qsp/prosthetic-joint-infection/pji_mrgsolve_model.R at main · pipetcpt/qsp

Repository navigation

History

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

542

543

544

545

546

547

548

549

550

551

552

553

554

555

556

557

558

559

560

561

562

563

564

565

566

567

568

569

570

571

572

573

574

575

576

577

578

579

580

581

582

583

584

585

586

587

588

589

590

591

592

593

594

595

596

597

598

599

600

601

602

603

604

605

606

607

608

609

610

611

612

613

614

615

616

617

618

619

620

621

622

623

624

625

626

627

628

629

630

631

632

633

634

635

636

637

638

639

640

641

642

643

644

645

646

647

648

649

650

651

652

653

654

655

656

657

658

659

660

661

662

663

664

665

666

667

668

669

670

671

672

673

674

675

676

677

678

679

680

681

682

683

684

685

686

687

688

689

690

691

692

693

694

695

696

697

698

699

700

701

702

703

704

705

706

707

708

709

710

711

712

713

714

715

716

717

718

719

720

721

722

723

724

725

726

727

728

729

730

731

732

733

734

735

736

737

738

739

740

741

742

743

744

745

746

747

748

749

750

751

752

753

754

755

756

757

758

759

760

761

762

763

764

765

766

767

768

769

770

771

772

773

774

775

776

777

778

779

780

781

782

783

784

785

786

787

788

789

790

791

792

793

794

795

796

797

798

799

800

801

802

803

804

805

806

807

808

809

810

811

812

813

814

815

816

817

818

819

820

821

822

823

824

825

826

827

828

829

830

831

832

833

834

835

836

837

838

839

840

841

842

843

844

845

846

847

848

849

850

851

852

853

854

855

856

857

858

859

860

861

862

863

864

865

866

867

868

869

870

871

872

873

874

875

876

877

878

879

880

881

882

883

884

885

886

887

888

889

890

891

892

893

894

895

896

897

898

899

900

901

902

903

904

905

906

907

908

909

910

911

912

913

914

915

916

917

918

919

920

921

922

923

924

925

926

927

928

929

930

931

932

933

934

935

936

937

938

939

940

941

942

943

944

945

946

947

948

949

950

951

952

953

954

955

956

957

958

959

960

961

962

963

964

965

966

967

968

969

970

971

972

973

974

975

976

977

978

979

980

981

982

983

984

985

986

987

988

989

990

991

992

993

994

995

996

997

998

999

1000

## =============================================================================

## pji_mrgsolve_model.R

## 인공관절 감염 (Prosthetic Joint Infection, PJI) QSP 모델

## Implant-associated Staphylococcus aureus osteomyelitis

## 53 ODEs · 6 antibiotics with explicit bone penetration · 5 bacterial

## phenotypic states · 2 resistant lineages · local elution from an

## antibiotic-loaded cement spacer · innate immunity with frustrated

## phagocytosis · RANKL/OPG osteolysis · 2018 ICM biomarkers · drug toxicity

## ---------------------------------------------------------------------------

## WHAT THIS MODEL IS ORGANISED AROUND

## ---------------------------------------------------------------------------

## PJI doctrine looks like a list of empirical rules ("you must debride",

## "rifampicin is special", "never give rifampicin alone", "DAIR only works

## early"). This model is written so that all four fall out of arithmetic

## rather than being coded in as assertions.

## PILLAR 1 — THE FOREIGN BODY MOVES THE INFECTIVE DOSE, NOT THE ORGANISM.

## Two mechanisms, both explicit: (a) FBFACT scales down phagocyte killing

## next to metal, and (b) KATT moves planktonic cells onto a surface where

## phagocytosis is "frustrated" (FRUST = 0.03). Neither is a threshold

## parameter. Simulated: an implant-free niche clears up to ~10^7 CFU;

## with an implant, ~10^1 CFU establishes. That ~10^5-10^6 fold shift is

## the model's version of Elek & Conen's suture experiment (1957) and of

## Zimmerli's tissue-cage model (1982).

