Metabolism of complement factor D in renal failure

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1 Kidney Interntionl, Vol. 34 (1988), pp Metbolism of complement fctor D in renl filure MANUL PASCUAL, GRTRAUD STIGR, JURK STRICHR, KVIN MACON, JOHN. VOLANAKIS, nd JURG A. SCHIFFRLI Déprtement de Medecine, nd Division d'informtique, Hôpitl Cntonl Universitire, Genev, Switzerlnd; nd Deprtment of Medicine, University of Albm, Birminghm, Albm, USA Metbolism of complement fctor D in renl filure. Fctor D is n essentil enzyme of the lterntive pthwy of complement. Its plsm concentrtion increses pproximtely tenfold in end-stge renl filure (SRF). To nlyze its metbolism in humns, we injected purified rdiolbelled fctor D into 5 helthy individuls nd 12 ptients with vrious renl diseses or renl filure. Frctionl metbolic rtes (FMR) nd extrvsculr/intrvsculr distributions (V/IV) were clculted using comprtmentl model. The FMR ws very rpid in norml individuls (men 59.6 %/hr; rnge 74.1 to 5.5), significntly diminished in the five ptients with SRF (5.7 %/hr; 7. to 2.8; P <.4), nd correlted well with the cretinine clernce (r =.89; P <.1). The extrrenl ctbolic rte ws not modified in renl filure. Despite significnt inverse correltion between plsm levels of fctor D nd cretinine clernce r =.68; P <.2], fctor D levels were not sensitive indictor of renl function becuse the synthesis rte (SR) vried widely from one individul to nother (men SR: 62.9 sg/kgihr; 14.9 to 136.5). Fctor D synthesis ws not significntly ltered by renl function, nd did not correlte with C-rective protein, suggesting tht fctor D is not n cute phse protein. The proportion of intct fctor D elimintion in the urine ws incresed in ptients with tubulr dysfunction (up to 15% compred to <.2% in norml individuls) confirming tht under norml circumstnces fctor D is filtered through the glomerulus nd ctbolized by tubulr cells. In renl filure the V/IV rtio ws diminished (in normls: 8.5, 1.8 to 7.1; in SRF: 2.4, 3.1 to 1.9; P <.4), indicting tht ccumultion of fctor D ws more pronounced intrvsculrly. These findings suggest tht in humns, fctor D ctbolism is very rpid nd essentilly renl. In renl filure the severl-fold increse in fctor D concentrtion my be responsible for enhnced lterntive pthwy ctivtion, prticulrly in the circultion. In norml serum fctor D is the rte limiting enzyme of the lterntive pthwy of complement ctivtion [1]. A sensitive rdioimmunossy to mesure fctor D in plsm hs been developed only recently, nd initil mesurements indicted tht fctor D levels were incresed in renl filure. Ptients with end-stge renl filure (SRF) hd n pproximtely 1-fold increse in fctor D [2, 31. No fctor D ws detectble in the urine of norml individuls or ptients with nephrotic syndrome, however, lrge mounts of fctor D were present in the urine of ptient with the Fnconi syndrome. These findings suggested tht fctor D is filtered through the glomerulus nd ctbolized in the proximl renl tubules. In vivo nephron microperfusion nd in vitro perfusion of rt kidneys indicted Received for publiction August 12, 1987 nd in revised form My 17, by the Interntionl Society of Nephrology lso tht fctor D is hndled initilly vi glomerulr filtrtion nd ctbolized fter rebsorption by the proximl tubule [4}. In vitro mcrophges nd humn heptocyte-derived cell line hve been shown to synthesize fctor D [5, 61. In vivo fctor D ppered not to be protein of the cute phse response [7, 8]. However synthesis, distribution nd ctbolism of fctor D hve not been defined, thus limiting the understnding of serum fctor D levels in inflmmtory sttes. Further, these prmeters my give insight into the biologicl role of fctor D under physiologicl nd pthologicl conditions such s renl filure. The im of this work ws to define the metbolism of fctor D by injecting purified rdiolbelled fctor D in humns with norml, decresed or bsent renl function. The min findings were tht fctor D ws rpidly eliminted by the