Spatiotemporal dynamics of clotting and pattern formationin human blood

Fazoil I. Ataullakhanov *, Georgii T. Guria, Vasilii I. Sarbash, Rimma I. VolkovaNational Research Center for Hematology, Russian Academy of Medical Sciences, Novozykovskii 4a, Moscow 125167, Russia

Received 9 July 1998; accepted 27 August 1998

Abstract

We examined the spatial dynamics of in vitro clot growth in human blood and plasma and found that initially, a clot growsat a constant speed, then abruptly stops and becomes surrounded by an `inhibition zone' in which coagulation is stronglysuppressed. We also observed the formation of `stratified structures' (target patterns) in which solid layers alternated withliquid plasma. These and other spatial regimes of clotting are explained in terms of two interacting concentration wavespropagating without attenuation. The experimental results are consistent with a hypothesis that blood is a bi-excitablemedium, a new type of excitable medium. ß 1998 Elsevier Science B.V. All rights reserved.

Keywords: Autowave; Excitable medium; Blood coagulation; Thrombin; Clot growth termination

1. Introduction

The process of blood coagulation is characterizedby a complex time and spatial organization. To stopbleeding, the damaged area must be quickly andcompletely sealed with a clot. It is equally dangerousfor an organism to form an insu¤ciently sized clot orto form a clot accidentally in another area or at thewrong time. The spatiotemporal dynamics of clotformation remains poorly understood, because size,

shape and structure of clots are usually studied afterthey have already been formed. These studies do notanswer the important dynamic questions concerningthe rate of clot growth and nature of the factors thatdetermine this rate and control the clot size, i.e. fac-tors responsible for the clot growth and its termina-tion. Although studies with the in vitro stirred sys-tems cannot directly address the spatiotemporalaspects of clot growth, they provided the main con-tribution to our knowledge of the molecular mecha-nisms of coagulation [1,2].

Clotting occurs when ¢brinogen, an abundantplasma protein, is converted to ¢brin in a proteolyticreaction catalyzed by thrombin. Fibrin polymerizesrapidly to form the clot fabrics. The kinetics of clot-ting is largely de¢ned by the kinetics of thrombinproduction and inactivation [1,2]. Thrombin is thelast enzyme in the cascade of proteolytic reactionsinitiated in blood by the appearance of activatorsof clotting. The nature of these activators (tissue

0304-4165 / 98 / $ ^ see front matter ß 1998 Elsevier Science B.V. All rights reserved.PII: S 0 3 0 4 - 4 1 6 5 ( 9 8 ) 0 0 1 0 2 - 0

Abbreviations: AMC, 4-methyl-7-aminocoumarin; substrateS, BOC-Ala-Pro-Arg-AMC, or t-N-butoxycarbonyl-alanyl-prol-yl-arginyl-4-methyl-7-aminocoumarin; CPD, citrate-phosphate-dextrose solution; FITC, £uorescein isothiocyanate; FITC-thrombin, thrombin conjugated to £uorescein isothiocyanate;PPP, platelet-poor plasma; PFP, platelet-free plasma; PRP,platelet-rich plasma; PVC, polyvinyl chloride

* Corresponding author. Fax: +7 (95) 212-8870;E-mail : Fazli@bioscience.msk.su

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factor or a foreign surface) de¢ne the initial set ofreactions, i.e. the extrinsic or intrinsic pathway eachof which eventually leads to the generation of throm-bin [1^3]. The characteristic feature of both pathwaysis an explosive production of thrombin after somelag-period (see [1^6] and references therein). Highlynon-linear production of thrombin depends on sev-eral positive feedbacks in the coagulation cascade,such as the activation of factors V and VIII bythrombin [1^6]. The self-sustained production ofthrombin should greatly increase the rate of clotgrowth not only in the damaged area, but in everypart of the blood where the thrombin spreads bydi¡usion or through the blood £ow. It is not com-pletely clear why clotting remains limited to the dam-aged area and does not spread over all of the bloodvolume.

Experimental observations provide evidence that,normally, blood clotting is not associated with sig-ni¢cant depletion of the precursors of coagulationfactors [2]. Therefore, it is di¤cult to attribute thetermination of clot growth to the depletion of factorsinvolved in its formation. The decisive role in limit-ing the clot growth is thought to be played by inhib-itors of coagulation. All known inhibitors can be putinto two main categories: those that are constantlypresent in the blood in their active forms and thosethat are activated during clotting. The former (an-tithrombin III, K2-macroglobulin, etc.) are veryabundant compared to the maximum thrombin con-centration possible in the blood [2,7,8]. Despite theirabundance, they can stop thrombin production onlyfor activations below some threshold [6,9]. Thisthreshold is vitally important for preventing sponta-neous clotting. The only known inhibitor of the sec-ond type is protein C, which is produced enzymati-cally from its precursor by thrombin [10,11]. ProteinC, in turn, inhibits the thrombin production by en-zymatically inactivating factors Va and VIIIa. Thisdisrupts positive feedbacks in the cascade of throm-bin production, preventing its explosive accumula-tion. In the in vitro stirred systems, it leads to a pulseof thrombin activation: thrombin concentration risesexponentially and then falls rapidly [4,5]. It remainsto be seen whether these inhibitors are su¤cient forlimiting clot growth to the damaged area or thisprocess also depends on the local environment. Thelocal processes that limit clot growth may include the

localization of protein C cofactor thrombomodulinto the vascular wall, thrombin binding to the ¢brinclot, platelet adhesion, and some others.

