BRIEF COMMUNICATION Simulting the Effect of Exercise on Ure Clernce in Hemodily si s STEPHEN W. SMYE,* ELIZABETH J. LINDLEY,* nd ERIC J. WILLt Deprtments of *Medicl Physics nd Renl Medicine, St. Jmes s University Hospitl, Leeds, United Kingdom. Abstrct. A two-comprtment model of ure kinetics during hemodilysis is used to predict the effect of exercise on hemodilysis dose. It is ssumed tht the two comprtments represent tissues tht re perfused by low nd high blood flows (initilly I. 1 L/min nd 3.8 L/min). The effect of chnging the distribution of flows between the comprtments, emulting the effect of exercise, is simulted using the model equtions for rnge of dilyzer clernces. Comprtmentl volumes re ssumed constnt (33.4 L nd 8.6 L for low- nd high-flow comprtments, respectively). The nlysis identifies muscle perfusion s rte-limiting fctor during the lter stges of hemodilysis nd illustrtes the benefit of exercise during this phse in incresing dilysis efficiency. The model suggests tht the postdilysis rebound in the blood ure concentrtion is eliminted by incresing flow to the low-flow comprtment from 1. 1 L/min to 7. 1 L/min nd sustining this for t lest 3 mm of 15-mm dilysis session, independent of the dilyzer clernce. Additionl exercise will not increse the dilysis dose. Experimentl studies re required to confirm the nlysis. (J Am Soc Nephrol 9: 128-132, 1998) In prctice, there re limited mens of incresing the dilysis dose. Dilyzers hve become more efficient, nd incresing blood flows help increse clernce. Extending dilysis s is less cceptble to ptients nd is costly. Ure kinetic modeling is firmly estblished s n pproch to the problem of estimting hemodilysis dequcy. However, the importnce of developing nd pplying kinetic model lies not only in the determintion of dilysis dose, but lso in providing insights to the underlying solute trnsport mechnisms. These insights my be expected to inform novel dilysis strtegies nd help to optimize clinicl procedures. Ure kinetic models were first developed by Dedrick nd Bischoff (1). Gotch nd Srgent subsequently pplied single pool model to clinicl popultion when nlyzing the results of the Ntionl Coopertive Dilysis Study (2). This nlysis used vrible volume single pool model of ure distribution nd expressed hemodilysis dose s Kt/V, where K nd t denote dilyzer clernce nd tretment, respectively, nd V denotes the ntionl single pool ure distribution volume. Models of ure kinetics were subsequently extended to explin the postdibysis ure rebound (3), which increses with reltive clernce K/V. In two-comprtment model, the ure is ssumed to be distributed in two well-mixed comprtment, initilly identified s intr- nd extrcellulr comprtments (3), between which ure trnsport occurs by diffusion. Schneditz et Received April 15, 1997. Accepted June 3, 1997. Correspondence to Dr. Stephen W. Smye. Medicl Physics Deprtment, St. Jmes s University Hospitl, Leeds LS9 7TF, United Kingdom. 