kidney in norml subjects, nd tht fctor D primrily ccumulted in the circultion in renl filure becuse of striking modificiton in its bodily distribution. In contrst, extr-renl ctbolic rte nd synthesis of fctor D were not modified by impired renl function. Methods Control subjects nd ptients Metbolic studies were performed in 5 helthy volunteers (Tble 1, subjects 1 to 5) nd 12 ptients with diminished or bsent renl function (ptients 6 to 17). Ptients 8, 12, 13 nd 17 presented with tubulr dysfunction s ttested by lysozymuri. Ptients 6 to 1 (Tble 2) were on long-term dilysis for end-stge renl filure (SRF); three were totlly nuric. Ptient 1 hd membrno-prolifertive glomerulonephritis type II,.29 g/liter of C3 (norml rnge:.7 to 1.3), nd circulting nephritic fctor. Ptients 11 to 17 (Tble 3) hd intermedite degrees of renl filure nd were not on dilysis. Ptient 12 hd received gentmicin (2 x 8 mg/dy) for two weeks becuse of sepsis, but his renl filure ntedted the infectious episode. Ptients 13 nd 17 hd osteomyelitis nd were on gentmicin (3 X 8 mg/dy) for more thn three weeks t the time of study. Both hd norml renl function before initition of ntibiotic therpy. Ptient 14 ws severely nephrotic (membrnous nephritis): his urinry protein excretion rnged from 4.2 to 9 g/24 hr. Ptients 15 nd 16 were studied three months fter renl trnsplnttion, nd both were treted with cyclosporine t the time of study. 529

2 53 Pscul et!: Metbolism of fctor D in renl filure Subject Cretinine clernce ml/min Tble 1. Fctor D metbolism in control subjects Synthesis T Serum D FMR V/IV rte q q g/ml % per hr rtio p.g/kglhr X 1O3/min q51 Glomerulr sieving coeff , Men Ptient Cretinine clernce ml/min Tble 2. Fctor D metbolism in end-stge renl filure; ptients on dilysis Synthesis 12 Serum D FMR V/IV rte q pg/ml % per hr rtio 1Lg/kg/hr q21 T q51 x 13/min < < < Men < Ptient Cretinine clernce mi/mm Tble 3. Fctor D metbolism in ptients with reduced renl function (non-dilyzed) Synthesis 12 Serum D FMR V/IV rte q jig/mi % per hr rtio jig/kg/hr q21 T q51 X 1O3/min Preprtion of rdiolbeiled fctor D Fctor D ws purified from the urine of ptient with Lowe's syndrome (incomplete Fnconi's syndrome with congenitl eye nd centrl nervous system defects) employing three successive chromtogrphic steps: Bio-Rex 7 (Bio Rd Lbortories, Richmond, Virgini, USA), hydroxyptite HPLC nd reversephse HPLC [9]. Cultures of purified fctor D were negtive for cytomeglovirus nd humn immunodeficiency virus (HIV). This ptient hd no detectble heptitis-b surfce ntigen nd no HIV ntibodies in his plsm. The protein ws rdiolbelled with '251-Bolton Hunter regent (Amershm, Buckinghmshire, UK) [1] to specific ctivity of 3.7 pc/jig of fctor D. Free iodine ws removed by gel filtrtion (Sephdex G-25, Phrmci, Uppsl, Sweden) fter hving dded 1% humn serum lbumin (HSA, Armour, stbourne, UK) to the rdiolbelled fctor D; the column ws prewshed with phosphte buffer ph 7.2 contining NCl 9 g/liter (PBS) nd 1% HSA. Trichlorocetic cid precipittion (TCA, finl concentrtion 1%) showed tht more thn 95% of the rdioctivity ws protein-bound. After Millipore filtrtion fctor D ws dilyzed ginst PBS, liquots were snp frozen in liquid nitrogen nd stored t 8 C until used. All mnipultions were done using sterile buffers nd equipment. Sterility testing ws done using stndrd bcteriologicl methods nd the preprtion ws pyrogen-free when tested in four rbbits, Purity nd functionl ctivity of rdiolbelled fctor D Sodium dodecylsulfte polycrilmide gel electrophoresis (SDS-PAG) [11] followed by utordiogrphy reveled only one bnd of pproximtely 23 Kd (Fig. 1). In lterntive pthwy hemolytic ssys using guine pig erythrocytes nd D-depleted serum, the functionl ctivity of fctor D ws unchnged by the lbelling procedure [12, 13]. The plsm disppernce curves of fctor D were studied in three rbbits fter the injection of 1 to 2 jici '251-D. The nimls were given drinking wter contining.1% of potssium iodide for three dys before the experiments. Blood smples were collected into DTA 1 m over four hours. After centrifugtion, the protein bound (TCA precipitble) nd free rdioctivity were mesured in plsm. In order to ssess whether the fctor D preprtion ws dentured (or ggregted), studies were performed in two rbbits to compre the elimintion rtes of unscreened nd biologiclly screened '251-D. The first rbbit injected with 21 pci 1251D ws bled fter 3 minutes (1 ml); 5 ml of its plsm