## PILLAR 2 — EVERY ANTIBIOTIC IS ITS FREE BONE CONCENTRATION DIVIDED BY MBEC.

## The model never uses the planktonic MIC to decide what happens in the

## biofilm. For each drug it carries (i) AUC_bone/AUC_plasma, (ii) the free

## fraction, (iii) the planktonic MIC and (iv) the biofilm MBEC. The ratio

## C_bone,free / MBEC at standard adult doses then computes to:

## vancomycin 1 g q12h ...... 2.083 / 512 = 0.0041

## cefazolin 2 g q8h (MSSA).. 2.500 / 256 = 0.0098

## daptomycin 8 mg/kg qd .... 0.350 / 32 = 0.0109

## linezolid 600 mg q12h .... 2.218 / 128 = 0.0173

## levofloxacin 750 mg qd ... 1.203 / 64 = 0.0188

## RIFAMPICIN 450 mg q12h ... 0.281 / 1.0 = 0.281 (week 1)

## 0.141 / 1.0 = 0.141 (autoinduced steady state)

## (all six computed by ratio_table(); rifampicin sits 7-68x above every

## other option and is still BELOW 1 -- which is why it needs surgery.)

## Rifampicin is not a better antibiotic. It is the only one whose target

## (RNA polymerase) is still load-bearing in an adherent, barely dividing

## cell, so its MBEC never climbs 2-3 logs above its MIC the way a cell-wall

## agent's does. The entire clinical special status of rifampicin in

## implant infection is that one column of the table.

## PILLAR 3 — SURGERY IS A MUTANT-SUPPLY OPERATION, NOT A CLEANING OPERATION.

## P(a pre-existing rpoB mutant) = 1 - exp(-mu * N), mu ~ 1e-8 per division.

## A mature biofilm carries N ~ 1e9-1e10, so that probability is ~1.000 and

## rifampicin is dead on arrival. A 4-log debridement takes N to ~1e5-1e6

## and the probability to ~1e-3-1e-2. Debridement does not "help" the

## antibiotic; it is the step that makes Pillar 2 usable at all. The model

## computes this probability continuously (PRPOB) and also carries explicit

## rpoB (RP/RB) and gyrA (QP/QB) lineages generated from the growth flux.

## PILLAR 4 — BIOFILM MATURITY IS A CLOCK, AND DAIR RACES IT.

## PHIB (the MBEC/MIC tolerance multiplier) is not constant: it is scaled by

## matrix maturity EPS, which rises with a ~28-day time constant. A biofilm

## debrided at day 7 is still ~40% "planktonic-like"; at day 90 it is fully

## tolerant. The 3-week symptom-duration rule for DAIR is therefore an

## emergent property of an EPS differential equation, not an if-statement.

## ---------------------------------------------------------------------------

## IMPORTANT MODELLING CHOICES / LIMITATIONS (read before using numbers)

## ---------------------------------------------------------------------------

## * Bone compartments are FREE-concentration effect compartments driven by

## total plasma concentration through FB_x = (AUC_bone/AUC_plasma) x fu.

## They do not feed mass back to plasma (periprosthetic mass is negligible).

## * The model is deterministic. Bacterial extinction is enforced by a smooth

## sub-single-cell decay term (KEXT below 1 CFU) so the ODE cannot carry

## "0.001 of a bacterium" and regrow it, which is the classic deterministic

## artefact in antimicrobial models. Stochastic outcomes (ID50, resistance

## emergence, cure) are reported as PROBABILITIES computed from the

## deterministic burden, not as trajectories.

## * Cure probability is mapped as PCURE = exp(-PSEED * N_end): a surviving

## burden of ~1 CFU next to a retained implant is roughly a coin flip.

## * Cytokine and biomarker units are local/periprosthetic, not serum, except

## CRP / ESR / PLT / SCR / ALT which are systemic.

