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Simulation: An underutilized approach to coagulation system model evaluation
THROH.ROSISRESEARCH 23; 223-236, 1982 0049-3848/82/200223-24$03.00/O Printed Copyright (c) 1982 Pergamon Press Ltd.
David
;I.
Cocchetto
in All
the CSA. rights reserved.
and T’norir
2.
ajornsson
Division or Clinicai Pharmacology, bepartments oli Pharmacology Dutte University Kedical Center, Durham, ;rorth Caroiiua 27710
ani
Xeciicine,
(Received Received
in
15.3.1982; in revised form 21.5.1982. Accepted by Editor H.C. Hemker. final form by Executive Editorial Office 3.8.1982)
ASSIXACT
Sirnulatior. anti mathesatical inoueiing have proven useful in evsiucclnj diverse physioioaical and biocheiiical systems. Rowever, sir‘ulatlon nethoas nave tiad extremely limlceci appllcrtion in t&e formulation and evaluation oi’ modeis of the blood coagulation system. The utiiity of this tool is illustrated via a mathematical motel of the rnteracrions of t‘hronbin with fibrinogen, ant ithronbin III, acd he?ar inant ithrombin III complex. This model provides exemplary simulations consistent with several empirical characteristics of the anticoagulant effect of heparin. Simulation methods provide a powerful tool with which to characterize the dynamic nature ot the hemostatic process ane its responsiveness CO endogenous perturbations or pharmacolo~icni interventions.
I \ INTPODUCTICF Simulation is the process of computerized quantitative exploration of rhe properties of a mathenaticaily deEinable dynamic system. Unlike ex?erinental simulation cannot create new information, science, but it can greatiy manipulation, facilitate more effective visualization, and application of a Shrager et al. siven body of knowledge (1). (2) stated concisely tne necessary precautions regarding simuiation rzethods: “Any set of data can be iir by many models. A gooo model is one tht fits a broad set of data vithoui virh nodei parameters that suggest some neeciess complexiry, and, preferably, physical meaning.” This notion of a good nooel will be used here. Simulation methoas have been combined t:ith mathematical mocei buiicic& >rocelures ana control tIieory to echance understanding of numerous aiverse physioio~ical, biocbecical, and ptarcacokinetic processes, incluciinb respiration (3,4), blood glucose regulation (5,6), copper metaboiism (7), slycolysis (S), mitochondriai metabolism cl), purine catabolism t9), crug
Xej/
vords :
coaguiation
system,
heparin,
simulation 223
224
COAGULATIONSYSTEX SIMEL,ATIOS
Vo1.28,
No.2
t0nce_tratFon-~e~erle3: i;laszct ?roteir. an< tissluf iiztii::; ., (IO), an-, zssess~~et; 92 s pite of ph3rcacokiae t ic Tararneter estizatioo pr0ceJurzs (11,123. its proven suluc, sinulatior. ttthotioiogy has haG Oni:._ Lizitei; agpiicacioc ia t’ke rorc,uiation anti evaluation oi zoaels L of tl:e blo3c cca;uiatFon s;;sten. i1 1965, :ier.;ker ?t ai. (13) Cerivec iLaChecatic;i Lilr:;:ulae basei: 0; cizsslcsl ,4 1 : i:c! La ii Lcnzyr>e kinetic analysis of tke results of 51001 c;otcing tests. . slsulati0r. ar:c: sathc:aCical tI:e s a 7.:e L3 s 1 c zoieiing 3e:;ici.s ::ere c3: ;1se5, a3tnecacicsl agprozcf: was later used to constrcc: aotieis of frbrinoiys~s (14,15) and ~ibri:iit~ezolysis 12). Sic.ilar netticcs n3ve cl50 bee:: csec; il0 0Ftrcize cOSii!;; of oral anticoagulants (16), forkclace 3 -ode1 of clit groxth (17), ancL fibrinolytic systems ;;rupose an integrated 1;10uei of the coaguiatiuz (1:,!9), ai?ci construct 2nd Valldaie 3 aode? Of ?rOttiio;.;:JiE 3CtiVation (26). In order to illustrate tke utility of sicluiation xethods ir. ey;aiu;tin, nodeis of coag,ulation processes, a simple and p3rsi;3onious nsthematical r.iodei for the anticoagulnnt effect of beparin was cons:ructed based on the kno:;r; Kepa; ir, apparently nolecuiar ;;iechanisc of action, of neparin. exerrs ap. , Inclrect snticoagulauc eifect by catalyziu; formation of COi.\ple%es ke:::een antitilro::ibin III (AT-III) and various serinc prote zses, ir.cliiding c;~zo::bir. (21-24). c:ld other seritit‘ Since the imitea proteolytic fucctioi: of throrzbin proteases is essentiai to ulci.~:,, qte formatior. cf a fibrin clot, iieTari:iinhibits severai faciiitated couplexation oti these proteases wi:l: AT-III Of ilepsiin uI1 recctions in the coagilation tkese effects cascaae. Ciinicaliy, coagulation are evaluated intiirectly by in vitr0 coagulation tests wi:lcL detect fibrin se1 fornation 35 the endpoint. Since the initiili report in investigators have observed au 3pparer.tly linear 1937 (25), nc:Gierous relationship between the logariti;n,s of various jg vilrro coagulation tines anti iri pl;sca or wi;ole blooa fronaninals anc! T.ian (26-41). tcparin activity Furthermore, recent studies in r:,i?n i:ave revealed 3 i:i&Liiy si;ni.fical;t poslcive baseline coagulatloc tir.ies (APE, i.e., correiativn betveer: activatfc partrai tGrocboplastin til;ie) acd the slopes of the log APTT versus plasma hepariu activity reiationships (32,35,41). Iieither the G;?arectiy lo&-lmear nature of tl!e relationship between in vitro k2pari.n effect ani pLasr.3 heparix 3cLivity nor the sigcific;nc positive correlation between the slope vaiues 01 the 106 ant ice azulant effect vs i:eparin acclvily relatiocship act bsselicc-: coa;ulatiun Lir.lcs Could bc precictec fror.: ClaSSiCa! i;har;kacodynar;ic equations (42-45) for the kinetics oi’ direct pharri,acoiogical effects. Lcte that a fet, lnVesCl;ators 11: SOi.!e cases h3VC Ilot observei 3Fpare:ltly linear reiacionsillps A ;ir;cticsi Detween 10~ APTT verses plssna keparin activity (30,32,33,46). sensitivity of experii.lectal consideration to be awsre 0: is tiiat tiie in vitro ti:e APTT to hepclrin varies greatly dependini; on the specific activ3tec tl~ro~.~‘uoplasciii reagent (36,32,33,47), buffer capacity of the reagent (461, anL coagulation tiner (47) usecl, but ileparin sensitivity appears to be independent of tte particulsr contact activating agent iucludcc: iii the reagent (30,32,33). Considcracioc of tnese moleculsr and clinicai characteristics of ct:e action of tieparin eca’oled cocstruccion of a iel3tiveiy sln?le and izarsinonious This enablec; cathe=aticai its anticoagulant effect. model node1 for slzulat1on of: the apparerit log -linear relationship between the iildirect, in proiongint effect of heparir on clotting tinle scd plasma heparin activity. sddicion, the heciostatic effects of perturbatiocs in thronbin activity, S1nulat LOI? fibriuogcn concentraticjn, and AT-III concentration vere slzuiated. - . Loois Giti: anti oicltherzaticai zoceling rethods are cewons:rated to be powerrul ?.*nLch to ev3luate ai:c ciiar3cterize a tiyL1a:.;lc coa;ula:ion syster.; i.iO
V01.28,
SO-2
COAGLZATIO~
FJrr.clstion
of
Zzsec; as
a
seriiic
cnroabin
ixio:;led~:-elz enzyrAe
proteases)
needed
realer
is tC
iieF;arir, tnat
(54).
of
The
t=ztc;eea
AT-
catalyze
tiirocbiu
0;i
hepzri::
activator anti
to
rer;iincec
cepsrln
22 5
nocei.