All current models of the time and spatial organ-ization of clot growth are highly speculative, becausedirect observations of the distribution of coagulationfactors during clot growth are still lacking. In thiswork, we studied and analyzed the dynamics ofthrombin production and ¢brin polymerization dur-ing clot growth in vitro. We found that initially aclot grows at a constant rate, but then abruptlystops. The resulting thickness of the clot is almostindependent of the initiating signals. The clot be-comes surrounded by an `inhibition zone' in whichcoagulation is strongly inhibited. Sometimes, thenormal pattern of clot growth is disrupted and`strati¢ed structures' are formed in which solid layersalternate with liquid blood. We also observed con-tinuous clot growth. The `inhibition zones' and the`strati¢ed structures' cannot be easily explained fromthe classic point of view on the clotting process.However, these data agree well with a hypothesisinvolving two interacting concentration autowavespropagating from the damaged area [12,13]. Thesewaves propagate without attenuation like a neuronalimpulse [14,15] or excitation in the Belousov^Zhabo-tinsky reaction [16]. In contrast to the classic auto-waves, the autowaves of blood coagulation can stopin a homogeneous medium as a result of their inter-action. The distance that the wave propagates de-pends mainly on the kinetic parameters of clotting,implying that this system has an autoscaling. Theinteracting waves produce patterns by a novel, essen-tially non-Turing mechanism. These results led us tosuggest that blood is a new type of excitable medium,i.e. a bi-excitable medium.

2. Materials and methods

2.1. Plasma and reagents

Experiments were performed on platelet poor plas-ma (PPP), unless speci¢ed otherwise. PPP was iso-lated from random-donor blood units by convention-al blood bank centrifugation techniques. Tripleblood pack systems (Baxter, S.A. de C.V., Morelos,Mexico) containing CPD in the primary bag were

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used for blood collection. Whole-blood units werecentrifuged at 1000Ug for 10 min at 22³C. The plate-let rich plasma (PRP) was transferred into a satellitebag and centrifuged at 2400Ug for 20 min at 22³C.The supernatant platelet poor plasma (PPP) con-tained less than 2U1010 platelets/l. It was stored inPVC transfer bags at room temperature for nolonger than 2 days. Platelet free plasma (PFP) wasobtained by centrifuging the PPP in 4-ml plastictubes at 10 000Ug for 20 min at 22³C. PRP, PFP,and whole blood were used on the day of bloodcollection. A speci¢c £uorogenic substrate for throm-bin BOC-Ala-Pro-Arg-AMC (substrate S) [17] wassynthesized by Dr. V. Pozdnev at the Institute ofBiological and Medical Chemistry, Russian Academyof Medical Sciences, Moscow. Substrate S was pre-pared as a 10 mM stock solution in dimethyl sulf-oxide and added to plasma or blood to a ¢nal con-centration of 0.05 mM.

2.2. Visualization and registration of clot growth

Aliquots (1 ml) of PPP, PFP, PRP, or whole bloodwere placed into 35-mm polystyrene Petri dishes(Medpolimer, St. Petersburg, Russia) to obtain liquidlayers 0.5 mm thick. The pH values of PPP and PFPwere stabilized at 7.2^7.6 by adding lactic acid to a¢nal concentration of 0.1^0.15% directly to thedishes and incubating them at 37³C for 40^60 minprior to the experiments. The Petri dish was thenplaced into a temperature-controlled chamber (Fig.1) and incubated for 1^5 min to equilibrate the tem-perature. Plasma (or blood) was recalci¢ed with 1 Mcalcium chloride (Sigma, USA) to achieve free calci-um concentration of 1.5^2.0 mM. The concentrationof free calcium in plasma (or blood) was determinedwith a Model 93-20 calcium electrode in a pair with aModel 90-02 reference electrode attached to a EA920 meter (Orion Research AG, Kusnacht, Switzer-land).

To initiate clotting, we added a glass bead approx-imately 0.6 mm in diameter (Fig. 1, 4) to the centerof a petri dish (Fig. 1, 1). In some experiments, col-lagen ¢bers (Serva, Germany) or thrombin (Merck,Germany) were used instead of glass beads. Precau-tions were made to leave the plasma almost undis-turbed. The dish was then sealed with a thermostat-ing lid with a quartz window (Fig. 1, 5 and 6). These

temperature-controlling conditions excluded conden-sation of moisture on the window (this allowed directvisualization of clot growth) and minimized the con-vective heat £ows. No vertical and horizontal mixingcould be detected during 2 h after addition of 50 Wlof Indian ink or some chalk powder into the plasma.The spreading patterns were determined by their dif-fusion, whereas convective £ows contributed insignif-icantly.

Fluorescence was excited with a 250 W DRSh-250-2 mercury lamp (Photon, Zelenograd, Russia; Fig. 1,12). The excitation light was ¢ltered using an aque-ous heat-blocking ¢lter and a UV ¢lter (300^380 nm;Fig. 1, 13 and 14). For light scattering experiments,we used a 50-W halogen lamp (Osram, Hungary;Fig. 1, 15). A set of aqueous heat-blocking and glass¢lters allowed light passage in the 600^700 nm range(Fig. 1, 16 and 17). The heat-blocking ¢lters pro-tected blood and plasma specimens from heat emis-sion of the lamps and, thereby, prevented the appear-ance of convective £ows. Both lamps were connectedto a voltage stabilizer with a stabilization coe¤cientof 0.1%.

