Intro to BME (Spring 2005) 2.1 Topic 2. Mass Balancing and Kinetics in Living Systems

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1 4-101 Intro to BME (Sprng 005).1 Topc. Mass Balancng and Knetcs n Lvng Systems TOPICS - general mass balance (conservaton of mass) equaton - specal cases of the general mass balance - types of systems (batch=closed and contnuous=open) and applcaton of the general mass balance - general method of soluton of mass balance problems - blood tracer example - cellular stochometry example - processes wth recycle, bypass and purge - human ron nventory example - netcs - mcrobal growth Systems Engneerng n Bology At a basc lfe level, lfe processes convert certan raw materals, such as glucose, nto other materals, such as protens or DNA. Processes that happen n one locaton, say nsde one type of organelle, are lned to processes elsewhere, say other organelles n the same cell, other cells, or other organs. To understand how these processes are coordnated, t s necessary to eep trac of all the materal that comes and goes n a bologcal system. Ths leads to the concept of a mass balance. A mass balance s an accountng of the mass wthn a system that has one or more nlets or outlets. The mass wthn a system may change wth tme, n whch case we perform an unsteady-state mass balance, or the mass wthn a system may be constant over tme, n whch case we perform a steady-state mass balance. system The concept of a mass balance s based on the smple statement that mass s conserved. Except for nuclear processes, mass s nether created nor destroyed. Ths s the Law of Conservaton of Mass. The molecules that mae up the mass may change f chemcal reactons occur, but the total mass does not change. If one or more chemcal reactons do occur n the system, stochometrc relatonshps derved from balanced chemcal reactons may be used to connect the consumpton of one speces wth the producton of another speces.

2 General Mass Balance Equaton Intro to BME (Sprng 005). Topc. Mass Balancng and Knetcs n Lvng Systems The general mass balance on a system s constructed as follows. Consder a system wth multple nlets, = 1 # nlets, and outlets, = 1 # outlets, and multple components, = 1 # chemcal speces, that pass through the varous nlets and outlets: system Let s consder how the mass (or moles) of the th speces n the system can change wth tme: Let m &, represent the mass (or molar) flowrate of speces n nput, and m &, represent the mass (or molar flowrate) of speces n outlet. Then, mathematcally we wrte the mass balance on the total mass (or moles) of speces contaned n the system as dm dt all nlets all outlets = m& m, &, + R where the term on the left s dervatve of the mass (or moles) of wth respect to tme. The last term, R s the rate at whch speces s produced nsde the system. (If speces s consumed nsde the system, then R s a negatve number.) Ths s the mathematcal equvalent of Amount of Accumulaton of th speces = Total Amount of n Total Amount of out + Amount Generated by Reacton

3 4-101 Intro to BME (Sprng 005).3 Topc. Mass Balancng and Knetcs n Lvng Systems Specal Cases of the General Mass Balance If we have the specal case of steady-state operaton, then the dervatve wth respect to tme s zero, and we have steady-state all nlets all outlets 0 = m& m R, &, + If we have the specal case of a non-reactng system, then R = 0, and we have non-reactng system dm all nlets = m& dt, all outlets m&, and a steady-state, non-reactng system follows steady-state, non-reactng all nlets all outlets 0 = m& m&,,

4 4-101 Intro to BME (Sprng 005).4 Topc. Mass Balancng and Knetcs n Lvng Systems Types of Systems Two mportant classfcatons of systems are batch (closed) and contnuous (open). batch: contnuous: everythng s added to the system at some tme t o, a process taes place for some amount of tme, and then the system s empted (e.g., bang a cae n an oven) materal flows n and/or out of the system on a contnuous bass (e.g., a runnng car engne) Industrally, there are many examples of each. Many pharmaceutcal manufacturng processes are batch. Many petrochemcal producton processes are contnuous. Many physologcal examples exst for each as well. Dgeston s approxmately a batch process. If you eat a bg meal, some dgeston has started before you fnsh eatng. Ths s consdered a sem-batch process. Batch processes are also ey to the feld of pharmaconetcs, the study of how drugs are dstrbuted throughout the body over tme ths s essental to defnng dosage regmens. Some examples of contnuous processes n the body nclude dney dalyss, and blood crculaton. NOTE, careful selecton of the system s very mportant: performng mass balances for dfferent (but related) systems wll gve you dfferent types of nsght to a problem. In some cases, careful selecton of system boundares may permt a prevously un-solvable mass balance to become solvable. Suppose we want to thn about delverng a drug to the body by havng a patent swallow a pll. The pll s dgested completely after the nput (ngeston) s done. Ths s a batch process f you consder that (the system) = (the stomach n between fll and empty cycles). What f (the system) = (the pll)? What dfferent types of nformaton could be obtaned?