07. cLrreilt
con-essential
other
3s
t:ie
SfM_!LLXTfOX
SYSTEN
a
the
the
i:l
Gnis
reaction
r2sc:Lts
direct
ant
(znl
T i-; e
bmcinb
0:
e:feci
0;
ttror4bin,
forr.iatlor.
for
iibric.
anticoag.ulo7.t
AT-Iii ic
vlewec!
competes to
tk;t
i’r.e
be
thronjir.
EiJri3ogen
betbee::
tiepsrin,
can
becwern
hysor;hesis
cl:ct:t
recctior.
ar,i~
. of
alternative
hepark
reactiofi
corversion
icicial
COLi~lk2X2tl~il AT-Ii1
phsrr.!scology, tte
2L+,!b’%-j3j b
(72 _)
11;
recei:t
is
i9
0:
5s
welt
ulnolecuiar
coc;)icxes (24,48). For sate or slr.;r?Lci.r_y, c’.?v irlnoiecclai Conplexation Trol;uC:s mcikei; in h aore coxpiicaceu thru;..i/ir. - AT-i;1 interactive sChe:.~e (23) viii ni)t be coxsiaereo i:ere. Joss~3le rfgeceratlou of hepsrln fro;. ilc?Lirir:-1;T-IiI-tiiro~~;:.bin conplex is i>Ot i:;C;uded in ttie ;.iodel. The cor.plecc _;occi iS iliustrsted in Figure 1. The I;tility of this c.oael Lies ii: 1;s
si:.!plicity r,uc:ber of
ma i:c~iel
parsiclor,y psrazetezs
due to needed
frugai inclusiur. of L;?e to describe the relevant I;cor:n co ociur ir: :.ix. t‘rc cociulatioi:/fijr;i:oi~sis
:;ocei parxncter governs a reaction of ;(or;ei is similar to c1 ;crtion by Iliatiis et al. (1s).
k2
kl THROMBIN
+
Fl8RlNOGEN
-4
T-F
k4
AT:III
--
antithronbin ;
t-l,
model III,
for anti
t
-l.
t-l,
k-1 ;;4
nather-lstlcai rG,el propooei
r------i
T + i FIERIN
L------J (Fn)
:
‘*‘*A%.
H
FIG. Sctiezntic
Tnis
-.
.. . +
t::i:2Lr.:urk ZSCli
rextrons.
k_:
(F)
HEPARIN
-
absolctr
1
reactions of thror,bic viti; fibrinogen, Ilote heparin-antitkror;lbin 111 cor~plex. -1 , icj = 0.01 -1 t ) = 0.301 r-1, k2 = 0.200 t
= 0.260
+,
k-4
= 0.0~04
t-1,
kg
= 1.
t-l,
and
that
k-g
=
k5
=
226
COAGLILATIO?:SYSTM SIM_lLATION
‘131.23.
No.2
., i;i Figure 1 1s Lhroii~noct this ?resexiaticx, the r.ocecclarire descrioec . usecl. Co=lbii:atio;?s of two or ti?ree letters tencte tile reievnnf 1 ‘JlZ;uleCclar or triizolecular coxpiex. If encioseci in parectheses, these cesidnatiocs refer to concentratiocs. Tile (t) is defined on 6u aro::rariiy seiected relative scale. This schematic r;.adel of coagulation is described sinultaueous first-ortier differential rate equatloas: a(T)/dt
= k_liTF) -k3
- k1 (T)(F)
(T)(A) (TF)
+ k-5
d(F)/ac
= k-1
- xl
ci(:i)/ot
= 1:_3(TA)-k3(P)(A)
d(;i)/Gt
= k_4W-::4(1NA)
+
(TAZ)
by the
foliowiok
- 1:5 (T)iAii)
+ k_4(AH)-k4(ki)(.4)
(Q.
1)
(Eq.
2)
(Eq.
3)
(3q.
4)
ti(Fn)/tit
= k$TF)
( Eq.
5)
d(TF)/dt
= kl(T)(F)-i~_l(TF)-~2(TF)
(Xc.
6)
d(Ati)/dt
= k4(;;)(A)-k_4(Ati)
(’I_%. 7)
d(TA)/ut
= k 3 (A)(T)+
d(TAiii,)/dt
=
icj(Ti(A:i)-:1C_j(TAli)
of
it;-3 (‘la)
(T)(F)
-3 (Tk)
set
+~_5(T”t;)-~~j(Ti)(AI!)
izq.
3)
izq.
9)
iii&hOu~h a few of the rate constants iu this il;ooei ili;Vi: oeen characterized via _ir. vitro studies usin; gel electrophoresis and fluorescence tiet’hodoiogies (53,55-571, the various rate constants in this nocei have hot been characterized to the extent needed fcrr direct substitution in the above equations. Estinates of the rate constants for the three reactions cou:petiug for thrornbi:; were chosen so that k >ic >ki.e., throtibin re3cts more rapidiy 5 1. J.’ with heparin - AT-III cor~plex than %Jlth fibriuog,eh, wfiic’h in turn reacts with tiirombin nore rapidly than does AT-III. Kucer ical values of ail rate constants were ciioseh such that there was at least a Z-ioid cifference between any given pair of rate constants. This ninimun difference improves the abiiity of the sisulation me:‘nods to resolve the ucique characteristics of the wodel (5&,59). Muezerical vaiues of rate constants are reported iu the figure the rate constant for forxatior, of a given cor,,plex iegends. In every case, (thronbin-AT-III, heparin-AT-III, heparin-AT-III-throcbin) was chosen to be 100 to 506 titles larger that1 the respective rate constant for dissociation. In additiou, fibricogenesis was treated as a sequential 2-step process in This fibrinogenesis w’hich the ii;itial step was rate-limiting (i.e. kl
vo1.23, No.2
COAGLXATION SYSTE?fSMlJLATION
227
Conpxa: ional nethccs and ;;.ocal valiaation. After specliying estiraies of the various iate constants anti initlai concentrations of AT-III, fibrinogen, heparin, and throlr,bin, a solution, tc the set of differential equations was obtaiaeti via the fourth-order Lunge-Kutta met‘noc through use of CS:IP (Continuous Systcr;: ilodeling Proprar), a digital computer prograa package eaabling siculation of continuous processes via severai nur;.erical integration methods (61,621. All calculations were done on an 13:: 4341 ci&ital COnpUtei. Ou:pu: or‘ interest consisted of the relationship between initial plasrra heparin activity and the tine (“cfottlng tine”) to generate the ;.lini.r:.uc. fi’orin concentration (chosen as 7.5 pFI based 01; research by Blozback and Okada i63i) needeti for r‘ibrin gel formation. The relationsnip ‘between the slope of clotting tine versus heparin activitj profiles (“clotting; tiae-heparin slope”) and baseline (pre-heparin) clotting trees iu’as also of interest. It shoulc be coted that the overall relatiocships incepencent of a Vlde range of fibrin concentration chosen as the cre endpoine.
eight
A nathenaticti experkentaliy
uodel of observed
the anticoagulact characteristics
actioi-, of of heparin
hepnrm effect:
c;ust
sikluiate
1.
potentially Large intersubject and intrasubject tine and 3-foid ran&es) in baseline clotting slope (31,39,41,64,65).
