest volume the property of whch can represent the property of the porous medum. 3

45 Fgure.3. Schematc dagram of representatve elementary volume and the varaton of the medum property wth the sze of representatve elementary volume..4. Local volume averagng method (LVA) Averagng propertes of the medum over REV s called local volume average propertes of the medum. If a property of one of the phases of the REV s averaged over the volume of the phase, t s called ntrnsc phase averaged property. Averagng the governng equatons of mass, heat and momentum transfer over the REV wth the applcaton of local volume average propertes s called the LVA method to descrbe transport phenomena n porous meda..4.3 Length scales n porous meda and LVA valdty condton In order to apply the concept of LVA method to mathematcal modelng of transport phenomena n a porous medum, the length scales of the medum must satsfy the valdty condton of LVA method. Fgure.4 llustrates the characterstc lengths of a typcal porous medum. The characterstc lengths of a porous medum are: the lnear dmenson of the porous medum system (L), the REV characterstc length (l), the pore sze (d p ), and Brnkman screenng dstance (K / ), whch s an ndcaton of the boundary layer thckness of the flud flowng n pores of the medum. 3

46 Fgure.4. Schematc dagram of a porous medum and the assocated length scales. The valdty condton of LVA method s: K / <<dp<l<l (.3) where K s the permeablty of the porous medum. Permeablty s an mportant property and a measure of the flow conductvty n the porous medum. Accordng to Darcy law, the flud velocty s lnearly related to the pressure gradent across the porous medum wth a proportonalty constant of the rato of permeablty to the flud vscosty (Darcy, 856). A porous medum s also characterzed by tortuosty whch s defned as the rato of the length of the tortuous path to the straght length between both ends of the porous medum. A plan medum (a non-porous medum) possesses a tortuosty of whle for a porous medum the tortuosty s greater than. Tortuosty has a great mpact on the molecular dffuson and heat conducton across a porous medum..4.4 Applcaton of LVA method to bologcal systems LVA method has been used for developng mechanstc models of numerous bologcal systems. Izadfar et al. (0) appled LVA method to develop a mathematcal model for mass transfer of nutrents and chondrocyte prolferaton n a cylndrcal cartlage scaffold. These 33

47 researchers developed two sets of partal dfferental equatons for glucose transfer across and cell prolferaton porous scaffold based on the LVA method. The governng equatons were smultaneously solved usng numercal methods to analyze and predct the requred tme for cell seedng n bomanufacturng fabrcaton and superfcal cell seedng methods (Izadfar et al., 0). Izadfar and Bak (00) developed a dynamc mathematcal model of heat transfer to descrbe the rado frequency enhanced etracton of podophylloton from a porous bologcal packed bed based on LVA method. Ther model had very good agreement wth the epermental data ndcatng that the LVA based model could be successfully used for optmzng the boprocess. Sanyal and Ma (00) treated three layers of the skn and subcutaneous regon as porous meda n an attempt to fnd the analytcal and numercal soluton of boheat transfer equaton to descrbe the human thermoregulaton system under varable physologcal parameters and atmospherc condtons. Ncholson (00) developed a mass transfer model based on porous meda prncples to descrbe glucose and oygen transfer from the vascular system to the bran cells as well as the drug delvery to the bran. He characterzed the bran tssue and dffusongenerated concentraton dstrbuton by the porosty and tortuosty of the bran. Hs model revealed that an ncrease n the tortuosty and a decrease n the porosty have sgnfcant effects n reducng the effectve mass dffusvty of molecules n the bran (Ncholson, 00). Le et al. (998) developed a comple model for mass transfer phenomenon n transvascular echange and etravascular transport of flud and macromolecules for a sphercal tumor. They treated both tssue and the tumor as porous meda wth a Darcy velocty of the blood flow whle the ntersttal flud was assumed to obey Starlng law. Ther model proved to have good agreement wth observatons. 34

48 The lver s a hghly perfused organ wth a porous structure of heagonal unts called acn. The blood flow conductvty of the lver tssue s determned by the nterconnected snusods n the lver. Although hepatc drug elmnaton has been mathematcally descrbed by dfferent physologcal models, as dscussed n Secton.3, the mass transfer phenomena n the lver has not been descrbed from a porous meda vewpont usng LVA method. The nherently porous structure of the lver allows us to descrbe the hepatc drug elmnaton process based on a porous meda approach..5 Research obectve The man obectve of ths study was to develop and valdate a porous meda model based on local volume averagng (LVA) method for descrbng the tme dependent drug concentraton gradent across the lver as well as drug hepatc elmnaton rate, hepatc clearance and hepatc boavalablty. LVA method wll ntegrate the propertes and mass transfer equatons as well as the hepatocellular metabolsm characterstcs over the REV of the lver tssue. The man dfference between LVA method and the mult-dmensonal method proposed by harles et al. (989) s that LVA method wll nclude the aal/radal dsperson, local volume equlbrum between tssue and blood, and tme-space dependent hepatocellular metabolsm n the porous meda model whle the model proposed by harles et al. (989) ecluded these mportant events. In hapter, the theoretcal background of the porous meda approach wll be eplored. The lver structure wll be characterzed based on four characterstc length scales and LVA valdty condton wll be evaluated. Then, the mathematcal modelng procedure wll be presented from fundamental begnnngs where the mechansms of mass transport are elaborated n a dfferental element of the lver tssue. The governng partal dfferental equaton of drug transport and 35

