research article massive exploration of perturbed...

Research ArticleMassive Exploration of Perturbed Conditions of the BloodCoagulation Cascade through GPU Parallelization

Paolo Cazzaniga123 Marco S Nobile24 Daniela Besozzi235

Matteo Bellini4 and Giancarlo Mauri24

1 Dipartimento di Scienze Umane e Sociali Universita degli Studi di Bergamo 24129 Bergamo Italy2 SYSBIO Centre for Systems Biology 20126 Milano Italy3 Istituto di Analisi dei Sistemi ed Informatica ldquoAntonio Rubertirdquo Consiglio Nazionale delle Ricerche 00185 Roma Italy4Universita degli Studi di Milano-Bicocca Dipartimento di Informatica Sistemistica e Comunicazione 20126 Milano Italy5 Universita degli Studi di Milano Dipartimento di Informatica 20135 Milano Italy

Correspondence should be addressed to Paolo Cazzaniga paolocazzanigaunibgit

Received 21 February 2014 Revised 7 May 2014 Accepted 18 May 2014 Published 16 June 2014

Academic Editor Horacio Perez-Sanchez

Copyright copy 2014 Paolo Cazzaniga et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The introduction of general-purpose Graphics Processing Units (GPUs) is boosting scientific applications in BioinformaticsSystems Biology and Computational Biology In these fields the use of high-performance computing solutions is motivated bythe need of performing large numbers of in silico analysis to study the behavior of biological systems in different conditionswhich necessitate a computing power that usually overtakes the capability of standard desktop computers In this work we presentcoagSODA a CUDA-powered computational tool that was purposely developed for the analysis of a large mechanistic modelof the blood coagulation cascade (BCC) defined according to both mass-action kinetics and Hill functions coagSODA allowsthe execution of parallel simulations of the dynamics of the BCC by automatically deriving the system of ordinary differentialequations and then exploiting the numerical integration algorithm LSODA We present the biological results achieved with amassive exploration of perturbed conditions of the BCC carried out with one-dimensional and bi-dimensional parameter sweepanalysis and show that GPU-accelerated parallel simulations of this model can increase the computational performances up to a181times speedup compared to the corresponding sequential simulations

1 Introduction

Mathematical modeling and computational analysis of bio-logical systems nowadays represent an essential methodol-ogy complementary to conventional experimental biologyto achieve an in-depth comprehension of the functioning ofthese complex systems [1 2] Given a model that describesthe physical or logical interactions between the componentsof a biological system different algorithms can be exploitedto make predictions on the way this system behaves in bothphysiological and perturbed conditions For instance start-ing from distinct parameterizations of the model simulationalgorithms can be used to devise the different emergentbehaviors that the system can present the massive explo-ration of high-dimensional parameter spaces allows us to

better understand the system functioning across a wide spec-trum of natural conditions as well as to derive statisticallymeaningful properties Indeed standard investigations ofbiological systems usually rely on computational methodsthat require the execution of a large number of simulationssuch as parameter sweep analysis [3] sensitivity analysis [4]structure and parameter identifiability [5] parameter estima-tion [6ndash8] and reverse engineering of model topologies [9ndash13]

In this context the use of general-purpose GraphicsProcessing Units (GPUs) has recently boosted many applica-tions in scientific computing where CPUs have traditionallybeen the standard workhorses As a matter of fact whenseveral batches of simulations need to be executed the neces-sary computing power can rapidly overtake the capabilities

Hindawi Publishing CorporationBioMed Research InternationalVolume 2014 Article ID 863298 20 pageshttpdxdoiorg1011552014863298

2 BioMed Research International

of standard desktop computers therefore requiring high-performance computing solutions After the introductionof general-purpose GPUs and of Compute Unified DeviceArchitecture (CUDA Nvidiarsquos GPU programming language)the adoption of these graphics engines largely increasedin research fields as Bioinformatics Systems Biology andComputational Biology (see an overview in [14ndash16]) Anywaydespite the remarkable advantages concerning the computa-tional speedup computing with GPUs usually requires thedevelopment and the implementation of ad hoc algorithmssince GPU-based programming substantially differs fromCPU-based computing as a consequence scientific appli-cations of GPUs might undergo the risk of remaining aniche for few specialists To avoid such limitations severalpackages and software tools have recently been released(see eg [16ndash18]) so that also users with no knowledgeof GPUs hardware and programming can access the high-performance computing power of graphics engines

To investigate the dynamics of biological systems eitherdeterministic or stochastic approaches can be exploited [19]which are based on numerical integration (eg Eulerrsquos orRunge-Kutta methods [20]) or on Markov processes (egGillespiersquos algorithm [21]) respectively To date the most effi-cient algorithms to integrate a system of ordinary differentialequations (ODEs) or to perform stochastic simulations ofreaction-basedmodels are LSODA [22] and tau-leaping [23]respectively In [24ndash26] we previously presented cupSODAand cuTauLeaping the GPU-powered implementation ofLSODA and tauleaping respectively cupSODA allows torun parallel deterministic simulations of a given mass-actionbased system of biochemical reactions using the LSODAalgorithm cuTauLeaping represents a novel restructuring ofthe tau-leaping workflow that fits the GPU architecture andavoids any inefficiency drawback for coarse-grain massiveparallel stochastic simulations

In this work we introduce coagSODA an extension ofcupSODA that was specifically designed for the analysis ofa model of the blood coagulation cascade (BCC) Blood is thesubject of an intense scientific research thanks to its key roleinmaking diagnosis of numerous diseases [27] Humans haveevolved a complex hemostatic system that is able to maintainblood in a fluid state and allow the circulation through anintricate network of vessels in particular the presence ofseveral fine-tuned feedback mechanisms in the BCC allowskeeping all blood components within appropriate concen-tration ranges The BCC consists in a complex network ofcellular reactions which under physiological conditions invivo are inhibited by the presence of intact endothelium [28]Anyway in response to any vascular injury the hemostaticsystem is able to stop the blood leakage by rapidly sealing thedefects in the vesselsrsquo wall [29]

In order to investigate the variations in blood coagu-lation components among individuals and to understandthe corresponding response of the system to perturbedconditions we consider here a computational perspectiveto study the BCC we analyze the alterations (prolongationor reduction) of the time required to form the clot (iethe clotting time) by exploiting a reduced version of themathematical model defined in [30] This model describes

the intrinsic extrinsic and common pathways of the BCCand more importantly it accounts for platelets activationas well as the presence of several inhibitors (eg the TissueFactor Pathway Inhibitor antithrombin III and C1-inhibitor)[31]

Numerous mathematical models of blood coagulationwere developed in the last years as they represent a useful toolfor systematic studies of the intricate network of the coagu-lation cascade and allow obtaining a suitable reconstructionof empirical observations (see eg [32ndash34]) The earliestmodels considered only simple steps of the whole BCC suchas the conversion of the clot fibrin by thrombin [35] latelyHockin et al [36] developed a comprehensive ODE-basedmodel of the extrinsic blood coagulation system This modelwas then considered as reference by several research groupsto investigate the thrombotic risk in healthy and ill popula-tions [37 38] or to understand other complex biochemicalprocesses for instance in [39] the roles of protein C proteinS and phospholipid surface actions were considered whilein [40] the influence of trace amounts of key coagulationproteases on thrombin generation was investigated Recentlyother works modeling the blood clotting process in a com-prehensive manner have been published besides the alreadymentioned model developed by Chatterjee et al [30] wemention the model defined by Wajima et al [32] whichsimulates the intrinsic extrinsic and common pathways thevitamin K cycle the therapy with the anticoagulant drugswarfarin and heparin and the laboratory tests PT and aPTTas well as the Taipan snake bite which causes coagulopathy

coagSODA the GPU-accelerated simulator that wepresent and exploit in this work for the analysis of the BCCmodel is a user-friendly and efficient tool that circumventsthe need of manually defining the system of ODEs thatdescribe the blood coagulation network More preciselycoagSODA is able to automatically derive the system of(mass-action and Hill function-based) ODEs and then per-form their numerical integration starting from the given set of96 biochemical reactions which fully describe the molecularinteractions between all the species involved in the BCCin vivo We show that coagSODA allows us to efficientlyexecute a large number of parallel deterministic simulationsof the BCC at a considerable reduced computational costwith respect to CPUs In particular we exploit coagSODAto carry out one-dimensional and bi-dimensional parametersweep analysis of the BCC to the purpose of investigatingthe prolongation and the reduction of the clotting time inresponse to perturbed values of some reaction constants andof the initial concentration of somemolecular species chosenaccording to their meaning within the whole pathway

The paper is structured as follows In Section 2 we fullydescribe the mechanistic model of the BCC used in thiswork and present the simulation method at the basis ofcupSODA tool to introduce the coagSODA simulator InSection 3 we present the results obtained from the parametersweep analysis of the BCC model as well as a comparison ofthe computational performance of coagSODAwith respect toaCPU-based implementation of LSODA Finally in Section 4we conclude the paper with a discussion of the presentedwork and future research directions

BioMed Research International 3

2 Materials and Methods

21 A Mechanistic Model of the Blood Coagulation CascadeBlood is an essential component in human life whoseprimary functions are to feed cells by delivering a multitudeof nutrients such as oxygen and to carry away the cellularwastes such as carbon dioxide Specialized cells and fluidsin blood perform many physiological functions and can beisolated and analyzed through specific laboratory tests givingthe opportunity to settle a personrsquos health condition Allblood components are kept within appropriate concentrationranges by means of fine-tuned regulatory mechanisms ruledby several feedback controls the constancy of blood com-position is maintained thanks to the circulation through anintricate network of vessels In particular humans evolved acomplex hemostatic system that under physiological condi-tions maintains blood in a fluid state however in responseto any vascular injury this system is able to rapidly react andseal the defects in the vesselsrsquo wall in order to stop the bloodleakage [29] Indeed the circulatory system is self-sealingotherwise a continuous blood flow from even the smallestwound would become a threat for the individualrsquos life

To allow blood coagulation in humans there exist 13blood clotting proteins called coagulation factors which areusually designated by Roman numerals I through XIII Asa consequence of a vascular injury platelets become activeand the Tissue Factor (TF also called factor III) is exposedin the subendothelial tissue starting the blood coagulationcascade (BCC) The ultimate goal of the BCC is to convertprothrombin (factor II) into thrombin (factor IIamdashie theactive factor II) the enzyme that catalyzes the formation ofa clot Traditionally the BCC is divided into the extrinsic andintrinsic pathways both leading to the activation of factor X[41] The last part of the cascade downstream of this factoris called the common pathway and leads to the formationof fibrin monomers whose polymers finally constitute thebackbone of the clot

Excluding thrombin all the enzymes involved in bloodclotting are characterized by a low activity which increasesupon binding to a specific protein cofactor (eg factors Vand VIII) or to appropriate phospholipid surfaces (eg theplasmamembranes of active platelets) [41] Even calcium ionshave a central role in coagulation since they are essentialto start and enhance numerous reactions without calciumions the blood coagulation cannot occur [42] In the BCCpathways the activity of the various active proteases is limitedby several inhibitory factors which allow regulating thewhole cascade When the hemostatic system is unregulatedthrombosis (ie the formation of a blood clot obstructing theblood flow in vessels) may occur due to impairment in theinhibitory pathway or because the functioning of the naturalanticoagulant processes is overwhelmed by the strength of thehemostatic stimulus [29]

The BCC model we consider in this work is a slightlyreduced version of the ldquoPlatelet-Plasmardquo deterministic modeldefined in [30] built upon a previous model [36] whichdescribes all parts of blood coagulation the plateletsactivation and aggregation the extrinsic intrinsic andcommon pathways (with the exception of factor XIII)

the action of several inhibitory molecules (Tissue Fac-tor Pathway Inhibitor antithrombin III C1-inhibitor 1205721-antitrypsin and 1205722-antiplasmin) In addition to simulate thecoagulation process in vitro Chatterjee et al also modeledthe action of corn trypsin inhibitor (CTI) which inhibits theactivation of the so-called contact system as well as the actionof the fluorogenic substrate Boc-VPR-MCA widely used inlaboratories for thrombin titration [30] The role of calciumwas not explicitly included in the model but considered asa nonlimiting factor as in living organisms this ion veryrarely drops to such low levels able to alter the kinetics of theformation of the fibrin clot [43]

Since the aim of this work is the investigation of bloodcoagulation in vivo we exclude a small set of reactions givenin [30] that have no effect on the clotting time To be moreprecise we do not consider the reactions occurring in vitro(namely entries 28 and 35 in Table 1 in [30]) as well asthe reactions downstream the fibrinogen conversion that isthe interactions between the fibrin polymers thrombin andantithrombin III (namely entries 55 56 and 57 in Table 1 in[30])

For the sake of completeness we describe in Table 1 theBCC model considered in this work overall consisting in 96reactions among 71 molecular species A graphical sketch ofthemainmolecular interactions among the BCC componentsis given in Figure 1

The model can be partitioned into four functional mod-ules

(1) Thefirstmodule corresponds to the extrinsic pathwaywhich consists in

(i) the formation of a complex between TissueFactor and factor VII (modeled by reactions1199031 119903

4)

(ii) the activation of factor VII by the complexbetween Tissue Factor and factor VIIa (reaction1199035)

(iii) the activation of factor IX by the complexbetween Tissue Factor and factor VIIa (reac-tions 119903

13 11990314 11990315) and by factor VIIa (reactions

11990378 11990379 11990380)

(iv) the activation of factor X by the complexbetween Tissue Factor and factor VIIa(1199038 119903

12) and by factor VIIa (119903

81 11990382 11990383)

(2) The second module corresponds to the intrinsic path-way which consists in

(i) the formation of a complex between factor VIIIaand factor IXa (modeled by reactions 119903

18and

11990319)

(ii) the activation of factor X by the complexbetween factor VIIIa and factor IXa (reactions11990320 11990321 11990322) and by factor IXa (reactions 119903

72 11990373

11990374)

(iii) the activation of factor XII by factor XII itself(reaction 119903

44) by factor XIIa (reactions 119903

45 11990346

11990347) and by kallikrein (reactions 119903

51 11990352 11990353)


Table 1 Reaction-based model of the blood coagulation cascade (reduced version of the ldquoPlatelet-Plasmardquo model described in [30]) Themodel consists in 96 reactions among 71 molecular species With the exception of reaction 119903

44(see Section 31) the values of all reaction

constants were taken from [30]