2.
the apparently heparin plasna unit/n1 (26-41).
^
the apparently licear relationship between clotting time-heparln slope ano baseline clotting time over ranges of approxkately 2 to i/2 fold and 1 l/2 KO 2-fold, respectively (38,39,41).
3.
linear relationship between 10L clotting activity over 3 ranje of approxixateiy
4.
the nonlinear, versus thrombin
5.
the nonlinear, slope versus
6.
the nonlinear, concordant relationship of clotting time-heparin slope versus AT-III
7.
thronbin activity tine (13,18,41).
8.
thronbin activity and AT-III cor.centration of clotting tine-heparin slope (18,41,67).
The ability of pharmacological validation.
the
variability (1 l/2 clotting tine-heparic
discordant activity
relationships and fibrinogen
of baseline concentration
discoraant relationships of chrombin activity anti fibrinogen
inust
be
a major
proposed model to effect of heparin
tine and 0.1 to 1
clotting (13,1S,66).
Lust
simulate the above as the was used
2
time
clotting tine-heparin concentration (1s).
baseline clotting time concentration (18).
determinant
to
of
baseline
be major
and
ciotting
detercinants
characteristics criterion for
of the node1
EESULTS Clotting tine varies with initial plasma hegarin activity over a 1 l/2 to 2-fold range of baseline clotting tines similar to the previously observed of simulated range (38,39,41). As illustrated in Figure 2, the logarithms times can exhibit an apparently linear relationship to initial plasrtia clotting
;:
;:
D.ASMA
Fi;.
.;/
: z 012 05
7
z 6 Z : 5 T z 4I -z
between clotting tixe tieparin and ir,it ial plasha sii:lulstea by v&ryl:l:, activity tllrol::bin COItCe;:CiatlOX initiai (ui_Fer left), varyin; i:liti&l f ibrii:06;en concentration (upper vnry In;; ar.d lnitisl right), 3 I’.t 1 t ,: r 0 t: 0 1 c concentration ( i 0 1; e r “lc’ i’i). I :I i c i a i
P
. ;E5 0.10 ~ 1.0 IZ I4 1.6 II . I*ftl,NtCLOImm TlMl .
s
PO5 2
E Y 5
concentration
(
zcijacent
t
relatiocship
s
syzbois)
plcsL;P T ile
,
/!,!//!lilll 02 PLASMA
04
0.6
HEPARIN
oa
io
ACTIVITY
10,
corres~onain~
(U/ml)
cte
ciottin:,
tir;-.e-
5aselir.e
a 11c
apparenti.; cite
VeiSUS
activiiry -2rofiles. solid ii::ezr stiFeill..~OSeC
> values antithr0z:'bi.n r F11e
clottin;
th e
l:epsriri
regression 12
T i?e
(uslnb;
tll.;e for
lir,e;r
noted
sh 0 V
slope
clotcic~
is
curv5.
insets tetveen
heparin
1
ii:
each
to
respective
;2 E 6
‘? j;/m!i
2
2
23 ;
13
ACTIV!lY
LelatiCai:ship
P d’
2 &lb IO :, F 9 ZYOI‘ 8 g
08
7-3
HE?ARlN
s i z:f- ;y
iices
are
S f;Cm>;i; ( 3 j,i
C.9SG). III
In
iilset,
ccnnects
ihe
;
the casiifll
t.;e ;;oints.
SYST’E?!
COAGULATION
SI?ITY’LXTiON
229
_._
The sizulateti increasing and lecreasir.5 coagulability __. . 2ecreasi::j .AT-III concen:ration and decreaslzb ilorlcoaen c3nceC;rat1oC, respectiveiy, are consistent with present ‘kn0wledg.e of the hemostatic roles of ihese txo substances. The obvious nonlinear nature of these effect-activity profiles for the lotier initial chrocbir: concentrations, hig’ner AT-III concer.tr3tioris, and lover fibrinojen concentrazions zay be a result of the underlying Rorkiinearity of this aodel systes or failure to sr:*.ulat? a cmticuous input of throrzbin. QCiiIi
t iOrlCOg22.
associz:ei:
with
The relationships betcieer. clotting tine-he?arir. slope and baseline ciottin; tioe for the apparentiy linear cecibers of each faziiy of curves are illustracec in ihe insets of Figure 2. Consistmt with previous observations (3; ,39,&l), the clotting ci=e-Xeparin slope is apparently linearly related to Jaselxe clotti2; tic;? for the cases of varying initial thror!bin or fibrinogen concentrntloc. Kowever, varying initial AT-III concentration vas associated with a nonlinear relationship between clottin;; cizt-aeparin slope an6 ‘baseline clotting tinie. Such a nonlinear relationship was anLlcipaLeci based on the 2 u r e Ly inuirect on the clottin, effect of AT-III eadpoint-detercining iriteraction ‘aetveen tiironbin ana fibrinogen. The posslbie associations between clotting tine-‘he?arin slope, baseline AT-III, and fibrinogen clotting time, acd initial concentrations of thro:.;i)in, 1ncrensir:g initial concentrations of 5ie r e 3oc;ht (Fi,ure 3). Clearly,
INITIAL THROM8lN CONCENTRATION (0,pMl I 8
I 9
I 10
1 11
1 12
1 13
1 14
1 15
INITIAL FIBRINOGEN CONCENTRATION (n,pM) i I
I
I 2
b
1 3
1
1 4
1
INITIAL ANTITHROMBINCONCENTRATION (Q,M)
P,eiatior.st:ip ntigarin -.. riorinogen,
ciotcrng
between sLo?e ant
title
baseline
versus
antithrozbin
versus
clottin,;
initial
plaszc
III heparin
1 5
Ill
trr.;e
;n6
clottin,
concentrzrions
for the acLivity
apparently profiles.
