Mean transit time and mean residence time for linear diffusion ...

Computational and Mathematical Methods in Medicine,Vol. 8, No. 1, March 2007, 37–49Mean transit time and mean residence time forlinear diffusion–convection–reaction transportsystemJACEK WANIEWSKI†‡{*†Institute of Biocybernetics and Biomedical Engineering PAS, ul. Trojdena 4, 02-109 Warsaw, Poland‡Division of Baxter Novum K56, Department of Clinical Sciences, Karolinska Institutet, Stockholm, Sweden{Interdisciplinary Centre for Mathematical and Computation Modelling, Warsaw University, ul. Pawinskiego 5a,Warsaw, Poland(Received 22 May 2006; revised 13 September 2006; in final form 15 February 2007)Characteristic times for transport processes in biological systems may be evaluated as meantransit times (MTTs) (for transit states) or mean residence times (MRT) (for steady states).It is shown in a general framework of a (linear) reaction–diffusion–convection equation thatthese two times are related. Analytical formulas are also derived to calculate moments of exittime distribution using solutions for a stationary state of the system.Keywords: Mean transit time; Linear diffusion; Transit states; Steady states; Exit timedistribution function; Higher moments1. IntroductionTransport processes in various structures are investigated by the injection of a bolus amountof a marker at the inlet to the system and the measurement of the injected marker fraction atthe outlet from the system as a function of time. For example, physiological experiments arecarried out for the whole isolated organs, in which exchange of water and solutes betweenblood and the tissue involves a complex network of membranes and distributed transportbarriers [1–3]. Another system of this kind is applied for the penetration of infrared photonsinto biological tissues [4–6].Mean transit time (MTT) is usually defined for experiments with a momentary input to thesystem of a known amount of substance, M 0 (often a small amount of a labelled form of theinvestigated substance), and the recording of its concentration, C, at the outlet fromthe system as a function of time. MTT may then be calculated as Ð 10 tCðtÞdt=Ð 10CðtÞdt [1].The theoretical evaluation of MTT needs a solution of the model equations for the system,which is generally described by ordinary differential equations (compartment systems) orpartial differential equations (distributed systems). The transport may be convective,diffusive or combined. It was shown that for pure diffusive or pure convective transport insome simple transport systems this definition of MTT is equivalent to another one that isformulated for the steady state of the system [7–9]. In this case, for a steady input of thesolute, the system reaches after some time a steady state of the concentration distribution and* Email: jacekwan@ibib.waw.plComputational and Mathematical Methods in MedicineISSN 1748-670X print/ISSN 1748-6718 online q 2007 Taylor & Francishttp://www.tandf.co.uk/journalsDOI: 10.1080/17486700701298293

Taking into account that under the stated assumptions: F out (t) ¼ 2dM(t)/dt, M(t) ! 0 fort !1, and therefore Ð 10 F outðtÞdt ¼ M 0 , one gets:hðtÞ ¼2 1 dMð3ÞM 0 dtand, using formula (1):Transit time for transport systems 39t ¼ 1 M 0ð 10MðtÞdt ð4Þwhere it is assumed that tM(t) ! 0ift !1.If the system is in the steady state then the solute concentration, C S , within the system, thesolute mass, M S , in the system, and the solute inflow, F Sin , to the system are constant. TheMRT, t res , i.e. the mean time the particle that enters the system spends in it, is given as [8]:t res ¼ M SF Sinð5ÞThe equality of t and t res was initially shown for convective transport of a flow indicatorfor measurements of blood flow and volume [7]. Next, the Hardt theorem stated that for thesystems with pure diffusion and the inflow and outflow only through the boundary, t ¼ t res[8]. It was mentioned by Hardt that this theorem is also valid for the systems with thedistributed source/sink of the solute [8]. We show that this result is valid for a general lineardiffusion–convection—reaction system (described by elliptic differential operator), and thathigher moments of the transit time distribution function may also be calculated using asteady state solution for the system.2. Formulation of the modelThe basic equation for C(x, t) in a connected bounded region V of n-dimensional Euclidianspace R n , n $ 1; is:›C›twhere L is a linear operator defined as¼ LC þ S ð6ÞLC ¼ 2div F 2 ACð7ÞandF ¼ 2D grad C þ JCð8Þfor D ¼ðD ij ÞðxÞ; J ¼ðJ i ÞðxÞ; A ¼ AðxÞ $ 0; and S ¼ SðxÞ $ 0; i; j ¼ 1; ...n and symbols inbold describe n-dimensional vectors or n £ n matrices. Furthermore, it is assumed thatdiffusivity matrix is symmetric, D ij ¼ D ji , and that j·D(x)j $ uj·j for some u . 0, almost allx [ V, and all j [ R n , where R n is n-dimensional Euclidean space, and R ¼ R 1 is the set ofreal numbers. The last condition states that L is an elliptic operator, and guarantees thatdiffusion goes along the concentration gradient [18].