119899 Reactants Products Constant (119896)1199031

TF + fVII TF-fVII 320 sdot 106Mminus1 sminus1

1199032

TF-fVII TF + fVII 310 sdot 10minus2 sminus1

1199033

TF + fVIIa TF-fVIIa 230 sdot 107Mminus1 sminus1

1199034

TF-fVIIa TF + fVIIa 310 sdot 10minus5 sminus1

1199035

TF-fVIIa + fVII TF-fVIIa + fVIIa 440 sdot 105Mminus1 sminus1

1199036

fXa + fVII fXa + fVIIa 130 sdot 107Mminus1 sminus1

1199037

fIIa + fVII fIIa + fVIIa 230 sdot 104Mminus1 sminus1

1199038

TF-fVIIa + fX TF-fVIIa-fX 250 sdot 107Mminus1 sminus1

1199039

TF-fVIIa-fX TF-fVIIa + fX 105 sdot 10minus2 sminus1

11990310

TF-fVIIa-fX TF-fVIIa-fXa 600 sminus1

11990311

TF-fVIIa-fXa TF-fVIIa + fXa 1900 sminus1

11990312

TF-fVIIa + fXa TF-fVIIa-fXa 220 sdot 107Mminus1 sminus1

11990313

TF-fVIIa + fIX TF-fVIIa-fIX 100 sdot 107Mminus1 sminus1

11990314

TF-fVIIa-fIX TF-fVIIa + fIX 240 sminus1

11990315

TF-fVIIa-fIX TF-fVIIa + fIXa 180 sminus1

11990316

fII + fXa fIIa + fXa 750 sdot 103Mminus1 sminus1

11990317

fIIa + fVIII fIIa + fVIIIa 200 sdot 107Mminus1 sminus1

11990318

fVIIIa + fIXa fIXa-fVIIIa 100 sdot 107Mminus1 sminus1

11990319

fIXa-fVIIIa fVIIIa + fIXa 100 sdot 10minus4 sminus1

11990320

fIXa-fVIIIa + fX fIXa-fVIIIa-fX 100 sdot 108Mminus1 sminus1

11990321

fIXa-fVIIIa-fX fIXa-fVIIIa + fX 100 sdot 10minus5 sminus1

11990322

fIXa-fVIIIa-fX fIXa-fVIIIa + fXa 820 sminus1

11990323

fVIIIa fVIIIa1-L + fVIIIa2 600 sdot 10minus5 sminus1

11990324

fVIIIa1-L + fVIIIa2 fVIIIa 220 sdot 104 Mminus1 sminus1

11990325

fIXa-fVIIIa-fX fVIIIa1-L + fVIIIa2 + fX + fIXa 100 sdot 10minus3 sminus1

11990326

fIXa-fVIIIa fVIIIa1-L + fVIIIa2 + fIXa 100 sdot 10minus3 sminus1

11990327

fIIa + fV fIIa + fVa 200 sdot 107Mminus1 sminus1

11990328

fXa + fVa fXa-fVa 400 sdot 108Mminus1 sminus1

11990329

fXa-fVa fXa + fVa 02 sminus1

11990330

fXa-fVa + fII fXa-fVa-fII 100 sdot 108Mminus1 sminus1

11990331

fXa-fVa-fII fXa-fVa + fII 10300 sminus1

11990332

fXa-fVa-fII fXa-fVa + fmIIa 6350 sminus1

11990333

fXa-fVa + fmIIa fXa-fVa + fIIa 150 sdot 107Mminus1 sminus1

11990334

fXa + TFPI fXa-TFPI 900 sdot 105Mminus1 sminus1

11990335

fXa-TFPI fXa + TFPI 360 sdot 10minus4 sminus1

11990336

TF-fVIIa-fXa + TFPI TF-fVIIa-fXa-TFPI 320 sdot 108 Mminus1 sminus1

11990337

TF-fVIIa-fXa-TFPI TF-fVIIa-fXa + TFPI 110 sdot 10minus2 sminus1

11990338

TF-fVIIa + fXa-TFPI TF-fVIIa-fXa-TFPI 500 sdot 107Mminus1 sminus1

11990339

fXa + ATIII fXa-ATIII 150 sdot 103Mminus1 sminus1

11990340

fmIIa + ATIII fmIIa-ATIII 710 sdot 103Mminus1 sminus1

11990341

fIXa + ATIII fIXa-ATIII 490 sdot 102Mminus1 sminus1

11990342

fIIa + ATIII fIIa-ATIII 710 sdot 103Mminus1 sminus1

11990343

TF-fVIIa + ATIII TF-fVIIa-ATIII 230 sdot 102Mminus1 sminus1

11990344

fXII fXIIa 500 sdot 10minus4 sminus1

11990345

fXIIa + fXII fXIIa-fXII 100 sdot 108Mminus1 sminus1

11990346

fXIIa-fXII fXIIa + fXII 75000 sminus1

11990347

fXIIa-fXII fXIIa + fXIIa 330 sdot 10minus2 sminus1

11990348

fXIIa + PKal fXIIa-PKal 100 sdot 108Mminus1 sminus1

11990349

fXIIa-PKal fXIIa + PKal 360 sdot 103 sminus1


Table 1 Continued


fXIIa-PKal fXIIa + Kal 4000 sminus1

11990351

fXII + Kal fXII-Kal 100 sdot 108Mminus1 sminus1

11990352

fXII-Kal fXII + Kal 4530 sminus1

11990353

fXII-Kal fXIIa + Kal 570 sminus1

11990354

PKal + Kal Kal + Kal 270 sdot 104Mminus1 sminus1

11990355

Kal Kal119894

110 sdot 10minus2 sminus1

11990356

fXIIa + C1inh fXIIa-C1inh 360 sdot 103Mminus1 sminus1

11990357

fXIIa + ATIII fXIIa-ATIII 2160Mminus1 sminus1

11990358

fXI + fIIa fXI-fIIa 100 sdot 108Mminus1 sminus1

11990359

fXI-fIIa fXI + fIIa 500 sminus1

11990360

fXI-fIIa fXIa + fIIa 130 sdot 10minus4 sminus1

11990361

fXIIa + fXI fXIIa-fXI 100 sdot 108Mminus1 sminus1

11990362

fXIIa-fXI fXIIa + fXI 200 sminus1

11990363

fXIIa-fXI fXIIa + fXIa 570 sdot 10minus4 sminus1

11990364

fXIa + fXI fXIa + fXIa 319 sdot 106Mminus1 sminus1

11990365

fXIa + ATIII fXIa-ATIII 320 sdot 102Mminus1 sminus1

11990366

fXIa + C1inh fXIa-C1inh 180 sdot 103Mminus1 sminus1

11990367

fXIa + A1AT fXIa-A1AT 100 sdot 102Mminus1 sminus1

11990368

fXIa + A2AP fXIa-A2AP 43 sdot 103Mminus1 sminus1

11990369

fXIa + fIX fXIa-fIX 100 sdot 108Mminus1 sminus1

11990370

fXIa-fIX fXIa + fIX 4100 sminus1

11990371

fXIa-fIX fXIa + fIXa 770 sminus1

11990372

fIXa + fX fIXa-fX 100 sdot 108Mminus1 sminus1

11990373

fIXa-fX fIXa + fX 064 sminus1

11990374

fIXa-fX fIXa + fXa 700 sdot 10minus4 sminus1

11990375

fXa + fVIII fXa-fVIII 100 sdot 108Mminus1 sminus1

11990376

fXa-fVIII fXa + fVIII 210 sminus1

11990377

fXa-fVIII fXa + fVIIIa 0023 sminus1

11990378

fVIIa + fIX fVIIa-fIX 100 sdot 108Mminus1 sminus1

11990379

fVIIa-fIX fVIIa + fIX 090 sminus1

11990380

fVIIa-fIX fVIIa + fIXa 360 sdot 10minus5 sminus1

11990381

fVIIa + fX fVIIa-fX 100 sdot 108Mminus1 sminus1

11990382

fVIIa-fX fVIIa + fX 21000 sminus1

11990383

fVIIa-fX fVIIa + fXa 160 sdot 10minus6 sminus1

11990384

Fbg + fIIa Fbg-fIIa 100 sdot 108Mminus1 sminus1

11990385

Fbg-fIIa Fbg + fIIa 63600 sminus1

11990386

Fbg-fIIa Fbn1 + fIIa + FPA 8400 sminus1

11990387

Fbn1 + fIIa Fbn1-fIIa 100 sdot 108Mminus1 sminus1

11990388

Fbn1-fIIa Fbn1 + fIIa 74260 sminus1

11990389

Fbn1-fIIa Fbn2 + fIIa + FPB 740 sminus1

11990390

Fbn1 + Fbn1 (Fbn1)2 100 sdot 106Mminus1 sminus1

11990391

(Fbn1)2 2Fbn1 640 sdot 10minus2 sminus1

11990392

(Fbn1)2 + fIIa (Fbn1)2-fIIa 100 sdot 108Mminus1 sminus1

11990393

(Fbn1)2-fIIa (Fbn1)2 + fIIa 70100 sminus1

11990394

(Fbn1)2-fIIa (Fbn2)2 + fIIa + FPB 4900 sminus1

11990395


11990396

Fbn2-fIIa Fbn2 + fIIa 100 sdot 103 sminus1


fXIITF-fVII

TFfVIIa

fVII

TF-fVIIa

Fbg

FPA

Fbn1

FPB

Fbn2

Fbn1

(Fbn1)2

(Fbn2)2 FPB

fXa

fIIa

TF-fVIIa-fXa

fIX

fIXa

fX

fmIIa

fII

PKal

KalifXI

fXIa

fIX

fIXa

fVIII

fVIIIa

fIXa-fVIIIa

fVIIIa1 -L

fVIIIa2

fVIIIa2

fIXa

fV

fVa

fXa-fVa

fXa-TFPI

TFPI

TF-fVIIa-fXa-TFPI

fXa-AT

AT

AT

fXIa-AT

C1inh

fXIIa-C1inh

A1AT

A2AP

fXIa-A2APfXIa-A1AT

TF-fVIIa

fXIIa Kal

fVIIIa1 -L

Figure 1 Graphical representation of the blood coagulation cascade model considered in this work Legend Blue box coagulation factor redbox inhibitor and related complexes Black arrow complex formation green arrow catalytic activation violet arrow activation red arrowinhibition The reaction is reversible if the arrow tail consists in a dot


(iv) the activation of prekallikrein by factor XIIa(reactions 119903

48 11990349 11990350) and by kallikrein (reac-

tion 11990354)

(v) the activation of factor XI by factor XIIa (reac-tions 119903

61 11990362 11990363) and by factor XIa (reaction

11990364)

(vi) the activation of factor IX by factor XIa (reac-tions 119903

69 11990370 11990371)

(vii) the dissociation of free factor VIIIa (reactions11990323and 11990324)

(viii) the dissociation of factor VIIIa in complex withother factors (reactions 119903

25and 11990326)

(3) The third module corresponds to the common path-way which consists in

(i) the activation of factor II by factor Xa (modeledby reaction 119903

16) and by the complex between

factor Xa and factor Va through the formationof the intermediate meizothrombin (reactions11990330 119903

33)

(ii) the activation of factor VII by factor Xa (reac-tion 1199036) and by factor IIa (reaction 119903

7)

(iii) the activation of factor VIII by factor Xa (reac-tions 119903

75 11990376 11990377) and by factor IIa (reaction 119903

17)

(iv) the formation of a complex between factor Xaand factor Va (reactions 119903

28and 11990329)

(v) the activation of fibrinogen by factor IIa (reac-tions 119903

84 119903

96)

(vi) the activation of factor V by factor IIa (reaction11990327)

(vii) the activation of factor XI by factor IIa (reac-tions 119903

58 11990359 11990360)

(4) The fourth module describes the inhibition of coagu-lation carried out through the main inhibitors of theBCC (antithrombin Tissue Factor Pathway InhibitorC1-inihibitor 1205721-antitrypsin and 1205722-antiplasmin)These are tight binding inhibitors belonging to theserpin superfamily which form irreversible com-plexes This module consists in

(i) the inhibition of factor Xa by Tissue FactorPathway Inhibitor (modeled by reactions 119903

34and

11990335)

(ii) the inhibition of the complex between TissueFactor factor VIIa and factor Xa by TissueFactor Pathway Inhibitor (reactions 119903

36 11990337 11990338)

(iii) the inhibition of factor IIa by antithrombin(reactions 119903

40and 11990342)

(iv) the inhibition of factor Xa by antithrombin(reaction 119903

39)

(v) the inhibition of factor IXa by antithrombin(reaction 119903

41)

(vi) the inhibition of factor XIa by antithrombin(reaction 119903

65) and by C1-inhibitor (reaction 119903

66)

Table 2 Initial concentrations of molecular species in the bloodcoagulation cascade model (values taken from [30])

Species Symbol Concentration (M)1205721-Antitrypsin A1AT 450 sdot 10

minus5

1205722-Antiplasmin A2AP 100 sdot 10minus6

Antithrombin III ATIII 340 sdot 10minus6

C1-inhibitor C1inh 250 sdot 10minus6

Fibrinogen Fbg 900 sdot 10minus6

Factor II fII 140 sdot 10minus6

Factor V fV 200 sdot 10minus8

Factor VII fVII 100 sdot 10minus8

Active factor VII fVIIa 100 sdot 10minus10

Factor VIII fVIII 700 sdot 10minus10

Factor IX fIX 900 sdot 10minus8

Factor X fX 16 sdot 10minus7

Factor XI fXI 310 sdot 10minus8

Factor XII fXII 340 sdot 10minus7

Prekallikrein Pkal 450 sdot 10minus7

Tissue Factor TF 500 sdot 10minus12

Tissue Factor Pathway Inhibitor TFPI 250 sdot 10minus9

(vii) the inhibition of factor XIIa by C1-inhibitor(reaction 119903

56) by antithrombin (reaction 119903

57)

by 1205721-antitrypsin (reaction 11990367) and by 1205722-

antiplasmin (reaction 11990368)

(viii) the inhibition of the complex between TissueFactor and factor VIIa by antithrombin (reac-tion 11990343)

(ix) the inhibition of kallikrein (reaction 11990355)

The values of the initial concentrations of the molecularspecies occurring in the BCC model are given in Table 2According to [30] the concentrations of complexes and activefactors were set to 0 except for the active factor VII which isphysiologically present in the blood circulation even in theabsence of damage in a concentration that is approximatelyequal to 1 of the corresponding inactive factor [42]

The system of ordinary differential equations (ODEs)needed to carry out the simulations and the parametersweep analysis presented in Section 3 was derived fromthe reactions given in Table 1 according to the mass-actionlaw with the exception of 14 reactions belonging to the set119878120576= 11990319 11990321 11990323 11990329 11990331 11990335 11990346 11990349 11990352 11990362 11990370 11990373 11990376 11990382

The mass-action law is the fundamental empirical law thatgoverns biochemical reaction rates which states that in adilute solution the rate of an elementary reaction is propor-tional to the product of the concentration of its reactantsraised to the power of the corresponding stoichiometriccoefficient [44] The reactions in 119878

120576are instead influenced by

a specifically defined variable 120576 depending on aHill functionfit against experimental data which quantify the plateletactivation status which is used to model the physiologicallevels of thrombin concentration as a function of plateletactivation as thoroughly described in [30] More preciselythe value of 120576 influences the formation of some complexes


0 0 100 200 300 400 500 600 700

Fact

or II

a con

cent

ratio

n (M

)