Growing clots were photographed using a Practi-ca-TTL camera (Zeiss, Germany) with an 8U6-mm¢eld of view (Fig. 1, 20). The camera was positionedbehind the UV-light blocking ¢lters (Fig. 1, 19). Themercury lamp was used to measure the AMC £uo-rescence (emission maximum 440 nm) derived fromthe substrate S cleaved by thrombin, while the halo-gen lamp was used to measure the light scatteredfrom clots (red region of the spectrum). When bothlamps were on, the AMC £uorescence and clotgrowth were visualized simultaneously. More quanti-tative information was obtained using a color CCDcamera (World Precision, Sarasota, FL), which wasmounted in place of the photocamera (Fig. 1, 7). Theimage of a small area (0.1U0.1 mm) was projectedonto an RGB matrix of the CCD camera (Fig. 1, 18).The electrical output signal proportional to the aver-age illumination of this area was formed by the ma-trix photodiodes (30U30 pixels). The signals propor-tional to the intensity of £uorescent and scatteredlight were recorded by the B (blue) and R (red) chan-nels, respectively. The interference between the chan-nels was less that 5%. The signal was applied to a Y-input of a two-coordinate recorder. To obtain spatialdistribution of the £uorescent and/or scattered light

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during clot growth, a base plate with the mountedPetri dish was driven horizontally at 0.1 mm/s with amotor connected to a micrometer screw (Fig. 1, 9and 10). The signal proportional to the base plate'sdisplacement was applied to the X-input of a record-er. These scanning conditions circumvented the prob-lems caused by a non-uniform illumination of aspecimen, because the signals were recorded from asmall area (0.1U0.1 mm) with constant illuminationrelative to which the specimen was driven.

3. Results

3.1. Dynamics of clot growth

To examine the dynamics of clot growth in vitro, aPetri dish with a thin layer of whole blood supple-mented with £uorogenic substrate S was placed in a

specially designed thermostated chamber (see Section2). Two to three minutes after addition of glass beadsto initiate clotting, one can see the appearance ofnarrow, bright £uorescent zones around the beads.This £uorescence is produced by AMC cleaved fromsubstrate S by thrombin, which is generated duringclotting. The £uorescent zones grow quickly untilthey reach 0.5 mm in thickness and then stop (Fig.2a). While growing, the edges of the £uorescentzones are very sharp, implying that the rate ofgrowth is faster than that of di¡usion of the £uores-cent product. The di¡usion becomes apparent afterthe clots stop growing, leading to slow fading of thebright areas. If the liquid blood was removed fromthe dish after the clots had stopped growing, theclots were exposed (Fig. 2b). As expected, their sizewas identical to the size of the bright zone. Thus, clotgrowth in vitro has the characteristic features of theclotting in vivo: the clots are produced around theactivating centers and do not spread all over theliquid blood.

The abundance of red blood cells makes it di¤cultto monitor clot growth continuously. When red cellsare removed, the remaining plasma behaves identi-cally to the blood and clot growth can be easilymonitored by light scattering of ¢brin-polymer(Fig. 3) or £uorescence of AMC (Fig. 4). At 18^20³C, the clot growth in plasma continues for 15^20 min (Fig. 3a^l), but then stops. The analogousdynamics was observed for thrombin activation asvisualized by AMC £uorescence (Fig. 4). Initially,the bright £uorescent zones of AMC grow fast(Fig. 4a^i). These zones have sharp edges, butwhen the clots stop growing, AMC di¡usion be-comes apparent. The AMC leaves the clots andspreads all over the Petri dish, so that 1 h after theclot stopped growing, the £uorescence in the clotarea becomes very weak (Fig. 4i).

Fig. 5a shows the spatial distribution of the £uo-rescent product of the substrate cleavage by throm-bin in plasma. Di¡erent curves represent di¡erentmoments of time after the initiation of clot growth.The intensity of AMC £uorescence is proportional tothe total AMC concentration; therefore, thrombinactivity is proportional to the rate of AMC accumu-lation. Thus, from the kinetics of AMC accumula-tion, one can see that initially thrombin activity risesquickly near the activating surface, but then abruptly

Fig. 1. Diagram of the experimental set-up: 1, polystyrene Petridish; 2, plasma or blood; 3, temperature-controlling jacket(37³C); 4, glass bead (0.6 mm in diameter) ; 5, temperature-con-trolling lid with 6, a quartz window; 7, CCD camera; 8, opticsystem; 9, motor to move (11) the stage with the petri dish at0.1 mm/s; 10, micrometer screw (arrows show the direction ofspecimen movement); 12, UV-light source (mercury lamp); 13and 16, aqueous heat-absorbing ¢lters; 14, UV glass ¢lter witha bandpass of 300^380 nm; 15, light source (halogen lamp) forlight scattering measurements ; 17, glass ¢lters with a bandpassof 600^700 nm; 18, RGB matrix of the CCD camera; 19, glass¢lters blocking UV-light; and 20, camera (arrow shows that ei-ther a camera or CCD camera can be mounted). B and R de-note output signals of the Blue and Red channels of the RGBmatrix, respectively. The signal was applied to the Y-input of atwo-coordinate recorder. The signal proportional to the dis-placement of a specimen (X) was applied to the X-input of therecorder.

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falls. The wave of AMC £uorescence starts propagat-ing from the surface, so that its front remains almostidentical for di¡erent points in time up until the wavestops. Fig. 5c shows how the distance from the glasssurface to the point where £uorescence is 0.5 (curve1) or 0.2 (curve 2) of the maximum changes overtime. The rates of propagation determined at di¡er-ent points of the wave front are almost identical.This explains the similarity of the concentration pro-¢les for di¡erent times, up until the speed of thefront drops to almost zero. Simultaneous registrationof clot growth by light scattering (Fig. 5b) has re-

vealed that ¢brin polymerization mimics the dynam-ics of thrombin generation after some lag-period(Fig. 5c,d). Fig. 5d illustrates the movement of theclot's edge, which is characterized by the same fea-tures as the propagation of the wave of £uorescence.The initial speed of thrombin front propagation andclot growth is about 0.05^0.1 mm/min under theseconditions.