49 nonlnear/lnear drug elmnaton processes wll be derved followed by the detals of the numercal soluton of the model. The stablty and consstency of the numercal soluton wll be eamned and smulaton results wll be compared to epermental data as well as the predcted values from other models. At the end of hapter, t wll be shown how the proposed model can be appled to perform the senstvty analyses of hepatc drug elmnaton as well as the drug dstrbuton n the lver wth respect to dfferent physologcal condtons. Ths study wll assess two maor hypotheses. The frst hypothess s that characterstc length scales of the lver tssue ncludng snusodal dameter (pore sze), acnus dameter and tssue equvalent length scale can satsfy LVA valdty condton such that LVA method can be appled for modelng the hepatc drug elmnaton process. The second hypothess s that the hepatc drug elmnaton process can be successfully descrbed by a mechanstc model whch s based on the porous meda approach of LVA. The valdty of the hypothess wll be assessed usng observatons reported by other researchers for eght dfferent drugs wth a wde range of ntrnsc clearance and unbound fracton. References: Anderson V., J. Sonne, S. Slettng and A. Prp Volume of the lver n patents correlates to body weght and alcohol consumpton. Alcohol & Alcoholsm 35: Anssmov Y.G., A.J. Bracken and M.S. Roberts Interconnected-tubes model of hepatc elmnaton. J. Theor. Bol. 88:89-0. Anssmov Y.G., A.J. Bracken and M.S. Roberts Interconnected-tubes model of hepatc elmnaton: steady state consderaton. J. Theor. Bol 99: Bass L., P. Robnson and A.J. Bracken Hepatc elmnaton of flowng substrates: the dstrbuted model. J. Theor. Bol 7:

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52 Nro R., J.P. Byers, R.L. Fourner and K. Bachmann Applcaton of a convectvedsperson model to predct n vvo hepatc clearance from n vvo measurements utlzng cryopreserved human hepatocytes. urr. Drug Metab. 4: Ortz de Montellano P.R ytochrome P-450: Structure, Mechansm and Bochemstry, nd ed. New York, Plenum Press. Pang, K.S. and J.R.Gllette.978. Knetcs of metabolte formaton and elmnaton n the perfused rat lver preparaton: dfferences between the elmnaton of preformed acetamnophen and acetamnophen formed from phenacetn. J. Pharmacol. Ep. Ther. 07: Pang K.S., M. Wess and P. Macheras Advanced Pharmacoknetc Models Based on Organ learance, rculatory, and Fractal oncepts. AAPS J. 9: E68-E83. Park B.K., N.R. Ktterngham, J.L. Maggs and et al The role of metabolc actvaton n drug-nduced hepatotocty. Annu. Rev. Pharmacol. Tocol. 45:77-0. Rdgway D., J.A. Tuszynsk and Y.K. Tam Reassessng models of hepatc etracton. J. Bol. Phys. 9:-. Roberts M.S. and M. Rowland A dsperson model of hepatc elmnaton:. Formulaton of the model and bolus consderatons, J. Pharmacoknet. Bopharm. 4: 7-6. Roberts M.S. and M. Rowland A dsperson model of hepatc elmnaton, based on the resdence tme dstrbuton of blood elements wthn the lver. J. Pharmacoknet. Bopharm. 4: Roberts M.S. and Y.G. Anssmov Dstrbuton Knetcs wth the Etended. onvecton- Dsperson Model. J. Pharmacoknet. Pharmacodyn. 7: Rowland M, L.Z. Benet and G.G. Graham learance concepts n pharmacoknetcs. J. Pharmacoknet. Bopharm. :

53 Sanyal D.. and N.K. Ma. 00. Thermoregulaton through skn under varable atmospherc and physologcal condtons. J. Theor. Bol. 08: Sherar M.D., A.S. Gladman, S.R.H. Davdson, J. Trachtenberg and M.R. Gertner. 00. Helcal antenna arrays for ntersttal mcrowave thermal therapy for prostate cancer: tssue phantom testng and smulatons for treatment. Phys. Med. Bol. 46: Shtara, Y., H. Sato and Y. Sugyama Evaluaton of drug drug nteracton n the hepatoblary and renal transport of drugs. Annu. Rev. Pharmacol. Tocol. 45: Sun H., K. S. Pang. 00. Physologcal modelng to understand the mpact of enzymes and transporters on drug and metabolte data and boavalablty estmates. Pharmaceut. Res. 7: Teutsch H The modular mcroarchtecture of human lver. Hepatology Phladelpha, PA. Trona R.G., W. Lee, B.F. Leake and et al The orphan nuclear receptor HNF4α determnes PXR- and AR-medated enobotc nducton of YP3A4. Nature Med. 9:0-4. Wlknson G.R., and D.G. Shand ommentary: a physologcal approach to hepatc drug clearance. ln. Pharmacol. Ther. 8: Yamazak M., H. Suzuk and Y. Sugyama. 996a. Recent advances n carrer-medated hepatc uptake and blary ecreton of enobotcs. Pharm. Res. 3: Yann S. and D.R. Thakker Prodrugs: Absorpton, Dstrbuton, Metabolsm, Etracton (ADME) Issues, Sprnger, New York. 40

54 HAPTER A POROUS MEDIA APPROAH FOR MEHANISTI MODELING OF DRUG ELIMINATION BY THE LIVER M. Izadfar a, O.D. Bak b, J. Alcorn c a Dvson of Bomedcal Engneerng, ollege of Engneerng, Unversty of Saskatchewan, Saskatoon, SK, anada b Department of hemcal and Bologcal Engneerng, ollege of Engneerng, Unversty of Saskatchewan, Saskatoon, SK, anada c Dvson of Pharmacy, ollege of Pharmacy and Nutrton, Unversty of Saskatchewan, Saskatoon, SK, anada Abstract Applyng local volume averagng method and local equlbrum to the lver as a porous medum, a governng equaton takng nto account lver porosty, tortuosty, permeablty, unbound drug fracton and hepatc tssue partton coeffcent, drug-plasma dffusvty, aal/radal dsperson and hepatocellular metabolsm parameters was developed. The governng equaton was numercally solved to predct changes n dug concentraton wth tme and poston across the lver and the hepatc clearance and boavalablty followng an ntravenous drug admnstraton. The predcted values of hepatc clearance and boavalablty had good agreement wth the reported observatons for hgh and low clearance drugs. As well, the model was able to successfully predct an unsteady state of hepatc drug elmnaton wth concentraton dependent ntrnsc clearance. When statstcally compared to the well-strred, parallel tube and dsperson models the proposed model suggested a smaller mean squared predcton error and very good 4