Time (s)

1eminus7

2eminus7

3eminus7

4eminus7

5eminus7

6eminus7

7eminus7

8eminus7

9eminus7

Figure 2 Dynamics of thrombin (factor IIa) in physiological condition

occurring on the plateletsrsquo surface by modifying the activityof reactions of the form

119860 + 119861

1198961

999445999468

1198962

119860119861 (1)

Namely 120576 intervenes with the dissociation constant of thesereactions so that the corresponding standard ODEs arechanged to yield new equations of the form

119889 [119860119861]

119889119905

= 1198961sdot [119860] sdot [119861] minus

1198962sdot [119860] sdot [119861]

120576

(2)

The value of 120576 in (2) depends on the following HillfunctionH which quantifies the state of platelets activationaccording to the thrombin concentration (here denoted as[fIIa]) that is the factor catalyzing the formation of the fibrinclot

H ([fIIalowast (119905)]) =[fIIalowast (119905)]16123

[fIIalowast (119905)]16123 + (24279 sdot 10minus9)16123

(3)

where [fIIalowast(119905)] = max1199051015840isin[0119905][fIIa(1199051015840)] for a chosen time

interval [0 119905] of simulationThe value [fIIalowast(119905)] represents the maximum transient

thrombin concentration and is needed to simulate the factthat in physiological conditions the thrombin concentrationstarts decreasing after rising to a peak (Figure 2) So doing[fIIalowast(119905)] never decreases once it reaches its maximummagnitude [fIIalowast(119905)] is equivalent to [fIIa(119905)] until the con-centration of factor IIa reaches the peak while thereafter itremains constant at that value which is the maximum in theconsidered time interval [0 119905] FunctionH allows simulatingthe physiological condition whereby platelets remain activealso when the thrombin concentration decreases

For a given concentration of factor IIa the maximumplatelets activation state 120576max is defined as

120576max = 120576max0

+ (1 minus 120576max0

) sdotH ([fIIalowast (119905)]) (4)

where 120576max0

defines the basal activation state of the plateletsat simulation time 119905 = 0 The value 120576max

0

is initially set to 001assuming a basal 1 binding strength of coagulation factorsto the resting plateletsrsquo surface When the full activationof platelets is reached 120576max is equal to 1 and the complexdissociation constants are minimized (see [30] for moredetails)

22 CUDA Architecture and the coagSODA Simulator Intro-duced by Nvidia in 2006 Compute Unified Device Architec-ture (CUDA) is a parallel computing platform and program-ming model that provides programmers with a frameworkto exploit GPUs in general-purpose computational tasks(GPGPU computing) GPGPU computing is a low-cost andenergy-wise alternative to the traditional high-performancecomputing infrastructures (eg clusters of machines) whichgives access to the tera-scale computing on common work-stations of mid-range price However due to the innovativearchitecture and the intrinsic limitations of GPUs a directporting of sequential code on the GPU is most of the timesunfeasible therefore making it challenging to fully exploit itscomputational power and massive parallelism [45]

CUDA combines the single instruction multiple data(SIMD) architecture with multithreading which automat-ically handles the conditional divergence between threadsThe drawback of such flexibility is that any divergence ofthe execution flow between threads causes a serialization ofthe execution which affects the overall performances UnderCUDArsquos naming conventions the programmer implementsthe kernel that is a CC++ function which is loaded fromthe host (the CPU) to the devices (one or more GPUs)and replicated in many copies named threads Threads canbe organized in three-dimensional structures named blockswhich in turn are contained in three-dimensional grids(a schematic description is given in Figure 3 left side)Whenever the host runs a kernel the GPU creates thecorresponding grid and automatically schedules each blockon one free streaming multiprocessor available on the GPUallowing a transparent scaling of performances on different


Host

Kernel 1

Kernel 2

Device

Grid 1

Grid 2

Block(0 0)

Block(1 0)

Block(2 0)

Block(0 1)

Block(1 1)

Block(2 1)

Block (1 1)

Thread(0 0)

Thread(1 0)

Thread(2 0)

Thread(3 0)

Thread(4 0)

Thread(0 1)

Thread(1 1)

Thread(2 1)

Thread(3 1)

Thread(4 1)

Thread(0 2)

Thread(1 2)

Thread(2 2)

Thread(3 2)

Thread(4 2)

Global memory

Texture memory

Constant memory

Shared memory Shared memory

Block (0 0) Block (1 0)

Localmemory

Localmemory

Localmemory

Localmemory

Thread(0 0)

Thread(1 0)

Registers Registers Registers Registers

Thread(0 0)

Thread(1 0)

Grid

Figure 3 Schematic description of CUDArsquos architecture in terms of threads and memory hierarchy Left Side Threads organization a singlekernel is launched from the host (the CPU) and is executed in multiple threads on the device (the GPU) Threads can be organized in three-dimensional structures named blocks which can be in turn organized in three-dimensional grids The dimensions of blocks and grids areexplicitly defined by the programmer Right Side Memory hierarchy threads can access data from many different memories with differentscopes Registers and local memories are private for each thread Shared memory lets threads belonging to the same block communicate andhas low access latency All threads can access the global memory which suffers high latencies but it is cached since the introduction of theFermi architecture Texture and constant memory can be read from any thread and are equipped with a cache as well in this work we exploitthe constant memory Figures are taken from Nvidiarsquos CUDA programming guide [60]

devices Threads within a block are executed in groups of 32threads named warps

The GPU is equipped with different kinds of memoryIn this work we exploit the global memory (accessible fromall threads) the shared memory (accessible from threadsof the same block) the local memory (registers and arraysaccessible from owner thread) and the constant memory(cached and not modifiable) A schematic representation ofthis memory hierarchy is shown in Figure 3 right side Toachieve the best performances the sharedmemory should beexploited as much as possible however it is very limited (ie49152 bytes for each multiprocessor since the introductionof the Fermi architecture) and it introduces restrictions onthe blocksrsquo size On the other hand the global memory isvery large (thousands ofMBs) but suffers from high latenciesA solution to this problem was implemented on the Fermiarchitecture where the global memory is equipped with aL2 cache This architecture also introduced the possibility tobalance 64KB of fast on-chip memory between the sharedmemory and L1 cache using two possible configurations48KB for the shared memory and 16KB for L1 cache or16 KB for the shared memory and 48 for L1 cacheThe Keplerarchitecture used in this paper allows a third and perfectly

balanced configuration where shared memory and L1 cacheobtain the same amount of memory (32KB) See Figure 4 fora schematization of the memory architecture

The systematic analysis of models of biological systemsoften consists in the execution of large batches of simulationsOne of the standard analyses that can be executed on suchkind of models regards an intensive search within the param-eters space which requires large numbers of independentsimulations In order to reduce the computational burdenwe previously implemented on the CUDA architecture oneof the most efficient numerical integration algorithms forODEs LSODA that is able to automatically recognize stiffand nonstiff systems and to dynamically select between themost appropriate integration procedure (ie Adams methodin the absence of stiffness and the Backward DifferentiationFormulae otherwise) [22] This GPU-powered tool is calledcupSODA [24 25] and exploits CUDArsquos massive parallelismto execute different and independent simulations in eachthread thus reducing the computational time required by astandard CPU counterpart of LSODA

Besides being very efficient for the simulation of manyindependent simulations cupSODA is also user-friendlyLSODA was originally designed to solve ODEs systems


Thread

Shared memory L1 cache Read-onlydata cache

L2 cache

DRAM

Figure 4 Schematic description of memory hierarchies in Fermi and Kepler architectures GPUs relying on these architectures are equippedwith a two-level data cache and a read-only data cache Shared memory and L1 cache share the same on-chip 64KB memory banks theamount of memory can be reconfigured by the user according to the specific needs of the application Figure taken from Nvidiarsquos KeplerGK110 whitepaper [57]

written in the canonical form but the user is supposed tospecify the system of ODEs by implementing a custom Cfunction that is passed to the algorithmmoreover in order tospeed up the computation when dealing with stiff systems inLSODA the Jacobianmatrix associatedwith theODEs systemmust be implemented as a custom C function as well On thecontrary cupSODA was conceived as a black-box simulatorthat can be easily used without any programming skillscupSODA consists in a tool to automatically convert a genericmechanistic reaction-based model of a biological system intothe corresponding set of ODEs to comply with the mass-action kinetics [46] and to directly encode the obtainedsystem along with the corresponding Jacobian matrix as Carrays

This fully automaticmethodology can be exploited for thesimulation of models whose chemical kinetics is based on themass-action law The BCC model described in Section 21though includes a set 119878

120576of reactions that do not follow

the mass-action kinetics [30] the platelets activity is notexplicitly modeled by biochemical reactions but it is realizedby modulating the rate of the dissociation of the complexesformed on a plateletrsquos surface by means of the variable120576 which is calculated with a special equation during theintegration steps For this reason cupSODA was used asa starting point for the development of a new tool ableto compute at run-time the specific kinetics of this set ofreactions

This new CUDA-powered simulation tool named coag-SODA is specifically tailored for the simulation of the BCC

model developed in [30] In particular coagSODA realizesthe run-time calculation of the 120576 value required to correctlysimulate the activity of platelets which is determined accord-ing to the equations described in Section 21 The plateletsactivation state 120576 is calculated at each time instant by solvingthe following differential equation

119889120576

119889119905

= 119896 (120576max minus 120576) (5)

where the constant 119896 is inversely proportional to the timescale of platelets activation and is set to 0005 This isconsistent with the fact that platelets do not instantly achievetheir maximum attainable activation state (120576max) but theyreach it on a physiologically relevant timescale [30]

Dealing with the 14 reactions in the set 119878120576that are

influenced by the Hill functionH (see Section 21) the valueof [fIIalowast(119905)]must be stored on the GPU because during eachintegration step coagSODA recalculates (4) which exploitsthe value [fIIalowast(119905)] to determine a new 120576 value Since CUDArsquosarchitecture does not offer static variables the informationfor each thread has to be memorized in the global memoryThe accesses to the global memory and the computationalcosts due to these additional calculations slow down theintegration process with respect to the integration of ODEsperformed according to strictly mass-action based kinetics(as in the cupSODA simulator) Nevertheless in Section 33we show that our parallel implementation largely outper-forms a sequential counterpart of the LSODA algorithm


0 100 200 300 400 500 600 700

Fibr

inog

en co

ncen

trat

ion

(M)

Time (s)

30 of fibrinogen

0

1eminus6

2eminus6

3eminus6

4eminus6

5eminus6

6eminus6

7eminus6

8eminus6

9eminus6

Figure 5 Dynamics of fibrinogen in physiological condition The clotting time is defined as the time necessary to convert the 70 of thefibrinogen into fibrin

As in the case of cupSODA [24 25] coagSODA exploitsthe shared memory to improve performances by storing thecurrent state and time of the simulations in these low-latencymemory banks Despite the improvement of performancesensured by this solution it strongly affects the occupancyof GPUrsquos multiprocessors and therefore it represents thelimiting factor for the number of blocks that can be executedsimultaneously as a matter of fact coagSODA is limited to2 blocks per streaming multiprocessor on GPUs based onthe Fermi architecture with a reduced exploitation of theGPU with respect to the theoretical 8 blocks allowed by thisarchitecture

3 Results and Discussion

In this section we discuss the results of the parameter sweepanalysis (PSA) carried out on the BCC model to the aimof investigating either the prolongation or the reduction ofthe clotting time in response to perturbed values of somereaction constants and of the initial concentration of somemolecular species chosen according to their meaning withinthe whole pathway PSA was performed by generating aset of different initial conditions corresponding to differentparameterizations of the model and then automaticallyexecuting the deterministic simulations with coagSODATheuse ofGPU technology is fundamental in this type of analysisespecially for large biological systems as the BCC is becauseit drastically reduces the computational time

The sweep analysis for single parameters (PSA-1D) wasperformed considering a logarithmic sampling of numericalvalues of each parameter under investigation (reaction con-stant or initial molecular concentration) within a specifiedrange with respect to its physiological reference value Thesweep analysis over pairs of parameters (PSA-2D) was per-formed by simultaneously varying the values of two param-eters within a specified range considering a logarithmicsampling on the resulting lattice The logarithmic samplingallows uniformly spanning different orders of magnitude ofthe parameters value using a reduced and fine-grained set

of samples therefore efficiently analyzing the response of thesystem in a broad range of conditions

To determine the response of the BCC to perturbed con-ditions we chose the clotting time (CT) as output of the PSAThe CT is defined as the time necessary to convert the 70of the fibrinogen into fibrin (see Figure 5) conventionallyassumed to correspond to the time required to form the clot[47] and it is generally used in laboratory tests formonitoringthe therapywith anticoagulant drugs According to themodeldefined in [30] the reference value of CT is around 300seconds in physiological conditions We investigate herethe response of the BCC by evaluating the CT in variousconditions corresponding to different values of the reactionconstants varying over six orders of magnitude with respectto their physiological values (ie three below and three abovethe reference values if not otherwise specified) as well asto different values of the initial molecular concentrationsand varying over twelve orders of magnitude with respect totheir physiological values (ie six below and six above thereference values if not otherwise specified)

The total number of parallel simulations executed tocarry out these analyses was 100 for PSA-1D over reactionconstants 200 for PSA-1D over initial concentrations and1600 for PSA-2D

Finally we present the comparison of the performanceof the CPU and GPU to run an increasing number ofsimulations of the BCC model to prove the efficiency ofcoagSODA

31 PSA-1D of the BCC Model

311 Reaction 11990344 Thefirst PSA was performed to determine

the value of the kinetic constant for the autoactivation offactor XII (reaction 119903

44in Table 1) which corresponds to an

upstream process in the intrinsic pathway This analysis wasmotivated by two considerations Firstly by using the fullparameterization given in [30] the action of the intrinsicpathway turns out to be fundamental for the BCC in vivowhich is in contrast to experimental observations which


324

326

328

330

332

334

336

338

340

Clot

ting

time (

s)

1eminus

10

1eminus

9

1eminus

8

1eminus

7

1eminus

6

1eminus

5

1eminus

4

1eminus

3

k44 (sminus1)

Figure 6 Clotting time according to PSA-1D over constant 11989644of reaction 119903

44 XII rarr XIIaThe reference value of 119896

44used in [30] is 5sdot10minus4sminus1

the sweep range is [5 sdot 10minus10 5 sdot 10minus4]

280

300

320

340

360

380

400

1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)

k27 (Mminus1sminus1)


27 IIa + V rarr IIa + Va The reference value is 2 sdot 107Mminus1 sminus1 the

sweep range is [2 sdot 104 2 sdot 1010]

indicate that the extrinsic pathway is the main responsibleof clot formation Secondly in [30] all constants values havea reference to experimental measurements except for theconstant of this reaction (which corresponds to entry 29 inTable 1 in [30])

Figure 6 shows the results of this PSA-1D where 11989644was

varied over six orders of magnitude considering the valuegiven in [30] as the upper limit of the sweep interval We canobserve that by decreasing the value of 119896