Since £uorescent zones do not remain localized tothe clot area for a long time, it seems unlikely thatAMC or substrate S could bind or become trappedin the clot. Nevertheless, we examined this directly in

Fig. 2. Clot formation around objects of di¡erent size, shape and composition in (a,b) blood or (c^g) platelet-poor plasma as visual-ized by (a,f) £uorescence, (b,d,e,g) light scattering or (c) transmitted light: (a) clots 20 min after addition of glass beads (0.5^0.6 mmin diameter) and a dumb-bell (indicated by an arrow) into blood; (b) the same clots after the liquid blood had been removed (21 minafter addition of the activating objects) ; (c) clotting around a glass bar (3U15U2.5 mm); (d) clots around glass beads and a collagen¢ber (indicated by the arrow) and four spontaneous clots that appear smaller in size because they started to grow later; (e) and (f)clots around a small pinch (100^200 Wg) of thrombin powder; and (g) clots around a small pinch (100^200 Wg) of thrombin powderand glass beads. The thickness of plasma or blood was 0.5 mm in all experiments except in c, where it was 3 mm.

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the following experiment. PPP supplemented with 5WM AMC (or substrate S) was placed in a £uorom-eter cuvette and coagulated by thrombin taken at aconcentration su¤cient for complete hydrolysis ofthe substrate. The clots were sedimented by centrifu-gation, and the £uorescence of the supernatant se-rum was recorded and compared with the initial sig-nal from 5 WM AMC in plasma. The signals in thesera were 96% of the initial level. Hence, the bindingof AMC or substrate S to the clot accounted for nomore than 4% of the initial signal. The £uorescencefrom these clots placed in a saline bu¡er was too lowto be detected. Therefore, we concluded that the spa-tial distribution of AMC £uorescence during clotgrowth could not be explained by the binding ofAMC or substrate S to the forming clots.

3.2. Role of platelets in the dynamics of clot growth

It is well known that platelets participate in theformation of primary clots in vivo, so we examined

how they in£uence the clot growth in our assay. Theclotting in plasma containing various concentrationsof platelets was initiated with glass beads as de-scribed above. After the clots stopped growing, thedishes containing platelet free plasma (Fig. 6A),platelet poor plasma (Fig. 6B), and platelet richplasma (Fig. 6C) were examined. We found thatthe clot size around the glass bead did not dependon the amount of platelets present. However, highplatelet concentrations greatly increased the numberof spontaneously forming clots (Fig. 6C).

Fig. 7 shows the propagation of £uorescent zonesat initial stages of clot growth around the glass beadsas a function of platelets concentration. The initia-tion of clotting and the rate of clot growth are sim-ilar in PPP and PRP. We observed a slightly slowerrate of clot growth in PFP, but the di¡erence is stat-istically non-signi¢cant. We, therefore, concludedthat the platelets stimulate the formation of clots invitro, but they do not in£uence signi¢cantly the dy-namics of clots growth.

Fig. 3. Transmitted light pictures of a clot growing around a glass bead in platelet-poor plasma. The pictures were taken every 3 minafter addition of the glass bead.

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3.3. The initial speed of clot growth and clot's ¢nalsize are almost independent of the activatingsignals

In order to examine clot growth around objects ofdi¡erent shape and size, we initiated clotting in bloodor plasma by addition of a glass dumb-bell (Fig.2a,b) or a glass bar (Fig. 2c, note the di¡erence inscales). Surprisingly, we found that the rate of clotgrowth and the clot thickness were independent ofthe size and shape of activating objects. Nor did itdepend on the thickness of a plasma layer, which wassix times larger in Fig. 2c, than in Fig. 2d. We theninitiated clot growth from di¡erent activating mate-

rials, such as a collagen ¢ber (Fig. 2d) or powderedthrombin (Fig. 2e^g). We found that the smaller theactivation, the longer the lag period, but the rate ofclot growth and the thickness of clots were almostidentical in all cases examined. The powdered throm-bin initiates the wave of active thrombin immediatelyupon contact with plasma. The bright area expandsquickly, but then abruptly stops (Fig. 2f), so that theresulting thickness of the clot is almost identical tothat of the clot formed around the glass bead (Fig.2g). Same e¡ect of thrombin is seen in PPP withoutcalcium added (free calcium concentration 6 50WM). Paradoxically, the amount of added thrombinvastly exceeded the amount necessary to cause com-

Fig. 4. Fluorescent images of thrombin production in platelet-poor plasma, as visualized with AMC cleaved from substrate S (see Sec-tion 2). Clotting was induced by addition of a glass bead. The frames are separated by 3-min intervals except for the last image,which was taken 1 h later. Note that after the clot has stopped growing, the £uorescent product slowly dissipates.

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Fig. 5. Dynamics of in vitro clot growth in platelet-poor plasma: (a) spatial distribution of AMC cleaved from substrate S by throm-bin; (b) light scattering material (¢brin-polymer); (c) the distance from the glass surface to the point where AMC concentration is 0.5(curve 1) or 0.2 (curve 2) of the maximum plotted against time; (d) same dependence as in (c) but for optical density. In a and b, dif-ferent curves represent di¡erent times after initiation of clot growth: F, 13 min; b, 18 min; R, 23 min; S, 28 min; 8, 33 min; and+, 38 min. Clotting was activated in plasma by addition of a glass bar (as in Fig. 2c), so that clot was growing as a £at front in thedirection parallel to the bottom of the dish. The amount of the light scattering material was detected by its optical density. Fluores-cence was registered simultaneously by switching from a halogen light source to the UV one. Spatial distributions were obtained byscanning the specimen at a speed of 0.1 mm/s. Other conditions were as indicated in Fig. 2c. The intensity of AMC £uorescence isproportional to the total AMC concentration; therefore, thrombin activity is proportional to the increase in AMC £uorescence.