To proof formula (12) let us defineTransit time for transport systems 41ð t ð ›CAðx; tÞ ¼ Gðx; y; t 2 uÞs V›u 2 L yC ðy; uÞþ ›G ›u þ L* y G ðx; y; t 2 uÞCðy; uÞdyduð13ÞBecausethen, using equation (6),›G›Gðx; y; t 2 uÞ ¼2›u ›t ðx; y; t 2 uÞ ¼2L* yGðx; y; t 2 uÞð t ðAðx; tÞ ¼ Gðx; y; t 2 uÞSðyÞdydu:sVð14ÞOn the other hand, equation (13) may be rearranged as follows:ðAðx; tÞ ¼Cðx; tÞ 2Vð t ðGðx; y; t 2 sÞCðy; sÞdy þ Gðx; y; t 2 uÞF n ðy; uÞdydus ›Vð t ðþ ðDðyÞgrad Gðx; y; t 2 uÞÞ n Cðy; uÞdydus ›Vð15Þusing the Gauss–Ostrogradzki theorem Ð V div B dy ¼ Ð ›V B ndy, and the identity div(AB) ¼A div B þ B·grad A for scalar function A(y) and vector function B(y). The application of theboundary conditions for C to the integrals over ›V in equation (15) and the comparison ofequations (14) and (15) yield formula (12).3. Mean transit times for impulse inputsWe assume that the system has a (stable) steady state C 1 (x) with total massM 1 ¼ Ð V C 1ðxÞdx, and this state is disturbed at t ¼ 0 by a pulse function concentrated atx ¼ z : M 0 dðx 2 zÞdðtÞ. Let C(x,t;z) be the solution of equation (6) with the initial conditionC 0 (x;z) ¼ C 1 (x) þ M 0 d(x 2 z), and denote Mðz; tÞ ¼ Ð VCðx; t; zÞdx. The MTT is defined as(c.f. equation (4))tðzÞ ¼2 1 ð 1tdmðz; tÞM 0 0ð16Þwhere m(z,t) ¼ M(z,t) 2 M 1 .Integrating by parts one getstðzÞ ¼ 1 ð 12ðtmðz; tÞÞj 1 0M þ mðz; tÞdt0 0ð17Þ

42J. Waniewskiand, assuming that tm(z, t) ! 0 for t !1and z [ Vn{›V W > ›V D };tðzÞ ¼ 1 ð 1mðz; tÞdt ¼ 1 ð 1 ððCðx; t; zÞ 2 C 1 ðxÞÞdtdx:M 0 0 M 0 0 VThe solution C(x,t;z) may be found using formula (12) for the initial state C 0 (x;z)ðCðx; t; zÞ ¼M 0 Gðx; z; tÞþð t ð2sVð t ðGðx; y; t 2 sÞC 1 ðyÞdy þ Gðx; y; t 2 uÞSðyÞdydus VðDðyÞgrad Gðx; y; t 2 uÞÞ n wðyÞdydu›V Dð t ð2 Gðx; y; t 2 uÞcðyÞdydus ›V Nð18Þð19Þfor any s [ [0,t ].Note, that formula (12) applied for the steady state C 1 (x) implies that the terms withintegrals in equation (19) sum up to C 1 (x). ThusandFrom equation (9):Cðx; t; zÞ ¼M 0 Gðx; z; tÞþC 1 ðxÞðmðz; tÞ ¼M 0 Gðx; z; tÞdxVð20Þð21Þ›m›t ðz; tÞ ¼ L* z m ðz; tÞ ð22Þwith(1) “no diffusive flux” Neumann boundary condition on ›V W < ›V N :(D grad m) n (z,t) ¼ 0for z [ ›V W < ›V N ; t . 0,(2) Dirichlet boundary condition for ›V D :m(z,t) ¼ 0 for z [ ›V D ; t . 0,(3) Initial condition for m:m(z,0) ¼ M 0 for z in Vn›V.From equations (18) and (20) it follows thatðtðzÞ ¼ Gðx; z; tÞdtdxð 10Now, we may formulate an ordinary differential equation for t(z). Let us considerð 1 ðð 1 ðL * z tðzÞ ¼ L * z G ›Gðx; z; tÞdtdx ¼ ðx; z; tÞdtdx0 V0 V ›tð¼ ½Gðx; z; 1Þ 2 Gðx; z; 0ÞŠdxVVð23Þð24Þ