of
time-
throtzbin,
linear
i0.g
Vo1.28,
COAGULATIONSYSTW SI?iUL.A-t’IOX
230
x0.2
trnrOiiitui anC i iDrL;o,ec r;ere both tssociatfa wi:h ti2creasir.g DaseiiLe ciocti.nL clxes azd cecreaslng tize-keparin slopes. Sir,ce trrrot.:bin ant cioctinb -.. tlorinogen are ti:e proceiza:eoLs reactailts in the endpoint-prolucicg re3ctLor., incr2asir.g rkelr iRitia1 concentrations shotiici indeed iias:en ti,e clottind > process bott ill the 3bsei:ce auc ijresence or ke?arin, thtis cecreaslag Dotn bsselice clottir:g tine and clottin; clzr-teparin slope. iiirsi; 2~ ai. (64) 03SeiVZtiUIl of fibricogen acrichcent increasing tne ilave reporteil a sick;c APTT of control hC:r_an pl3s~a, while Raoerser. ~CL Godal (65) aid zot observe any change ir, tbro-;Sin tine with fibrinogen: concentratioc varyic; over the norCal rang2 of 2 to 4 g/l either in the Fresence or absence of scaii aizounts of heparin (2 ci.25 IU/i.;l). These are the only observaiioas report25 tkac contilicr: with the sizulacea results. Gase1ir.e clocriq time ~5s 3;;roxizateiy eqoolly sensitive LO the sibiulated cilan2es in initial thrombin afiG fibrkogen concentrations. Lowever, clotting Lirfiie-i;e;!arin siope variea over a Z-folc r:ilile slope o&s far less range wiLii varying mitiai. chroc:bin concentration, sensitive (varyins over a i l/3-fold range) to changes ic initiai flbricogen IncreasiaL the inrtiai concectration of AT-III, an ecdobenous coccentration. . . prote!n that inhibits tllror;.bin activity, ~82s assocrateci wltli ;r.creasxb t irie-heparin slopes (Figure bssellne clottin tices anc increasing ciottinb seilsitive (varying over a 2Clotting tine-te?arin siope was relativei:, 3). This sensitive folti range) to changes in initiai AT-I11 concentration. facilitates AT-III-tilrombic relationship is expected since heparin readily vith throxbin in tiie interaction. Since AT-III interacts ir.uch less tiia2 was effected lit tie by chaqing ATabsence oE heparin, baseline clotting Studies (36,39,41) neither oaseline III concentration. As iii previous exhibited a statistically clottin& tines nor cioLtin& tir;ie-hep arin slopes significact linear relationship to icicial coccencratioc of fibrinojer. or dTFigure 3 illustrates tile III. i! 0 w e v e r , nithou6h not linear in uatu:e, ckiscordant and coniinear concordant naiu:‘e Of the respective r:onlii;ear relationships betwe2n both clotting paraneters (baseLil;e ciotting tine, ciotting time-heparin slope) and initial concentrations of fibrinogen and ATIII. sirzulations summarized in Figure 3 illustrate that baseline Therefore, thought to be an index of hy?ercoagulability, is determined ciottin;; tirse, largeiy by the throcbin activity and fibrinogen concentration, tliiilt clotting an index of heFarin cosege requiren.enc, is ceterolned tine-heparin slope, primarily by throxbin activity ana AT-III concentration. 1
4
_..
DISCUSSIO:: coabulation As early as 1937, ecpirical equations were used to predict concentration and several experinerltally tine 3s a function of heparin The observation by numerous investigators that an determined constants (25). apparently linear relationship exists between the lodarittcs of various & vitro cl0ttir.g tiaes an< heparia activity in plasaa or whole blood has not oociei of direct or iildirect been explained to date using any availabie phareacologic effect. The present skulatior. study has denonstrate? chat such an apparently linear reiationship can be preiicced frocl a relativeiy simple natheuatical node1 of the known molecular interactions arzong throwbin, fibrinogen, AT-III, This rsodel illustrates that clotting time, and hepar in. being the secondary expression of the primary pharmacologic effect of heparin, cannot be treateci in the same manner as directly observed prinary pharmacologic effects such as tubocurarine-induced neuromuscular blockade. clotting time represents the time it takes to reach a givea effect, Rather, namely, a certain L;ibrin or gel formation in a coagulation test tizer. The
model
was
validated
by
virtue
of
its
ability
co
simulate
eight
Vo1.28,
No.2
COAGL%ATIONSYSTM SIWLATION
231
5se r v E c expericcnta; char3cceriscics cf hepaiin ertect. Future _i charscterizacion of tze rhte consrauts 'usec iC this lode1 and insrtiaSin,v :zechznis:ic knowledge of t.he koiecuiar blxoio;y 0:_ bl~'oC co3gu;ation c;Lii enable experinentai challengin; aad refinetenL of this noael. ?hrous;n this ccnstant process of chailen;ing and subsequent refinenect of the l;.odel, acd nathezaticai a:odeling methods Frovide insight si=uiation into tile Ir,echazis=s of the observed ghenonena and ~;ay ulc;rateiy yield rocels useful to both the molecular biologist ar.d ciinician.
0
like provided by these sinulations is broad. The scope of inforG3tion potentiaiiy large variabiliLy in clotting tixe effect associatec I:ith 3 c;iveu The sinulaced relative plasc;a iieparin activity Flds illustrated. hypofiorinogenenia 3r.d relative hypocoagulability associ ated with associated wit‘fl decreaseo AT-111 concettration are hgpercoagulability consistent with t\he cliuical hemostatic defects associate.2 5ich these perLurbations (64,69,70). In addition to sinulacin& Liiese chnracteriscics, tiie model also sinul3tes cite apparently licear iel3tionshi.2 betiieer,clottin; tize-;;eparin slope acd baseline clottin& tiae. As expected b3sel oc previous research (li;), clotting parar.eters (baseliue clottin tine acd ciottins tl:.iehepnrih slope) were related in 3 nonlinear discordant zanner Lo throzbin sctivity ar.u fibrinogen concentration, while clotting parxaeters i;ere related ?lodel simul-tions .z III 3 noniinear concordant c:anner to A'i-III ccncentration. showed that baseline clotting time was primarily cecerniuer: by thrombin time-heparii; slope IjLtS activity and f ibrlnogen concentration, c;iiile clottin& acd AT-i11 concentratioi:. primarily determined by thrombin activity Large intersuojecc F,ecent studies in normal subjects have cemons tiaCC?d variability ir,both baseline clotting time (XPTT) and clotting Liae-heparin Was related LO activity slope. however, tieither of these parameters, concentrations of fibrinogen or AT-ITT; the concentration of these proteins Cn the basis of c.ne nocel for showed little intersubject variability (39,411. it nay be the anticoagulant action of heparin presented in this study, proposec that 3 major determinant of these coagulation para:deters, at least in normal subjects, is the initial thrombin concentration or the rat;? of throtibin generation. iu addition LO reconciling an observed non-ciasslcal doseThus, the response relationship with known characteristics of hepar in pharmacology, a clinicnily relevant c.xplanation for the large motel presented suggests T’r.e intersubject variability in the experimentaliy obtained data (39,411. various perturbations of 2 0 c c 1 p r e s e r. t e d ma y also be useful in studyrn& heparin effect, such as interac:ions between hep3ri.n and other axticoat;uiants, and in studying the hemostatic impact of chan&es in concentration or activity Simulac ion siethods (71of individual components of the coagulation syscer... biochemical theories 73) have proven useful in evaLuatin& numerous mechanistic and may provide a means by which the dynamic character of hemostatic processes in perturbations LO subtle can be visualized and its responsiveness coagulation factor activities, endob;enous inhibitors, and pharmacological interventions evaluated. ACiiilO\-ILEDG EXEDTS A preliminary report of this work was presented at toe eighty-third for Clinical Pharmacology ana annual meeting of the American Society This wori was supported in part by XE Grant iio. Therapeutics io :;arch 1982. liL-24343. T.D. Bjornsson is a recipient of a Pharmaceutical Xanufacturers Association Foundation Faculty Development Award in Clinicai Pharmacology. The authors gratefully acknowledge the skilled typing assistance of ?:s. Zanda Seynour
*
232
vo1.25,
x0.2
3.
I: V.
7.
L.
9.