3 Pscul et!: Metbolism of fctor D in renl filure p A Fig. 1. SDS-PAG of'251-lbelled fctor Dfollowed by utordiogrphy. Mrkers for moleculr weights re indicted (Kd). () Purified '251-D, nd (b) 1-fold concentrted urine from norml subject: the protein bound 1251 ws intct fctor D. contining.6 pci of '251-D were injected into second rbbit nd the two disppernce curves were compred. Humn metbolic studies These studies were pproved by the ethicl committee of the Hôpitl Cntonl Universitire of Genev. After informed consent hd been obtined, norml volunteers nd ptients were injected with 7 to 15 Ci '251-D. Two milliliters of potssium iodide 2% were given twice dily from the dy before the strt of the study until its end. Blood smples were collected in DTA t 3 minutes, 5, 1, 2, 3, 6 minutes, nd then t hourly intervls during the first six hours; lter smpling times were chosen ccording to the rte of disppernce of '251-D, tht is, for 24 hours in normls or two to four dys in ptients with renl filure. In dilysis ptients (ptients 6 to 1), rdiolbelled fctor D ws injected two hours fter the end of dilysis session. xcept for ptient 6, the plsm disppernce curves were not followed for more thn 5 hours, tht is, they were studied between two dilysis sessions in dilysis fluid ws miniml (<.1% of the injected dose), nd thus did not ffect the nlysis of the results. The plsm elimintion curve ws unffected by dilysis in ptient 6. The protein-bound rdioctivity in plsm smples ws mesured fter TCA precipittion. Urine collections were done for three to five dys. Urinry protein-bound rdioctivity ws determined by TCA precipittion of 4 ml smple supplemented B with 2 mg/liter IgG s crrier. Two hundred milliliters of urine were concentrted to volume of 2 ml by ultrfiltrtion using membrne with cut-off of 1 Kd (Amicon). To show tht fctor D ws present in the retined rdioctivity, smples of concentrted urines were nlyzed by SDS-PAG followed by utordiogrphy. Fctor D concentrtions in blood nd urine were determined by rdioimmunossy s previously described [2] nd by hemolytic ssy in ser tht ws sensitive enough to build dose response curve of ctivity [12, 13]. The specific hemolytic ctivity of fctor D ws obtined from the rtio of hemolytic/ ntigenic mesurements. C-rective protein (CRP) ws mesured by single rdil immunodiffusion. Spontneous in vitro ctivtion of C3 ws mesured in seven ser of ptients included in the present study: 5 l of serum ws mixed with 5 pi of brbitone buffer contining NCI.145 M, MgCI.83 mm, nd CC1.25 ms, ph 7.2 (CFD, Oxoid), nd incubted for 2 minutes t 37 C. The spontneous ctivtion of C3 ws mesured by crossed immunoelectrophoresis using monospecific got nti-humn C3 ntiserum (Miles) [13]. In control experiments, 5 pi of monoclonl mouse nti-humn fctor D ntibody (M. Pscul, unpublished) ws dded to the mixtures before incubtion. Comprtmentl tnode! used for the nlysis of fctor D metbolism The results were nlyzed using model with five comprtments for '251-D; 1) extrvsculr nd 2) intrvsculr comprtments for '251-D; 3 nd 4) urinry comprtments for '251-D nd 125J, respectively; 5) extr-urinry elimintion comprtment (feces, sliv, etc.). The model ws expressed in terms of system of five ordinry differentil equtions, with five unknown prmeters to be estimted (minly the exchnge rtes between comprtments) (Fig. 2 nd Appendix). Differentil equtions of the model: n S1(t) = q1i S(t), i = 1,..., n, j= I where /n (=5) is the number of comprtments, S1 the stte functions, q the