Transit time for transport systems 43Note, that G(x,z,t) ! 0 for t !1, because of the boundary conditions for G, and thatG(x,z,0) ¼ d(x 2 z). ThereforeL * z t ðzÞ ¼21ð25ÞThis PDE is supplemented by the boundary conditions that are of the same type as forG(x,z,t):(1) (›t/›n)(z) ¼ 0 for z [ ›V W < ›V N ;(2) t(z) ¼ 0 for z [ ›V D .Remark. Defining tðz; tÞ ¼ Ð t Ð0 VGðx; z; uÞdudx one can derive an equation for t(z,t):4. Mean residence time›t›t ðz; tÞ ¼ L* z t ðz; tÞþ1: ð26ÞLet CðxÞ be a steady state solution for equation (6) for some boundary conditions, perhapsdifferent from those already specified, which will be defined later on. Then, by equation (25),ðCðzÞ L * z t ðzÞdz ¼ 2 Mð27ÞVwhereðM ¼VCðzÞdzð28ÞOn the other hand,ððCðzÞ L * z t ðzÞdz ¼VVL zC ððzÞtðzÞdz þ›VCðD grad tÞ n þ t F nðzÞdzð29ÞBecause L zC ðzÞ ¼2SðzÞ by the definition of the stationary state, one gets, comparingequations (27) and (29) and using the boundary conditions for t,ððM ¼ 2 ð CðD grad tÞ n ÞðzÞdz 2 t F n ðzÞdz þ SðzÞtðzÞdz ð30Þ›V Dð›V N VLet us assume now the boundary conditions for C, c.f. [8],(1) F n ðzÞ ¼0 for z [ ›V W ,(2) F n ðzÞ ¼2 cðzÞ, cðzÞ $ 0 for z [ ›V N , with ›V N considered as the inlet to the system,(3) CðzÞ ¼ wðzÞ, wðzÞ $ 0 for z [ ›V D ,with›V D considered as the outlet from the system.The total inflow through the boundary, F inlet , is thenðF inlet ¼ cðzÞdz ð31Þ›V N

44J. WaniewskiThe inflow-averaged mean time t inlet for the solute entering the system through ›V N isdefined ast inlet ¼ 1 ððt cÞðzÞdz ð32ÞF inlet ›V NThe total inflow from the source is F source ¼ Ð VSðzÞdz, and the inflow-averaged mean timet source for the solute entering the system from the source is defined ast source ¼ 1 ðtSÞðzÞdzF sourceðVEquations (30)–(33) together with the boundary conditions for C yield the formulaðt inflow 2 ð wðD grad tÞ n ÞðzÞdz ¼›V DMF inlet þ F sourceð33Þð34Þwhere M and F inlet are calculated for the steady state C, and t inflow is defined ast inflow ¼F inletF sourcet inlet þt source :F inlet þ F sourceF inlet þ F sourceð35ÞNote that the residence time, t res , for this system is defined ast res ¼MF inlet þ F sourceð36Þc.f. equation (5). Thus, for w ; 0 one gets from equation (34) thatt inflow ¼ t resð37Þi.e. the Hardt theorem [8]. Formula (34) may be therefore considered as a generalization ofthe Hardt theorem to arbitrary Dirichlet conditions at the outlet from the system.The inflow-averaged mean time t inflow for the solute entering the system through ›V Ndepends on the boundary condition cðzÞ at the inlet. However, if cðzÞ does not depend onz [ ›V N , thent inflow ¼ 1 ðtðzÞdz ð38Þj›V N j ›V Nt inflow , given by equation (38), may be calculated directly if equation (25) is solved.5. Higher moments of exit functionUsing similar methods we can derive recursive equations and formulas for higher momentsof the exit time distribution function defined ast k ðzÞ ¼2 1 ð 1t k dmðt; zÞM 0 0ð39Þ