5 4-101 Intro to BME (Sprng 005).5 Topc. Mass Balancng and Knetcs n Lvng Systems Applcaton of the general mass balance equatons to dfferent types of process systems. Batch dm dt all nlets all outlets = m& m, &, + R Between t o and t fnal, there are no nlet or outlet flows, the system ust reacts. Can a batch system be operated at steady-state? 0 = R sys, Ths would ust mean that nothng happens. For the cae-bang analogy, we d add our flour, sugar, etc., but then we wouldn t turn the oven on. How to solve the batch materal balance equaton? Ths s a dfferental equaton that must be ntegrated by the separaton of varables technque. dm dt dm = R = R dt m m ( t = t fnal dm = ( t = t ) o ) t fnal t o R dt The left-hand sde s readly ntegrated: m ( t = t ) m ( t = t ) = fnal o t fnal t o R dt If speces does not partcpate n the reacton, then R = 0 for that speces and ts mass does not change. To solve such materal balance problems, we need to have mathematcal expressons for the reacton rate R. Specfyng such relatonshps s called netcs.

6 Contnuous Systems Intro to BME (Sprng 005).6 Topc. Mass Balancng and Knetcs n Lvng Systems dm dt all nlets all outlets = m& m, &, + R If we have the specal case of steady-state operaton, then the dervatve wth respect to tme s zero, and we have steady-state all nlets all outlets 0 = m& m R, &, + If we have the specal case of a non-reactng system, then R = 0, and we have non-reactng system dm all nlets = m& dt, all outlets m&, and a steady-state, non-reactng systems balance follows steady-state, non-reactng all nlets all outlets 0 = m& m&,, Termnology note: steady-flow operaton means that all of the flow rates n the system are constant,.e. the flow rates, m&, don t change over tme. For the purposes of Intro to BME, we wll typcally assume that all of our problems nvolvng contnuous processes are operated under steady-flow condtons.

7 4-101 Intro to BME (Sprng 005).7 Topc. Mass Balancng and Knetcs n Lvng Systems

8 4-101 Intro to BME (Sprng 005).8 Topc. Mass Balancng and Knetcs n Lvng Systems

9 4-101 Intro to BME (Sprng 005).9 Topc. Mass Balancng and Knetcs n Lvng Systems Blood Tracer Experment example (MMD example.4.4): A tracer s used to determne a patent s blood flowrate. For safety purposes, ths tracer could be a normally occurrng metabolte. The normal concentraton of ths tracer n the blood s 100 parts per mllon, where 1 ppm = 1 mg/g. Inect 1000 mg of tracer nto the blood over a perod of fve mnutes and measure the downstream concentraton. We fnd the downstream concentraton s 4000 ppm. Determne the mass flowrate of blood.. tracer n 1. blood + tracer n system 3. blood + tracer out

10 4-101 Intro to BME (Sprng 005).10 Topc. Mass Balancng and Knetcs n Lvng Systems Cellular Stochometry example: Consder the contnuous aerobc growth of a bacterum on glucose and ammona accordng to the stochometry: 0.5 C 6 H 1 O NH O CH 1.67 N 0.0 O CO H O A botech company s nterested n producng 100 g/hr of bactera (on a dry bass). A feed stream contanng 0.7 g/l glucose and g/l ammona s avalable. What feed stream flow rate, n g/hr, and O flow rate, n g/hr, wll be requred? Assume all lqud streams have a densty smlar to water and that all O s consumed. What wll the composton of the product stream be?. oxygen 1. feed 3. product boreactor