55 agreement to reported observatons for eght drugs. A senstvty analyss revealed that an ncrease n lver porosty results n a slght decrease n the drug concentraton gradent across the lver whle hgher tssue partton coeffcent values ncrease the concentraton gradent. The model also suggested that the boavalablty was senstve to the nteracton between unbound fracton and ntrnsc clearance. Ths study ndcates that the lver and the hepatc drug elmnaton can be successfully eplored from a porous meda vewpont and may provde better mechanstc predctons to drug elmnaton processes by the lver.. Introducton The lver plays a very mportant role n the elmnaton of drugs, toc substances, and harmful bochemcal products produced by the body. The lver receves nutrent rch but poorly oygenated blood from the ntestnes va the portal ven and oygenated blood from the hepatc artery accountng for 75% and 5% of total blood supply, respectvely. Both blood supples perfuse to each heagonal functonal unt called acnus n whch portal and arteral blood are med n the smallest acn vessels called snusods and n whch mass echange takes place between blood and hepatocytes. The lver s essental role n the mantenance of homeostass as well as drug and ton elmnaton n the body demands a detaled understandng of lver functon. Mechanstc models that effectvely descrbe lver functon can play an mportant role n understandng and predctng drug concentraton and hepatc metabolc performance. Dfferent physologcal models have been developed for the lver based on dfferent degrees of smplfcatons and assumptons. The well-strred (WS) model and the parallel-tube (PT) model are the two most commonly used models descrbng drug elmnaton by the lver (Pang and Rowland, 977). These models are 4

56 based on dealzed stuatons of blood flow and drug dstrbuton n the lver wth an mplct assumpton that the partton rato of free drug between snusodal blood and hepatocytes s constant. In the well-strred model, the drug s assumed to be nstantaneously and homogeneously med wth the blood n the lver resultng n very unform drug concentraton across the lver. Bass et al. (977) and Forker and Luon (977) developed a dstrbuted model representng blood flow n parallel tubes where each tube transports a volumetrc fracton of total blood flow as determned by a dstrbuton functon. Roberts and Rowland (986) developed a physologcal-based dsperson (DP) model whch was based on the resdence tme dstrbuton of the drug n the lver where the dstrbuton of resdence tmes depended upon the aal dsperson. The analytcal soluton of the DP model requres a general assumpton whereby no parameter vares along the length of snusods and ntrnsc clearance due to metabolc enzyme actvty or transporter functon are ndependent of drug concentraton. In addton, t assumes a steady state appromaton of unbound drug concentraton n hepatocytes. Although the DP model takes nto account hepatocellular permeablty of drugs, t fals to consder the hepatc tssue partton coeffcent and tssue structural characterstcs such as porosty and tortuosty. Hsaka and Sugyama (998) solved the fundamental equaton of DM at nonlnear and unsteady state hepatc elmnaton of substances usng eplct fnte dfference method whch s less accurate than the mplct method. They also ncorporated ther method nto a nonlnear leastsquares fttng algorthm to estmate PK parameters. Ther numercal model was a resemblance of a seres of m compartments correspondng to the free or bound drug n the vascular space, blood cells, or Dsse space, or n the cells at varous radal dstances from the vasculature; however, the model was unable to nclude the nstantaneous equlbrum condtons (.e. tssue partton coeffcent) between the tssue cells and the blood. In addton, the structural propertes 43

57 of the lver tssue (.e. porosty, tortuosty) were not ncluded n the model. harles et al. (989) ntroduced a three dmensonal fnte element model for the flud flow and mass transfer n the lver based on the prncples of porous meda. Assumng a Reynolds number smaller than, they adopted the prncples of porous meda where the blood flow n the lver was a creepng flow descrbed by Darcy s law. They developed the momentum equaton accordng to the hydraulc permeablty of the tssue. For the mass transfer equaton, they assumed the Peclet number to be greater than where the dsperson could be neglected due to the domnant convecton. The mass transfer governng equaton was unable to descrbe the hepatocellular drug metabolsm whch can be lnear or nonlnear. In addton, the model was unable to nclude the tssue-blood equlbrum of the drug speces but the drug transfer from the blood to the cells was descrbed based on the cell membrane permeablty. The lack of nformaton of tssue partton coeffcent, the aal dsperson and hepatocellular metabolsm n the model were sgnfcant downsdes of the proposed model. The man obectve of ths study s to develop and valdate a porous meda model based on local volume averagng (LVA) method for descrbng the tme dependent drug concentraton gradent across the lver as well as drug hepatc elmnaton rate, hepatc clearance and boavalablty. Unlke the DP model, the proposed model takes nto account an unsteady state drug concentraton n the lver as well as concentraton dependent hepatocellular metabolsm whch vares wth tme and poston along the lver. In addton, unlke WS, PT and DP models, the proposed mechanstc model ncludes structural characterstcs of the lver (.e. porosty and tortuosty). Fnally, the model consders transport propertes such as aal/radal dsperson, molecular dffuson, and hepatc tssue partton coeffcent. The model was used for predcton of hepatc clearance and boavalablty as well as the drug concentraton gradent across the lver 44

58 wth tme. Also, a senstvty analyss was performed usng the model to nvestgate the nfluence of dfferent parameters on drug dstrbuton n the lver and hepatc elmnaton. The fleblty of the model was demonstrated through ts couplng wth an absorpton model to smulate the dynamc changes n drug dstrbuton n the lver assocated wth the gastrontestnal absorpton process.. Theory.. Local Volume Averagng (LVA) method A porous medum s a sold matr consstng of a sold phase and spaces whch can be flled wth a flud. As a hghly perfused tssue the lver can be treated as a porous medum consstng of snusodal spaces flled wth blood and a matr of hepatocytes. In porous meda when a sold matr cannot be descrbed wthn pore sze, a representatve elementary volume (REV) wth a characterstc length of l and volume of V l s defned to represent the structure of the sold matr. A REV s defned as the smallest dfferental volume resultng n statstcally meanngful average propertes of the porous medum. As shown n Fgure., the lver tssue s composed of repeatng heagonal unts of acn. Each heagonal unt, or acnus, conssts of hepatocytes lnng blood-flled snusods of dameter, d p. Blood from branches of the portal ven and hepatc artery s med and flows through the snusods to subsequently dran nto the central ven. Substances (e.g. oygen, drugs) are transferred from the blood to the hepatocytes durng flow through the snusods. For the lver an acnus can be consdered as a REV where averagng a property over the volume of an acnus (V l ) s called local volume averaged property defned as: 45