Transit time for transport systems 45for integer k $ 0. Note that t 0 (z) ¼ 1. In the following, we assume that t k mðt; zÞ ! 0 andt k21 Gðx; z; tÞ ! 0 for t !1, z [ Vn{›V W > ›V D }, and all integer k $ 1. Then, for k $ 1t k ðzÞ ¼ k ð 1 ð 1 ðt k21 mðt; zÞdt ¼ k t k21 Gðx; z; tÞdtdxð40ÞM 0 00c.f. equations (17) and (23) and their derivations. The PDE for t k (z) is obtained in the sameway as equation (25):VL * z t kðzÞ ¼2ktk21 ðzÞ ð41Þand the boundary conditions are the same as those for equation (25).The theorem by Hardt about the description of MTT by MRT [8], section 4, may begeneralized for higher moments of exit functions. Let us define the following parameters:t k;source ¼ 1 ðt k SÞðzÞdzF sourceðVð42Þt k;inlet ¼ 1F inletð›V Nðt kcÞðzÞdz ð43Þfor the steady state C. Thent k ¼ 1 MðVð Ct k ÞðzÞdzðt k;inflow 2 ð wðD grad t k Þ n ÞðzÞ ¼›V Dk M t k21F inlet þ F sourceð44Þð45Þwhere M and F inlet are calculated for the steady state C, and t k,inflow is defined ast k;inflow ¼F inletF sourcet k;inlet þt k;sourceF inlet þ F sourceF inlet þ F sourceð46ÞFor k ¼ 1, formulae (42)–(46) reduce to those given in section 4.6. ExamplesLet us consider the transport equation with constant coefficients on interval ½0; LŠ›C›T ¼ D ›2 C›X 2 2 J ›C›XAfter rescaling with x ¼ X/L and t ¼ DT=L 2 we get›C›t ¼ ›2 C›x 2 2 j ›C›x2 AC þ S ð47Þ2 aC þ s ð48Þ

46J. Waniewskiwith the coefficients j ¼ JL=D, a ¼ AL 2 =D, s ¼ SL 2 =D. Equation (25) is now› 2 t›z 2 þ j ›t 2 at ¼ 21 ð49Þ›zwith (›t/›z)(0) ¼ 0 and t(1) ¼ 0. The solution of equation (49) for j, a – 0istðzÞ ¼ 1 u coshðuz=2Þþj sinhðuz=2Þ1 2 expðjð1 2 zÞ=2Þau coshðu=2Þþj sinhðu=2Þpwhere u ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij 2 þ 4a.Important special cases of solution (50) areð50Þ(1) for j – 0, a ¼ 0(2) for j ¼ 0, a . 0(3) for j ¼ 0, a ¼ 0tðzÞ ¼ 1 j1 2 z 2 expð2jÞðexpðjð1 2 zÞÞ 2 1ÞjtðzÞ ¼ 1 a1 2 coshð pffiffia zÞpffifficoshð a ÞtðzÞ ¼ 1 2 ð1 2 z 2 Þð51Þð52Þð53Þand, fort 2 ðzÞ ¼ 1 52 6 2 z 2 1 2 z 2 6ð54ÞvarðzÞ ¼2 1 ð 1ðt 2 tÞ 2 dmðt; zÞ ¼t 2 2 t 2 ;M 0 0varðzÞ ¼ 1 6 ð1 2 z 4 Þ:ð55ÞFor the Dirichlet boundary conditions at both boundaries for the main equation (49) onehas the boundary conditions for t:t(0) ¼ 0 and t(1) ¼ 0. Then, the solution of equation (49)for j, a – 0ispwhere u ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij 2 þ 4a.tðzÞ ¼ 1 a1 2 eðu2jÞz=2 þðe u=2 2 e j=2 2jz=2 sinhðuz=2ÞÞesinhðu=2Þð56Þ