146-6673/91-1 28$3./ Journl of the Americn Society of Nephrology Copyright 1998 by the Americn Society of Nephrology l. (4,5) hve since shown tht vritions in regionl blood flow could result in n effective multicomprtmentl distribution nd on this bsis ccount for the postdilysis ure rebound. In this pproch, the two-comprtment distribution rises from two tissue comprtments perfused by either low or high blood flows, nd the mixing of blood with different ure concentrtions between these comprtments ccounts for mjor prt of the postdibysis rebound. This rticle uses the flow model of ure kinetics to ssess the likely effect of chnging blood flow on dilysis efficiency nd considers the impliction of these findings for dilysis strtegies. In this nlysis, the solute considered is ure, lthough similr pproch my be developed for other sobutes. Model Development The bsic model (4) comprises high-flow comprtment (smll orgns, lungs, blood, hert, brin, nd portl system; references 4 nd 6) in prllel with low-flow comprtment (bone, muscle, skin, nd dipose tissue; references 4 nd 6) (see Figure I ). If VL nd CL denote the ure distribution volume nd concentrtion in the low-flow comprtment nd VH nd CH denote corresponding prmeters in the high-flow comprtments, ssuming tht diffusion of ure between tissue nd blood is rpid gives: d(vlcl) - dt = (C CI)QL + C1 d(v11 CH) dt = (C CFI)Qn + CH where QL nd QH denote the flow to the low- nd high-flow comprtments, respectively, C 5 the ure concentrtion in the rteril dvl dv
Effect of Exercise on Ure Clernce in Hernodilysis 129 ( I cl ch Figure 1 Flow model cv cdo where = QH/(QS + Kd) nd b = QL(Qs + Kd). Assuming tht the vrious blood flows nd K1 re constnt throughout the dilysis. equtions 1 through 3 give (7): Cit = Ae + Bek z (4) Low Flow High Flow Pool Pool VL. cl VH. ch #{176}L c Qs QjHA oc J C Dilyzer Figure 1. Two-comprtment ure kinetic model used (from Schneditz ci!. [4]) for simultion of effects of exercise. Nottion s in text. CL CeA1 + Dc -A I (5) where A, B, C, D, A. nd A re constnts derived from the initil condition CH () CL () 1 nd given in detil in Appendix I. The ure concentrtion is expressed s frction of the initil concentrtion, nd the comprtmentl volumes re ssumed to be constnt. The mss (iii) of ure removed during dilysis is given by:??1 = ( VH + V1) - ( V11C11( T) + VLCL( T)) (6) with the postdilysis equilibrium ure concentrtion C given by: Cc = 1 - in blood, nd ure genertion nd ultrfiltrtion hve been neglected. By conservtion of mss, the concentrtion of ure in the mixed venous pool is: C, = tic11 + ( 1 - u)c1 where ii = QH/QS nd Q denotes the systemic flow (Q + Q1). Neglecting ny delys in mixing due to the finite required for blood to pss long the different vessels, the venous blood mixes in the hert with blood from the ccess nd dilysis venous line giving: QcC Q5C, + QtC + QhCd,. where Q. Q1, nd Q, represent the crdic output. fistul bypss, nd dilyzer blood flows, respectively. nd C(1() is the concentrtion of ure t the dilyzer outlet. Neglecting ccess recircultion, if the dilyzer ure clernce is Kd, then: C C11 + b C1 Becuse the comprtmentl volumes re ssumed to be constnt (VH nd V = 8.6 L nd 33.4 L, respectively: references 4 nd 6) nd ure genertion hs been neglected, the true dilysis dose is given by Kt/(V + VL), nd the pprent dose given by the single-comprtment model is log(l/c11(tfl. The difference (D) between the pprent nd ctul dilysis doses is therefore given by: D = e;;] - F 1.] Kt + V1) The postdilysis percentge rebound in ure concentrtion is defined s: 1C - C11(T) R 1L- = The effect of chnges in the distribution of blood flow on the ( 3 ) rebound nd difference between the pprent nd ctul dilysis doses 1 8 6 QL mi/mm 4 2 1 5 1.4.5 1 1.2 5.8 f.6 Figure 2. Vrition of perfusion rte Q1 to low-flow comprtment (muscle, bone, skin) with nd exercise prmeter (j).