Fig. 6. Role of platelets in the formation of clots in the absence of stirring: (a) platelet-free plasma; (b) platelet-poor plasma; (c) plate-let-rich plasma. Pictures were taken 70 min after initiation of clotting.

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plete clotting of the plasma in the dish. Indeed, clot-ting quickly proceeds to completion if the content ofthe dish is stirred continuously. We also found thatthe same amount of powdered thrombin could causeclotting in the entire dish without stirring, if ¢brino-gen solution was substituted for plasma (not shown).These results strongly suggest that special mecha-nism(s) limiting the clot growth exist in plasma, butnot in ¢brinogen solution. They also imply that thismechanism(s) cannot be attributed to the binding ofthrombin to the clot, since clot grows uninterruptedin ¢brinogen solutions.

In vitro clotting may occur spontaneously, asshown in Figs. 2d anf 6. These spontaneous centersof activation occur later, but the clots grow by thesame scenario and stop growing when they reach thesame thickness as the non-spontaneous clots (notshown). The number of spontaneous centers stronglydepends on the platelet concentration (Fig. 6). It islikely that platelet aggregates serve as the origins ofspontaneous clotting. Eventually, the entire dish be-comes covered with clots which fuse if they growclose together. Surprisingly, if the distance betweenthe clots is approximately twice the characteristic sizeof a single clot, the clots do not fuse, but, instead,inhibit each other's growth (Fig. 8a). Between theclots, one can see a narrow zone of blood that re-mains liquid for many hours, implying that clottingis inhibited in this area. These `inhibition zones'

eventually cover the entire dish by a network ofliquid blood channels (Fig. 8b). The long straight`inhibition zones' between the clots suggest that theclots grow with the same speed towards each other,but cannot get closer than a certain distance (the`inhibition zone' width).

Fig. 7. Dynamics of clot growth in vitro for di¡erent plateletconcentrations: the distance from the glass surface to the pointwhere AMC concentration is 0.5 of the maximum as a functionof time (see Fig. 5). Spatial distributions were obtained by scan-ning the specimen moving it at 0.1 mm/s relative to the ¢eld ofview of the camera.

Fig. 8. `Inhibition zones' : (a) 20 min; and (b) 120 min after ad-dition of glass beads. All conditions are as in Fig. 2.

Fig. 9. Formation of `strati¢ed structures:' (a) 0 min; (b) 12min; (c) 30 min; and (d) 60 min. All other conditions are as inFig. 2.

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3.4. Formation of the strati¢ed structures

Sometimes, the normal pattern of clot growth isdisrupted (Fig. 9). Soon after the normal clot growthhas stopped, we observed the growth of a secondlayer. This second layer starts growing at some dis-tance from the ¢rst clot, approximately equal to thewidth of the `inhibition zone' (Fig. 9c). The `inhibi-tion zones' around di¡erent clots fuse together shap-ing the start line of the second layer. This growthproceeds similarly to normal clot growth and whenit stops, the third layer may start growing after somelag (Fig. 9d). In these `strati¢ed structures', whichlook similar to the so-called target patterns, solidlayers alternate with areas in which blood remainsliquid for many hours. It is easier to observe theformation of strati¢ed structures if spontaneousclot growth is suppressed. Experimentally, this canbe achieved by lowering the concentration of freecalcium in plasma. The second layer is often muchweaker than the original clot, so that it cannot beobserved by light scattering, but it can be visualizedby staining with the protein binding dye crystal violet(not shown). In these cases, the third layer is virtu-ally undetectable.

Plasma of freshly collected blood is usually hyper-activated during several hours after its preparation.In this plasma, the fast initial linear phase of clotgrowth is followed by the phase of steady slowgrowth. The density of the slower growing layer isnoticeably lower than that of the original clot.

4. Discussion

4.1. Experimental results

In previous theoretical works, we have shown thatcertain dynamic aspects of clotting could not be ad-dressed in experiments with stirring [18,19]. Here, wedirectly examined the spatial dynamics of clot growthin an in vitro non-stirred system using a £uorogenicsubstrate for thrombin. The results obtained sug-gested that in this in vitro system, the dynamic char-acteristics of clot growth resembles, in general, thoseobserved in vivo. The £uorogenic substrate allowssimultaneous registration and comparison of the dy-namics of clot growth and the distribution of active

Fig. 10. Fibrin concentration pro¢les for di¡erent regimes ofclotting as computed from the model: (a) a compact clot(p1 = 3 nM); (b) strati¢ed structures (p1 = 1.5 nM); and (c) con-tinuous growth (p1 = 0.5 nM). The parameters and constantsused in computations: a = 2.0 min31, b = 0.0015 min31, c = 2.8min31, k1 = 0.05 min31, k2 = 5.0 min31 nM31, k3 = 0.35 min31,vm = 5.0 nM, p2 = 0.05 nM, D1 = 1037 cm2 min31, D2 = 1037

cm2 min31.

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thrombin. The clot starts growing 10^12 min aftercontact activation with a glass bead (Fig. 3). It growscontinuously with approximately constant speed for20^30 min, but then stops. The dynamics of clotgrowth is very similar to the dynamics of generationand propagation of thrombin (Figs. 4 and 5). Thethrombin wave propagates 2^3 min earlier than the¢brin gel. The rate of clot growth is signi¢cantlyhigher than the rate of di¡usion of small AMC mol-ecules. AMC, which is cleaved from substrate S bythrombin, spreads as a concentration wave with asteep front (Figs. 4 and 5). The edges of the £uores-cent spot of AMC become fuzzy due to the AMCdi¡usion only after the clot has stopped growing(Fig. 4). Additional experiments demonstrated thatthe sharpness of the edges does not result from theAMC binding to the clot.