Some special cases of solution (56) are:Transit time for transport systems 47(1) for j – 0, a ¼ 0tðzÞ ¼ 1 j1 2 expð2jzÞ1 2 expð2jÞ 2 zð57Þ(2) for j ¼ 0, a . 0tðzÞ ¼ 1 pffiffiffia1 2 expð pffiffipffiffisinhð a zÞa zÞþðexpð a Þ 2 1Þ pffiffiffisinhð a Þð58Þ(3) for j ¼ 0, a ¼ 0tðzÞ ¼ 1 zð1 2 zÞ:2 ð59ÞFor the Neumann boundary conditions at both boundaries for the main equation (49) onehas the boundary conditions for t :(dt/dz)(0) ¼ 0 and (dt/dz)(1) ¼ 0. Then, the solution ofequation (49) for a – 0ist ðzÞ ¼ 1 að60ÞHowever, there is no solution if a ¼ 0.7. DiscussionThe equivalence of the passage times for transient and steady linear transport processes isshown for the general linear reaction–diffusion–convection equation in R n , n . 1. It wasdescribed previously as the identity of MTT and MRT for pure convective and pure diffusivetransport separately [7,8]. This identity is now proved for combined diffusive–convectivetransport with a possible linear chemical reaction. However, the “reaction” term may alsodescribe an internal sink of the solute, extending considerably (e.g. due to the presence ofblood microvessels and lymphatics) the range of systems under consideration. Furthermore,formula (46) demonstrates how higher moments of the distribution of passage times may becalculated from a steady state solution of the transport equation. The presented proof is basedon the theory of linear parabolic differential equations, but physical arguments may refer tothe assumption that the motion of any single molecule is independent of the motion of allother particles, as reflected by the linearity of the system. Therefore, the proof might be alsobased on the theory of stochastic processes, as it was used in Ref. [8].Although the transport-reaction systems are in general non-linear, the linear equations aresometimes applicable for some of these systems. There may be several reasons for usefulnessof linear systems in biological applications. First of all, some solutes are chemically “inert” andtransported only by passive diffusion and convection, as for example, urea and creatinine, thebasic markers for the clinical status of uremic patients and the efficiency of dialysis procedures,inert gases and some drugs, etc. [1,3,12,14–16]. Also, linear approximations for some generalmodels are found to be accurate and practically useful descriptions, as in near-infrared