I 3 Journl of the Americn Society of Nephrology % rebound % rebound 14 12 the bow-flow comprtment (initilly ssumed to be 1.1 L/min; references 4 nd 6) due to chnges in crdic output (8), wheres QH remins constnt (3.8 L/min) (6). Given tht exercise my produce up to 1-fold increse in the muscle perfusion rte (8), but tht this increse my not be chieved in ptients with renl filure, it is ssumed tht QL increses smoothly from I. I to 7. 1 L/min over 2- to 3-mm period nd then remins constnt. This is consistent with regulr periods of exercise of, for exmple, 1 mm durtion (9) interspersed with short recovery period (severl minutes), during which the high flow to the muscle comprtment is sustined. This nlysis lso ssumes tht the ccess blood flow remins constnt throughout the dilysis session nd is independent of exercise. The effect of this exercise regimen will depend on the into dilysis when the increse in QL occurs. For the purpose of the clcultion, this is defined by it, the midpoint of the 2-mm period of incresing flow. f therefore represents the frctionl into dilysis t which this exercise occurs. f = suggests tht the ptient begn exercising before strting dilysis. If f > I, the first exercise period occurred close to the end of, or fter, dilysis nd will hve little or no effect on the ure kinetics. The flow QL t t is represented mthemticlly by: 1 e 2( t -[1)/TI QL I. 1 + 6.[ 1 + e)2o(t_th] nd the dependence of QL on f nd t is shown digrmmticlly in Figure 2 for < t < 15 nd.5 <f< 1.5. In this cse, the solutions to equtions 4 nd 5 no longer pply, nd numericl solution is required. The effect of chnges in the distribution of blood flow nd the dilyzer clernce (K) on the rebound (R) nd difference (D) between the pprent nd ctul dilysis doses is b Figure 3. (A) Vrition of postdilysis percentge rebound with exercise prmeter (J) nd dilyzer clernce ure (K). (B) Vrition of postdilysis percentge rebound with exercise prmeter (J) nd dilyzer clernce ure (K =.3 L/min). my be simulted by vrying QL nd QH during the tretment. It is ssumed tht exercise produces n increse in the perfusion rte QL of simulted using equtions I through 3 nd 8 through 1 for.5 <f< 1.5 nd.15 < K <.35 L/min nd T = 15 mm. Numericl solutions to the equtions were obtined using Mthemtic (version 2.2. Wolfrm Reserch, Inc., Chmpign, IL) on PC pltform (13 MHz Pentium [Intel], Ebonex, Ltd., London, United Kingdom). Results The results of the simultion re illustrted in Figures 3 nd 4, which show the effect of vritions infnd K on percentge rebound (R) nd the difference (D) between the true nd ctul dilysis dose. Figures 5 nd 6 demonstrte the vrition in KT/V difference.15..1 K (1/mm) Figure 4. Vrition of the difference between the true nd pprent dilysis dose with exercise prmeter (J) nd dibyzer clernce ure (K).
Effect of Exercise on Ure Clernce in Hemodilysis 131.9.8 OQ#{149}7.-4 (4.6.5.4 SSSSSSSS#{149}.3 -- - SSSSSSS. 2 4 6 8 1 12 14 S S Figure 5. Vrition of the rteril ure concentrtion during dilysis for K =.25 L/min nd I = 1.5 (no exercise during dilysis)..9.8 U c: 7 (1 <.6.5.4 #{149}- #{149} -... -. -..... -. 2 4 6 8 1 12 14 Figure 6. Vrition of the rteril ure concentrtion during dilysis for K =.25 L/min nd f =.5 (exercise during dilysis). rteril ure concentrtion throughout dilysis for K =.25 L/min, f = 1.5 (no exercise during dilysis), nd f =.5 (exercise during dilysis), respectively. Discussion The results of this nlysis suggest tht postdilysis ure rebound cn be virtully eliminted if high blood flow to the tissues cn be mintined for t lest the lst 3 mm of 15-mm dilysis session. Figure 3, A nd B, nd Figure 4 illustrte tht, for short periods of exercise (f > 1) during which mximum flow to the muscle tissue my not be chieved, the rebound, nd therefore the difference, between the ctul nd pprent dose increses linerly with dilyzer clernce. This is consistent with the previous observtions by Dugirds (1). Protrcted exercise (f <.8), in eliminting the rebound, lso elimintes the dependence of the rebound on K nd thereby suggests cution in pplying the Dugirds rte eqution (1) for the true Kt/V to ptients in whom exercise hs occurred. The simultion results my be understood by noting tht during the erly stges of hemodilysis, clernce of ure from tissues perfused by reltively high blood flow occurs