11 Kdney Functon example Intro to BME (Sprng 005).11 Topc. Mass Balancng and Knetcs n Lvng Systems The ob of the dney s to flter out toxc metabolc products such as urea from the blood. Ths happens n tny structures called nephrons. The frst part of the nephron s the glomerulus, where a tufted networ of capllares s surrounded by a capsule called the Bowman s capsule. Blood flowng nto the tuft of capllares, s laden wth byproducts that are to be fltered. Blood flowng out of the capllares s free of byproducts. The fltraton s accomplshed by the glomerular wall that serves as a sem-permeable membrane t has fne pores that do not allow cells or protens to pass out of the blood, but small molecules (such as water and urea) do pass. The lqud that passes through the glomerular wall s called the glomerular ultrafltrate, that s destned to become urne. The rest of the nephron regulates the composton of the urne to mantan the body s water and salt balance. In the Bowman s capsule, the composton of the glomerular ultrafltrate s dentcal to that of blood, mnus the cells and protens. efferent arterole hgh P glomerulus low P blood flow afferent arterole Bowman s capsule The blood flowng n the capllares s pressurzed by the heart so that t has a hgher pressure than the glomerular ultrafltrate. Ths hydrostatc pressure dfference P between the capllares and the glomerular ultrafltrate s what drves fluds across the glomerular wall for fltraton. Flud flows from regons of hgh hydrostatc pressure to regons of low hydrostatc pressure. If the hydrostatc pressure dfference were too large, t could be possble for too much flud to lea out of the capllares. The flud flow drven by the hydrostatc pressure dfference s counteracted by an osmotc pressure dfference Π that favors flow n the other drecton (from the glomerular ultrafltrate nto the capllares).

12 4-101 Intro to BME (Sprng 005).1 Topc. Mass Balancng and Knetcs n Lvng Systems Flud tends to flow from regons of low osmotc pressure to regons of hgh osmotc pressure. The net flow across the glomerular wall s determned by the competton between these two opposng pressure dfferences. The glomerular wall as a sze-selectve, sem-permeable membrane. blood glomerular ultrafltrate Water, salts, urea, other small solutes protens Osmotc pressure depends on the concentraton of solutes n the flud. A hgh solute concentraton corresponds to a hgh osmotc pressure. Snce the glomerular wall passes water and all small solutes, but blocs all protens, the glomerular ultrafltrate has a low proten concentraton (low osmotc pressure), whle the blood n the capllares has a hgh proten concentraton (hgh osmotc pressure). Why does flud flow from low to hgh osmotc pressure? The reason les n thermodynamcs, but n essence, there s a drvng force for the solvent (water) to dlute the flud wth a hgh solute concentraton. The solvent flows from the low concentraton flud to the hgh concentraton flud n order to equalze the solute concentratons. The nterestng ssue s that the solutons wll never have equal concentraton, snce the protens can never pass through the membrane. To frst order, we can approxmate the osmotc pressure by Π = a C + a C where the a constants are called vral coeffcents, and C s the 1 concentraton of solute. Do not confuse the captal Gree letter Π used for osmotc pressure wth the lower case Gree letter π = 3.14

13 4-101 Intro to BME (Sprng 005).13 Topc. Mass Balancng and Knetcs n Lvng Systems The concentraton of salt n the blood n the glomerulus capllares C salt glom s equal to the salt concentraton n the glomerular ultrafltrate n the Bowman s capsule, C salt bc. C salt glom = C salt bc = C salt Blood n glomerulus Proten concentraton = C Π = ac Proten concentraton = C Water flow glomerular ultrafltrate n Bowman s capsule Proten concentraton = 0 Π = 0 Salt concentraton = C salt Salt concentraton = C salt so the salts do not contrbute to the osmotc pressure dfference. As blood flows further and further through the capllares n the glomerulus, t loses more and more water. Ths has the effect of mang the proten concentraton n the blood hgher and hgher. So, the osmotc pressure dfference becomes hgher and hgher. Eventually, the osmotc pressure dfference exactly equals the hydrostatc pressure dfference, and no more flud passes out of the blood. Quanttatve Model for Glomerular Ultrafltraton Our goal s to predct how the proten concentraton changes n the blood between the nlet and the outlet of the glomerulus. A byproduct of ths analyss wll be a method to measure how leay the glomerular wall s. We start by defnng a volumetrc flux of flud across the glomerular wall J v. Ths s the volume of flud that crosses the wall per unt area per unt tme: J v = ( P Π) [ = ] volume area tme The proportonalty constant s the hydraulc permeablty of the glomerular wall. It depends on the structure and health of the nephron tssue.