44with respect to

the value considered in [30] the CT increases with respect toits reference value however for values of this constant lowerthan 100 sdot 10minus6 sminus1 the CT remains unaltered and this canbe intuitively explained by the fact that in this situation thefibrinogen is mainly activated by the extrinsic pathway

Consequently we assigned the value 700 sdot 10minus6 sminus1 to11989644 achieving a CT that is comparable to the experimental

observations of the BCC in vivo This new value was used inall PSA discussed in what follows

312 Reactions 11990327

and 11990358 In the next PSA we investigated

the effect of the perturbation of the kinetics of two pivotal

reactions of the BCC model Reaction 11990327 which describes

the catalytic activation of factor V by factor IIa was chosenbecause it represents the main positive feedback within thecommon pathway reaction 119903

58 which is involved in the

activation of factor XI by binding to factor IIa was chosenbecause it represents the main positive feedback in theintrinsic pathway (Figure 1) Moreover preliminary PSA overall reaction constants of the BCC model given in Table 1evidenced that these two reactions are among the mostrelevant steps of the coagulation network

The PSA-1D over 11989627(Figure 7) shows that the CT is very

sensitive to the perturbation of the rate of this reaction whenthe reference value of its constant (ie 119896

27= 2 sdot 10

7Mminus1 sminus1)is either increased or decreased in particular when 119896

27is

very low a plateau in CT is reached since the strength ofthe positive feedback exerted by factor IIa is largely reduceda condition where the contribution of the amplification ofthe hemostatic stimulus (due by the common pathway) tothe formation of the clot is basically not effective On theother side the PSA-1D over 119896

58(Figure 8) shows that while

a decrease of the constant with respect to its reference value


310

320

330

340

350

360

370

1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)



58 IIa + XI rarr IIa-XI The reference value is 1 sdot 108Mminus1 sminus1 the


220

240

260

280

300

320

340

360

380

400

Clot

ting

time (

s)

Factor VIII (M)

Seve

re h

aem

ophi

lia

Mild

hae

mop

hilia

Prot

hrom

botic

stat

e

1eminus

16

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

Figure 9 Clotting time at different initial concentrations of factor VIII The reference value is 700 sdot 10minus10M (dashed black line) the sweeprange is [700 sdot 10minus16 700 sdot 10minus4] The blue area indicates a condition in which factor VIII concentration is less than 1 of its physiologicalconcentration corresponding to a situation of severe haemophilia the red area indicates a condition in which factor VIII concentration isbetween 1 and 30 of its physiological concentration corresponding to a situation of mild haemophilia the green area indicates a conditionin which factor VIII concentration is greater than 130 of its physiological concentration corresponding to a situation of hypercoagulability

(ie 11989658= 1 sdot 10

8Mminus1 sminus1) does not have any substantialeffect an increase of its value leads to a progressive increasein the CT This increase is due to the fact that factor IIa issequestered in the formation of a complex with factor XI andhence it is no longer available as a free component in bloodto participate in other reactions especially those reactionsof the extrinsic pathway which principally lead to the clotformation in vivo This behavior highlights that the intrinsicpathway has a secondary role in blood coagulation in vivocompared with the extrinsic pathway as also evidenced byvarious experimental observations [48]

313 Factors VIII IX and II The next set of PSA-1D wasrealized by varying the initial concentrations of factors VIIIIX and II These factors were selected since both an excess

and a deficiency of their concentrations lead to diseasesrelated to blood clotting

The PSA-1D over factor VIII (Figure 9) shows thatincreasing the initial concentration of this factor resultsin decreasing the CT suggesting the possible presence ofhypercoagulable states in these perturbed conditions As amatter of fact high levels of factor VIII cause an increasedrisk of deep vein thrombosis and pulmonary embolism[49] On the other hand individuals with less than 1of the average concentration of factor VIII show a severehaemophilia A characterized by higher CT (Figure 9) andrequire infusions of plasma containing the deficient factorotherwise frequent spontaneous bleeding would occur [50]When the concentration of this factor is between 5 and 30of the average concentration individuals still risk bleeding incase of trauma [50]


250

300

350

400

450

500

550

600

650

700

Clot

ting

time (

s)

Factor IX (M)

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0

Figure 10 Clotting time at different initial concentrations of factor IXThe reference value is 900sdot10minus8M the sweep range is [900sdot10minus14 900sdot10minus2]

0

100

200

300

400

500

600

700

Clot

ting

time (

s)

Factor II (M)

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0

1e2

Figure 11 Clotting time at different initial concentrations of thrombinThe reference value is 140sdot10minus6M the sweep range is [140sdot10minus12 140sdot100]

The PSA-1D over factor IX (Figure 10) shows only aslight decrease of the CT as the initial concentration offIX increases this is in contrast to recent studies thatdemonstrated how the excess of factor IX leads to anincreased risk of deep vein thrombosis [51] This result canbe explained by considering that (i) a high concentrationof factor IX is not sufficient to bring about coagulationproblems though when the concentration of other factorsis above the average value (yet not at pathological levels)prothrombotic states can be observed (ii) in this model weconsider average values as initial concentrations of factorshowever individuals are characterized by different (balanced)combinations of procoagulant and anticoagulant factor levelsthat altogether contribute to define a unique coagulationphenotype that reflects the developmental environmentalgenetic nutritional and pharmacological influences of eachindividual [52] On the contrary the lack of factor IX causes

haemophilia B characterized by higher CT with respect tothe reference value (Figure 10)

Furthermore by comparing the PSA-1D of factors VIIIand IX (Figures 9 and 10) it is clear that haemophilia A ismore serious than haemophilia B since the CT achieved inconditions of factor VIII deficiency is higher than the CTobtained in the case of factor IX deficiency

In both PSA-1D over factors VIII and IX we observedafter the initial decrease of the CT an unexpected increaseof the CT as the factor concentration increasesThis counter-intuitive behavior arises at very high concentrations of thesefactors (with respect to the average physiological levels) andto the best of our knowledge it was never observed in vivoNonetheless it would be interesting to verify by means of adhoc laboratory experiments if the model correctly describesthe behavior of the BCC even in these conditions or on thecontrary it is not predictive in these extreme situations


The PSA-1D over factor II (Figure 11) shows a dramaticdecrease of the CT as the initial concentration of this factorincreases (with respect to the average physiological level)This behavior resembles the effects of hypercoagulability (orthrombophilia) a disease caused by mutation G20210A inthe prothrombin gene [53] that causes an increase of theprothrombin level (factor II) in the blood flow resulting inan excessive formation of the active form of this factor thusheightening venous thrombosis risks [54] Hypercoagulabil-ity is usually treated with warfarin therapy or with otheranticoagulants with a similar effect These drugs decrease thecapacity of coagulation factors to become active preventingthe formation of unwanted thrombi

On the other hand when the initial concentration offactor II is low we achieved the effects of prothrombindeficiency a rare autosomal recessive disease that causes atendency to severe bleeding [29 55] As shown in Figure 11 aconcentration equal to 10of the physiological value of factorII (ie 14 sdot 10minus6 M) leads to clotting effects similar to severehaemophilia A

32 PSA-2D of the BCCModel We present here the results ofthe PSA-2D on the BCCmodel where couples of parameterswere varied to analyze the possible effects arising from thecombined perturbation of their values


and 11990358 Figure 12 shows the effect of

the simultaneous variation of constants 11989627

and 11989658

(overthe same sweep ranges considered in the two PSA-1DSection 31) This result remarks that reaction 119903

27 involved

in the common pathway has a stronger influence on theBCC and that there is a synergic interplay between thesetwo reactions In particular when the value of 119896

27is low and

the value of 11989658

is high the CT is higher than the valuesachieved when only a single constant is changed because inthis condition both the intrinsic and the common pathwaysare simultaneously inhibited

322 Factor VIII Factor IX and Tissue Factor In the last twoPSA-2Dwe varied the initial concentration of factor VIII andTissue Factor and the initial concentration of factor IX andTissue Factor respectively

The initial concentrations of factors VIII and IX were var-ied over four orders of magnitude using their physiologicalvalues as upper limit for the sweep ranges the concentrationof Tissue Factor was varied over four orders of magnitudetwo above and two below its reference value (see Table 2)The rationale behind this choice is to observe how the BCCmodel in conditions corresponding to different states ofhaemophilia (obtained by decreasing the concentrations offactors VIII and IX) behaves with different initial concentra-tions of the Tissue Factor which is the upstream factor of theextrinsic pathway that is the most important element of theBCC

The results of these PSA-2D show that with respect tothe condition of haemophilia B in the case of haemophiliaA the amount of Tissue Factor (below its reference value)has a negligible influence on the CT as indicated by the

presence of a plateau in Figure 13 on the contrary concerninghaemophilia B a deficiency of Tissue Factor leads to anincrease of the CT especially when factor IX is present in lowconcentrations (Figure 14)

The different results achieved in the two PSA-2D are dueto the presence in the BCC model of a direct interactionbetweenTissue Factor and factor IX bymeans of theTF-fVIIacomplex (see reactions 119903

13 119903

15in Table 1) Indeed the lack

of Tissue Factor directly affects the concentration of activefactor IX which results in a strong alteration of the CT withrespect to physiological conditions

33 CPU versus GPU Performance Comparison In orderto show the relevant speedup achieved by coagSODA wepresent here the comparison of the computational effortrequired by GPU and CPU for the simulation of the BCCmodel The performances of our GPU simulator were com-pared with those obtained using the LSODA algorithmimplemented in the software COPASI [56] executing onthe CPU the same set of simulations that were run on theGPU COPASI is single-threaded and does not exploit thephysical and logical cores of the CPU therefore it representsa good benchmark as a single-node CPU-bound simulator ofbiological systems

In all simulations executed on both GPU and CPU westored 100 samples uniformly distributed in the time intervalconsidered for each simulation that is [0 700] seconds of thedynamics of all chemical species involved in the BCCmodelThe settings for the LSODA algorithm were the followingrelative error 1 sdot 10minus7 absolute error 1 sdot 10minus13 and maximumnumber of internal steps set to 20000

331 Benchmark Details The GPU used for the simulationsis a Nvidia Tesla K20c equipped with 2496 cores organized in13 streamingmultiprocessors GPU clock 706MHz and 5GBofDDR5RAM In all tests coagSODAwas compiled and exe-cuted with version 55 of CUDA libraries Even though thisGPU has compute capability 35 and is based on the GK110Kepler architecture currently coagSODA does not exploitany of the new features (eg Dynamic Parallelism Hyper-QRemote DMA) with the exception of the new ISA encodingwhich allows threads to exploit an increased number ofregisters (255 instead of 63) reducing register spilling intoglobal memory and increasing performances [57] Moreoveras described in Section 22 the Kepler architecture offers thepossibility to reconfigure the 64KB on-chip cache balancingbetween L1 cache and shared memory Since coagSODAexploits the shared memory to reduce the latencies duringthe access to the concentration values of the BCC model weopted for the following configuration 48KB for the sharedmemory 16 KB for the L1 cache

The performance of the GPU was compared against apersonal computer equippedwith a dual-core CPU Intel Corei5 frequency 23 GHz 4GB of DDR3 RAM running theoperating systemMacOSXLion 1075The software used as acoagSODA-equivalent single-threaded CPU implementationis COPASI version 48 (build 35)


280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )


Figure 12 Clotting time according to PSA-2D over reaction constants 11989627and 119896

58 The sweep ranges are [2 sdot 104 2 sdot 1010] and [1 sdot 105 2 sdot 1011]

respectively

100

200

300

400

Tissue factor (M) Factor VIII

(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10

Figure 13 Clotting time at different initial concentrations of Tissue Factor and factor VIII The sweep ranges are [500 sdot 10minus14 500 sdot 10minus10]and [700 sdot 10minus14 500 sdot 10minus10] respectively

A direct comparison of the performances between thesetwo different architectures and implementations is not aneasy task since CPUs are optimized for single-thread exe-cution and exploit a large number of optimizations (eghigher clock frequency instruction caching pipelining outof order execution and branch prediction) whereas GPUsare optimized for graphics processing and parallel exe-cution of identical code relying only on multilevel datacaching For this reason in this paper we propose anempirical comparison of the performances based on therunning times of identical batches of simulations of the BCCmodel

332 Benchmark Results In Figure 15 we show the compar-ison of the running times required to run several batchesof simulations executed to carry out the PSA-1D over thereaction constant 119896

27 The choice of comparing the perfor-

mances of the GPU and CPU implementations by executingbatches of simulations that are related to a PSA instead ofrunning 119899 independent but identical simulations (ie allcharacterized by the same parameterization of the model)is due to the fact that these results represent a more realisticscenario in the computational analysis of biological systemswhereby it is useful to investigate large search spaces ofparameters corresponding to different perturbed conditionsof themodel [26]Moreover for the evaluation of the running

time the execution of a batch of 119899 identical deterministicsimulations would be futile The figure clearly shows thatcoagSODA always performs better than the CPU counter-part In particular while the CPU performance increaseslinearly with the number of simulations the running timesare strongly reduced on the GPU in this case the overallrunning time roughly corresponds to the time required forthe execution of the slowest simulations This is due to thefact that different parameterizations may require differenttime steps for LSODA to converge and the execution ofa block terminates as long as all the threads it containsterminate In turn the execution of a kernel terminates whenall blocks terminate for this reason a single simulationwhose dynamics requires more steps than the others mayaffect the overall running time

In Table 3 we report the running times of all batchesof simulations along with the speedup achieved on theGPU In particular these results highlight that the advantageof exploiting the GPUs for the simulation of the BCCmodel becomes more evident as the number of simula-tions increases with a 181times speedup when a PSA with10000 different parameterizations is executed Therefore theGPU accelerated analysis of the BCC model with coag-SODA represents a novel relevant computational means toinvestigate the behavior of this complex biological systemunder nonphysiological conditions and could be exploited


0

200

400

600

Tissue factor (M) Factor IX (M

)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9

Figure 14 Clotting time at different initial concentrations of Tissue Factor and factor IXThe sweep ranges are [500 sdot 10minus14 500 sdot 10minus10] and[900 sdot 10

minus12 900 sdot 10

minus8] respectively

5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)

Number of simulations

CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7

Figure 15 Comparison of the computational time required to execute an increasing number of simulations of the BCCmodel on CPU (IntelCore i5 23 GHz) and GPU (Nvidia Tesla K20c GPU clock 706MHz) The computational time is expressed in seconds The time valuesrelated to 1000 5000 and 10000 simulations on the CPU were estimated by regression (see also Table 3)