rpidly, nd the clernce from skeletl muscle, skin, nd bone occurs more slowly. The dependence of blood ure concentrtion during this phse (see Figures 5 nd 6) reflects the dilyzer clernce rte, becuse the required to cler ure through the dilyzer is much longer thn the s ssocited with clernce from the high flow rte tissues. Subsequently, the contribution of ure from the well-penfused tissues to the blood level of ure becomes negligible, nd the dominnt fctor is the trnsport of ure from the poorly perfused skeletl muscle, skin, bone, nd ft, nd subsequent ure clernce through the dibyzer. The rte-limiting step becomes the perfusion of skeletl muscle, skin, nd bone. If muscle blood flow remins constnt during this stge, nd it is ssumed tht clernce through the dilyzer is reltively rpid. then the rte of decrese in blood ure concentrtion reflects the muscle perfusion rte. After dilysis, the rebound in ure concentrtion is due to blood from high flow tissues with reltively low ure concentrtion, mixing with tht from skeletl muscle, bone, nd skin, which hve higher ure concentrtion. Exercise during hemodilysis results in n increse in the rte of skeletl muscle, bone, nd skin comprtment ure clernce by fctor of up to 6 nd reduction of the concentrtion grdient between the two comprtments. The rteril ure concentrtion rises fter onset of exercise (compre Figures 5 nd 6 for I > 6 mm), reflecting the increse in ure trnsport from the low-flow comprtment. The nlysis emphsizes tht n improvement in dilysis clermice is likely to be chieved by improving muscle perfusion during the lter stges of dilysis, rther thn simply incresing dilyzer clernce. In this model, there is theoreticl limit beyond which no further increse in dilysis dose is chieved. The dose of dilysis my thus be incresed by clinicl procedures designed to optimize tissue blood flow during the ppliction of ny prticulr dilyzer nd ccess flow rte. There my be phrmcologicl mens of prepring nd sustining tissue perfusion to tke dvntge of the reltively nrrow rnge of convenient nd cost-effective dilyzer clernces. The study gives theoreticl bsis for developing nd interpreting further experimentl studies designed to study the effect of exercise on dilysis dose. It lso suggests bsis for optimizing these regimens to ensure tht the mximum dose of dilysis is delivered with minimum ptient effort. Appendix 1 In the min text, the solutions of the model equtions 1 nd 2 under conditions of constnt volume nd flow re given by equtions 4 nd 5 (see references 4 nd 7), with the constnts defined by: b-, 1-l, - 1, - A = /2 4pr - 2p = - -, [1 -b-(vla+/ql)] [1 b (VLAJQ3]
132 Journl of the Americn Society of Nephrology ndp= QL ( I - - b). Ure concentrtions hve been expressed s frction of the initil concentrtion. References I. Dedrick RL, Bischoff KB: Phrmcokinetics in pplictions of the rtificil kidney. Chem Eng Prog Symp Ser 64: 32-44, 1968 2. Gotch F, Srgent I: A mechnistic nlysis of the Ntionl Coopertive Dilysis Study (NCDS). Kidney Int 28: 526-534, I 985 3. Pedrini LA, Zereik 5, Rsmy 5: Cuses, kinetics nd clinicl implictions of post-hemodilysis ure rebound. Kidney hit 34: 817-824, 1988 4. Schneditz D, Vn Stone IC, Dugirds IT: A regionl blood circultion lterntive to in-series two comprtmentl ure kinetic modelling. ASAIO J 39: M573-M577, 1993 5. Schneditz D, Friyike B, Osheroff R, Levin NW: Is intercomprtmentl ure clernce during hemodilysis perfusion term? A comprison of two pooi ure kinetic models J Am Soc Nephro/ 6: 136-137, 1995 6. Willims LR, Leggett RW: Reference vlues for resting blood flow to orgns of mn. C/in Phys Physio/ Mes 1: 187-217, 1989 7. Smye SW, Evns JHC, Will E, Brocklebnk IT: Peditric hemodilysis: Estimtion of tretment efficiency in the presence of ure rebound. C/in Phys Physiol Mes 13: 51-62, 1992 8. Dowell RT: Chpter 1. In: Exercise Medicine: Physiologicl Principles nd Clinicl Applictions. edited by Bove AA, Lowenthb DT, Orlndo, Acdemic, 1983, pp 2-26 9. Ronco C, Crepldi C, Brendoln A, L Grec G: Intrdilytic exercise increses effective dilysis efficiency nd reduces rebound [Abstrct]. J Am Soc Nephro/ 6: 612, 1995 1. Dugirds IT, Schneditz D: Overestimtion of hemodilysis dose depends on dilysis dose by regionl blood flow model but not by conventionl two-pool ure kinetic nlysis. ASAIO J 41: M7l9-M724, 1995