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COAGULATION SYSTM
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?LlSLA,
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The effect TRZPLETP, D.A., UAP.XS, C.S. and XGZ?X?:z, J.A. ;I ?;.,e ~. r J. -I Clin. t&e activated partial thronhoplastin cl-e. 553, 197s.
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39
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antitnrorzbix III 329-336, 1977. 65.
kel
York:
due
Prentice-
E:l~lewood
John
K il2%* I
David
;I.
Cocchetto
in All
the CSA. rights reserved.
and T’norir
2.
ajornsson
Division or Clinicai Pharmacology, bepartments oli Pharmacology Dutte University Kedical Center, Durham, ;rorth Caroiiua 27710
ani
Xeciicine,
(Received Received
in
15.3.1982; in revised form 21.5.1982. Accepted by Editor H.C. Hemker. final form by Executive Editorial Office 3.8.1982)
ASSIXACT
Sirnulatior. anti mathesatical inoueiing have proven useful in evsiucclnj diverse physioioaical and biocheiiical systems. Rowever, sir‘ulatlon nethoas nave tiad extremely limlceci appllcrtion in t&e formulation and evaluation oi’ modeis of the blood coagulation system. The utiiity of this tool is illustrated via a mathematical motel of the rnteracrions of t‘hronbin with fibrinogen, ant ithronbin III, acd he?ar inant ithrombin III complex. This model provides exemplary simulations consistent with several empirical characteristics of the anticoagulant effect of heparin. Simulation methods provide a powerful tool with which to characterize the dynamic nature ot the hemostatic process ane its responsiveness CO endogenous perturbations or pharmacolo~icni interventions.
I \ INTPODUCTICF Simulation is the process of computerized quantitative exploration of rhe properties of a mathenaticaily deEinable dynamic system. Unlike ex?erinental simulation cannot create new information, science, but it can greatiy manipulation, facilitate more effective visualization, and application of a Shrager et al. siven body of knowledge (1). (2) stated concisely tne necessary precautions regarding simuiation rzethods: “Any set of data can be iir by many models. A gooo model is one tht fits a broad set of data vithoui virh nodei parameters that suggest some neeciess complexiry, and, preferably, physical meaning.” This notion of a good nooel will be used here. Simulation methoas have been combined t:ith mathematical mocei buiicic& >rocelures ana control tIieory to echance understanding of numerous aiverse physioio~ical, biocbecical, and ptarcacokinetic processes, incluciinb respiration (3,4), blood glucose regulation (5,6), copper metaboiism (7), slycolysis (S), mitochondriai metabolism cl), purine catabolism t9), crug
Xej/
vords :
coaguiation
system,
heparin,
simulation 223
224
COAGULATIONSYSTEX SIMEL,ATIOS
Vo1.28,
No.2
t0nce_tratFon-~e~erle3: i;laszct ?roteir. an< tissluf iiztii::; ., (IO), an-, zssess~~et; 92 s pite of ph3rcacokiae t ic Tararneter estizatioo pr0ceJurzs (11,123. its proven suluc, sinulatior. ttthotioiogy has haG Oni:._ Lizitei; agpiicacioc ia t’ke rorc,uiation anti evaluation oi zoaels L of tl:e blo3c cca;uiatFon s;;sten. i1 1965, :ier.;ker ?t ai. (13) Cerivec iLaChecatic;i Lilr:;:ulae basei: 0; cizsslcsl ,4 1 : i:c! La ii Lcnzyr>e kinetic analysis of tke results of 51001 c;otcing tests. . slsulati0r. ar:c: sathc:aCical tI:e s a 7.:e L3 s 1 c zoieiing 3e:;ici.s ::ere c3: ;1se5, a3tnecacicsl agprozcf: was later used to constrcc: aotieis of frbrinoiys~s (14,15) and ~ibri:iit~ezolysis 12). Sic.ilar netticcs n3ve cl50 bee:: csec; il0 0Ftrcize cOSii!;; of oral anticoagulants (16), forkclace 3 -ode1 of clit groxth (17), ancL fibrinolytic systems ;;rupose an integrated 1;10uei of the coaguiatiuz (1:,!9), ai?ci construct 2nd Valldaie 3 aode? Of ?rOttiio;.;:JiE 3CtiVation (26). In order to illustrate tke utility of sicluiation xethods ir. ey;aiu;tin, nodeis of coag,ulation processes, a simple and p3rsi;3onious nsthematical r.iodei for the anticoagulnnt effect of beparin was cons:ructed based on the kno:;r; Kepa; ir, apparently nolecuiar ;;iechanisc of action, of neparin. exerrs ap. , Inclrect snticoagulauc eifect by catalyziu; formation of COi.\ple%es ke:::een antitilro::ibin III (AT-III) and various serinc prote zses, ir.cliiding c;~zo::bir. (21-24). c:ld other seritit‘ Since the imitea proteolytic fucctioi: of throrzbin proteases is essentiai to ulci.~:,, qte formatior. cf a fibrin clot, iieTari:iinhibits severai faciiitated couplexation oti these proteases wi:l: AT-III Of ilepsiin uI1 recctions in the coagilation tkese effects cascaae. Ciinicaliy, coagulation are evaluated intiirectly by in vitr0 coagulation tests wi:lcL detect fibrin se1 fornation 35 the endpoint. Since the initiili report in investigators have observed au 3pparer.tly linear 1937 (25), nc:Gierous relationship between the logariti;n,s of various jg vilrro coagulation tines anti iri pl;sca or wi;ole blooa fronaninals anc! T.ian (26-41). tcparin activity Furthermore, recent studies in r:,i?n i:ave revealed 3 i:i&Liiy si;ni.fical;t poslcive baseline coagulatloc tir.ies (APE, i.e., correiativn betveer: activatfc partrai tGrocboplastin til;ie) acd the slopes of the log APTT versus plasma hepariu activity reiationships (32,35,41). Iieither the G;?arectiy lo&-lmear nature of tl!e relationship between in vitro k2pari.n effect ani pLasr.3 heparix 3cLivity nor the sigcific;nc positive correlation between the slope vaiues 01 the 106 ant ice azulant effect vs i:eparin acclvily relatiocship act bsselicc-: coa;ulatiun Lir.lcs Could bc precictec fror.: ClaSSiCa! i;har;kacodynar;ic equations (42-45) for the kinetics oi’ direct pharri,acoiogical effects. Lcte that a fet, lnVesCl;ators 11: SOi.!e cases h3VC Ilot observei 3Fpare:ltly linear reiacionsillps A ;ir;cticsi Detween 10~ APTT verses plssna keparin activity (30,32,33,46). sensitivity of experii.lectal consideration to be awsre 0: is tiiat tiie in vitro ti:e APTT to hepclrin varies greatly dependini; on the specific activ3tec tl~ro~.~‘uoplasciii reagent (36,32,33,47), buffer capacity of the reagent (461, anL coagulation tiner (47) usecl, but ileparin sensitivity appears to be independent of tte particulsr contact activating agent iucludcc: iii the reagent (30,32,33). Considcracioc of tnese moleculsr and clinicai characteristics of ct:e action of tieparin eca’oled cocstruccion of a iel3tiveiy sln?le and izarsinonious This enablec; cathe=aticai its anticoagulant effect. model node1 for slzulat1on of: the apparerit log -linear relationship between the iildirect, in proiongint effect of heparir on clotting tinle scd plasma heparin activity. sddicion, the heciostatic effects of perturbatiocs in thronbin activity, S1nulat LOI? fibriuogcn concentraticjn, and AT-III concentration vere slzuiated. - . Loois Giti: anti oicltherzaticai zoceling rethods are cewons:rated to be powerrul ?.*nLch to ev3luate ai:c ciiar3cterize a tiyL1a:.;lc coa;ula:ion syster.; i.iO
V01.28,
SO-2
COAGLZATIO~
FJrr.clstion
of
Zzsec; as
a
seriiic
cnroabin
ixio:;led~:-elz enzyrAe
proteases)
needed
realer
is tC
iieF;arir, tnat
(54).
of
The
t=ztc;eea
AT-
catalyze
tiirocbiu
0;i
hepzri::
activator anti
to
rer;iincec
cepsrln
22 5
nocei.