Table 3 CPU versus GPU performance comparison

Number ofsimulations CPU time (sec) GPU time (sec) Speedup

5 3808 1278 298times10 10604 2329 456times50 51704 8058 642times100 116528 10977 1061times500 536058 17394 3081times1000 1073582lowast 19982 5373times5000 5363507lowast 30960 1732times10000 10724975lowast 58951 1819timeslowastValues estimated by regression

to efficiently determine the response of the BCC to differenttherapeutical drugs

4 Conclusions

Thanks to their high-performance computing capabilitiesand the very low costs GPUs nowadays represent a suitable

technology for the development and the application of par-allel computational methods for in silico analysis of complexbiological systems However the implementation of efficientcomputational tools able to fully exploit the large potentialityof GPUs is still challenging since good programming skillsare required to implement GPU-based algorithmic methodsand to handle specific features as an optimal usage ofmemoryor the communication bandwidth between GPU and CPUMoreover algorithms cannot be directly ported on the GPUbecause of the limited programming capabilities allowed byGPUkernels as amatter of fact they need to be redesigned tomeet the requirements of the underlying SIMD architecture

In this work we presented coagSODA a GPU-poweredsimulator specifically developed to carry out fast parallelsimulations of the BCC model coagSODA was designed tooffer a black-box solution usable by any final user in aneasy way It relies on cupSODA [24] a numerical integratorfor ODEs that we previously implemented for the GPUarchitecture based on the LSODA algorithm and capableof automatically translating a reaction-based model into aset of coupled ODEs In addition to mass-action kinet-ics coagSODA implements specific functions to compute


the kinetics of Hill function based reactions such as thoseinvolved in the platelets activity of the BCC model coag-SODA exploits the massive parallel capabilities of modernGPUs and our results demonstrated that it can achieve arelevant reduction of the computational time required toexecutemany concurrent and independent simulations of theBCC dynamics The mutual independence of the simulationsallows fully exploiting the underlying SIMD architecturemoreover coagSODA benefits from an additional speedupthanks to our choice of storing the state of the systeminto the low-latency shared memory (a solution that wasalready implemented in cupSODA) Since the BCC modelis large (96 reactions 71 chemical species) a large amountof shared memory was assigned to each thread stronglyreducing the theoretical occupancy of the GPU that is theratio of active warps with respect to the maximum numberof warps supported by each streaming multiprocessor of theGPU However the results of the analysis presented in thiswork performed on the BCC model show that coagSODAachieves a relevant boost with respect to a reference CPUimplementation For instance in the case of 10000 simula-tions we achieved a noticeable 181times speedup Interestinglythe performances of coagSODA are better than COPASI evenfor small batches of simulations These results indicate thatfor biological models consisting in many reactions and manyspecies our GPU implementation of LSODA becomes moreprofitable than the CPU counterpart as the number of con-current simulations increases making it suitable especiallywhen performing demanding computational analysis suchas for example parameter sweep parameter estimation orsensitivity analysis

As a test case in this paper we presented several parame-ter sweep analyses over reaction constants and initial concen-trations of factors involved in the BCC model Other com-putational analyses on mathematical models of this pathwaywere previously presented For instance in [58] a parameterestimation of the reaction constants of different modelsinvolving the activation of factor X factor V prothrombinand the inactivation of factors was performed The aim ofthis analysis was to discriminate between different modelsto unravel the mechanisms on the basis of the BCC Similaranalyses were executed on a model describing thrombingeneration in plasma since the reaction mechanism thereaction constants and initial concentrations were unknown[59] In [34] a sensitivity analysis of a model consisting of44 species over 34 chemical species was presented reactionconstants were varied between 10 and 1000 of theirreference value to the aim of identifying the most influentialfactors of the BCC

All these analyses were performed bymeans of sequentialsimulations of the models under investigation thereforein general only small batches of simulations could be run(for instance in [34] only 836 simulations were executed toexecute the sensitivity analysis of the model) On the otherhand in this paper we presented the results of different PSAefficiently executed bymeans of coagSODA overall providinguseful information regarding unknownparameters and inter-esting insights into the functioning of the BCC A thoroughsensitivity analysis of the whole BCC model consisting in

around 5 sdot 105 simulations is currently in progress on GPUsby our group exploiting the great computational efficiency ofcoagSODA

The results of our computational analyses should benow validated by means of specifically designed laboratoryexperiments In particular we identified a plausible valuefor the constant of the reaction describing the autoactivationof factor XII (since no reference values can be found inliterature) moreover the results of the PSA over factor IXsuggest that a deficiency of this factor is not enough to causesevere bleeding disorders as haemophilia B but an alterationof other factors seems to be necessary for the occurrence ofsuch condition (eg the lack of Tissue Factor as suggestedby the PSA-2D over Tissue Factor and factor IX) Finallythe PSA over factors VIII and IX showed in a situationcharacterized by very high concentrations of these factorsa counterintuitive behavior in which the clotting time isincreased with respect to the value obtained in physiologicalconditions Indeed it would be interesting to design ad hoclaboratory experiments to verify if the BCCmodel is actuallypredictive in such extreme situations

The coagSODA software and the SBML version of theBCC model used in this work are available from the authorsupon request

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is partially supported by SYSBIO Centre forSystems Biology

References

[1] B B Aldridge JM BurkeDA Lauffenburger andP K SorgerldquoPhysicochemicalmodelling of cell signalling pathwaysrdquoNatureCell Biology vol 8 no 11 pp 1195ndash1203 2006

[2] I-C Chou and E O Voit ldquoRecent developments in parameterestimation and structure identification of biochemical andgenomic systemsrdquoMathematical Biosciences vol 219 no 2 pp57ndash83 2009

[3] D Besozzi P Cazzaniga D Pescini G Mauri S Colomboand EMartegani ldquoThe role of feedback control mechanisms onthe establishment of oscillatory regimes in the RascAMPPKApathway in S cerevisiaerdquo EURASIP Journal on Bioinformaticsand Systems Biology vol 2012 no 10 2012

[4] A Saltelli M Ratto T Andres et al Global Sensitivity AnalysisThe Primer Wiley-Interscience 2008

[5] A Raue C Kreutz T Maiwald et al ldquoStructural and practicalidentifiability analysis of partially observed dynamical modelsby exploiting the profile likelihoodrdquo Bioinformatics vol 25 no15 pp 1923ndash1929 2009

[6] A Drager M Kronfeld M J Ziller et al ldquoModeling metabolicnetworks in C glutamicum a comparison of rate laws incombination with various parameter optimization strategiesrdquoBMC Systems Biology vol 3 article 5 2009


[7] D Besozzi P Cazzaniga G Mauri D Pescini and L Van-neschi ldquoA comparison of genetic algorithms and particle swarmoptimization for parameter estimation in stochastic biochem-ical systemsrdquo in Proceedings of the 7th European ConferenceEvolutionary Computation Machine Learning and Data Miningin Bioinformatics (EvoBIO rsquo09) C Pizzuti M Ritchie and MGiacobini Eds vol 5483 of Lecture Notes in Computer Sciencepp 116ndash127 Springer 2009

[8] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA aGPU-basedmulti-swarmPSOmethod for param-eter estimation in stochastic biological systems exploitingdiscrete-time target seriesrdquo in Proceedings of the 10th EuropeanConference Evolutionary Computation Machine Learning andData Mining in Bioinformatics (EvoBIO rsquo12) M Giacobini LVanneschi and W Bush Eds vol 7246 of Lecture Notes inComputer Science pp 74ndash85 Springer

[9] M S Nobile D Besozzi P Cazzaniga D Pescini and GMaurildquoReverse engineering of kinetic reaction networks by meansof Cartesian Genetic Programming and Particle Swarm Opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo13) pp 1594ndash1601 June 2013

[10] M S Nobile ldquoEvolutionary inference of biochemical inter-action networks accelerated on Graphics Processing Unitsrdquoin Proceedings of the 11th International Conference on HighPerformance Computingamp Simulation 2013 (HPCS rsquo13) pp 668ndash670 Helsinki Finland 2013

[11] K Kentzoglanakis and M Poole ldquoA swarm intelligence frame-work for reconstructing gene networks searching for bio-logically plausible architecturesrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 9 no 2 pp 358ndash371 2012

[12] J R Koza W Mydlowec G Lanza Y Jessen and M A KeaneldquoAutomatic computational discovery of chemical reaction net-works using genetic programmingrdquo inComputationalDiscoveryof Scientific Knowledge S Dzeroski and L Todorovski Edsvol 4660 of Lecture Notes in Computer Science pp 205ndash227Springer 2007

[13] S Ando E Sakamoto and H Iba ldquoEvolutionary modeling andinference of gene networkrdquo Information Sciences vol 145 no3-4 pp 237ndash259 2002

[14] L Dematte and D Prandi ldquoGPU computing for systemsbiologyrdquo Briefings in Bioinformatics vol 11 no 3 Article IDbbq006 pp 323ndash333 2010

[15] J L Payne N A Sinnott-Armstrong and J HMoore ldquoExploit-ing graphics processing units for computational biology andbioinformaticsrdquo Interdisciplinary Sciences Computational LifeSciences vol 2 no 3 pp 213ndash220 2010

[16] M J Harvey and G De Fabritiis ldquoA survey of computationalmolecular science using graphics processing unitsrdquoWiley Inter-disciplinary Reviews Computational Molecular Science vol 2no 5 pp 734ndash742 2012

[17] Y Zhou J Liepe X Sheng M P H Stumpf and C BarnesldquoGPU accelerated biochemical network simulationrdquo Bioinfor-matics vol 27 no 6 Article ID btr015 pp 874ndash876 2011

[18] M Vigelius A Lane and B Meyer ldquoAccelerating reaction-diffusion simulations with general-purpose graphics processingunitsrdquo Bioinformatics vol 27 no 2 Article ID btq622 pp 288ndash290 2011

[19] D J Wilkinson ldquoStochastic modelling for quantitative descrip-tion of heterogeneous biological systemsrdquo Nature ReviewsGenetics vol 10 no 2 pp 122ndash133 2009

[20] J C Butcher ldquoTrees and numerical methods for ordinarydifferential equationsrdquo Numerical Algorithms vol 53 no 2-3pp 153ndash170 2010

[21] D T Gillespie ldquoStochastic simulation of chemical kineticsrdquoAnnual Review of Physical Chemistry vol 58 pp 35ndash55 2007

[22] L Petzold ldquoAutomatic selection of methods for solving stiffand nonstiff systems of ordinary differential equationsrdquo SIAMJournal of Scientific and Statistical Computing vol 4 no 1 pp136ndash148 1983

[23] Y Cao D T Gillespie and L R Petzold ldquoEfficient step sizeselection for the tau-leaping simulation methodrdquo The Journalof Chemical Physics vol 124 no 4 2006

[24] M S Nobile D Besozzi P Cazzaniga GMauri andD PescinildquocupSODA a CUDA-powered simulator of mass-action kinet-icsrdquo in Proceedings of the 12th International Conference onParallel Computing Technologies (PaCT rsquo13) V Malyshkin Edvol 7979 of Lecture Notes in Computer Science pp 344ndash357

[25] M S Nobile P Cazzaniga D Besozzi and G Mauri ldquoGPU-accelerated simulations of mass-action kinetics models withcupSODArdquoThe Journal of Supercomputing 2014

[26] M S Nobile P Cazzaniga D Besozzi D Pescini and GMauri ldquocuTauLeaping a GPU-powered tau-leaping stochasticsimulator for massive parallel analyses of biological systemsrdquoPLoS ONE vol 9 no 3 2014

[27] K Rogers Blood Physiology and CirculationTheHuman BodyBritannica Educational Publishing 2010

[28] A D Michelson Platelets Elsevier 2nd edition 2007[29] R W Colman J Hirsh V J Marder A W Clowes and J

N George Hemostasis and Thrombosis Basic Principles andClinical Practice Lippincott Williams ampWilkins 2006

[30] M S Chatterjee W S Denney H Jing and S L DiamondldquoSystems biology of coagulation initiation Kinetics of thrombingeneration in resting and activated human bloodrdquo PLoS Com-putational Biology vol 6 no 9 Article ID e1000950 2010

[31] M Bellini D Besozzi P Cazzaniga GMauri andM S NobileldquoSimulation and analysis of the blood coagulation cascadeaccelerated on GPUrdquo in Proceedings of the 22nd EuromicroInternational Conference on Parallel Distributed and Network-Based Processing (PDP) pp 590ndash593 IEEE Computer SocietyConference Publishing Services 2014

[32] T Wajima G K Isbister and S B Duffull ldquoA comprehensivemodel for the humoral coagulation network in humansrdquo Clini-cal Pharmacology and Therapeutics vol 86 no 3 pp 290ndash2982009

[33] K Lo W S Denney and S L Diamond ldquoStochastic modelingof blood coagulation initiationrdquo Pathophysiology of Haemostasisand Thrombosis vol 34 no 2-3 pp 80ndash90 2006

[34] C M Danforth T Orfeo K G Mann K E Brummel-Ziedins and S J Everse ldquoThe impact of uncertainty in a bloodcoagulation modelrdquo Mathematical Medicine and Biology vol26 no 4 Article ID dqp011 pp 323ndash336 2009

[35] S D Lewis P P Shields and J A Shafer ldquoCharacterizationof the kinetic pathway for liberation of fibrinopeptides duringassembly of fibrinrdquoThe Journal of Biological Chemistry vol 260no 18 pp 10192ndash10199 1985

[36] M F Hockin K C Jones S J Everse and K G Mann ldquoAmodel for the stoichiometric regulation of blood coagulationrdquoThe Journal of Biological Chemistry vol 277 no 21 pp 18322ndash18333 2002

[37] K Brummel-Ziedins C YVossen F R Rosendaal KUmezakiand K G Mann ldquoThe plasma hemostatic proteome thrombin


generation in healthy individualsrdquo Journal of Thrombosis andHaemostasis vol 3 no 7 pp 1472ndash1481 2005

[38] K E Brummel-Ziedins C Y Vossen S Butenas K G Mannand F R Rosendaal ldquoThrombin generation profiles in deepvenous thrombosisrdquo Journal of Thrombosis and Haemostasisvol 3 no 11 pp 2497ndash2505 2005

[39] S D Bungay P A Gentry and R D Gentry ldquoA mathematicalmodel of lipid-mediated thrombin generationrdquo MathematicalMedicine and Biology vol 20 no 1 pp 105ndash129 2003

[40] S Butenas T Orfeo M T Gissel K E Brummel and K GMann ldquoThe significance of circulating factor IXa in bloodrdquoThe Journal of Biological Chemistry vol 279 no 22 pp 22875ndash22882 2004

[41] D Voet and J G Voet Biochemistry Wiley 2011[42] A C Guyton and J E Hall Textbook of Medical Physiology

Saunders 2010[43] M F M James and A M Roche ldquoDose-response relationship

between plasma ionized calcium concentration and thrombe-lastographyrdquo Journal of Cardiothoracic and Vascular Anesthesiavol 18 no 5 pp 581ndash586 2004

[44] D L Nelson and M M Cox Lehninger Principles of Biochem-istry W H Freeman Company 2004

[45] R M Farber ldquoTopical perspective on massive threading andparallelismrdquo Journal of Molecular Graphics and Modelling vol30 pp 82ndash89 2011