trnsfer rtes (most of them ) if i j, nd = The initil conditions of the system were given s mesured initil dose into comprtment 2. The fct tht we expressed the model s closed comprtmentl system provided prcticl device for controlling the stbility of the numericl computtion, s the sum of ll stte functions hd to remin constnt (= initil dose). Clcultions were run using the progrm BWITCH (streicher J, unpublished dt) consisting of n optimizer bsed on the Guss-Newton-Mrqurdt lgorithm [14] nd n ordinry differentil equtions' solver bsed on the Runge-Kutt type method of Dormnd nd Prince [15]. Wheres the volumes of the comprtments nd the exchnge rtes were treted s unknown prmeters to be directly computed by the optimiztion methods, the extrvsculr! ji q1

4 532 Pscul et l: Metbolism of fctor D in renl filure >. 4-, xtrvsculr q5 1 I-. C, CD -n o.. q. 'II Intrvsculr q32 q42 4-, U, CD 1 Time, mm 2 3 Fig. 3. '251-D disppernce curve in experimentl nimls. Symbols re: screened nd unscreened U preprtions of '251-D showing comprble plsm disppernce curves. limintion in urine xtr urinry elimintion (res. sliv, stomch, etc.) Fig. 2. Comprtmentl model used for the nlysis offctor D metbolism; q exchnge rtes between comprtments (frction/mm); : comprtments in which mesures were mde. Becuse in most individuls q32 ws very low, the totl renl ctbolism (T) ws nlyzed: T = q32 + q42. Although this model hs five comprtments from the mthemticl point of view, it hs only two bodily comprtments (intrvsculr nd extrvsculr). intrvsculr (V/IV) rtios nd the frctionl metbolic rtes (FMR % of the intrvsculr pool of fctor D lost per hour by ll mechnisms, tht is, by renl nd extrrenl ctbolism) were deduced by elementry lgebr from the fitted vlues of the exchnge rtes, using the following formule tht pply to the prticulr context of the comprtment structure: V/IV rtio = q51 FMR T + 12 q21 + q5, V(q21 + q51 + T) 4 q21 q12 (q2, + q51 q1 + T) 2 q21 An pproximtion for the glomerulr sieving coefficient (USC) of fctor D in the five norml individuls ws clculted from the formul: fctor D clernce/cretinine clernce, where fctor D clernce = T (frction of the intrvsculr pool of fctor D disppering into the urine/mm) x plsm volume (ml) 116] D-125l Sttisticl nlysis When pproprite, liner correltion coefficients nd the Mnn-Whitney test were used. Results Initil experiments were designed to estblish tht fctor D ws not ltered fter the Bolton-Hunter rdiolbelling procedure; the functionl hemolytic ctivity of the rdiolbelled protein ws 1% compred to purified unlbelled fctor D. The turnover chrcteristics of the unscreened nd the screened preprtion of '251-D re shown in Figure 3; plsm disppernce curves were similr. The screened preprtion of '251-D ws clered t slightly reduced rte which could indicte smll frction of fctor D denturtion or smll individul vrition of fctor D ctbolism s observed in two other rbbits injected with the unscreened preprtion. Thirty minutes fter i,v. injection of '251-D remining in the circultion ws not modified s judged by SDS-PAG utordiogrphy. Fctor D metbolism in humns Metbolic prmeters of control subjects re shown in Tble 1. The very high FMR of fctor D (pproximtely 6%/br) ws explined essentilly by fst urinry elimintion (T). Most fctor D ws ctbolized in the kidney (75%) nd the rest (25%) ws eliminted by other routes (extr-urinry elimintion). The rte of diffusion of fctor D into the V spce ws rpid (q12), nd the clculted V/IV rtio indicted tht very lrge frction of the totl mount of fctor D ws in the LV comprtment. Using the results obtined for urinry elimintion, nd considering tht this elimintion ws due only to glomerulr filtrtion we clculted the USC for fctor D. The men USC for the five norml subjects ws.36, tht is, corresponding to high filtrtion rte pproximtely one third of the clernce of cretinine. The ptients' results re shown in Tbles 2 nd 3. Those with SRF (Tble 2) hd very different metbolic results from the control subjects; the FMR nd the V/IV rtios were significntly reduced, respectively 1-fold (P <.4) nd 4-fold (P C.4). Defective renl ctbolism of fctor D ws not compensted for by ny increse in the extr-renl elimintion rte (q5l). The ctbolism of fctor D in the ptient with mesngiocpillry UN type II (ptient 1) ws similr to tht in the other four ptients. The ptients with reduced renl function (Tble 3)