07. cLrreilt
con-essential
other
3s
t:ie
SfM_!LLXTfOX
SYSTEN
a
the
the
i:l
Gnis
reaction
r2sc:Lts
direct
ant
(znl
T i-; e
bmcinb
0:
e:feci
0;
ttror4bin,
forr.iatlor.
for
iibric.
anticoag.ulo7.t
AT-Iii ic
vlewec!
competes to
tk;t
i’r.e
be
thronjir.
EiJri3ogen
betbee::
tiepsrin,
can
becwern
hysor;hesis
cl:ct:t
recctior.
ar,i~
. of
alternative
hepark
reactiofi
corversion
icicial
COLi~lk2X2tl~il AT-Ii1
phsrr.!scology, tte
2L+,!b’%-j3j b
(72 _)
11;
recei:t
is
i9
0:
5s
welt
ulnolecuiar
coc;)icxes (24,48). For sate or slr.;r?Lci.r_y, c’.?v irlnoiecclai Conplexation Trol;uC:s mcikei; in h aore coxpiicaceu thru;..i/ir. - AT-i;1 interactive sChe:.~e (23) viii ni)t be coxsiaereo i:ere. Joss~3le rfgeceratlou of hepsrln fro;. ilc?Lirir:-1;T-IiI-tiiro~~;:.bin conplex is i>Ot i:;C;uded in ttie ;.iodel. The cor.plecc _;occi iS iliustrsted in Figure 1. The I;tility of this c.oael Lies ii: 1;s
si:.!plicity r,uc:ber of
ma i:c~iel
parsiclor,y psrazetezs
due to needed
frugai inclusiur. of L;?e to describe the relevant I;cor:n co ociur ir: :.ix. t‘rc cociulatioi:/fijr;i:oi~sis
:;ocei parxncter governs a reaction of ;(or;ei is similar to c1 ;crtion by Iliatiis et al. (1s).
k2
kl THROMBIN
+
Fl8RlNOGEN
-4
T-F
k4
AT:III
--
antithronbin ;
t-l,
model III,
for anti
t
-l.
t-l,
k-1 ;;4
nather-lstlcai rG,el propooei
r------i
T + i FIERIN
L------J (Fn)
:
‘*‘*A%.
H
FIG. Sctiezntic
Tnis
-.
.. . +
t::i:2Lr.:urk ZSCli
rextrons.
k_:
(F)
HEPARIN
-
absolctr
1
reactions of thror,bic viti; fibrinogen, Ilote heparin-antitkror;lbin 111 cor~plex. -1 , icj = 0.01 -1 t ) = 0.301 r-1, k2 = 0.200 t
= 0.260
+,
k-4
= 0.0~04
t-1,
kg
= 1.
t-l,
and
that
k-g
=
k5
=
226
COAGLILATIO?:SYSTM SIM_lLATION
‘131.23.
No.2
., i;i Figure 1 1s Lhroii~noct this ?resexiaticx, the r.ocecclarire descrioec . usecl. Co=lbii:atio;?s of two or ti?ree letters tencte tile reievnnf 1 ‘JlZ;uleCclar or triizolecular coxpiex. If encioseci in parectheses, these cesidnatiocs refer to concentratiocs. Tile (t) is defined on 6u aro::rariiy seiected relative scale. This schematic r;.adel of coagulation is described sinultaueous first-ortier differential rate equatloas: a(T)/dt
= k_liTF) -k3
- k1 (T)(F)
(T)(A) (TF)
+ k-5
d(F)/ac
= k-1
- xl
ci(:i)/ot
= 1:_3(TA)-k3(P)(A)
d(;i)/Gt
= k_4W-::4(1NA)
+
(TAZ)
by the
foliowiok
- 1:5 (T)iAii)
+ k_4(AH)-k4(ki)(.4)
(Q.
1)
(Eq.
2)
(Eq.
3)
(3q.
4)
ti(Fn)/tit
= k$TF)
( Eq.
5)
d(TF)/dt
= kl(T)(F)-i~_l(TF)-~2(TF)
(Xc.
6)
d(Ati)/dt
= k4(;;)(A)-k_4(Ati)
(’I_%. 7)
d(TA)/ut
= k 3 (A)(T)+
d(TAiii,)/dt
=
icj(Ti(A:i)-:1C_j(TAli)
of
it;-3 (‘la)
(T)(F)
-3 (Tk)
set
+~_5(T”t;)-~~j(Ti)(AI!)
izq.
3)
izq.
9)
iii&hOu~h a few of the rate constants iu this il;ooei ili;Vi: oeen characterized via _ir. vitro studies usin; gel electrophoresis and fluorescence tiet’hodoiogies (53,55-571, the various rate constants in this nocei have hot been characterized to the extent needed fcrr direct substitution in the above equations. Estinates of the rate constants for the three reactions cou:petiug for thrornbi:; were chosen so that k >ic >ki.e., throtibin re3cts more rapidiy 5 1. J.’ with heparin - AT-III cor~plex than %Jlth fibriuog,eh, wfiic’h in turn reacts with tiirombin nore rapidly than does AT-III. Kucer ical values of ail rate constants were ciioseh such that there was at least a Z-ioid cifference between any given pair of rate constants. This ninimun difference improves the abiiity of the sisulation me:‘nods to resolve the ucique characteristics of the wodel (5&,59). Muezerical vaiues of rate constants are reported iu the figure the rate constant for forxatior, of a given cor,,plex iegends. In every case, (thronbin-AT-III, heparin-AT-III, heparin-AT-III-throcbin) was chosen to be 100 to 506 titles larger that1 the respective rate constant for dissociation. In additiou, fibricogenesis was treated as a sequential 2-step process in This fibrinogenesis w’hich the ii;itial step was rate-limiting (i.e. kl
vo1.23, No.2
COAGLXATION SYSTE?fSMlJLATION
227
Conpxa: ional nethccs and ;;.ocal valiaation. After specliying estiraies of the various iate constants anti initlai concentrations of AT-III, fibrinogen, heparin, and throlr,bin, a solution, tc the set of differential equations was obtaiaeti via the fourth-order Lunge-Kutta met‘noc through use of CS:IP (Continuous Systcr;: ilodeling Proprar), a digital computer prograa package eaabling siculation of continuous processes via severai nur;.erical integration methods (61,621. All calculations were done on an 13:: 4341 ci&ital COnpUtei. Ou:pu: or‘ interest consisted of the relationship between initial plasrra heparin activity and the tine (“cfottlng tine”) to generate the ;.lini.r:.uc. fi’orin concentration (chosen as 7.5 pFI based 01; research by Blozback and Okada i63i) needeti for r‘ibrin gel formation. The relationsnip ‘between the slope of clotting tine versus heparin activitj profiles (“clotting; tiae-heparin slope”) and baseline (pre-heparin) clotting trees iu’as also of interest. It shoulc be coted that the overall relatiocships incepencent of a Vlde range of fibrin concentration chosen as the cre endpoine.
eight
A nathenaticti experkentaliy
uodel of observed
the anticoagulact characteristics
actioi-, of of heparin
hepnrm effect:
c;ust
sikluiate
1.
potentially Large intersubject and intrasubject tine and 3-foid ran&es) in baseline clotting slope (31,39,41,64,65).