[46] OWolkenhauerMUllahWKolch andK-HCho ldquoModelingand simulation of intracellular dynamics choosing an appropri-ate frameworkrdquo IEEETransactions onNanobioscience vol 3 no3 pp 200ndash207 2004

[47] B H Estridge A P Reynolds and N J Walters Basic MedicalLaboratory Techniques Cengage Learning 4th edition 2000

[48] T Renne A H Schmaier K F Nickel M Blomback and CMaas ldquoIn vivo roles of factor XIIrdquo Blood vol 120 no 22 pp4296ndash4303 2012

[49] P V Jenkins O Rawley O P Smith and J S ODonnellldquoElevated factor VIII levels and risk of venous thrombosisrdquoBritish Journal of Haematology vol 157 no 6 pp 653ndash663 2012

[50] C E Walsh and K M Batt ldquoHemophilia clinical gene therapybrief reviewrdquo Translational Research vol 161 no 4 pp 307ndash3122013

[51] A Van Hylckama Vlieg I K Van Der Linden R M Bertinaand F R Rosendaal ldquoHigh levels of factor IX increase the risk ofvenous thrombosisrdquo Blood vol 95 no 12 pp 3678ndash3682 2000

[52] K Brummel-Ziedins ldquoModels for thrombin generation and riskof diseaserdquo Journal ofThrombosis and Haemostasis vol 11 no 1pp 212ndash223 2013

[53] U Seligsohn and A Lubetsky ldquoGenetic susceptibility to venousthrombosisrdquoTheNew England Journal of Medicine vol 344 no16 pp 1222ndash1231 2001

[54] I Martinelli P Bucciarelli and P M Mannucci ldquoThromboticrisk factors basic pathophysiologyrdquo Critical Care Medicine vol38 no 2 pp S3ndashS9 2010

[55] K Brummel-Ziedins A Undas T Orfeo et al ldquoThrombingeneration in acute coronary syndrome and stable coronaryartery disease dependence on plasma factor compositionrdquoJournal ofThrombosis andHaemostasis vol 6 no 1 pp 104ndash1102008

[56] S Hoops R Gauges C Lee et al ldquoCOPASImdasha complexpathway simulatorrdquo Bioinformatics vol 22 no 24 pp 3067ndash3074 2006

[57] Nvidia NVIDIArsquos Next Generation CUDA Compute Architec-ture Kepler GK110 2012

[58] W Stortelder W Hemker and H Hemker ldquoMathematicalmodelling in blood coagulation simulation and parameterestimationrdquoModelling Analysis and Simulation vol 20 pp 1ndash111997

[59] S M C Orfao G Jank and K Mottaghy ldquoA new mathematicalapproach to modelling thrombin generationrdquo InternationalJournal of Artificial Organs vol 29 no 7 pp 703ndash708 2006

[60] Nvidia Nvidia CUDA C Programming Guide v50 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of


Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology


Molecular Biology International

GenomicsInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


BioinformaticsAdvances in

Marine BiologyJournal of



Signal TransductionJournal of


BioMed Research International

Evolutionary BiologyInternational Journal of



Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Genetics Research International


Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational



Enzyme Research



Microbiology


of standard desktop computers therefore requiring high-performance computing solutions After the introductionof general-purpose GPUs and of Compute Unified DeviceArchitecture (CUDA Nvidiarsquos GPU programming language)the adoption of these graphics engines largely increasedin research fields as Bioinformatics Systems Biology andComputational Biology (see an overview in [14ndash16]) Anywaydespite the remarkable advantages concerning the computa-tional speedup computing with GPUs usually requires thedevelopment and the implementation of ad hoc algorithmssince GPU-based programming substantially differs fromCPU-based computing as a consequence scientific appli-cations of GPUs might undergo the risk of remaining aniche for few specialists To avoid such limitations severalpackages and software tools have recently been released(see eg [16ndash18]) so that also users with no knowledgeof GPUs hardware and programming can access the high-performance computing power of graphics engines

To investigate the dynamics of biological systems eitherdeterministic or stochastic approaches can be exploited [19]which are based on numerical integration (eg Eulerrsquos orRunge-Kutta methods [20]) or on Markov processes (egGillespiersquos algorithm [21]) respectively To date the most effi-cient algorithms to integrate a system of ordinary differentialequations (ODEs) or to perform stochastic simulations ofreaction-basedmodels are LSODA [22] and tau-leaping [23]respectively In [24ndash26] we previously presented cupSODAand cuTauLeaping the GPU-powered implementation ofLSODA and tauleaping respectively cupSODA allows torun parallel deterministic simulations of a given mass-actionbased system of biochemical reactions using the LSODAalgorithm cuTauLeaping represents a novel restructuring ofthe tau-leaping workflow that fits the GPU architecture andavoids any inefficiency drawback for coarse-grain massiveparallel stochastic simulations

In this work we introduce coagSODA an extension ofcupSODA that was specifically designed for the analysis ofa model of the blood coagulation cascade (BCC) Blood is thesubject of an intense scientific research thanks to its key roleinmaking diagnosis of numerous diseases [27] Humans haveevolved a complex hemostatic system that is able to maintainblood in a fluid state and allow the circulation through anintricate network of vessels in particular the presence ofseveral fine-tuned feedback mechanisms in the BCC allowskeeping all blood components within appropriate concen-tration ranges The BCC consists in a complex network ofcellular reactions which under physiological conditions invivo are inhibited by the presence of intact endothelium [28]Anyway in response to any vascular injury the hemostaticsystem is able to stop the blood leakage by rapidly sealing thedefects in the vesselsrsquo wall [29]

In order to investigate the variations in blood coagu-lation components among individuals and to understandthe corresponding response of the system to perturbedconditions we consider here a computational perspectiveto study the BCC we analyze the alterations (prolongationor reduction) of the time required to form the clot (iethe clotting time) by exploiting a reduced version of themathematical model defined in [30] This model describes

the intrinsic extrinsic and common pathways of the BCCand more importantly it accounts for platelets activationas well as the presence of several inhibitors (eg the TissueFactor Pathway Inhibitor antithrombin III and C1-inhibitor)[31]

Numerous mathematical models of blood coagulationwere developed in the last years as they represent a useful toolfor systematic studies of the intricate network of the coagu-lation cascade and allow obtaining a suitable reconstructionof empirical observations (see eg [32ndash34]) The earliestmodels considered only simple steps of the whole BCC suchas the conversion of the clot fibrin by thrombin [35] latelyHockin et al [36] developed a comprehensive ODE-basedmodel of the extrinsic blood coagulation system This modelwas then considered as reference by several research groupsto investigate the thrombotic risk in healthy and ill popula-tions [37 38] or to understand other complex biochemicalprocesses for instance in [39] the roles of protein C proteinS and phospholipid surface actions were considered whilein [40] the influence of trace amounts of key coagulationproteases on thrombin generation was investigated Recentlyother works modeling the blood clotting process in a com-prehensive manner have been published besides the alreadymentioned model developed by Chatterjee et al [30] wemention the model defined by Wajima et al [32] whichsimulates the intrinsic extrinsic and common pathways thevitamin K cycle the therapy with the anticoagulant drugswarfarin and heparin and the laboratory tests PT and aPTTas well as the Taipan snake bite which causes coagulopathy

coagSODA the GPU-accelerated simulator that wepresent and exploit in this work for the analysis of the BCCmodel is a user-friendly and efficient tool that circumventsthe need of manually defining the system of ODEs thatdescribe the blood coagulation network More preciselycoagSODA is able to automatically derive the system of(mass-action and Hill function-based) ODEs and then per-form their numerical integration starting from the given set of96 biochemical reactions which fully describe the molecularinteractions between all the species involved in the BCCin vivo We show that coagSODA allows us to efficientlyexecute a large number of parallel deterministic simulationsof the BCC at a considerable reduced computational costwith respect to CPUs In particular we exploit coagSODAto carry out one-dimensional and bi-dimensional parametersweep analysis of the BCC to the purpose of investigatingthe prolongation and the reduction of the clotting time inresponse to perturbed values of some reaction constants andof the initial concentration of somemolecular species chosenaccording to their meaning within the whole pathway

The paper is structured as follows In Section 2 we fullydescribe the mechanistic model of the BCC used in thiswork and present the simulation method at the basis ofcupSODA tool to introduce the coagSODA simulator InSection 3 we present the results obtained from the parametersweep analysis of the BCC model as well as a comparison ofthe computational performance of coagSODAwith respect toaCPU-based implementation of LSODA Finally in Section 4we conclude the paper with a discussion of the presentedwork and future research directions













4)




11990378 11990379 11990380)



81 11990382 11990383)



18and

11990319)


72 11990373

11990374)



45 11990346


51 11990352 11990353)







1199032


1199033


1199034


1199035


1199036


1199037


1199038


1199039


11990310


11990311


11990312


11990313


11990314


11990315


11990316


11990317


11990318


11990319


11990320


11990321


11990322


11990323


11990324


11990325


11990326


11990327


11990328


11990329


11990330


11990331


11990332


11990333


11990334


11990335


11990336


11990337


11990338


11990339


11990340


11990341


11990342


11990343


11990344


11990345


11990346


11990347


11990348


11990349



Table 1 Continued



11990351


11990352


11990353


11990354


11990355

Kal Kal119894


11990356


11990357


11990358


11990359


11990360


11990361


11990362


11990363


11990364


11990365


11990366


11990367


11990368


11990369


11990370


11990371


11990372


11990373


11990374


11990375


11990376


11990377


11990378


11990379


11990380


11990381


11990382


11990383


11990384


11990385


11990386


11990387


11990388


11990389


11990390


11990391


11990392


11990393


11990394


11990395


11990396



fXIITF-fVII

TFfVIIa

fVII

TF-fVIIa

Fbg

FPA

Fbn1

FPB

Fbn2

Fbn1

(Fbn1)2

(Fbn2)2 FPB

fXa

fIIa

TF-fVIIa-fXa

fIX

fIXa

fX

fmIIa

fII

PKal

KalifXI

fXIa

fIX

fIXa

fVIII

fVIIIa

fIXa-fVIIIa

fVIIIa1 -L

fVIIIa2

fVIIIa2

fIXa

fV

fVa

fXa-fVa

fXa-TFPI

TFPI

TF-fVIIa-fXa-TFPI

fXa-AT

AT

AT

fXIa-AT

C1inh

fXIIa-C1inh

A1AT

A2AP

fXIa-A2APfXIa-A1AT

TF-fVIIa

fXIIa Kal

fVIIIa1 -L





tion 11990354)



11990364)


69 11990370 11990371)



25and 11990326)





33)


7)



17)


28and 11990329)


84 119903

96)



58 11990359 11990360)



34and

11990335)


36 11990337 11990338)


40and 11990342)


39)


41)



66)



minus5



















57)











0 0 100 200 300 400 500 600 700

Fact

or II

a con

cent

ratio

n (M

)

Time (s)

1eminus7

2eminus7

3eminus7

4eminus7

5eminus7

6eminus7

7eminus7

8eminus7

9eminus7



119860 + 119861

1198961

999445999468

1198962

119860119861 (1)


119889 [119860119861]

119889119905

= 1198961sdot [119860] sdot [119861] minus

1198962sdot [119860] sdot [119861]

120576

(2)




(3)





120576max = 120576max0

+ (1 minus 120576max0


where 120576max0


0





Host

Kernel 1

Kernel 2

Device

Grid 1

Grid 2

Block(0 0)

Block(1 0)

Block(2 0)

Block(0 1)

Block(1 1)

Block(2 1)

Block (1 1)

Thread(0 0)

Thread(1 0)

Thread(2 0)

Thread(3 0)

Thread(4 0)

Thread(0 1)

Thread(1 1)

Thread(2 1)

Thread(3 1)

Thread(4 1)

Thread(0 2)

Thread(1 2)

Thread(2 2)

Thread(3 2)

Thread(4 2)

Global memory

Texture memory

Constant memory



Localmemory

Localmemory

Localmemory

Localmemory

Thread(0 0)

Thread(1 0)


Thread(0 0)

Thread(1 0)

Grid








Thread


L2 cache

DRAM








119889120576

119889119905

= 119896 (120576max minus 120576) (5)





0 100 200 300 400 500 600 700

Fibr

inog

en co

ncen

trat

ion

(M)

Time (s)

30 of fibrinogen

0

1eminus6

2eminus6

3eminus6

4eminus6

5eminus6

6eminus6

7eminus6

8eminus6

9eminus6
















324

326

328

330

332

334

336

338

340

Clot

ting

time (

s)

1eminus

10

1eminus

9

1eminus

8

1eminus

7

1eminus

6

1eminus

5

1eminus

4

1eminus

3

k44 (sminus1)





280

300

320

340

360

380

400

1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)








44with respect to













27= 2 sdot 10


27is





310

320

330

340

350

360

370

1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)





220

240

260

280

300

320

340

360

380

400

Clot

ting

time (

s)

Factor VIII (M)

Seve

re h

aem

ophi

lia

Mild

hae

mop

hilia

Prot

hrom

botic

stat

e

1eminus

16

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2


(ie 11989658= 1 sdot 10






250

300

350

400

450

500

550

600

650

700

Clot

ting

time (

s)

Factor IX (M)

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0


0

100

200

300

400

500

600

700

Clot

ting

time (

s)

Factor II (M)

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0

1e2













and 11989658


27 involved


27is low and








13 119903








280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )




respectively

100

200

300

400


(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10









0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology













4)




11990378 11990379 11990380)



81 11990382 11990383)



18and

11990319)


72 11990373

11990374)



45 11990346


51 11990352 11990353)







1199032


1199033


1199034


1199035


1199036


1199037


1199038


1199039


11990310


11990311


11990312


11990313


11990314


11990315


11990316


11990317


11990318


11990319


11990320


11990321


11990322


11990323


11990324


11990325


11990326


11990327


11990328


11990329


11990330


11990331


11990332


11990333


11990334


11990335


11990336


11990337


11990338


11990339


11990340


11990341


11990342


11990343


11990344


11990345


11990346


11990347


11990348


11990349



Table 1 Continued



11990351


11990352


11990353


11990354


11990355

Kal Kal119894


11990356


11990357


11990358


11990359


11990360


11990361


11990362


11990363


11990364


11990365


11990366


11990367


11990368


11990369


11990370


11990371


11990372


11990373


11990374


11990375


11990376


11990377


11990378


11990379


11990380


11990381


11990382


11990383


11990384


11990385


11990386


11990387


11990388


11990389


11990390


11990391


11990392


11990393


11990394


11990395


11990396



fXIITF-fVII

TFfVIIa

fVII

TF-fVIIa

Fbg

FPA

Fbn1

FPB

Fbn2

Fbn1

(Fbn1)2

(Fbn2)2 FPB

fXa

fIIa

TF-fVIIa-fXa

fIX

fIXa

fX

fmIIa

fII

PKal

KalifXI

fXIa

fIX

fIXa

fVIII

fVIIIa

fIXa-fVIIIa

fVIIIa1 -L

fVIIIa2

fVIIIa2

fIXa

fV

fVa

fXa-fVa

fXa-TFPI

TFPI

TF-fVIIa-fXa-TFPI

fXa-AT

AT

AT

fXIa-AT

C1inh

fXIIa-C1inh

A1AT

A2AP

fXIa-A2APfXIa-A1AT

TF-fVIIa

fXIIa Kal

fVIIIa1 -L





tion 11990354)