14 4-101 Intro to BME (Sprng 005).14 Topc. Mass Balancng and Knetcs n Lvng Systems We wll defne the hydrostatc pressure of the blood nsde the capllares of the glomerulus as P g and the hydrostatc pressure of the ultrafltrate nsde the Bowman s capsule as P bc, and P = P P g bc Lewse, we defne osmotc pressures smlarly, and Π = Π Π = Π g bc g 0 In order to model the flux across the glomerular wall, we wll need to eep trac of the hydrostatc pressure gradent and the osmotc pressure gradent at all postons along the glomerulus. Snce the glomerulus s a complex, convoluted tuft of capllares, we wll approxmate t wth a much smpler geometry, a unform tube of length L and radus R. Ths tube s bathed by the flud n the Bowman s capsule. Flud enters the tube at the afferent (A) end and leaves at the efferent (E) end. Ths smplfyng approxmaton maes the problem tractable. R E A L Before begnnng our analyss, let s consder what nformaton s avalable. Mcropuncture technques mae t possble to sample flud from the afferent and efferent arteroles, but t s not possble to sample flud from the ponts n between. Thus, we now typcal values of the proten concentratons C A and C E at the afferent and efferent ends. C A ~ 90 mg/ml C E ~ 57 mg/ml We also now the values of the vral coeffcents a 1 and a for typcal blood protens, a mmhg 0.16 mg / ml mmhg a = mg / ml 1 = ( ) From these we can estmate Π A ~ 35 mmhg Π E ~ 19 mmhg Also usng mcropuncture, we can measure the hydrostatc pressure n the afferent and efferent arteroles. There s only a slght pressure drop between the afferent and efferent ends, so we can approxmate these pressures as beng equal:

15 4-101 Intro to BME (Sprng 005).15 Topc. Mass Balancng and Knetcs n Lvng Systems P A ~ P E ~ 45 mmhg The pressure nsde the Bowman s capsule s P E ~ 10 mmhg, so there s a hydrostatc pressure dfference P = (45 mmhg-10 mmhg) = 35 mmhg that drves ultrafltraton. Ths s roughly constant along the entre length of the glomerulus, but the flux does decrease further down the glomerulus due to the ncreasng osmotc pressure dfference. The other quanttes we can measure by mcropuncture are the afferent and efferent plasma flowrates Q A and Q E. Conservaton of Blood Plasma Mass Our mathematcal analyss begns wth a materal balance on the blood plasma n the capllares. (Plasma s the name for the flud porton of the blood mnus cells and mnus protens.) The densty of the blood ρ does not change as t flows through the capllares, so t wll be acceptable to perform a balance on the blood volumetrc flow rather than formally consderng the plasma mass flow snce Q = m& ρ. Consder an nfntesmally small cross-sectonal slce of the capllary from x to x+. The tube surface area correspondng to ths small slce s ds. The flowrate enterng s Q x and the flowrate leavng s Q x+. Q x+ s slghtly smaller than Q x due to the flux across the glomerular wall. The outward flow across the wall s J v ds. x x+ Q x Q x+d J v ds Startng from the general materal balance equaton for the volume V (equvalent to mass) of flud n our small cross-secton, we wll cut out un-needed terms. dv dt = Q x Q x+ J ds v + R gen

16 4-101 Intro to BME (Sprng 005).16 Topc. Mass Balancng and Knetcs n Lvng Systems The dney can be consdered to operate at steady-state, so the tme dervatve on the left-hand sde s zero, and there s no generaton of blood n the capllary, so R gen = 0. So, the plasma materal balance becomes Q x+ Q x = J ds v Let s dvde both sdes of the equaton by Q x+ Q x = J v ds. Snce the surface area S of a tube of length x s S = πrx, the dervatve ds = πr For a tube whose total length s L, then we see that ths dervatve s smply equal to S/L. Our equaton smplfes to Q Q x+ x S = J v L Tae the lmt as 0, and the term on the rght s seen to be the defnton of the dervatve of Q wth respect to x: dq S S K f = J = P v L L L ( P Π) = ( Π) dq K f = P L ( Π) [1] where K f =S s called the ultrafltraton coeffcent. Eqn. [1] descrbes the conservaton of blood plasma. Conservaton of Proten Mass Next we perform a materal balance on the proten n the blood plasma n the capllares. Remember that proten does not pass out of the glomerular wall the blood does not lose any proten. The total amount of proten flowng n the blood s equal to QC. Snce the total amount of proten does not change n the blood, the mass balance equaton can be reduced to d ( QC ) = 0 []