59 Fgure.. Schematc dagram of the lver mcrostructure wth the assocated length scales. V l V l dv (.) where s the property of nterest and s the local volume averaged property. The method that uses local volume averaged transport governng equatons and propertes over the REV s called local volume averagng (LVA) method. In order to apply LVA method, the valdty of LVA must be verfed by the followng condton as: k / phys d p l L (.) where k phys s the snusodal based permeablty (m ), d p s the average snusodal dameter (m), l s the length scale of REV (m), and L s the equvalent length of the lver tssue (m). Snce a lver appromately conssts of one mllon acn (Jones and Sprng-Mlls, 988), the REV characterstc length of a normal lver wth a volume of 3 ± 7 cm 3 wll be appromately 600 m (Zhou et al., 007). onsderng that the snusodal based permeablty s m (Smye et al., 007) and the snusod dameter of the lver tssue wth a length of ~0 cm s as small as a few cells (.e. <60 m), the valdty condton of LVA s satsfed as <<60-5 <60-4 <<

60 .. Mathematcal Modelng The geometry of the lver was smplfed so that a slab wth the same thckness and volume as the lver could represent the lver tssue (Fgure.). Havng an average volume and thckness of the lver, the length of the representatve slab was calculated as the equvalent length of the lver. Fgure.a llustrates the lver representatve slab wth an equvalent length of L where the blood enters the slab through the portal ven and hepatc artery and leaves the tssue through the hepatc ven. Fgure.b shows the schematc dagram of a porous dfferental element wth a length of. Transport of drug nto/out of the porous dfferental element s due to the molecular dffuson, aal/radal dsperson, and advecton. Fgure.. Schematc dagram of the smplfed geometry (a) and the porous dfferental element (b) of the lver. As the blood flows through the snusods (pores of the element) wth a Darcy velocty, the unbound drug n plasma s assumed to be locally n equlbrum wth the hepatocytes wthn whch the drug undergoes hepatocellular metabolsm at a metabolsm rate descrbed by defnton as: ˆ m met lnt_ nvvo fu( B) p (.3) where mˆ s the hepatocellular metabolsm rate normalzed by lver tssue volume (mgs - ml - ), met 47

61 48 f u(b) s the unbound fracton of the drug n the blood, l nt-n vvo s the average value of n vvo hepatc ntrnsc clearance (s - ), and P s the local volume averaged drug concentraton n plasma (mgml - ). In addton to Eq. (.3), the concentraton dependent hepatocellular metabolsm can be descrbed by Mchaels Menten equaton as: P B u M P B u met f K f V m ) ( ) ( ma ˆ (.4) where V ma and K M are the mamum metabolsm rate capacty (mgs - ml - ) and affnty term (mgml - ), respectvely. Applyng a transent mass balance over the dfferental element wth a thckness of results n: t A f K A A D D A D D m A u A u Af D Af D P P B u P d d P d d met P B P B P B u AB P B u AB ) ( *. ) ( ) ( ) ( (.5) where D AB s the molecular dffuson of the unbound drug n the plasma (m s - ), s the snusodal based porosty, s the snusodal based tortuosty, f u(b) s the unbound fracton of the drug n the blood, A s the cross sectonal area perpendcular to the hepatc blood flow nto the lver tssue representatve slab (m ), s the poston of the drug compound n the lver (m),. met m s the hepatocellular metabolsm rate (mgs - ), K * s the lver tssue partton coeffcent, t s tme (s), d D and d D are aal and radal dsperson coeffcents (m s - ), respectvely, and B u s the blood Darcy velocty (ms - ) whch can be gven as: A Q u h B (.6) where Q h s the hepatc perfuson rate (mls - ). In addton to Eq. (.6), havng the snusodal

62 permeablty of the lver tssue and the blood pressure drop across the lver, the blood Darcy velocty can be obtaned as: u B k phys P L (.7) where P/L s the lnear blood pressure gradent (Pas.m - ) across the lver and s the blood vscosty (Pas.s). The partal dfferental governng equaton of drug transfer, Eq. (.8), then, s derved by substtuton of the metabolsm term by Eq. (.3) and dvson of Eq. (.5) by the volume of the dfferental element followed by smplfcaton of the equaton and lettng and t go to zero: D P P AB ) D D fu( B) * ub mˆ met fu( B) ( K (.8) t P where mˆ met s the hepatocellular metabolsm rate normalzed by the lver tssue volume (mgml - s - ) gven by Eqs. (.3) and (.4) for a constant and nonlnear hepatocellular metabolsm, respectvely. Accordngly, the ntrnsc clearance functon can be defned as: l nt l K ntnvvo M f l f() Vma f l f() (.9) P nt f u(b) nt As ndcated n Eq. (.8), the tme dependent drug concentraton n both hepatocytes and plasma s descrbed by the accumulaton term on the rght sde of the equaton. The lver structural characterstcs of porosty and tortuosty are ncluded n the governng equaton whle the blood Darcy velocty and the aal dsperson coeffcent takes nto account the nfluence of the tssue permeablty. The ntal plasma drug concentraton of the lver tssue s zero and s defned as: 49