2.
the apparently heparin plasna unit/n1 (26-41).
^
the apparently licear relationship between clotting time-heparln slope ano baseline clotting time over ranges of approxkately 2 to i/2 fold and 1 l/2 KO 2-fold, respectively (38,39,41).
3.
linear relationship between 10L clotting activity over 3 ranje of approxixateiy
4.
the nonlinear, versus thrombin
5.
the nonlinear, slope versus
6.
the nonlinear, concordant relationship of clotting time-heparin slope versus AT-III
7.
thronbin activity tine (13,18,41).
8.
thronbin activity and AT-III cor.centration of clotting tine-heparin slope (18,41,67).
The ability of pharmacological validation.
the
variability (1 l/2 clotting tine-heparic
discordant activity
relationships and fibrinogen
of baseline concentration
discoraant relationships of chrombin activity anti fibrinogen
inust
be
a major
proposed model to effect of heparin
tine and 0.1 to 1
clotting (13,1S,66).
Lust
simulate the above as the was used
2
time
clotting tine-heparin concentration (1s).
baseline clotting time concentration (18).
determinant
to
of
baseline
be major
and
ciotting
detercinants
characteristics criterion for
of the node1
EESULTS Clotting tine varies with initial plasma hegarin activity over a 1 l/2 to 2-fold range of baseline clotting tines similar to the previously observed of simulated range (38,39,41). As illustrated in Figure 2, the logarithms times can exhibit an apparently linear relationship to initial plasrtia clotting
;:
;:
D.ASMA
Fi;.
.;/
: z 012 05
7
z 6 Z : 5 T z 4I -z
between clotting tixe tieparin and ir,it ial plasha sii:lulstea by v&ryl:l:, activity tllrol::bin COItCe;:CiatlOX initiai (ui_Fer left), varyin; i:liti&l f ibrii:06;en concentration (upper vnry In;; ar.d lnitisl right), 3 I’.t 1 t ,: r 0 t: 0 1 c concentration ( i 0 1; e r “lc’ i’i). I :I i c i a i
P
. ;E5 0.10 ~ 1.0 IZ I4 1.6 II . I*ftl,NtCLOImm TlMl .
s
PO5 2
E Y 5
concentration
(
zcijacent
t
relatiocship
s
syzbois)
plcsL;P T ile
,
/!,!//!lilll 02 PLASMA
04
0.6
HEPARIN
oa
io
ACTIVITY
10,
corres~onain~
(U/ml)
cte
ciottin:,
tir;-.e-
5aselir.e
a 11c
apparenti.; cite
VeiSUS
activiiry -2rofiles. solid ii::ezr stiFeill..~OSeC
> values antithr0z:'bi.n r F11e
clottin;
th e
l:epsriri
regression 12
T i?e
(uslnb;
tll.;e for
lir,e;r
noted
sh 0 V
slope
clotcic~
is
curv5.
insets tetveen
heparin
1
ii:
each
to
respective
;2 E 6
‘? j;/m!i
2
2
23 ;
13
ACTIV!lY
LelatiCai:ship
P d’
2 &lb IO :, F 9 ZYOI‘ 8 g
08
7-3
HE?ARlN
s i z:f- ;y
iices
are
S f;Cm>;i; ( 3 j,i
C.9SG). III
In
iilset,
ccnnects
ihe
;
the casiifll
t.;e ;;oints.
SYST’E?!
COAGULATION
SI?ITY’LXTiON
229
_._
The sizulateti increasing and lecreasir.5 coagulability __. . 2ecreasi::j .AT-III concen:ration and decreaslzb ilorlcoaen c3nceC;rat1oC, respectiveiy, are consistent with present ‘kn0wledg.e of the hemostatic roles of ihese txo substances. The obvious nonlinear nature of these effect-activity profiles for the lotier initial chrocbir: concentrations, hig’ner AT-III concer.tr3tioris, and lover fibrinojen concentrazions zay be a result of the underlying Rorkiinearity of this aodel systes or failure to sr:*.ulat? a cmticuous input of throrzbin. QCiiIi
t iOrlCOg22.
associz:ei:
with
The relationships betcieer. clotting tine-he?arir. slope and baseline ciottin; tioe for the apparentiy linear cecibers of each faziiy of curves are illustracec in ihe insets of Figure 2. Consistmt with previous observations (3; ,39,&l), the clotting ci=e-Xeparin slope is apparently linearly related to Jaselxe clotti2; tic;? for the cases of varying initial thror!bin or fibrinogen concentrntloc. Kowever, varying initial AT-III concentration vas associated with a nonlinear relationship between clottin;; cizt-aeparin slope an6 ‘baseline clotting tinie. Such a nonlinear relationship was anLlcipaLeci based on the 2 u r e Ly inuirect on the clottin, effect of AT-III eadpoint-detercining iriteraction ‘aetveen tiironbin ana fibrinogen. The posslbie associations between clotting tine-‘he?arin slope, baseline AT-III, and fibrinogen clotting time, acd initial concentrations of thro:.;i)in, 1ncrensir:g initial concentrations of 5ie r e 3oc;ht (Fi,ure 3). Clearly,
INITIAL THROM8lN CONCENTRATION (0,pMl I 8
I 9
I 10
1 11
1 12
1 13
1 14
1 15
INITIAL FIBRINOGEN CONCENTRATION (n,pM) i I
I
I 2
b
1 3
1
1 4
1
INITIAL ANTITHROMBINCONCENTRATION (Q,M)
P,eiatior.st:ip ntigarin -.. riorinogen,
ciotcrng
between sLo?e ant
title
baseline
versus
antithrozbin
versus
clottin,;
initial
plaszc
III heparin
1 5
Ill
trr.;e
;n6
clottin,
concentrzrions
for the acLivity
apparently profiles.