48J. Waniewskispectroscopy [4–6]. Linearized systems appear also if small quantities of labelled markers areapplied in experiments, because such small quantities do not change the state of the system [1].Finally, linear systems are studied just as an approximation to obtain an initial insight into thesystem characteristics [11,14]. The parameters estimated from the experimental data for linearsystems may later be used directly or by interpolation for solutes that are involved in, say, lineartransport and non-linear reaction, etc. Note also that transit and residence times may be definedin a dose/concentration independent manner only for linear systems.The theorem about the equality of MRT and MTT depends on the assumption that afterperturbation of the system the total solute mass in the system returns to its steady state levelasymptotically faster than t 21 , see equations (4) and (17)–(18). Furthermore, the derivedformulas for higher moments of the exit function are based on the assumption that theasymptotic rate of return of the system mass to its steady state value is faster than t 2k if the kmoment is considered (see section 5). These properties of the exit function, taken for grantedin previous studies [7,8], are difficult to prove for general elliptic operators. The problemmay be reduced to the investigation of the principal (maximal) eigenvalue of the ellipticoperator that describes the system. Operator L, Equation (7), is in general the non-symmetricif convective term J – 0. Nevertheless, it may often be shown that its principal eigenvalue isreal and strictly negative and all other eigenvalues have real parts not higher than theprincipal eigenvalue if A . 0, see a proof for the case of Dirichlet boundary condition in Ref.[18]. From such a characteristic of the operator’s spectrum a conclusion about the desiredasymptotic properties of the solute mass may easily be obtained.The observation that information about the exit function for the linear transport systemsmay be derived at large extent from a steady state solution may be helpful in obtaininganalytical formulas for systems with simple geometry, as shown in section 6 and Ref. [8]. Itmay potentially be useful for numerical analysis of the transport systems. It also allows forthe derivation of some conclusions about transit characteristics from the studies of the systemin the steady state.References[1] Bassingthwaighte, J.B. and Goresky, C.A., 1984, Modeling in the analysis of solute and water exchange in themicrovasculature. In: S.R. Geiger, C.C. Michel, J.R. Pappenheimer and E.M. Renkin (Eds.) Handbook ofPhysiology—The Cardiovascular System IV, Vol. 4 (Microcirculation. Bethesda, MD: American PhysiologicalSociety).[2] Beard, D.A. and Bassingthwaighte, J.B., 2000, Advection and diffusion of substances in biological tissues withcomplex vascular networks, Annals of Biomedical Engineering, 28(3), 253–268.[3] Roberts, M.S. and Anissimov, Y.G., 1999, Modeling of hepatic elimination and organ distribution kinetics withthe extended convection–dispersion model, Journal of Pharmacokinetics and Biopharmaceutics, 27(4),343–382.[4] Arridge, S.R. and Schweiger, M., 1995, Direct calculation of the moments of the distribution of photon time offlight in the tissue with finite element method, Applied Optics, 34(15), 2683–2687.[5] Liebert, A., Wabnitz, H., Grosenick, D., Moller, M., Macdonald, R. and Rinneberg, H., 2003, Evaluation ofoptical properties of highly scattering media by moments of distributions of times of flight of photons, AppliedOptics, 42(28), 5785–5792.[6] Liebert, A., Wabnitz, H., Steinbrink, J., Obrig, H., Moller, M., Macdonald, R., et al., 2004, Time-resolvedmultidistance near-infrared spectroscopy of the adult head: intracerebal and extracerebal absorption changesfrom moments of distribution of flight of photons, Applied Optics, 43(15), 3037–3047.[7] Meier, P. and Zierler, K., 1954, On the theory of the indicator-dilution method for measurement of blood flowand volume, Journal of Applied Physiology, 12(6), 731–744.[8] Hardt, S.L., 1981, The diffusion transit time; a simple derivation, Bulletin of Mathematical Biology, 45, 89–99.[9] Segel, L.A., 1980, Mathematical Models in Molecular and Cellular Biology (Cambridge: CambridgeUniversity Press).

Transit time for transport systems 49[10] Mellick, G.D., Anissimov, Y.G., Bracken, A.J. and Roberts, M.S., 1997, Metabolite mean transit times in theliver as predicted by various models of hepatic elimination, Journal of Pharmacokinetics andBiopharmaceutics, 25(4), 477–505.[11] Roberts, M.S. and Rowland, M., 1986, A dispersion model of hepatic elimination: 1. Formulation of the modeland bolus considerations, Journal of Pharmacokinetics and Biopharmaceutics, 14(3), 227–260.[12] Flessner, M.F., 2005, The transport barrier in intraperitoneal therapy, American Journal of Physiology RenalPhysiology, 288(3), F433–F442.[13] Flessner, M.F., 2001, Transport of protein in the abdominal wall during intraperitoneal therapy. I. Theoreticalapproach, American Journal of Physiology. Gastrointestinal and Liver Physiology, 281(2), G424–G437.[14] Waniewski, J., 2001, Physiological interpretation of solute transport parameters for peritoneal dialysis, Journalof Theoretical Medicine, 3, 177–190.[15] Waniewski, J., Werynski, A. and Lindholm, B., 1999, Effect of blood perfusion on diffusive transport inperitoneal dialysis, Kidney International, 56(2), 707–713.[16] Waniewski, J., 2002, Distributed modeling of diffusive solute transport in peritoneal dialysis, Annals ofBiomedical Engineering, 30(9), 1181–1195.[17] Waniewski, J., 2004, A mathematical model of local stimulation of perfusion by vasoactive agent diffusingfrom tissue surface, Cardiovascular Engineering An International Journal, 4(1), 115–123.[18] Evans, L.C., 1998, Partial Differential Equations (Providence, RI: American Mathematical Society).