63 P (, t 0) 0 (.0) If nstantaneous drug dstrbuton n the body occurs followng ntravenous (IV) bolus necton of drug then the rato of the IV dose (X 0 ) to the volume of dstrbuton (V d ) determnes the plasma drug concentraton at the blood entry to the lver. At the lver nlet and outlet boundares the convectve mass flow predomnates and dffuson and dsperson are assumed to be nsgnfcant. onsequently, the boundary condtons can be descrbed as: P X ( 0, t) Vd P 0 L, t 0 & & D D d d ( 0, t) 0 ( L, t) 0 (.) When the drug dstrbuton wthn the lver s complete, the model can predct the plasma unbound drug concentraton ust at the lver outlet n the hepatc ven. Knowledge of the drug concentraton at the nlet and outlet of the lver allows the calculaton of the hepatc clearance (l h-lva ) as: P P Qh u u 0 L l hlva P (.) u 0 where u P s the local volume averaged unbound drug concentraton (mgml - ), L s the equvalent length of the lver (m). In order to compare the proposed model to other models, the hepatc clearance was calculated for WS, PT and DP models, respectvely, as follows (Ito and Houston, 004): Qh f u l B nt nvvov ( ) t lh WS (.3) Q V f l h t u( B) nt 50

64 l hpt Q h ep f u( B) l Q V nt nvvo h t (.4) l hdp Q h a ep a D n 4a a a ep Dn (.5) where l h-ws, l h-pt and l h-dp (s - ) are the hepatc clearance suggested by WS, PT and DP models, the dsperson number of D n n Eq. (.5) s 0.7 and a s defned as (Robert and Rowland, 986): a 4 f u( B) Vtlnt nvvo Dn (.6) Q h where V t s the volume of the lver tssue (ml) and can be calculated for males and females by Eqs. (.7) and (.8), respectvely, as a functon of body weght as (Anderson et al., 000): V t W b 0.5 log(0ac ) (.7) V (.8) t W b where V t s the lver volume (ml), W b s the body weght (kg) and a c s the number of drnks per day. Because of a poorer correlaton between lver volume and body weght suggested by the allometrc model (Swft et al., 978), Eqs. (.7) or (.8) were used for calculaton of lver volume n ths study. For nonalcoholc males and females, the average values of the lver volume are 60 and 34 ml, respectvely (Anderson et al., 000). Wth estmates of hepatc clearance (l h ) and the hepatc perfuson rate (Q h ), the boavalablty s calculated as (Roberts and Rowland, 986): 5

65 5 h h H Q l F (.9)..3 Numercal Soluton of the model Numercal soluton of the model requred a number of smplfyng assumptons as follows: ) Etrahepatc clearance of the drug was neglgble; ) Blary ecreton of parent drug was neglgble; ) Followng IV admnstraton drug undergoes nstantaneous dstrbuton n the body such that plasma drug concentraton at the lver nlet could be assumed as the rato of IV dose to the volume of dstrbuton of the drug; v) Blood concentraton to plasma concentraton rato of the drug was unty; v) The unbound fracton of the drug n the blood remaned unchanged wth tme; v) Radal dsperson was neglgble compared to advecton and aal dsperson. The lver representatve slab was dvded nto N+ nodes where N was the node at the lver outlet and N+ was a fcttous node ust n the hepatc ven. Then, the governng equaton, Eq. (.8), was dscretzed usng mplct fnte dfference method. Rearrangng the fnte dfference equatons resulted n the followng system of algebrac equatons as: 3 0 (.0) N (.) N (.)

66 where s the node nde, s the tme step nde, represents the plasma unbound drug concentraton, and the coeffcents of,,, and are defned as: * K f u ( B) t d D D AB 6 f u( B) 3 lntnvvo f l Vma f l K M u B f u ( B) nt nt f() f() (.3) The mesh sze was determned based on the senstvty analyss of the drug concentraton gradent across the lver at the tme when drug dstrbuton n the lver reaches a dstrbuton equlbrum (.e. at 00 s) wth respect to the number of nodes. Tme step sze was determned based on the senstvty of the stablty, accuracy and the speed of soluton wth respect to the tme step sze at dfferent mesh sze. Gauss-Sedel teratve method wth a convergence crteron of 0-6 was used for solvng the system of algebrac equatons smultaneously...4 Pharmacoknetc and structural parameters for the smulaton Smulaton was performed for eght drugs, naloone, ldocane, metoprolol, verapaml, caffene, tmolol, dazepam, and phenacetn (Shbata et al., 00), at a hepatc perfuson rate of 500 mlmn - and a snusodal porosty of 0. (Smye et al., 007) for a tme-course of 00 s followng a 5 mg IV dose of each drug. The tortuosty of the lver tssue was calculated accordng to the tortousty of porous meda consstng of layer by layer parallel rods as (Perry and Green, 997): 53

67 ( ) (.4) Table. shows the pharmacoknetc propertes of each drug used for the smulaton. The unbound fracton and ntrnsc clearance of the drugs vary from 0.03 (dazepam) to (metoprolol), and from 0.3 (dazepam) to 54.9 (nalaone) mlmn - kg -, respectvely. Ldocane, wth a tssue partton coeffcent of 0.6 (Joseph et al., 00), was chosen as a canddate for the smulaton of drug dstrbuton across the lver as well as for senstvty analyses. Table.. Pharmacoknetc parameters of drugs used n the smulaton of hepatc clearance. Drug Unbound Intrnsc clearance Volume of dstrbuton fracton (mlmn - kg - ) (Lkg - ) Ldocane 0.65 a 9.8 g 3.00 h Nalaone b 54.9 b.64 Metoprolol c 7.8 c 4.5 Verapaml 0.80 d 3.0 d 4.63 k affene e.7 e.06 l Phenacetn f 7.5 f.55 m Tmolol o 7.7 o 3.5 n Dazepam 0.03 p 0.3 p.57 q a Jacob et al., 983; b Asal and Brown, 984; Holford, 998; c Regardh et al., 98; d Deshmukh and Harsch, 0; e Blanchard, 98; f, m Raaflaub and Dubach, 975; g Wng et al., 984; Remmel et al., 99; h Ikeda et al., 00; Glass et al., 994;; Hardman et al., 996; k McAllster and Krsten, 98; l Lelo et al., 986; n Else et al., 978; o Holford, 998; p Dvoll et al., 983; q Norman et al., Results and dscusson In order to determne the optmum mesh sze, the senstvty of the drug concentraton gradent across the lver to the grd sze at 00 s was nvestgated. Fgure.3 depcts the plasma drug concentraton wth the lver equvalent length for dfferent mesh grd numbers. Accordng to Fgure.3, the drug concentraton gradent across the lver ncreases sharply wth the number of nodes up to 00. Thereafter, the concentraton gradent ncreases margnally up to 350 nodes and then remans relatvely unchanged. A mesh sze of 0.63 mm equvalent to 350 nodes was 54