5 Pscul Cf 1 Metbolism of fctor D in renl filure U. (I) U d 3 2 :1 U). I Serum cretinine, mmo/ filter. I. o Cretin i ne clernce, rn//minute Fig. 4. Correltion between renl function nd the frctionl metbolic rte (FMR) of fctor D fr =.89, P <.1]. Symbols re: () control subjects; (U) ptients. > > LU U U 'U (42.2) hd metbolic prmeters intermedite between norml subjects nd ptients with SRF. The FMR nd the V/IV rtio correlted with cretinine clernce (Figs. 4 nd 5), however, the V/IV rtios were well bove the expected vlues in the two ptients in whom gentmicin nephrotoxocity ws very likely (ptients 13 nd 17). Fctor D synthesis ws very vrible from one individul to nother; this observtion would explin why fctor D levels in serum were not sensitive indictors of renl function in ptients with cretinine clernce between 5 nd 23 mllmin, lthough the generl correltion with serum cretinine ws excellent (Fig. 6). Fctor D synthesis did not correlte with plsm CRP concentrtion which ws incresed in five ptients. There ws no significnt chnge in the specific hemolytic ctivity of fctor D in renl filure. In norml subjects less thn.2% of the rdioctivity recov- I I I I I I Cretinine clernce, mi/minute Fig. 5. Correltion between renl function nd the extrvsculrl intrvsculr rtio (V/IV) of fctor Dir =.8, P <.1]. Symbols re: () control subjects; (U) ptients. F.! I Cretinine clernce, rn//minute Fig. 6. Reltion between cretinine clernce nd serum fctor D levels. Symbols re: (LI) control subjects; ( ) ptients (insert: reltion between fctor D nd serum cretinine; the white squre represents the five control subjects). ered in the urine ws protein-bound. When the urine smples of two norml subjects were concentrted 1-fold it ws possible to show by SDS-PAG utordiogrphy tht this miniml mount of protein-bound rdioctivity ws '251-D (Fig. 1). In contrst, the urine of ptients with lysozymuri (ptients 8, 12, 13, 14 nd 17) contined significntly more protein-bound rdioctivity (respectively 15%, 12%, 2.2%, 14.5% nd 1%) which ws shown to be intct '251-D s well. The presence of fctor D in the urine ws confirmed by rdioimmunossy in the three ptients with more thn 1% protein-bound I25J To see whether lterntive pthwy ctivtion of C3 might be enhnced in renl filure, we mesured the spontneous in vitro ctivtion of C3 in serum incubted t 37 C for 2 minutes. The percentge of C3 converted to C3c ws higher in four ser of ptients with renl filure thn in three norml ser (men conversion: 4% vs. 19%, Mnn-Whitney U test, P <.5). This conversion ws brogted (3%) in the presence of monoclonl mouse nti-fctor D ntibodies, indicting tht it ws due to lterntive pthwy ctivtion. Discussion This study defines the bsis for the mnyfold increse in fctor D concentrtion observed in ptients with renl filure. In norml individuls, in stedy stte the ctbolism of rdiolbelled fctor D ws minly renl (75%). In ptients without renl function the elimintion rte of fctor D ws reduced pproximtely 1-fold, thus explining the incresed plsm levels. Neither the extr-renl ctbolic rte (q51) nor the synthesis rte were modified significntly by renl filure. The results obtined indicted lso tht the synthesis rte of fctor D ws vrible from one individul to nother, precluding the use of fctor D level in blood s sensitive mrker for renl function in clinicl prctice. Finlly, in renl filure fctor D ccumultion ws minly intrvsculr. Fctor D is glycoprotein of 23.5 Kd with n isoelectric point of 7.4 [17, 18]. The elimintion of fctor D ws comprble to tht of other molecules of similr moleculr weight [16, 19 21].

