# PharmPK Discussion - Questions on PBPK organ models

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• On 20 Jan 2006 at 17:32:26, cedric.vinson.-a-.fr.netgrs.com sent the message
`The following message was posted to: PharmPKDear all,I have 2 questions concerning PBPK organ models* on the well-sirred model :CLh= Qh*fub*CLint/ (Qh+fub*CLint)- question 1 : is fub the fraction unbound in blood or in plasma?I was convinced it was fu in blood (=fu plasma/Rb), but the book ofM.Rowland and T.Tozer states p.166 that is fu in plasma. I'm a bitconfusednow... Could someone clarify this point?- question 2 : if the fub is fu in blood, it is possible in theory tohave afub superior to 1 (with for exemple with fup=1 and a Rb=0.7)I don't know if this case can happen in reality or not, but from amathematical point of view, it's possible In the well stirred equation,should I use in this case the value (fup/Rb)>1 or should I limit thefub to1?I ask that because for products in early development, you can have ahigh fu(predicted or measured) without any idea of the blood/plasma ratio(Rb). Andi've read somewhere that Rb is more often close to (1-haematocrit, solet'ssay 0.6) than to 1, so 0.6-0.7 is my default value when i don't know theRb...* on the parallel tubes model :Could someone know some references that contains this model expressed inordinary differential equations (to use in a numerical simulationsoftware)? In all papers I found, the model is expressed in the integrated/analyticalform (I'm not sure of the correct english word for that), but i didn'tmanaged to find it in ODE form...Best regardsC.Vinson[The text "Applied Pharmacokinetics and Pharmacodynamics", 4th ed.,page 133 has equations for CL(H) for both models. They look likeintegrated (analytical) equations but they are just 'defining' avalue for the parameter which can be used in the differentialequation. i.e. CL = ...If I have some parts of the recent clearance discussion correct youcan write the differential equation as:dX/dt = CL(H) x Cx (where Cx is concentration in blood or plasmadepending on the answers to question 1 and 2.)Since Cx = X/VxdX/dt = CL(H) x X/Vx (where Vx is ...). Be careful of your units andget a clear idea about the answers to questions 1 and 2 - db]`
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• On 26 Jan 2006 at 16:31:01, cedric.vinson.aaa.fr.netgrs.com sent the message
`The following message was posted to: PharmPKThanks for your answer DavidI think i have the answer for question 1 : it could be some sort oftypo inthe book of Rowland, because in the same book (chapter "Definition ofsymbols"), the fub is fup/Rb. So i'm quite confident again that Ishould usethe fu blood in the well-stirred model. But the question 2 is stillunresolved...Concerning your feeling on the differential form of PT model, i'm notsureyou're right, because in the differential form of the well-stirredmodel,dX/dt is not equal to  CL(H) x Cx = (Qh*fub*CLint/ (Qh+fub*CLint))*Cxwhat i have understood is that the formula Qh*fub*CLint/(Qh+fub*CLint) is anintegrated formula describing the *blood* clearance due to hepaticelimination (cf. chapter 9 of Gibaldi and perrier, S Bloodclearance : inthis chapter, the authors states that the formula is an integratedform of aPBPK model...)But after the long debate on clearance and elimination rate, i'm notsurethat someone want to discuss that sort of question anymore now... ^_^'[Interesting follow-up. If you are looking at the whole body you canuse CL(H) - model of your choice equation in a differential orintegrated equation.For example C = (Dose/V) x exp(-CL(H) x t/V) (did I get that right? iwould use the k version)or dC/dt = - CL(H) x C/VHowever if you are interested in the fine detail of how the CL(H)equation is derived you should look in the primary literature. Forexample the book by Kwon (http://www.boomer.org/pkin/book.html)p90-93 discusses three different models. I'm sure there are plenty ofdifferential equations included in those primary references ;-)As an aside I have started to develop an applet to display Cp versustime using either the well stirred or parallel tube model - thecurrent version seems to have a bug ;-( but should be fixed tonight db]`
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• On 30 Jan 2006 at 17:29:16, "Hans Proost" (j.h.proost.aaa.rug.nl) sent the message