68 Fgure.3. Senstvty analyss for determnaton of optmum mesh sze for the numercal soluton. adopted for the numercal soluton. Accordng to accuracy and speed of soluton, whch were performed for dfferent tme step szes of 0.,, 5, and 0 s, a tme step sze of s was adopted for the numercal soluton. Fgure.4 llustrates the correlaton between the predcted values of hepatc clearance and observed values reported by Shbata et al. (00) for eght drugs at a perfuson rate of 500 mlmn - for a 70 kg male subect. The coeffcent of determnaton (R ) of 0.9 ndcates a good agreement between predcted and epermental values. The proposed model seems to underestmate the hepatc clearance of tmolol and verapaml whle overestmatng phenacetn (Fgure.4). Ths may be due to varablty n the measured values of unbound drug fracton and ntrnsc clearance, whch wll enhance uncertanty n the predcted values of hepatc clearance. For nstance, the genetc polymorphsm assocated wth tmolol hepatocellular metabolsm segregates a porton of the populaton nto a poor metabolzer phenotype and estmates of tmolol hepatc clearance causng a sgnfcant varablty n the populaton. Snce the pharmacoknetc propertes used for the smulaton are not populaton based, the predcted values may not capture 55

69 Predcted values of L h (mlmn - kg - ) R = 0.9 Naloone Verapaml Phenacetn Metoprolol affen Ldocane Tmolol Dazepam Observed values of L h (mlmn - kg - ) Fgure.4. Predcted values of hepatc clearance from Porous Meda based model versus reported observatons (Shbata et al., 00) for eght drugs. the populaton varablty and, hence, are assocated wth greater uncertanty n the estmaton of the hepatc clearance of tmolol. Table. presents the observed values of the hepatc clearance of eght drugs as well as the predcted values from the proposed porous meda (PM) based model and the WS, PT and DP models. Table. ndcates that PM model predctons are mostly consstent wth PT and DP models and relatvely less consstent wth the WS model, lkely due to the oversmplfcatons assocated wth WS model. A comparson of observed values to the predcted values of the models ndcates that the PM model s less predctve of caffene hepatc clearance whle more predctve for dazepam hepatc clearance relatve to the WS, PT and DP models. Fgure.5 llustrates the correlaton between the predcted and the reported values of boavalablty for eght drugs at a hepatc perfuson rate of 500 mlmn - for a 70 kg male subect. The coeffcent of determnaton (R ) of 0.80 ndcates a relatvely good agreement between predcted and observed values of boavalablty, although PM model overestmates verapaml and tmolol and underestmates naloone, metoprolol, and phenacetn. Snce the boavalablty s calculated based on the hepatc clearance, Eq. (.9), the error assocated wth 56

70 Table.. omparson of model predctons wth observed values of the hepatc clearance assocated wth seven drugs. Model predctons of l H (ml/mn.kg) Observatons ompound PM WS PT DP reported a Naloone Verapaml ±5.0 Phenacetn ±4.5 Ldocane ±.5 Metoprolol ±.5 affene ±0.4 Tmolol ±. Dazepam ±0. a Shbata et al., 00 the predcted hepatc clearance accumulates n the predcted values of boavalablty. For nstance, predcted hepatc clearance accumulates n the predcted values of boavalablty. For nstance, underestmaton of hepatc clearance of verapaml and tmolol leads to the overestmaton of ther boavalablty. Furthermore, the lterature reports boavalablty followng oral admnstraton wth reductons n boavalablty resultng from ntestnal and hepatc mechansms. The PM model only accounts for drug loss by the lver n ts predcton of boavalablty. Predcted values of boavalablty R = 0.80 Naloone Verapaml Phenacetn Metoprolol affene Ldocane Tmolol Dazepam Observed values of boavalablty Fgure.5. Predcted values of boavalablty from Porous Meda based model versus reported observatons (Shbata et al., 00) for eght drugs. 57

71 Table. presents the observed values of boavalablty of eght drugs as well as the predcted values from the proposed porous meda (PM) based model and the WS, PT and DP models. Table.3 ndcates that boavalablty predctons from the PM model show good agreement wth the DP model whle a sgnfcant dscrepancy can be dstngushed between PM and WS models. The comparson of predcted values wth the reported data of boavalablty shows that the WS model substantally overestmates drug boavalablty. Table.3. omparson of model predctons wth observed values of boavalablty assocated wth seven drugs. Model predctons of Boavalablty (F H ) Reported ompound PM WS PT DM observatons a Naloone Verapaml ±0. Phenacetn ±0.03 Ldocane ±0.05 Metoprolol ±0. affene ±0.04 Tmolol ±0.06 Dazepam ±0. a Shbata et al., 00 Table.4 shows mean squared predcton errors (MSE) assocated wth PM, WS, PT and DP models for hepatc clearance and boavalablty. It ndcates that PM model results n smaller MSE for hepatc clearance predctons compared to the other models and the WS, whch s a physologcally oversmplfed model, leads to sgnfcantly larger MSE assocated wth hepatc clearance predctons. PM, PT and DP models result n smlar MSE whle WS causes larger MSE for boavalablty predctons. Unlke MSE values, all models have close coeffcent of determnaton values for hepatc clearance and boavalablty. Snce the proposed numercal model takes nto account more structural (.e. porosty, tortousty) and physco-chemcal parameters (.e. tssue partton coeffcent) compared to the other models, errors assocated wth the parameters are epected to propagate n the soluton resultng n larger uncertantes n 58