11990364)


69 11990370 11990371)



25and 11990326)





33)


7)



17)


28and 11990329)


84 119903

96)



58 11990359 11990360)



34and

11990335)


36 11990337 11990338)


40and 11990342)


39)


41)



66)



minus5



















57)











0 0 100 200 300 400 500 600 700

Fact

or II

a con

cent

ratio

n (M

)

Time (s)

1eminus7

2eminus7

3eminus7

4eminus7

5eminus7

6eminus7

7eminus7

8eminus7

9eminus7



119860 + 119861

1198961

999445999468

1198962

119860119861 (1)


119889 [119860119861]

119889119905

= 1198961sdot [119860] sdot [119861] minus

1198962sdot [119860] sdot [119861]

120576

(2)




(3)





120576max = 120576max0

+ (1 minus 120576max0


where 120576max0


0





Host

Kernel 1

Kernel 2

Device

Grid 1

Grid 2

Block(0 0)

Block(1 0)

Block(2 0)

Block(0 1)

Block(1 1)

Block(2 1)

Block (1 1)

Thread(0 0)

Thread(1 0)

Thread(2 0)

Thread(3 0)

Thread(4 0)

Thread(0 1)

Thread(1 1)

Thread(2 1)

Thread(3 1)

Thread(4 1)

Thread(0 2)

Thread(1 2)

Thread(2 2)

Thread(3 2)

Thread(4 2)

Global memory

Texture memory

Constant memory



Localmemory

Localmemory

Localmemory

Localmemory

Thread(0 0)

Thread(1 0)


Thread(0 0)

Thread(1 0)

Grid








Thread


L2 cache

DRAM








119889120576

119889119905

= 119896 (120576max minus 120576) (5)





0 100 200 300 400 500 600 700

Fibr

inog

en co

ncen

trat

ion

(M)

Time (s)

30 of fibrinogen

0

1eminus6

2eminus6

3eminus6

4eminus6

5eminus6

6eminus6

7eminus6

8eminus6

9eminus6
















324

326

328

330

332

334

336

338

340

Clot

ting

time (

s)

1eminus

10

1eminus

9

1eminus

8

1eminus

7

1eminus

6

1eminus

5

1eminus

4

1eminus

3

k44 (sminus1)





280

300

320

340

360

380

400

1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)








44with respect to













27= 2 sdot 10


27is





310

320

330

340

350

360

370

1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)





220

240

260

280

300

320

340

360

380

400

Clot

ting

time (

s)

Factor VIII (M)

Seve

re h

aem

ophi

lia

Mild

hae

mop

hilia

Prot

hrom

botic

stat

e

1eminus

16

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2


(ie 11989658= 1 sdot 10






250

300

350

400

450

500

550

600

650

700

Clot

ting

time (

s)

Factor IX (M)

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0


0

100

200

300

400

500

600

700

Clot

ting

time (

s)

Factor II (M)

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0

1e2













and 11989658


27 involved


27is low and








13 119903








280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )




respectively

100

200

300

400


(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10









0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology







1199032


1199033


1199034


1199035


1199036


1199037


1199038


1199039


11990310


11990311


11990312


11990313


11990314


11990315


11990316


11990317


11990318


11990319


11990320


11990321


11990322


11990323


11990324


11990325


11990326


11990327


11990328


11990329


11990330


11990331


11990332


11990333


11990334


11990335


11990336


11990337


11990338


11990339


11990340


11990341


11990342


11990343


11990344


11990345


11990346


11990347


11990348


11990349



Table 1 Continued



11990351


11990352


11990353


11990354


11990355

Kal Kal119894


11990356


11990357


11990358


11990359


11990360


11990361


11990362


11990363


11990364


11990365


11990366


11990367


11990368


11990369


11990370


11990371


11990372


11990373


11990374


11990375


11990376


11990377


11990378


11990379


11990380


11990381


11990382


11990383


11990384


11990385


11990386


11990387


11990388


11990389


11990390


11990391


11990392


11990393


11990394


11990395


11990396



fXIITF-fVII

TFfVIIa

fVII

TF-fVIIa

Fbg

FPA

Fbn1

FPB

Fbn2

Fbn1

(Fbn1)2

(Fbn2)2 FPB

fXa

fIIa

TF-fVIIa-fXa

fIX

fIXa

fX

fmIIa

fII

PKal

KalifXI

fXIa

fIX

fIXa

fVIII

fVIIIa

fIXa-fVIIIa

fVIIIa1 -L

fVIIIa2

fVIIIa2

fIXa

fV

fVa

fXa-fVa

fXa-TFPI

TFPI

TF-fVIIa-fXa-TFPI

fXa-AT

AT

AT

fXIa-AT

C1inh

fXIIa-C1inh

A1AT

A2AP

fXIa-A2APfXIa-A1AT

TF-fVIIa

fXIIa Kal

fVIIIa1 -L





tion 11990354)



11990364)


69 11990370 11990371)



25and 11990326)





33)


7)



17)


28and 11990329)


84 119903

96)



58 11990359 11990360)



34and

11990335)


36 11990337 11990338)


40and 11990342)


39)


41)



66)



minus5



















57)











0 0 100 200 300 400 500 600 700

Fact

or II

a con

cent

ratio

n (M

)

Time (s)

1eminus7

2eminus7

3eminus7

4eminus7

5eminus7

6eminus7

7eminus7

8eminus7

9eminus7



119860 + 119861

1198961

999445999468

1198962

119860119861 (1)


119889 [119860119861]

119889119905

= 1198961sdot [119860] sdot [119861] minus

1198962sdot [119860] sdot [119861]

120576

(2)




(3)





120576max = 120576max0

+ (1 minus 120576max0


where 120576max0


0





Host

Kernel 1

Kernel 2

Device

Grid 1

Grid 2

Block(0 0)

Block(1 0)

Block(2 0)

Block(0 1)

Block(1 1)

Block(2 1)

Block (1 1)

Thread(0 0)

Thread(1 0)

Thread(2 0)

Thread(3 0)

Thread(4 0)

Thread(0 1)

Thread(1 1)

Thread(2 1)

Thread(3 1)

Thread(4 1)

Thread(0 2)

Thread(1 2)

Thread(2 2)

Thread(3 2)

Thread(4 2)

Global memory

Texture memory

Constant memory



Localmemory

Localmemory

Localmemory

Localmemory

Thread(0 0)

Thread(1 0)


Thread(0 0)

Thread(1 0)

Grid








Thread


L2 cache

DRAM








119889120576

119889119905

= 119896 (120576max minus 120576) (5)





0 100 200 300 400 500 600 700

Fibr

inog

en co

ncen

trat

ion

(M)

Time (s)

30 of fibrinogen

0

1eminus6

2eminus6

3eminus6

4eminus6

5eminus6

6eminus6

7eminus6

8eminus6

9eminus6
















324

326

328

330

332

334

336

338

340

Clot

ting

time (

s)

1eminus

10

1eminus

9

1eminus

8

1eminus

7

1eminus

6

1eminus

5

1eminus

4

1eminus

3

k44 (sminus1)





280

300

320

340

360

380

400

1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)








44with respect to













27= 2 sdot 10


27is





310

320

330

340

350

360

370

1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)





220

240

260

280

300

320

340

360

380

400

Clot

ting

time (

s)

Factor VIII (M)

Seve

re h

aem

ophi

lia

Mild

hae

mop

hilia

Prot

hrom

botic

stat

e

1eminus

16

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2


(ie 11989658= 1 sdot 10






250

300

350

400

450

500

550

600

650

700

Clot

ting

time (

s)

Factor IX (M)

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0


0

100

200

300

400

500

600

700

Clot

ting

time (

s)

Factor II (M)

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0

1e2













and 11989658


27 involved


27is low and








13 119903








280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )




respectively

100

200

300

400


(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10









0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


Table 1 Continued



11990351


11990352


11990353


11990354


11990355

Kal Kal119894


11990356


11990357


11990358


11990359


11990360


11990361


11990362


11990363


11990364


11990365


11990366


11990367


11990368


11990369


11990370


11990371


11990372


11990373


11990374


11990375


11990376


11990377


11990378


11990379


11990380


11990381


11990382


11990383


11990384


11990385


11990386


11990387


11990388


11990389


11990390


11990391


11990392


11990393


11990394


11990395


11990396



fXIITF-fVII

TFfVIIa

fVII

TF-fVIIa

Fbg

FPA

Fbn1

FPB

Fbn2

Fbn1

(Fbn1)2

(Fbn2)2 FPB

fXa

fIIa

TF-fVIIa-fXa

fIX

fIXa

fX

fmIIa

fII

PKal

KalifXI

fXIa

fIX

fIXa

fVIII

fVIIIa

fIXa-fVIIIa

fVIIIa1 -L

fVIIIa2

fVIIIa2

fIXa

fV

fVa

fXa-fVa

fXa-TFPI

TFPI

TF-fVIIa-fXa-TFPI

fXa-AT

AT

AT

fXIa-AT

C1inh

fXIIa-C1inh

A1AT

A2AP

fXIa-A2APfXIa-A1AT

TF-fVIIa

fXIIa Kal

fVIIIa1 -L





tion 11990354)



11990364)


69 11990370 11990371)



25and 11990326)





33)


7)



17)


28and 11990329)


84 119903

96)



58 11990359 11990360)



34and

11990335)


36 11990337 11990338)


40and 11990342)


39)


41)



66)



minus5



















57)











0 0 100 200 300 400 500 600 700

Fact

or II

a con

cent

ratio

n (M

)

Time (s)

1eminus7

2eminus7

3eminus7

4eminus7

5eminus7

6eminus7

7eminus7

8eminus7

9eminus7



119860 + 119861

1198961

999445999468

1198962

119860119861 (1)


119889 [119860119861]

119889119905

= 1198961sdot [119860] sdot [119861] minus

1198962sdot [119860] sdot [119861]

120576

(2)




(3)





120576max = 120576max0

+ (1 minus 120576max0


where 120576max0


0





Host

Kernel 1

Kernel 2

Device

Grid 1

Grid 2

Block(0 0)

Block(1 0)

Block(2 0)

Block(0 1)

Block(1 1)

Block(2 1)

Block (1 1)

Thread(0 0)

Thread(1 0)

Thread(2 0)

Thread(3 0)

Thread(4 0)

Thread(0 1)

Thread(1 1)

Thread(2 1)

Thread(3 1)

Thread(4 1)

Thread(0 2)

Thread(1 2)

Thread(2 2)

Thread(3 2)

Thread(4 2)

Global memory

Texture memory

Constant memory



Localmemory

Localmemory

Localmemory

Localmemory

Thread(0 0)

Thread(1 0)


Thread(0 0)

Thread(1 0)

Grid








Thread


L2 cache

DRAM








119889120576

119889119905

= 119896 (120576max minus 120576) (5)





0 100 200 300 400 500 600 700

Fibr

inog

en co

ncen

trat

ion

(M)

Time (s)

30 of fibrinogen

0

1eminus6

2eminus6

3eminus6

4eminus6

5eminus6

6eminus6

7eminus6

8eminus6

9eminus6
















324

326

328

330

332

334

336

338

340

Clot

ting

time (

s)

1eminus

10

1eminus

9

1eminus

8

1eminus

7

1eminus

6

1eminus

5

1eminus

4

1eminus

3

k44 (sminus1)





280

300

320

340

360

380

400

1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)








44with respect to













27= 2 sdot 10


27is





310

320

330

340

350

360

370

1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)





220

240

260

280

300

320

340

360

380

400

Clot

ting

time (

s)

Factor VIII (M)

Seve

re h

aem

ophi

lia

Mild

hae

mop

hilia

Prot

hrom

botic

stat

e

1eminus

16

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2


(ie 11989658= 1 sdot 10






250

300

350

400

450

500

550

600

650

700

Clot

ting

time (

s)

Factor IX (M)

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0


0

100

200

300

400

500

600

700

Clot

ting

time (

s)

Factor II (M)

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0

1e2













and 11989658


27 involved


27is low and








13 119903








280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )




respectively

100

200

300

400


(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10









0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


fXIITF-fVII

TFfVIIa

fVII

TF-fVIIa

Fbg

FPA

Fbn1

FPB

Fbn2

Fbn1

(Fbn1)2

(Fbn2)2 FPB

fXa

fIIa

TF-fVIIa-fXa

fIX

fIXa

fX

fmIIa

fII

PKal

KalifXI

fXIa

fIX

fIXa

fVIII

fVIIIa

fIXa-fVIIIa

fVIIIa1 -L

fVIIIa2

fVIIIa2

fIXa

fV

fVa

fXa-fVa

fXa-TFPI

TFPI

TF-fVIIa-fXa-TFPI

fXa-AT

AT

AT

fXIa-AT

C1inh

fXIIa-C1inh

A1AT

A2AP

fXIa-A2APfXIa-A1AT

TF-fVIIa

fXIIa Kal

fVIIIa1 -L





tion 11990354)



11990364)


69 11990370 11990371)



25and 11990326)





33)


7)



17)


28and 11990329)


84 119903

96)



58 11990359 11990360)



34and

11990335)


36 11990337 11990338)


40and 11990342)


39)


41)



66)



minus5



















57)











0 0 100 200 300 400 500 600 700

Fact

or II

a con

cent

ratio

n (M

)

Time (s)

1eminus7

2eminus7

3eminus7

4eminus7

5eminus7

6eminus7

7eminus7

8eminus7

9eminus7



119860 + 119861

1198961

999445999468

1198962

119860119861 (1)


119889 [119860119861]

119889119905

= 1198961sdot [119860] sdot [119861] minus

1198962sdot [119860] sdot [119861]

120576

(2)




(3)





120576max = 120576max0

+ (1 minus 120576max0


where 120576max0


0





Host

Kernel 1

Kernel 2

Device

Grid 1

Grid 2

Block(0 0)

Block(1 0)

Block(2 0)

Block(0 1)

Block(1 1)

Block(2 1)

Block (1 1)

Thread(0 0)

Thread(1 0)

Thread(2 0)

Thread(3 0)

Thread(4 0)

Thread(0 1)

Thread(1 1)

Thread(2 1)

Thread(3 1)

Thread(4 1)

Thread(0 2)

Thread(1 2)

Thread(2 2)

Thread(3 2)

Thread(4 2)

Global memory

Texture memory

Constant memory



Localmemory

Localmemory

Localmemory

Localmemory

Thread(0 0)

Thread(1 0)


Thread(0 0)

Thread(1 0)