of
time-
throtzbin,
linear
i0.g
Vo1.28,
COAGULATIONSYSTW SI?iUL.A-t’IOX
230
x0.2
trnrOiiitui anC i iDrL;o,ec r;ere both tssociatfa wi:h ti2creasir.g DaseiiLe ciocti.nL clxes azd cecreaslng tize-keparin slopes. Sir,ce trrrot.:bin ant cioctinb -.. tlorinogen are ti:e proceiza:eoLs reactailts in the endpoint-prolucicg re3ctLor., incr2asir.g rkelr iRitia1 concentrations shotiici indeed iias:en ti,e clottind > process bott ill the 3bsei:ce auc ijresence or ke?arin, thtis cecreaslag Dotn bsselice clottir:g tine and clottin; clzr-teparin slope. iiirsi; 2~ ai. (64) 03SeiVZtiUIl of fibricogen acrichcent increasing tne ilave reporteil a sick;c APTT of control hC:r_an pl3s~a, while Raoerser. ~CL Godal (65) aid zot observe any change ir, tbro-;Sin tine with fibrinogen: concentratioc varyic; over the norCal rang2 of 2 to 4 g/l either in the Fresence or absence of scaii aizounts of heparin (2 ci.25 IU/i.;l). These are the only observaiioas report25 tkac contilicr: with the sizulacea results. Gase1ir.e clocriq time ~5s 3;;roxizateiy eqoolly sensitive LO the sibiulated cilan2es in initial thrombin afiG fibrkogen concentrations. Lowever, clotting Lirfiie-i;e;!arin siope variea over a Z-folc r:ilile slope o&s far less range wiLii varying mitiai. chroc:bin concentration, sensitive (varyins over a i l/3-fold range) to changes ic initiai flbricogen IncreasiaL the inrtiai concectration of AT-III, an ecdobenous coccentration. . . prote!n that inhibits tllror;.bin activity, ~82s assocrateci wltli ;r.creasxb t irie-heparin slopes (Figure bssellne clottin tices anc increasing ciottinb seilsitive (varying over a 2Clotting tine-te?arin siope was relativei:, 3). This sensitive folti range) to changes in initiai AT-I11 concentration. facilitates AT-III-tilrombic relationship is expected since heparin readily vith throxbin in tiie interaction. Since AT-III interacts ir.uch less tiia2 was effected lit tie by chaqing ATabsence oE heparin, baseline clotting Studies (36,39,41) neither oaseline III concentration. As iii previous exhibited a statistically clottin& tines nor cioLtin& tir;ie-hep arin slopes significact linear relationship to icicial coccencratioc of fibrinojer. or dTFigure 3 illustrates tile III. i! 0 w e v e r , nithou6h not linear in uatu:e, ckiscordant and coniinear concordant naiu:‘e Of the respective r:onlii;ear relationships betwe2n both clotting paraneters (baseLil;e ciotting tine, ciotting time-heparin slope) and initial concentrations of fibrinogen and ATIII. sirzulations summarized in Figure 3 illustrate that baseline Therefore, thought to be an index of hy?ercoagulability, is determined ciottin;; tirse, largeiy by the throcbin activity and fibrinogen concentration, tliiilt clotting an index of heFarin cosege requiren.enc, is ceterolned tine-heparin slope, primarily by throxbin activity ana AT-III concentration. 1
4
_..
DISCUSSIO:: coabulation As early as 1937, ecpirical equations were used to predict concentration and several experinerltally tine 3s a function of heparin The observation by numerous investigators that an determined constants (25). apparently linear relationship exists between the lodarittcs of various & vitro cl0ttir.g tiaes an< heparia activity in plasaa or whole blood has not oociei of direct or iildirect been explained to date using any availabie phareacologic effect. The present skulatior. study has denonstrate? chat such an apparently linear reiationship can be preiicced frocl a relativeiy simple natheuatical node1 of the known molecular interactions arzong throwbin, fibrinogen, AT-III, This rsodel illustrates that clotting time, and hepar in. being the secondary expression of the primary pharmacologic effect of heparin, cannot be treateci in the same manner as directly observed prinary pharmacologic effects such as tubocurarine-induced neuromuscular blockade. clotting time represents the time it takes to reach a givea effect, Rather, namely, a certain L;ibrin or gel formation in a coagulation test tizer. The
model
was
validated
by
virtue
of
its
ability
co
simulate
eight
Vo1.28,
No.2
COAGL%ATIONSYSTM SIWLATION
231
5se r v E c expericcnta; char3cceriscics cf hepaiin ertect. Future _i charscterizacion of tze rhte consrauts 'usec iC this lode1 and insrtiaSin,v :zechznis:ic knowledge of t.he koiecuiar blxoio;y 0:_ bl~'oC co3gu;ation c;Lii enable experinentai challengin; aad refinetenL of this noael. ?hrous;n this ccnstant process of chailen;ing and subsequent refinenect of the l;.odel, acd nathezaticai a:odeling methods Frovide insight si=uiation into tile Ir,echazis=s of the observed ghenonena and ~;ay ulc;rateiy yield rocels useful to both the molecular biologist ar.d ciinician.
0
like provided by these sinulations is broad. The scope of inforG3tion potentiaiiy large variabiliLy in clotting tixe effect associatec I:ith 3 c;iveu The sinulaced relative plasc;a iieparin activity Flds illustrated. hypofiorinogenenia 3r.d relative hypocoagulability associ ated with associated wit‘fl decreaseo AT-111 concettration are hgpercoagulability consistent with t\he cliuical hemostatic defects associate.2 5ich these perLurbations (64,69,70). In addition to sinulacin& Liiese chnracteriscics, tiie model also sinul3tes cite apparently licear iel3tionshi.2 betiieer,clottin; tize-;;eparin slope acd baseline clottin& tiae. As expected b3sel oc previous research (li;), clotting parar.eters (baseliue clottin tine acd ciottins tl:.iehepnrih slope) were related in 3 nonlinear discordant zanner Lo throzbin sctivity ar.u fibrinogen concentration, while clotting parxaeters i;ere related ?lodel simul-tions .z III 3 noniinear concordant c:anner to A'i-III ccncentration. showed that baseline clotting time was primarily cecerniuer: by thrombin time-heparii; slope IjLtS activity and f ibrlnogen concentration, c;iiile clottin& acd AT-i11 concentratioi:. primarily determined by thrombin activity Large intersuojecc F,ecent studies in normal subjects have cemons tiaCC?d variability ir,both baseline clotting time (XPTT) and clotting Liae-heparin Was related LO activity slope. however, tieither of these parameters, concentrations of fibrinogen or AT-ITT; the concentration of these proteins Cn the basis of c.ne nocel for showed little intersubject variability (39,411. it nay be the anticoagulant action of heparin presented in this study, proposec that 3 major determinant of these coagulation para:deters, at least in normal subjects, is the initial thrombin concentration or the rat;? of throtibin generation. iu addition LO reconciling an observed non-ciasslcal doseThus, the response relationship with known characteristics of hepar in pharmacology, a clinicnily relevant c.xplanation for the large motel presented suggests T’r.e intersubject variability in the experimentaliy obtained data (39,411. various perturbations of 2 0 c c 1 p r e s e r. t e d ma y also be useful in studyrn& heparin effect, such as interac:ions between hep3ri.n and other axticoat;uiants, and in studying the hemostatic impact of chan&es in concentration or activity Simulac ion siethods (71of individual components of the coagulation syscer... biochemical theories 73) have proven useful in evaLuatin& numerous mechanistic and may provide a means by which the dynamic character of hemostatic processes in perturbations LO subtle can be visualized and its responsiveness coagulation factor activities, endob;enous inhibitors, and pharmacological interventions evaluated. ACiiilO\-ILEDG EXEDTS A preliminary report of this work was presented at toe eighty-third for Clinical Pharmacology ana annual meeting of the American Society This wori was supported in part by XE Grant iio. Therapeutics io :;arch 1982. liL-24343. T.D. Bjornsson is a recipient of a Pharmaceutical Xanufacturers Association Foundation Faculty Development Award in Clinicai Pharmacology. The authors gratefully acknowledge the skilled typing assistance of ?:s. Zanda Seynour
*
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