72 predctons. However, lower MSE of hepatc clearance predctons and the same MSE of boavalablty predctons along wth smlar coeffcent of determnaton values as other models suggest an mprovement n mechanstc modelng of hepatc drug elmnaton usng the LVA method. Table.4. Mean squared predcton error (MSE) and coeffcent of determnaton (R) values of hepatc clearance and boavalablty for wellstrred, parallel tube, dsperson and porous meda models. MSE R Model Hepatc Boavalablty Hepatc Boavalablty clearance clearance Porous meda Well strred Parallel tube Dsperson Fgure.6 depcts the effect of the aal dsperson on the plasma unbound drug concentraton gradent across the lver at dfferent tmes. In the absence of aal dsperson, dffuson and convecton mass transfer are the only mechansms of drug transport n the lver. As observed from Fg..6a, drug s dstrbuted n the lver by the molecular dffuson and the advecton assocated wth the blood flow so that the drug can only be dstrbuted wthn cm of the lver n 0 s whle hepatocellular drug metabolsm s takng place n the lver. After 0 s the drug reaches the hepatc ven at the lver outlet and drug concentraton begns rsng untl t reaches steady state at 80 s. Unlke advecton and dffuson, aal dsperson causes the drug to stretch along the lver wthn seconds. As seen n Fg..6b, the drug reaches the hepatc ven at the lver outlet n less than 0 s. In other words, aal dsperson causes the drug to eperence less resdence tme durng whch hepatocellular metabolsm of the drug takes place n the lver. Less resdence tme can cause an nadequate tme for the drug to reach equlbrum wth hepatocytes, whch, n turn, can reduce the frst pass effect wthn the ntal dstrbuton tme. 59

73 Fgure.6. Plasma unbound drug concentraton gradent across the lver at dfferent tmes n the absence (a) and presence (b) of aal dsperson (Dn) for ldocane. Fgure.7 llustrates the varaton of plasma drug concentraton at the lver outlet n the hepatc ven wth tme and aal dsperson number. For low levels of aal dsperson, the resdence tme of the drug s about 7 s when the drug reaches the lver outlet followed by a sharp ncrease n the plasma drug concentraton. However, as the dsperson number ncreases the resdence tme of drug n the lver decreases such that at hgher dsperson numbers t takes only a few seconds for the drug to leave the lver whle the plasma drug concentraton gradually ncreases at the hepatc ven. The smulated resdence tme has good agreement wth the dsperson model predctons of resdence tme of a drug bolus n the lver reported by Roberts and Rowland (986). Fgure.8 llustrates the senstvty of the drug concentraton gradent across the lver to the porosty of the lver at an aal dsperson number of 0.7. An ncrease n the lver porosty causes the plasma drug concentraton to elevate across the lver. Lkewse, decreases n lver porosty results n reductons n the plasma drug concentraton across the lver due to the effect of the porosty on the snusodal blood velocty (pore velocty), whch s obtaned by the rato of 60

74 Fgure.7. Varaton of plasma unbound drug concentraton at the hepatc ven versus tme and aal dsperson number for ldocane. Fgure.8. The nfluence of porosty on plasma unbound drug concentraton gradent across the lver at dfferent tmes for ldocane for aal dsperson number (D n ) of 0.7:, =0.06;, =0.;, =0.8. 6

75 blood Darcy velocty to the porosty. Lower snusodal porosty leads to hgher pore velocty enhancng the effect of the advecton term n Eq. (.8). Larger advecton causes the drug to move forward faster so that the resdence tme s reduced and the drug concentraton at any poston across the lver ncreases durng the transent dstrbuton tme. Lkewse, hgher porosty reduces the plasma drug concentraton at any poston durng the unsteady state perod across the lver; however, the effect of porosty on the concentraton gradent becomes nsgnfcant at 80 s (Fg..8). Fgure.9 depcts the senstvty of the plasma drug concentraton gradent across the lver to the tssue partton coeffcent of ldocane. Ldocane s large tssue partton coeffcent results n smaller plasma drug concentratons across the lver at all dstrbuton tmes of the drug. Accordng to the defnton of the tssue partton coeffcent (the rato of the drug concentraton n hepatocytes to that n plasma), the larger tssue partton coeffcent mples that a larger fracton of unbound drug s parttoned nto hepatocytes where the drug s metabolzed. Therefore, for larger hepatc partton coeffcent, plasma unbound drug concentraton s epected to decrease across the lver durng the transent drug dstrbuton (Fg..9). However, once drug dstrbuton n the lver reaches steady state at 80 s, the effect of tssue partton coeffcent becomes nsgnfcant. Mathematcally ths occurs because the accumulaton term n the rght sde of the governng equaton (Eq. (.8)) becomes zero at steady state where the effect of the tssue partton coeffcent s canceled n the model. In other words, the magntude of the tssue partton coeffcent has an nfluence on how fast drug dstrbuton across the lver reaches equlbrum. Fgure.0 depcts the affect of ntrnsc clearance (Fg..0a) and drug unbound fracton (Fg..0b) on the plasma drug concentraton gradent across the lver for ldocane at the aal 6

76 Fgure.9. Senstvty of plasma unbound drug concentraton gradent across the lver to the partton coeffcent for ldocane for aal dsperson number (D n ) of 0.7:, K * =0.70;, K * =0.6;, K * =0.40. Fgure.0. Influence of ntrnsc clearance (a) and unbound fracton (b) on plasma unbound drug concentraton gradent across the lver for ldocane at an aal dsperson number of 0.7. dsperson number of 0.7. At an unbound fracton of 0.65, the concentraton gradent of ldocane across the lver s nsgnfcant for low ntrnsc clearance values from 0 to 0.05 Lmn -. Ths mples that hepatocellular metabolsm of the drug s so low that ts concentraton remans relatvely unchanged across the lver. However, as ntrnsc clearance s enhanced from 0.05 to 63