17 4-101 Intro to BME (Sprng 005).17 Topc. Mass Balancng and Knetcs n Lvng Systems Equatons [1] and [] are the materal balance equatons. The dervatve n equaton [] can be wrtten va the chan ruleof calculus as ( QC) d dc dq = Q + C Substtutng from eqn [], we can then wrte dc dq Q + C = 0 or dq Q dc = [3] C Substtutng eqn [1] nto eqn [3], we get Q dc C K f = P L ( Π) [4] Here we use our consttutve equaton for osmotc pressure Π = a 1 C+a C osmotc pressure n Bowman s capsule s zero, so Π = Π g ) Q dc C ( P a C a C ) K f = [5] L 1 (remember that the By solvng ths dfferental equaton, we wll be able to show how the proten concentraton C vares along the length of the glomerulus. But we need to do a lttle more wor. Q vares wth poston due to the loss of plasma by ultrafltraton, and we now that the rate at whch t s lost depends on C because of the changng osmotc pressure resstance to ultrafltraton. We do now (eqn. []) that the total amount of proten s a constant, or CQ = constant. We now Q A and C A are avalable from mcropuncture, so we now our constant s CQ = C A Q A. Thus we wrte C Q = Q A [6] A C For convenence, defne a dmensonless concentraton C* = C/C A. Thus, eqn [6] s wrtten as Q Q = A [7] C *

18 4-101 Intro to BME (Sprng 005).18 Topc. Mass Balancng and Knetcs n Lvng Systems We also defne a dmensonless length x* = x/l. Substtutng eqn [7] for Q, x = Lx* and C = C A C* nto eqn [5] we get mg/ml dc * = * K C * f Q A ( P a C C * a C C * ) 1 A A or dc * * = K C * f Q A P 1 a C a C 1 A C * A C * P P [8] For the sae of clarty, let s condense some of these collectons of constants as F = K P f Q A A = 1 a C 1 A P A = a C A P so eqn [8] can be wrtten more concsely as dc * * ( 1 A C * A C * ) = FC * [9] 1 You wll learn ths later, but to solve a dfferental equaton, you need to specfy a boundary condton. Here we defne an ntal condton at x* = 0 (the afferent arterole nlet to the glomerulus): at x* = 0 C* = 1 (because C = C A at the nlet) We can ntegrate [9] by separaton of varables dc * = F * [10] C * ( 1 A C * A C * ) 1 but the ntegral s actually qute large and cumbersome to wrte out. Snce we are not tryng to emphasze the mathematcal technque but nstead want to emphasze the modelng method and the predcted behavor, I have nstead solved eqn [9] numercally, usng the computaton software pacage Mathcad. The predcted behavor s shown here for Q A = 1 nl/sec, C A = 57 g/l, and P = 35 mm Hg, wth K f specfed to be , wth concentraton n mg/ml plotted versus dmensonless dstance x* = x/l:

19 4-101 Intro to BME (Sprng 005).19 Topc. Mass Balancng and Knetcs n Lvng Systems C (mg/ml) C () xstar() 1 Notce that the proten concentraton s ~ 85 mg/ml by the end of the glomerulus (x*=1), close to the typcally measured value. The osmotc pressure vares along the length of the glomerular capllares as Π (mm ΠHg) () xstar() 1 Notce that by the end of the glomerulus (x* = 1), the osmotc pressure dfference s 35 mmhg, matchng the hydrostatc pressure dfference between the glomerulus and Bowman s capsule. Once these two pressures equalze, there s no more flux of flud across the glomerular wall, so the proten concentraton ceases to ncrease at the same poston where the osmotc pressure reaches 35 mmhg.