6 534 Pscul et!: Metbolism of fctor D in renl filure '5 C 5 ' CD t n... i fl 4 2 Time, minutes Time, hours Fig. 7. Adeqution of the model to mesurements in subject 1. Symbols re: left, model curve (. ) nd blood mesurements D; right, model vlues nd urine mesurements. Such medium sized proteins re poorly retined by the glomerulr membrne, their USC being round.5. A clculted estimtion of the USC for fctor D ws.36, which is in the expected rnge. Indeed the correltion between the FMR of fctor D nd cretinine clernce ws excellent. Snders et l hve shown, using in vivo nephron microperfusion studies nd in vitro whole rt kidney perfusions, tht fctor D is rebsorbed by sturble process in the proximl tubules where the protein is degrded [4]. In humns, intct fctor D ws detectble in the urine of ptient with defective proximl tubulr function [3]. Here we could demonstrte tht smll frction of intct fctor D reched the urine in normls (.2%). Incresed fctor D elimintion in the urine ws demonstrted in ptients with lysozymuri, nd gin such fctor D ws intct. In rbbits, '25I-fctor D ws shown not to be frgmented in the circultion in vivo, suggesting tht there is no extrcellulr clevge of fctor D in vivo. Thus fctor D degrdtion ppers to be n intrcellulr process occurring in renl tubulr cells, nd defective tubulr function results in the urinry loss of the unmodified protein. Vrious uthors hve found good correltion between fctor D nd cretinine levels in blood, nd between fctor D levels nd cretinine clernce. It hs even been suggested tht fctor D ws better indictor of renl function thn cretinine in ptients with connective tissue diseses [7, 8]. While we found similr good correltions, we would like to emphsize the wide rnge of fctor D synthesis rtes which ws responsible for suboptiml correltion between fctor D levels nd cretinine clernce, prticulrly in ptients with reduced function (Fig. 6). We could not identify ny pthologicl process responsible for n increse or decrese in fctor D synthesis. No correltion ws found between fctor D synthesis nd CRP levels, indicting tht fctor D is probbly not n cute phse protein. Totl fctor D synthesis in norml individuls ws 1.33 mg/kg/dy, which is on the sme order of mgnitude s CIq, CS nd Cl inhibitor synthesis, nd 5- to 1-fold below C3, C4 nd fctor H synthesis [22 28]. In clinicl prctice, it might be beneficil to block complement ctivtion in vrious circumstnces, for exmple in septic shock before the onset of the dult respirtory distress syndrome [29]. Complement inhibitors, in prticulr inhibitors of fctor D function, hve been developed in in vitro models [3]. Any ttempt to block fctor D function in vivo should tke into ccount the high synthetic rte of fctor D. Fctor D ctbolism ws not ccelerted, tht is, fctor D ws not consumed, in the ptient with circulting nephritic fctor nd continuous fluid phse lterntive pthwy ctivtion; this observtion confirms the multiple in vitro dt tht hve indicted bsence of fctor D consumption t the time of lterntive pthwy ctivtion [1, 311. The nlysis of the V/IV rtio of fctor D indicted tht in norml subjects fctor D ws predominntly extrvsculr (V/IV rtio = 8.5); since the V/IV rtio of extrcellulr wter is lower (pproximtely 3 to 4) our result suggests tht fctor D concentrtion is higher in the extrvsculr spce thn in plsm. In contrst, in ptients with SRF the distribution of fctor D ws comprble to the distribution of extrcellulr wter (V/IV rtio = 2.4), tht is, the concentrtion of fctor D is probbly similr in plsm nd in the extrvsculr spce. These observtions my be explined by the fst filtrtion of fctor D through the glomerulus in norml subjects which continuously reduces plsm fctor D concentrtion, precluding equilibrtion of fctor D concentrtions cross the vsculr wlls. The unexpectedly high V/IV rtio found in the two ptients with gentmicin toxicity my be due to ccumultion of fctor D in the renl tubulr cells relted to decrese in the lysozoml degrdtion processes described in this condition [32, 33]. Indeed the model used to nlyze fctor D metbolism does not discriminte between different extrvsculr comprtments. These findings my be of physiologicl importnce. First, lterntive pthwy ctivtion my proceed differently in plsm nd extrcellulr fluids; the bsolute nd reltive concentrtions of the vrious proteins involved re different [23 27]. Indeed, from the metbolic studies of C3, B, nd H it cn be estimted tht the concentrtions of these proteins re lower in extrvsculr fluids thn in plsm. Since the opposite ws observed for fctor D it is likely tht there is reltive excess of fctor D in the extrvsculr fluid under norml circumstnces. Second, in renl filure the 1-fold increse in plsm fctor D