`The following message was posted to: PharmPKDear Cedric,You wrote: > * on the well-sirred model : > CLh= Qh*fub*CLint/ (Qh+fub*CLint) > - question 1 : is fub the fraction unbound in blood or in plasma? > I was convinced it was fu in blood (=fu plasma/Rb), but the book of > M.Rowland and T.Tozer states p.166 that is fu in plasma. I'm a bit > confused now... Could someone clarify this point?and in a next message: > I think i have the answer for question 1 : it could be some sort of > typo in > the book of Rowland, because in the same book (chapter "Definition of > symbols"), the fub is fup/Rb. So i'm quite confident again that I > should use the fu blood in the well-stirred model.In my opinion the equation in the book of Rowland and Tozer (ClinicalPharmacokinetics, in my view the best book in the field) is correct. Thewell-stirred model describes the situation with respect to blood, andallvolume terms refer to blood: CL_b,H and Q_H. Please note thataccording tothe definition on page xiii of the book of Rowland and Tozer, fub isdefinedas the ratio of the unbound concentration in plasma and the total drugconcentration in blood, thus fub = Cu / Cb. This may look strange,but makesperfectly sense in equations 13 and 14 on page 166. This can be shownaftermultiplying numerator and denominator of the right side of equation14 byCb, i.e. the concentration in the blood leaving the liver, yieldingE_H = Cu * CL_int  / ( Cb * Q_H  +  Cu * CL_int)CL_int is the intrinsic clearance that relates rate of metabolism tounboundconcentration at the enzyme site; in the well-stirred model this unboundconcentration is assumed to be the same as the unbound concentrationleavingthe liver.So, the product 'Cu * CL_int' is the rate (amount / time) ofelimination inthe liver. The product 'Cb * Q_H' is the rate (amount / time) of drugpassing the liver unaltered. The extraction ratio E_H is the ratio oftherate of elimination in the liver divided by the rate of drug enteringtheliver, which equals the sum of the rate of elimination and the rateof drugleaving the liver, as shown in figure 11-1 on page 159.In conclusion, Rowland and Tozer are fully correct. > - question 2 : if the fub is fu in blood, it is possible in theory to > have a > fub superior to 1 (with for exemple with fup=1 and a Rb=0.7) > I don't know if this case can happen in reality or not, but from a > mathematical point of view, it's possible In the well stirredequation, > should I use in this case the value (fup/Rb)>1 or should I limit the > fub to 1?As stated above, fub = Cu / Cb, and the value of fub can exceed 1,since Cbcan be smaller than Cu, in case of no or low plasma protein binding(so Cuclose to C), and no or low distribution into red blood cells (so, Cblowerthan C). > Could someone know some references that contains this modelexpressed in > ordinary differential equations (to use in a numerical simulation > software)and in a next message, answering to David: > dX/dt is not equal to  CL(H) x Cx = (Qh*fub*CLint/ (Qh+fub*CLint))*CxWhy is this equation not correct? Since CL_b,H is defined as a bloodclearance, this equation is correct only if Cx refers to the bloodconcentration. And since the extraction refers to the amoun entering theliver, Cx refers to the concentration of drug in the blood entering theliver. If you include this in your equation, I would say that youhave thecorrect formula.Best regards,Hans ProostJohannes H. ProostDept. of Pharmacokinetics and Drug DeliveryUniversity Centre for PharmacyAntonius Deusinglaan 19713 AV Groningen, The Netherlandstel. 31-50 363 3292fax  31-50 363 3247Email: j.h.proost.aaa.rug.nl`
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• On 1 Feb 2006 at 16:44:35, cedric.vinson.at.fr.netgrs.com sent the message
`The following message was posted to: PharmPKDear Hans,Thanks for your long and precise answer.First I would like to say that I never intended to mean that therewas anerror in the model described in the Rowland and Tozer's book (i fullyagreethat it is one of the best book in the field). It's just that thestatementp.166, second line after equation 13,  "and fub is the fractionunbound inplasma" was just confusing for me. But one of my colleagues (thanksPascal!)pointed me out after reading my message that there was anotherdefinition offub page xiii (fub = Cu / Cb which fully agree with mine since Cu/Cb=fuplasma /Rb).Concerning the question on differential equations, i will try toimplementyour suggestion in my simulation software.But i'm a bit worried to not be able to conciliate the equations ofthe wellstirred model i use on a daily basis and yours.I tried to manipulate on paper