We examined the clot growth in whole blood andin plasmas containing platelets at various concentra-tions. Removal of red and white blood cells did notin£uence signi¢cantly the clot growth characteristics(Fig. 2). While the dynamics of clot growth and the¢nal clot size did not change after the removal ofplatelets (Fig. 7), the number of spontaneously form-ing clots increased greatly with increasing plateletconcentrations (Fig. 6), suggesting that spontaneousclots are formed around platelet aggregates. Theplatelets may clump around some foreign particlesand thereby amplify the activating signal above theactivation threshold [6,18]. Once the threshold isovercome, the clotting proceeds apparently inde-pendent of the platelet concentration, suggestingthat even PFP is saturated with procoagulant lipidsurface.

The clot growth was also examined for activationwith powdered thrombin. The resulting thickness ofthe clot and the rate of its growth were almost iden-tical to those of the clot formed around the glassbead (Fig. 2). Unlike glass beads, thrombin initiatesthe clot growth immediately upon contact with plas-ma. We found that clot growth was continuous ifthrombin was added to a ¢brinogen solution. Theclot grew uninterruptedly and soon ¢lled up the en-tire Petri dish. These experiments are di¤cult to in-terpret because the physicochemical characteristics ofthrombin dissolution in plasma are not well known.However, the qualitative di¡erences in the dynamicsof clot growth observed in plasma and ¢brinogen

solutions are obvious. We examined thoroughly thespatial dynamics of clotting in specially designedchambers where thrombin solution was di¡usinginto a chamber with plasma (Sinauridze et al., sub-mitted). The results obtained are in a good agree-ment with the e¡ects seen for the thrombin powder.

The mechanisms of clot growth termination arenot well understood. There are three major hypoth-eses, which are based on current knowledge of theclotting processes.

(1) Clot growth may be terminated due to the de-pletion of one of the coagulation factors. This hy-pothesis contradicts the experiments in which we ob-served the formation of strati¢ed structures (Fig. 9).Indeed, if clot growth was terminated due to thedepletion of prothrombin or a precursor of any ofthe coagulation factors, how could the clotting re-sume some time later? It is possible that if the ¢bri-nogen became depleted in the adjacent to clot area,and thrombin continued its propagation, then theclotting could resume once the thrombin wavereached the area rich in ¢brinogen. In this case, thecontinuous registration of clotting by light scatteringand £uorescence of AMC should have shown theinterrupted clot growth, but uninterrupted propaga-tion of the wave of the AMC £uorescence. We havenot observed such continuous propagation of AMC£uorescence during the formation of strati¢ed struc-tures.

(2) The clot may be impermeable to one or severalactivated coagulation factors, e.g. thrombin. It isknown that thrombin can bind to ¢brinogen [20],so it seems possible that this binding could eventuallymake the growing clot impermeable to the activethrombin. In this case, one would expect to see aslow deceleration of clot growth. However, in ourexperiments we observed the growth at a constantspeed and the abrupt termination of the growth(Fig. 5), rather than slow deceleration. Moreover,this hypothesis also contradicts the observed forma-tion of strati¢ed structures, because if the clot be-came impermeable to thrombin, this would precludegrowth of the second layer. We examined the perme-ability of the growing clot to FITC-labeled throm-bin, and found that thrombin can penetrate the cloteasily (Sinauridze et al., submitted).

(3) Clot growth can be terminated due to a de-crease in the concentration of factor XIa below the

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threshold [18,19]. This factor is formed only on theactivating surface. It propagates away from the sur-face only by di¡usion, unlike other factors (throm-bin, factor Xa, etc.), which can be formed every-where due to the positive feedbacks in the clottingcascade. This hypothesis predicts essentially the samebehavior as the previous hypothesis and cannot ex-plain the formation of the strati¢ed structures.

Below, we propose a hypothesis of clot growthtermination, which, in our opinion, can explain allthe experimental results. This hypothesis is based onthe well-known existence of the positive feedbacks inthe cascade of clotting reactions and the followingobservations about the key factors that determinethe rate of clot's growth.

The area in which the thrombin activity is high is¢lled up quickly with polymerized ¢brin, so it ap-pears that the rate of clot growth is determined bythe kinetics of thrombin generation and propagation.The distribution of the £uorescent product of sub-strate S cleavage by thrombin suggests that thethrombin pulse is generated near the activating sur-face. Thrombin concentration rises quickly and thenabruptly falls, so that the cleavage of substrate S(Fig. 5a) and the formation of ¢brin (Fig. 5b) arealmost completely abolished. Initially, the generatedpulse of thrombin activity propagates from the acti-vating surface without changing its shape, with ap-proximately the same amplitude and at a constantspeed. It appears that the front of thrombin activitypropagates in a self-sustaining manner. The auto-catalytic character of thrombin generation at thisstage agrees well with the exponential productionof thrombin in the in vitro stirred systems [2^4].

The waves that propagate without attenuation arewell known. Examples of such waves include prop-agation of a neuronal impulse [14,15] and excitationin a Belousov^Zhabotinsky reaction [16]. The mediathat allow an excitation to propagate without dissi-pation are called active or excitable media [15,16]. Itis well known from the theory of active media thatnon-linear, non-dissipative waves (autowaves) canpropagate in the reaction^di¡usion systems with au-tocatalytic kinetics. Parameters of the autowaves,such as their speed and amplitude, are independentof the initiation signals and are de¢ned only by in-ternal kinetic parameters of the system [15,16].