Grid








Thread


L2 cache

DRAM








119889120576

119889119905

= 119896 (120576max minus 120576) (5)





0 100 200 300 400 500 600 700

Fibr

inog

en co

ncen

trat

ion

(M)

Time (s)

30 of fibrinogen

0

1eminus6

2eminus6

3eminus6

4eminus6

5eminus6

6eminus6

7eminus6

8eminus6

9eminus6
















324

326

328

330

332

334

336

338

340

Clot

ting

time (

s)

1eminus

10

1eminus

9

1eminus

8

1eminus

7

1eminus

6

1eminus

5

1eminus

4

1eminus

3

k44 (sminus1)





280

300

320

340

360

380

400

1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)








44with respect to













27= 2 sdot 10


27is





310

320

330

340

350

360

370

1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)





220

240

260

280

300

320

340

360

380

400

Clot

ting

time (

s)

Factor VIII (M)

Seve

re h

aem

ophi

lia

Mild

hae

mop

hilia

Prot

hrom

botic

stat

e

1eminus

16

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2


(ie 11989658= 1 sdot 10






250

300

350

400

450

500

550

600

650

700

Clot

ting

time (

s)

Factor IX (M)

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0


0

100

200

300

400

500

600

700

Clot

ting

time (

s)

Factor II (M)

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0

1e2













and 11989658


27 involved


27is low and








13 119903








280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )




respectively

100

200

300

400


(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10









0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology




tion 11990354)



11990364)


69 11990370 11990371)



25and 11990326)





33)


7)



17)


28and 11990329)


84 119903

96)



58 11990359 11990360)



34and

11990335)


36 11990337 11990338)


40and 11990342)


39)


41)



66)



minus5



















57)











0 0 100 200 300 400 500 600 700

Fact

or II

a con

cent

ratio

n (M

)

Time (s)

1eminus7

2eminus7

3eminus7

4eminus7

5eminus7

6eminus7

7eminus7

8eminus7

9eminus7



119860 + 119861

1198961

999445999468

1198962

119860119861 (1)


119889 [119860119861]

119889119905

= 1198961sdot [119860] sdot [119861] minus

1198962sdot [119860] sdot [119861]

120576

(2)




(3)





120576max = 120576max0

+ (1 minus 120576max0


where 120576max0


0





Host

Kernel 1

Kernel 2

Device

Grid 1

Grid 2

Block(0 0)

Block(1 0)

Block(2 0)

Block(0 1)

Block(1 1)

Block(2 1)

Block (1 1)

Thread(0 0)

Thread(1 0)

Thread(2 0)

Thread(3 0)

Thread(4 0)

Thread(0 1)

Thread(1 1)

Thread(2 1)

Thread(3 1)

Thread(4 1)

Thread(0 2)

Thread(1 2)

Thread(2 2)

Thread(3 2)

Thread(4 2)

Global memory

Texture memory

Constant memory



Localmemory

Localmemory

Localmemory

Localmemory

Thread(0 0)

Thread(1 0)


Thread(0 0)

Thread(1 0)

Grid








Thread


L2 cache

DRAM








119889120576

119889119905

= 119896 (120576max minus 120576) (5)





0 100 200 300 400 500 600 700

Fibr

inog

en co

ncen

trat

ion

(M)

Time (s)

30 of fibrinogen

0

1eminus6

2eminus6

3eminus6

4eminus6

5eminus6

6eminus6

7eminus6

8eminus6

9eminus6
















324

326

328

330

332

334

336

338

340

Clot

ting

time (

s)

1eminus

10

1eminus

9

1eminus

8

1eminus

7

1eminus

6

1eminus

5

1eminus

4

1eminus

3

k44 (sminus1)





280

300

320

340

360

380

400

1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)








44with respect to













27= 2 sdot 10


27is





310

320

330

340

350

360

370

1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)





220

240

260

280

300

320

340

360

380

400

Clot

ting

time (

s)

Factor VIII (M)

Seve

re h

aem

ophi

lia

Mild

hae

mop

hilia

Prot

hrom

botic

stat

e

1eminus

16

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2


(ie 11989658= 1 sdot 10






250

300

350

400

450

500

550

600

650

700

Clot

ting

time (

s)

Factor IX (M)

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0


0

100

200

300

400

500

600

700

Clot

ting

time (

s)

Factor II (M)

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0

1e2













and 11989658


27 involved


27is low and








13 119903








280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )




respectively

100

200

300

400


(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10









0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


0 0 100 200 300 400 500 600 700

Fact

or II

a con

cent

ratio

n (M

)

Time (s)

1eminus7

2eminus7

3eminus7

4eminus7

5eminus7

6eminus7

7eminus7

8eminus7

9eminus7



119860 + 119861

1198961

999445999468

1198962

119860119861 (1)


119889 [119860119861]

119889119905

= 1198961sdot [119860] sdot [119861] minus

1198962sdot [119860] sdot [119861]

120576

(2)




(3)





120576max = 120576max0

+ (1 minus 120576max0


where 120576max0


0





Host

Kernel 1

Kernel 2

Device

Grid 1

Grid 2

Block(0 0)

Block(1 0)

Block(2 0)

Block(0 1)

Block(1 1)

Block(2 1)

Block (1 1)

Thread(0 0)

Thread(1 0)

Thread(2 0)

Thread(3 0)

Thread(4 0)

Thread(0 1)

Thread(1 1)

Thread(2 1)

Thread(3 1)

Thread(4 1)

Thread(0 2)

Thread(1 2)

Thread(2 2)

Thread(3 2)

Thread(4 2)

Global memory

Texture memory

Constant memory



Localmemory

Localmemory

Localmemory

Localmemory

Thread(0 0)

Thread(1 0)


Thread(0 0)

Thread(1 0)

Grid








Thread


L2 cache

DRAM








119889120576

119889119905

= 119896 (120576max minus 120576) (5)





0 100 200 300 400 500 600 700

Fibr

inog

en co

ncen

trat

ion

(M)

Time (s)

30 of fibrinogen

0

1eminus6

2eminus6

3eminus6

4eminus6

5eminus6

6eminus6

7eminus6

8eminus6

9eminus6
















324

326

328

330

332

334

336

338

340

Clot

ting

time (

s)

1eminus

10

1eminus

9

1eminus

8

1eminus

7

1eminus

6

1eminus

5

1eminus

4

1eminus

3

k44 (sminus1)





280

300

320

340

360

380

400

1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)








44with respect to













27= 2 sdot 10


27is





310

320

330

340

350

360

370

1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)





220

240

260

280

300

320

340

360

380

400

Clot

ting

time (

s)

Factor VIII (M)

Seve

re h

aem

ophi

lia

Mild

hae

mop

hilia

Prot

hrom

botic

stat

e

1eminus

16

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2


(ie 11989658= 1 sdot 10






250

300

350

400

450

500

550

600

650

700

Clot

ting

time (

s)

Factor IX (M)

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0


0

100

200

300

400

500

600

700

Clot

ting

time (

s)

Factor II (M)

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0

1e2













and 11989658


27 involved


27is low and








13 119903








280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )




respectively

100

200

300

400


(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10









0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


Host

Kernel 1

Kernel 2

Device

Grid 1

Grid 2

Block(0 0)

Block(1 0)

Block(2 0)

Block(0 1)

Block(1 1)

Block(2 1)

Block (1 1)

Thread(0 0)

Thread(1 0)

Thread(2 0)

Thread(3 0)

Thread(4 0)

Thread(0 1)

Thread(1 1)

Thread(2 1)

Thread(3 1)

Thread(4 1)

Thread(0 2)

Thread(1 2)

Thread(2 2)

Thread(3 2)

Thread(4 2)

Global memory

Texture memory

Constant memory



Localmemory

Localmemory

Localmemory

Localmemory

Thread(0 0)

Thread(1 0)


Thread(0 0)

Thread(1 0)

Grid








Thread


L2 cache

DRAM








119889120576

119889119905

= 119896 (120576max minus 120576) (5)





0 100 200 300 400 500 600 700

Fibr

inog

en co

ncen

trat

ion

(M)

Time (s)

30 of fibrinogen

0

1eminus6

2eminus6

3eminus6

4eminus6

5eminus6

6eminus6

7eminus6

8eminus6

9eminus6
















324

326

328

330

332

334

336

338

340

Clot

ting

time (

s)

1eminus

10

1eminus

9

1eminus

8

1eminus

7

1eminus

6

1eminus

5

1eminus

4

1eminus

3

k44 (sminus1)





280

300

320

340

360

380

400

1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)








44with respect to













27= 2 sdot 10


27is





310

320

330

340

350

360

370

1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)





220

240

260

280

300

320

340

360

380

400

Clot

ting

time (

s)

Factor VIII (M)

Seve

re h

aem

ophi

lia

Mild

hae

mop

hilia

Prot

hrom

botic

stat

e

1eminus

16

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2


(ie 11989658= 1 sdot 10






250

300

350

400

450

500

550

600

650

700

Clot

ting

time (

s)

Factor IX (M)

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0


0

100

200

300

400

500

600

700

Clot

ting

time (

s)

Factor II (M)

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0

1e2













and 11989658


27 involved


27is low and








13 119903








280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )




respectively

100

200

300

400


(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10









0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


Thread


L2 cache

DRAM








119889120576

119889119905

= 119896 (120576max minus 120576) (5)





0 100 200 300 400 500 600 700

Fibr

inog

en co

ncen

trat

ion

(M)

Time (s)

30 of fibrinogen

0

1eminus6

2eminus6

3eminus6

4eminus6

5eminus6

6eminus6

7eminus6

8eminus6

9eminus6
















324

326

328

330

332

334

336

338

340

Clot

ting

time (

s)

1eminus

10

1eminus

9

1eminus

8

1eminus

7

1eminus

6

1eminus

5

1eminus

4

1eminus

3

k44 (sminus1)





280

300

320

340

360

380

400

1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)








44with respect to













27= 2 sdot 10


27is





310

320

330

340

350

360

370

1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)





220

240

260

280

300

320

340

360

380

400

Clot

ting

time (

s)

Factor VIII (M)

Seve

re h

aem

ophi

lia

Mild

hae

mop

hilia

Prot

hrom

botic

stat

e

1eminus

16

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2


(ie 11989658= 1 sdot 10






250

300

350

400

450

500

550

600

650

700

Clot

ting

time (

s)

Factor IX (M)

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0


0

100

200

300

400

500

600

700

Clot

ting

time (

s)

Factor II (M)

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0

1e2













and 11989658


27 involved


27is low and








13 119903








280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )




respectively

100

200

300

400


(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10









0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


0 100 200 300 400 500 600 700

Fibr

inog

en co

ncen

trat

ion

(M)

Time (s)

30 of fibrinogen

0

1eminus6

2eminus6

3eminus6

4eminus6

5eminus6

6eminus6

7eminus6

8eminus6

9eminus6
















324

326

328

330

332

334

336

338

340

Clot

ting

time (

s)

1eminus

10

1eminus

9

1eminus

8

1eminus

7

1eminus

6

1eminus

5

1eminus

4

1eminus

3

k44 (sminus1)





280

300

320

340

360

380

400

1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)








44with respect to













27= 2 sdot 10


27is





310

320

330

340

350

360

370

1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)





220

240

260

280

300

320

340

360

380

400

Clot

ting

time (

s)

Factor VIII (M)

Seve

re h

aem

ophi

lia

Mild

hae

mop

hilia

Prot

hrom

botic

stat

e

1eminus

16

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2


(ie 11989658= 1 sdot 10






250

300

350

400

450

500

550

600

650

700

Clot

ting

time (

s)

Factor IX (M)

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0


0

100

200

300

400

500

600

700

Clot

ting

time (

s)

Factor II (M)

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0

1e2













and 11989658


27 involved


27is low and








13 119903








280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )




respectively

100

200

300

400


(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10









0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


324

326

328

330

332

334

336

338

340

Clot

ting

time (

s)

1eminus

10

1eminus

9

1eminus

8

1eminus

7

1eminus

6

1eminus

5

1eminus

4

1eminus

3

k44 (sminus1)





280

300

320

340

360

380

400

1e4 1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)








44with respect to













27= 2 sdot 10


27is





310

320

330

340

350

360

370

1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)





220

240

260

280

300

320

340

360

380

400

Clot

ting

time (

s)

Factor VIII (M)

Seve

re h

aem

ophi

lia

Mild

hae

mop

hilia

Prot

hrom

botic

stat

e

1eminus

16

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2


(ie 11989658= 1 sdot 10






250

300

350

400

450

500

550

600

650

700

Clot

ting

time (

s)

Factor IX (M)

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0


0

100

200

300

400

500

600

700

Clot

ting

time (

s)

Factor II (M)

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0

1e2













and 11989658


27 involved


27is low and








13 119903








280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )




respectively

100

200

300

400


(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10









0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


310

320

330

340

350

360

370

1e5 1e6 1e7 1e8 1e9 1e10 1e11

Clot

ting

time (

s)





220

240

260

280

300

320

340

360

380

400

Clot

ting

time (

s)

Factor VIII (M)

Seve

re h

aem

ophi

lia

Mild

hae

mop

hilia

Prot

hrom

botic

stat

e

1eminus

16

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2


(ie 11989658= 1 sdot 10






250

300

350

400

450

500

550

600

650

700

Clot

ting

time (

s)

Factor IX (M)

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0


0

100

200

300

400

500

600

700

Clot

ting

time (

s)

Factor II (M)

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0

1e2













and 11989658


27 involved


27is low and








13 119903








280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )




respectively

100

200

300

400


(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10









0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


250

300

350

400

450

500

550

600

650

700

Clot

ting

time (

s)

Factor IX (M)

1eminus

14

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0


0

100

200

300

400

500

600

700

Clot

ting

time (

s)

Factor II (M)

1eminus

12

1eminus

10

1eminus

8

1eminus

6

1eminus

4

1eminus

2

1e0

1e2













and 11989658


27 involved


27is low and








13 119903








280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )




respectively

100

200

300

400


(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10









0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology








and 11989658


27 involved


27is low and








13 119903








280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )




respectively

100

200

300

400


(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10









0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


280

300

320

340

360

380

400

420

440

1e41e6

1e91e11 1e5

1e7

1e9

1e11

280 320 360 400 440

Clot

ting

time (

s)

k 58(M

minus1 sminus1 )




respectively

100

200

300

400


(M)

Clot

ting

time (

s)

100

150

200

250

300

350

400

450

1eminus131eminus11

1eminus91eminus14

1eminus12

1eminus10









0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


0

200

400

600


)

Clot

ting

time (

s)

0

100

200

300

400

500

600

700

1eminus12

1eminus10

1eminus8

1eminus131eminus11

1eminus9


minus12 900 sdot 10


5 10 50 100 500 1000 5000 10000

Runn

ing

time (

s)


CPUGPU

1e2

1e3

1e4

1e5

1e6

1e7






4 Conclusions













Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology










Acknowledgment


References







































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology

































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology

research article massive exploration of perturbed...

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