## * Calibration is to published trial-level endpoints (Zimmerli 1998 JAMA;

## OVIVA 2019; DATIPO 2021; Lora-Tamayo 2013) and to standard PK/bone

## penetration reviews (Landersdorfer 2009; Thabit 2019). It has NOT been

## fitted to individual patient data. Educational / research use only.

## ---------------------------------------------------------------------------

## VERIFICATION (all 53 ODEs were re-implemented independently in

## Python/scipy and run before this file was finalised). That pass exposed

## and fixed six defects, each of which had silently broken one of the

## pillars:

## 1. IMMUNE CAPACITY WRITTEN PER COMPARTMENT. A Michaelis-Menten sink on

## each bacterial state made a rare rpoB clone face the ENTIRE phagocyte

## capacity as first-order clearance at ~19 /h, sterilising exactly the

## subpopulation Pillar 3 is about. Rewritten as one shared capacity

## allocated in proportion to accessibility (NPHAG / SHR).

## 2. EXTINCTION FLOOR APPLIED PER STATE. Forcing every compartment below

## 1 CFU to zero annihilated the rpoB lineage on every dip. Extinction is

## now a property of the CLONE. Before the fix an untreated 1e10 lesion

## carried 0.01 mutants; after it, 63 -- against the analytic mu*N = 100.

## 3. DEBRIDEMENT RESET BIOFILM MATURITY. EPS was stripped in proportion to

## the surgical log-kill, so a 2.5-log DAIR dropped EPS 0.62 -> 0.026 and

## handed the residual biofilm near-planktonic susceptibility. Result:

## levofloxacin MONOTHERAPY cured. A retained implant keeps its adherent

## matrix (EPSCUT retention factor 0.55 -> 0.05).

## 4. NO CARRYING CAPACITY ON THE FREE POOL. Dispersed organisms grew to

## 1.8e10 and outnumbered the biofilm, making the infection spuriously

## vancomycin-curable. NPCAP (fluid/abscess) and NICCAP (bounded by host

## cell number) added.

## 5. UNBOUNDED LOOSENING AND OSTEOBLAST COLLAPSE. LOOSEN ran to 266 on a

## 0-100 scale and KAPOP drove osteoblasts to 8% of baseline.

## 6. PHAGOCYTE SATURATION CONSTANT INCONSISTENT WITH THE ID50. KMIMM = 1e4

## against a target ID50 of ~1e7 made clearance first-order over three

## logs with a large rate constant, so the immune system held the biofilm

## at 1e4.8 CFU and no PJI ever established.

## KNOWN CALIBRATION GAP (stated, not tuned away): the model's DAIR-timing

## crossover falls between 3 and 6 months of infection duration, whereas the

## clinical rule is ~3 weeks of symptoms. Debridement at day 21 sterilises on

## day 60 of an 84-day course; at day 90 it sterilises on day 84 of 84 (zero

## margin); at day 180 it fails outright. The direction and the mechanism

## (EPS maturity ~10 d, sequestrum shielding ~3 mo) are right; the absolute

## clock is too slow because the model has no soft-tissue abscess or

## mechanical-loosening channel, which is how late DAIR actually fails.

## Author: QSP Disease Model Library (Claude Code Routine)

## =============================================================================

library(mrgsolve)

library(dplyr)

pji_code <- '

$PROB

# Prosthetic Joint Infection (PJI) QSP model

# Implant-associated Staphylococcus aureus osteomyelitis

# 53 ODEs | biofilm PK/PD | mutant supply | surgery | osteolysis

$GLOBAL

// Emax kill helper: EMAX * C / (EC50eff + C)

#define EKILL(EMX, CC, EC) ((EMX)*(CC)/((EC)+(CC)))

// Biofilm tolerance multiplier scaled by matrix maturity (Pillar 4).

// EPSM in [0,1); a bare, freshly attached cell is nearly planktonic.

#define PHIMAT(PHI, EPSM) (1.0 + ((PHI) - 1.0)*(EPSM))

$PARAM @annotated

// ---------------------------------------------------------------- SETUP ----

IMPL : 1 : 임플란트 존재 (1 = prosthesis in situ, 0 = implant-free bone)

MSSA : 0 : 메티실린 감수성 (1 = MSSA, cefazolin active; 0 = MRSA)

IMSUP : 1 : 숙주 면역 계수 (1 = 정상, 0.5 = 당뇨/스테로이드/RA)

INOC0 : 100 : CFU : 접종 균량 (유주균으로 t=0에 투입)