7 Pscul et!: Metbolis,n of fctor D in renl filure 535 concentrtion ws not mtched by similr increse in fctor D concentrtion in the extrvsculr spce. Thus, enhncement of lterntive pthwy ctivity by excess fctor D would occur predominntly in the circultion in these ptients. The preliminry experiments presented here indicted tht spontneous lterntive pthwy ctivtion of C3 ws enhnced in ser of ptients with renl filure. However, further studies re necessry to clrify the role of excess fctor D on lterntive pthwy function. Acknowledgments This work ws presented in prt t the Xth Interntionl Nephrology Congress, London, UK, July This work hs been supported by the Fonds Ntionl Suisse de I Recherche Scientifique ( ) nd by U.S. Public Helth Service Grnts AI-2167 nd AR M. Pscul is supported by Sir Jules Thorn Chritble Trust overses, nd J. Schifferli is the recipient of Mx Cloëtt Creer Development Awrd. The uthors thnk Dr. Mrtine Louis for ssistnce in the clinicl studies nd Professor M. Leski for llowing us to study ptients under his cre. Reprint requests to Dr. Mnuel Pscul, Clinique Médicle, Hôpitl Cntonl Universitire, 1211 Genev 4, Switzerlnd. Appendix Mthemticl model We tried to nlyze the metbolism of fctor D by mens of comprtmentl model which hd to meet the following requirements: uniqueness (the sme model structure hd to be pplied to ll cses); globlity (for ech cse, ll observtions hd to be simultneously tken into ccount by single set of equtions nd unknown prmeters); dequcy (ll observtions hd to be stisfctorily fitted by the model curves); minimlity (the size of the model hd to be s smll s possible, regrding the number of comprtments s well s the number of unknown prmeters). Wht is more, we impertively required the model to describe s closely s possible the lredy-known biologicl fetures of fctor D. We strted our ttempt to fulfill these requirements with the smllest possible model, incresing its complexity step-wise in order to improve the dequcy of the dt, until obtining model fitting properly the mesurements. Our criterium of proper model ws chi-squre vlue computed on the bsis of n ssumed Poisson-distribution of the rdioctive countings. As n illustrtion, Figure 7 shows model curves fitted to mesured dt, respectively in blood nd urine, for subject 1. Mthemticl methods Our problem is s follows: given system y'=f1(t,y1,ym,xi x),il m of m differentil equtions with n unknown prmeters x1 x, nd series of dt points (ti, t,, pj),j 1 N, where t represents the observed vlue t time t, with error ) nd function expressing the model vlue to be compred to the corresponding observed vlue, we hve to determine the prmeters x1, x in wy such s G(x1 = N j= I x1): {[/ij(tj, Yi y,,, x1 x) J]/o)}2 = minimum. In the prticulr cse of Poisson-distribution, crj=\/ ndg=xmn2. Where the minimum is reched, the prtil derivtives must vnish: Hk: = --- = {[(tj, y1,..., y, X x) js(tj, y1 ym' X1,..., x) k=l n. /) = If we now fix rbitrrily initil vlues to the prmeters x1,. x, the bove eqution llows to determine the x1 by the Newton's method in n vribles: = x1 w, where w re solutions of the liner eqution n 1=1 ox1 = Hk. Formlly, the prtil derivtives of the Hk re esy to formulte: N N OHk = + Ojj / [(pj )/o] / 3, Ox1 Ox, OXk j=i but the computtion of the second term bove is numericlly time consuming if some precision hs to be reched, since it involves second derivtive. We followed Mrqurt's ide, which ws simply to discrd this second term, s the whole method is known to work well only in the neighborhood of the solution, where the second term is negligible in comprison with the first one. Additionlly nd surprisingly, the method thus described is more stble in generl, s the mtrix (OHk \, oxi is now symmetric nd cn therefore be hndled with more powerful numeric liner methods thn the generl ones.

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