We propose that propagation of thrombin gener-

ation during clotting has an autowave nature. Theobserved constancy of the speed and amplitude ofthe thrombin wave at the initial stages of clot for-mation supports our hypothesis. As we have shown(Fig. 2), the ¢nal clot thickness is almost completelyindependent of the size, shape, and material of acti-vating objects. This means that the dynamics of clotgrowth is mainly determined by the molecular-kineticproperties of the biochemical reactions of clotting,but not by the activating signals. The independenceof the wave's parameters from the initiation signals isa characteristic feature of active media.

However, if the thrombin wave is an autowave,how can it stop in the isotropic space? In all activemedia known to date, autowaves propagate withoutattenuation until they reach the boundaries of themedia [14^16]. The existence of special mechanism(s)that stop propagation of the thrombin wave is im-plied by the `inhibition zones' in which plasma be-comes highly `anticoagulant'. We used mathematicalmodels to analyze the possible role of coagulationinhibitors in this process and found that the completestoppage or abrupt deceleration of the wave at amacroscopic distance from the point of initiation inthe isotropic space cannot be caused by the con-stantly present inhibitors. Therefore, the inhibitorthat can stop clot growth must be produced denovo (or activated) during clotting. For example,its production or activation might be induced by ac-tive thrombin itself (inhibitors of the second type,like protein C). If the rate of inhibitor productionis proportional to the thrombin concentration andtheir di¡usion coe¤cients are similar, the inhibitoraccumulation will always be delayed. Thus, the in-hibitor will fail to catch up with the activator(thrombin) wave, a situation typical of other excit-able media [15,16]. However, if the inhibitor is pro-duced autocatalytically, the situation changes drasti-cally. In such medium, two active interacting wavesof thrombin and its inhibitor can propagate. Thisanalysis allowed us to formulate a simple hypothesisconcerning a possible mechanism whereby clots growand stop growing [12].

4.2. The hypothesis

We hypothesize that blood is a bi-excitable me-dium. Thrombin generation propagates from the

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damaged area as an autowave. This ¢rst autowavecreates the conditions for autocatalytic generation ofa thrombin inhibitor, which after some delay startsits propagation as a second autowave, i.e. a wave ofthrombin inactivation. Moving faster than the ¢rst,the second autowave catches up with the ¢rst andstops it. The absence of thrombin terminates the gen-eration of the inhibitor and, therefore, the propaga-tion of the second autowave. The process of clottingends with the formation of a compact clot localizedto the damaged area.

To compare the hypothesis with the experimentaldata, we constructed a simple mathematical model[13].

DuDt� au2

u� p13k1u3k2uv�D1vu �1�

DvDt� bu�13 v

vm� �1� v2

p22

�3k3v�D2vv �2�

DwDt� cu �3�

where u = u(r,t), v = v(r,t), and w = w(r,t) are the con-centrations of thrombin, inhibitor, and ¢brin, respec-tively, at the point r = (x,y,z) and time t ; v is theLaplace operator; a, b, c, k1, k2, k3, p1, p2, vm arethe parameters; and D1, D2 are the di¡usion coe¤-cients for thrombin and inhibitor, respectively.

The ¢rst term in Eq. 1 describes the kinetics ofthrombin generation, with a being the activity factorand p1 being the coe¤cient. The second term de-scribes thrombin inactivation by inhibitors that areconstantly present in the blood, with k1 being thecatalytic constant for this inactivation. These ¢rsttwo terms determine the threshold behavior ofthrombin. The threshold thrombin concentration is

ucr � k1p1=�a3k1�:As long as the thrombin concentration u6 ucr, thesecond term is larger than the ¢rst and autocatalyticgeneration of thrombin does not occur. Whenus ucr, the ¢rst term becomes larger than the secondand thrombin concentration grows explosively. Thethird term in Eq. 1 describes thrombin inactivationby the inhibitor (v), whose production is catalyzed bythrombin. Here, k2 is a catalytic constant for the

inhibitor-dependent inactivation. The last term inEq. 1 describes thrombin di¡usion, with D1 beingthe di¡usion coe¤cient.

In the second equation, the ¢rst term describes thekinetics of inhibitor production, which is di¡erentfrom that of thrombin. The inhibitor generation isproportional to the thrombin concentration: whenthrombin is absent, inhibitor is not produced. Here,b is the rate constant for this production, which islimited by vm, the maximum concentration of theinhibitor. In contrast to thrombin, the inhibitor canbe produced even when its concentration is zero.With the growing inhibitor concentration, the rateof its production increases. The second term in Eq.2 re£ects the inhibitor's inactivation with the rateconstant k3. The autocatalytic production of the in-hibitor begins only when its concentration exceedssome threshold. The threshold inhibitor concentra-tion is described by a cubic equation; for us 0 andvs p2, this concentration approximately equals

vcr � k3p22=bu:

The last term in Eq. 2 describes the inhibitor dif-fusion (D2 is the inhibitor di¡usion coe¤cient). Eq. 3re£ects the production of ¢brin with the rate con-stant c. We assume that polymerization of ¢brin pro-ceeds so fast that its di¡usion is negligible.

Analysis of the model revealed that the zero, spa-tially uniform, stationary state of the system is stablerelative to small perturbations. However, if the ex-ternal stimulus causes a local rise in thrombin con-centration above the threshold p1, a thrombin auto-wave can be generated. At each point where the frontof the thrombin wave propagates, a rise in thrombinignites the production of the inhibitor (Eq. 2). Theproduction of the inhibitor becomes explosive whenits concentration reaches the threshold p2. This al-lows the propagation of the inhibitor as a secondautowave. The inhibitor autowave can propagateonly where the thrombin concentration is above thethreshold, i.e. where the thrombin wave has passedthrough.