your equation (EQ. 1) to find theequation iuse (EQ.2), but without any success :dX / dt = (Qh*fub*CL_int/ (Qh + fub*CL_int)) * Cart		EQ.1dX/dt = Qh*Cart - Qh*Ctb - CL_int*fub*Ctb				EQ.2with	Cart = concentration in blood entering the liver	Ctb = concentration in blood leaving the liver (= totalconcentration in liver / Kp)	fub = fu plasma / Rb	CL_int and Qh = as defined beforeQh*Cart = rate of input in the liverQh*Ctb  = rate of output from the liverCL_int*fub*Ctb = rate of eliminationI thought the (Qh*fub*CLint/ (Qh + fub*CLint)) equation was anintegratedform of EQ.2, used to describe the systemic clearance caused by thelivermetabolism... but after the explanations given by you and David, i'm nowwondering if i'm completely wrong. I'm also wondering was clear onthe factthat my questions concern full PBPK models. The equation 2 is used todescribe the variations of the compound's amounts/concentrations in theliver compartment only, not in the blood compartment.Best regards, and thanks again for answering by basic questionsCedric`
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• On 1 Feb 2006 at 13:20:52, Jorge Duconge (jduconge.-at-.yahoo.com.mx) sent the message
`Dear Cedric,Concerning your query on differential equation for hepatic clearance(well-stirred model), please try this way:Starting from your equation2dX/dt = Qh*Cart - Qh*Ctb - CL_int*fub*Ctb [EQ.2]assuming steady-state conditions:dX/dt =0; so the right-side of your equation will become:Qh*Cart - Qh*Ctb - CL_int*fub*Ctb = 0and after a straightforward manipulation,Cart = Ctb*(CL_int*fub +Qh)/Qhthereafter and using the well-known expression CLh = Qh*Ehwhere, Eh means hepatic extraction ratio = (Cart - Ctb)/Cartsubstituting and canceling out,Eh=CL_int*fub/(CL_int*fub +Qh),Finally, you will get equation1 componentClh=(Qh*fub*CL_int/ (Qh + fub*CL_int))I hope this help you,Regards,Jorge Duconge`
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• On 3 Feb 2006 at 09:34:35, cedric.vinson.at.fr.netgrs.com sent the message
`The following message was posted to: PharmPKDear Jorge,Thanks for your clear demonstration. So the equation 1 is aparticular case(steady-state) of equation 2, isn't it?My main error was to think EQ.2 was an integrated form of EQ.1, butit is infact equivalent to EQ.1 in the particular case of steady-state (sinceyoufix dX/dt=0).To get back to my original question concerning the parallel tubemodel : i'dlike to express it the same way than EQ.1. , i.e. dX/dt = Rate_in -Rate_out- Rate_elimination--> a form that does not imply steady state (am I wrong on this point?)I will try to express it this way starting from David and Hans'sequation(dX/dt=CLh*Cart, with CLh=Qh*[1-exp-(fub*CLint/Qh)]) and doing yourreasoning in an inverse way. I will get back on the forum if my limitedskills in mathematics  prevent me to do this ^_^'Thanks for your helpBest regardsCedric`
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• On 3 Feb 2006 at 14:27:03, "Hans Proost" (j.h.proost.-at-.rug.nl) sent the message
`The following message was posted to: PharmPKDear Cedric,Your equations > dX/dt = (Qh*fub*CL_int/ (Qh + fub*CL_int)) * Cart      EQ.1 > dX/dt = Qh*Cart - Qh*Ctb - CL_int*fub*Ctb      EQ.2are indeed different.Eq. 1 describes the loss of drug from the arterial blood flow throughtheliver. The only variable at the right side is Cart, as the drivingforce. Itdoes not describe how the concentration in the blood changes, and so Ctbremains to be calculated from:Eh  =  fub*CL_int/ (Qh + fub*CL_int))  = (Cart - Ctb) / CartPlease note that this approach does not take into account the volumeor Kpof the liver.Eq. 2 is a mass balance equation. If the volume of the liver is nottakeninto account, dX/dt = 0 at any time. As pointed out by Jorge Duconge,thisequation can be rearranged to the equation for Eh and CLb,h. > The equation 2 is used to > describe the variations of the compound's amounts/concentrationsin the > liver compartment only, not in the blood compartment.If you want to take the volume and Kp of the liver into account tomake afull PBPK model, you need additional equations to describe the change ofCart to Ctb, in order to solve Eq. 2, and the condition dX/dt = 0cannot beassumed.Best regards,Hans ProostJohannes H. ProostDept. of Pharmacokinetics and Drug DeliveryUniversity Centre for PharmacyAntonius Deusinglaan 19713 AV Groningen, The Netherlandstel. 31-50 363 3292fax  31-50 363 3247Email: j.h.proost.-at-.rug.nl`
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