77 0.5 Lmn -, the plasma drug concentraton gradent dramatcally ncreases. If the ntrnsc clearance s ncreased further, the plasma drug concentraton n the second half of the lver s sgnfcantly reduced so that for ntrnsc clearance values greater than 0. Lmn - the drug concentraton at the lver outlet s zero mplyng that all drug s elmnated across the lver. Unlke the second half of the lver, the frst half of the lver demonstrates more senstvty to hgher values (>0. Lmn - ) of the ntrnsc clearance. Mathematcally ths s eplaned by Eq. (.3) where hepatocellular metabolsm s a functon of both ntrnsc clearance and unbound drug concentraton such that hgh ntrnsc clearance values result n very low unbound drug concentratons n the second half of the lver. Hence, n the second half of the lver, despte hgh ntrnsc clearance values the rate of metabolsm s very low due to very small plasma drug concentratons and remans nsenstve to ntrnsc clearance. However, the frst half of the lver s subected to drug enrched blood of the portal ven and hepatc artery and the plasma drug concentraton s suffcently hgh to be senstve to the hgher values of ntrnsc clearance. Fgure.0b llustrates the senstvty of the plasma unbound drug concentraton across the lver to the unbound fracton of ldocane at an ntrnsc clearance of 9.8 mlmn - kg -. Wth hgher values of unbound drug fracton, the plasma drug concentraton gradent ncreases so that for unbound fractons greater than 0.8 the plasma drug concentraton at the last cm of the lver s very small. omparng Fg..0a to Fg..0b ndcates that the drug concentraton gradent across the lver dramatcally ncreases wth ncreases n ntrnsc clearance whle the unbound fracton has a relatvely neglgble effect on the drug concentraton gradent along the lver compared to the ntrnsc clearance. In addton, no sgnfcant change n drug concentraton occurs n the frst cm of lver length when the ntrnsc clearance and unbound fracton are ncreased. The dsperson of the drug enrched blood at the regon close to the lver nlet 64

78 suppresses the mpact of hgher ntrnsc clearance and unbound fracton values on the concentraton gradent n ths porton of the lver. Fgure. llustrates the varaton of hepatc clearance wth respect to the hepatc perfuson rate and ntrnsc clearance. At low values of ntrnsc clearance from 0.0 to 0. Lmn - kg - hepatc perfuson rate has no sgnfcant effect on hepatc clearance when perfuson rate ncreases from 00 to 800 mlmn - ; however, small changes n ntrnsc clearance nduce sharp ncreases n hepatc clearance. Ths suggests that for low ntrnsc clearance values the lmtng factor of hepatc clearance s the hepatocellular enzyme actvty represented by the ntrnsc clearance. For values of the ntrnsc clearance greater than 0. Lmn - kg - the effect of enzyme actvty on hepatc clearance becomes nsgnfcant whle hepatc clearance ncreases wth Fgure.. Varaton of hepatc clearance wth ntrnsc clearance and hepatc perfuson rate for ldocane at a snusodal porosty of 0. and perfuson rate of 500 mlmn -. hepatc perfuson rate. Ths suggests that at hgher values of ntrnsc clearance the speed of drug presentaton to the hepatocytes that s determned by perfuson rate s the lmtng factor for hepatc clearance. 65

79 Fgure. depcts the nfluence of the hepatc perfuson rate and ntrnsc clearance on the hepatc boavalablty of ldocane. At very low ntrnsc clearance values (less than 0.05 Lmn - kg - ) hepatc boavalablty s hgh for all hepatc perfuson rates; the effect of the ntrnsc clearance predomnates although, ncreasng perfuson rates result n slght ncreases n boavalablty at low ntrnsc clearance values. Ths s lkely due to shorter resdence tmes of the drug n the lver at hgher perfuson rates, whch reduces the equlbraton tme between the plasma and hepatocytes. As ntrnsc clearance values ncrease boavalablty decreases dramatcally such that for ntrnsc clearance values greater than 0. Lmn - kg - boavalablty approaches zero mplyng all drug s metabolzed by the lver durng ts frst pass across the lver. Fgure.. Varaton of boavalablty wth ntrnsc clearance and hepatc perfuson rate for ldocane at a snusodal porosty of 0. and perfuson rate of 500 mlmn -. Fgure.3 llustrates the senstvty of boavalablty to the ntrnsc clearance and the unbound drug fracton. For very low values of unbound fracton (<0.) or ntrnsc clearance (<0. Lmn - kg - ) boavalablty gradually decreases wth any ncreases n ether unbound fracton or ntrnsc clearance. Accordng to Eq. (.3) whch relates the metabolsm rate to the 66

80 ntrnsc clearance and unbound fracton, when unbound fracton and ntrnsc clearance are very low, an ncrease n only one does not produce a substantal drop n boavalablty due to the suppressng effect of the low value of the other varable. However, for hgher values (>0.5) of unbound fracton and ntrnsc clearance, an ncrease n ether one causes a sgnfcant drop n boavalablty mplyng that metabolsm rate s suffcently senstve to the magntude of the values of ether unbound fracton or ntrnsc clearance. Boavalablty approaches zero when unbound fracton and ntrnsc clearance gan values greater than 0.5 and 0. Lmn - kg -, respectvely. Fgure.3. Varaton of boavalablty wth ntrnsc clearance and unbound fracton for ldocane at a snusodal porosty of 0. and perfuson rate of 500 mlmn -. The proposed PM model was eecuted to predct the varaton of concentraton dependent ntrnsc clearance across the lver for '-hydroymdazolam (OH-MDZ) wth an affnty term (K m ) of 0.75 mgl -, mamum hepatocellular metabolsm rate (V ma ) of mg L - s - and tssue partton coeffcent of.5 (Wllmann and Edgnton, 007). Fgure.4 depcts the varaton of ntrnsc clearance and plasma unbound drug concentraton wth poston and tme across the lver. In Fg..4a, an eponentally ncreasng trend of the ntrnsc clearance can be 67