Our model is able to simulate all the regimes ofclotting observed in vitro. The formation of a local-ized and compact clot (Fig. 10a) is obtained forp1 s 2.4, where p1 is the parameter that sets thethrombin threshold. Initially, the inhibitor autowavepropagates faster than the thrombin wave does and,

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when it catches up with the thrombin wave, throm-bin concentration falls below its threshold leveleverywhere in space. As a result, both waves attenu-ate. The area a¡ected by the thrombin wave ¢lls upwith polymerized ¢brin. The clot's density is deter-mined by the amplitude of the thrombin autowave.The calculated distance of the wave propagation, i.e.the clot size, is almost independent of the amplitudeand duration of the initial stimulus, and is deter-mined by the ratio of the rates of the thrombinand inhibitor autowaves. The distance of wave prop-agation appears to be the internal characteristic ofthe system (its autoscale).

The numerical calculations reveal that the devel-oped clot is surrounded by an area in which coagu-lation is inhibited due to the inhibitor's passive dif-fusion into this area. The capacity for clotting isrestored only after some time necessary for degrada-tion of the inhibitor. Thus, the model describes wellthe formation of inactivation zones.

For 1.36 p1 6 2.3, the model's behavior changesdrastically and now it describes the formation ofstrati¢ed structures (Fig. 10b). First, the originalclot develops as described earlier: the inhibitor'swave severely reduces thrombin activity and, as aresult, the inhibitor concentration falls and ¢brinproduction is almost terminated. However, the weak-ened thrombin wave remains above the threshold forthese p1 and keeps propagating. Because of its lowamplitude, the thrombin wave does not cause signi¢-cant clotting, but once it reaches the area where theinhibitor concentration is below the critical value, thethrombin concentration rises explosively and clottingbegins in a new layer. When thrombin concentrationincreases to the critical level for which the local in-hibitor concentration is above its threshold, the anti-clotting wave is triggered. The process repeats, gen-erating the second, the third and so on layers ofpolymerized ¢brin.

When the thrombin threshold is very low(p1 6 1.2), the clotting proceeds continuously (Fig.10c). After the ¢rst clot is formed, both waves con-tinue steady propagation and eventually acquire thesame low speed. Their amplitudes become stabilizedafter a short period of oscillations. The movingfronts of these waves behave identically to the knownautowaves [14^16]. The speed of the front dependson the internal kinetics of the system and is inde-

pendent of the initial conditions. The area aroundthe original clot solidi¢es with ¢brin-polymer of alower density. There are no inhibition zones forthis solution.

5. Conclusions

With the help of mathematical modeling, we ana-lyzed the hypothesis that blood is a bi-excitable me-dium. Our model describes well all experimental ob-servations: the high and constant rate of clotgrowth; the abrupt termination of clot growth; the`inhibition zones'; and the independence of the clotthickness of the shape, size and material of activatingobjects. The model can also simulate the complexmodes of clot growth observed in vitro, such as con-tinuous clot growth, interaction between clots grow-ing close to one other and the formation of `strati¢edstructures'. These structures are formed by a succes-sive addition of new layers. This theoretically pre-dicted and experimentally observed regime of clot-ting is particularly interesting, because themechanisms leading to the formation of `strati¢edstructures' in vitro may also be responsible for thecomplex system disorders of blood clotting, such asdisseminated intravascular coagulation syndrome.

Although we cannot state that the proposed hy-pothesis is the only possible explanation of our ex-perimental results, we were unable to ¢nd anotherreasonable model for the observed phenomena.Analysis of current information on molecular mech-anisms of clot growth termination [2,4^10] has notrevealed the exact reactions or molecules that mightbe responsible for the autowave nature of this postu-lated inhibitor. Protein C, which is directly activatedby thrombin, would be the best candidate for playingthis role if the kinetics of its generation were auto-catalytic. However, the inhibitor displaying thepostulated kinetics of its formation have not yetbeen identi¢ed.

In a bi-excitable medium, the compact and spa-tially regular dissipative structures can be formedby the autowave mechanism, which is essentially dif-ferent from the known mechanisms of pattern for-mation [21^29]. Our theoretical model di¡ers signi¢-cantly from the well known basic models ofspontaneous pattern formation and self-organization

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in non-equilibrium systems of di¡erent nature[15,21^28]. For example, it is easy to see that inour model, the patterning occurs by the essentiallynon-Turing mechanism: it is impossible in model(1)^(3) to lose the stability in the sense of ¢nitewave perturbations (k = 0) as exp(3ikox). The respec-tive dispersive curves assume the form of V=const3Dk2.

The bi-autowave mechanism of patterning is alsodi¡erent from the mechanism responsible for the for-mation of stationary lamellar patterns in chemicalsystems [30]. In the latter, the initial and growingstructures are topologically equivalent at every pointin time, while the structure formation by the bi-wavemechanism occurs through successive addition ofnew structural elements to the ones formed earlier.Finally, our model di¡ers from those studied by thetheory of biological pattern formation [23,26^28]mainly because the latter have a single autocatalyticvariable. We also suggest that the autowave mecha-nisms similar to that proposed for blood clottingmay be involved in a step-by-step pattern formationduring biological development processes.

Acknowledgements

We thank A.V. Pokhilko, E.I. Sinauridze, and A.Yu. Safroshkina for technical assistance and helpfuldiscussions, E. Grishchuk for translating the manu-script, A.I. Vorob'ev for useful discussions of medi-cal aspects, and L. Berliner for comments on themanuscript. This work was supported in part bythe Russian Foundation for Basic Research (Project95-03-09052).

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