// ------------------------------------------------- VANCOMYCIN PK (IV) ------

VCVAN : 40 : L : VAN 중심 분포용적

VPVAN : 40 : L : VAN 말초 분포용적

QVAN : 8 : L/h : VAN 구획간 청소율

CLVAN : 4 : L/h : VAN 전신 청소율 (CrCl ~100 mL/min)

FUVAN : 0.50 : : VAN 유리 분율

PENVAN : 0.20 : : VAN AUC_bone / AUC_plasma

KEQVAN : 0.30 : 1/h : VAN 골 구획 평형 속도

// ------------------------------------------------ RIFAMPICIN PK (PO) -------

KARIF : 1.20 : 1/h : RIF 흡수속도

FRIF : 0.90 : : RIF 생체이용률

VCRIF : 55 : L : RIF 분포용적

CLRIF0 : 12 : L/h : RIF 기저 청소율 (자가유도 전)

FURIF : 0.20 : : RIF 유리 분율

PENRIF : 0.50 : : RIF AUC_bone / AUC_plasma

KEQRIF : 0.50 : 1/h : RIF 골 구획 평형 속도

KENZ : 0.0072: 1/h : CYP3A4 유도 전환속도 (t1/2 ~ 4일)

EIND : 1.00 : : RIF 자가유도 최대 배수 (청소율 x2)

// ---------------------------------------------- LEVOFLOXACIN PK (PO) -------

KALVX : 1.50 : 1/h : LVX 흡수속도

FLVX : 0.99 : : LVX 생체이용률

VCLVX : 90 : L : LVX 분포용적

CLLVX : 9 : L/h : LVX 청소율

FULVX : 0.70 : : LVX 유리 분율

PENLVX : 0.50 : : LVX AUC_bone / AUC_plasma

KEQLVX : 0.50 : 1/h : LVX 골 구획 평형 속도

// ------------------------------------------------ DAPTOMYCIN PK (IV) -------

VCDAP : 10 : L : DAP 분포용적

CLDAP : 0.80 : L/h : DAP 청소율

FUDAP : 0.08 : : DAP 유리 분율 (92% 알부민 결합)

PENDAP : 0.15 : : DAP AUC_bone / AUC_plasma

KEQDAP : 0.25 : 1/h : DAP 골 구획 평형 속도

// ------------------------------------------------- LINEZOLID PK (PO) -------

KALZD : 1.80 : 1/h : LZD 흡수속도

FLZD : 1.00 : : LZD 생체이용률

VCLZD : 45 : L : LZD 분포용적

CLLZD0 : 7 : L/h : LZD 기저 청소율

FULZD : 0.69 : : LZD 유리 분율

PENLZD : 0.45 : : LZD AUC_bone / AUC_plasma

KEQLZD : 0.50 : 1/h : LZD 골 구획 평형 속도

RIFLZD : 0.50 : : RIF에 의한 LZD 청소율 유도 강도 (AUC -32%)

// -------------------------------------------------- CEFAZOLIN PK (IV) ------

VCCFZ : 15 : L : CFZ 분포용적

CLCFZ : 4 : L/h : CFZ 청소율

FUCFZ : 0.20 : : CFZ 유리 분율

PENCFZ : 0.20 : : CFZ AUC_bone / AUC_plasma

KEQCFZ : 0.50 : 1/h : CFZ 골 구획 평형 속도

// --------------------------------- LOCAL ELUTION (ALBC / SPACER) -----------

VJOINT : 0.05 : L : 유효 관절강·계면 용적

KFAST : 0.05 : 1/h : 표면층 폭발 방출 속도

KSLOW : 0.0015: 1/h : 기질 확산 방출 속도 (t1/2 ~ 19일)

KCLJ : 0.35 : 1/h : 관절액 약물 소실 속도

FLOCP : 0.50 : : 국소 약물의 유주균 접근도

FLOCB : 0.15 : : 국소 약물의 잔존 바이오필름 접근도

// -------------------------------------------- MIC / MBEC (S. aureus) -------

MICVAN : 1.0 : mg/L : VAN 유주균 MIC

MICRIF : 0.012: mg/L : RIF 유주균 MIC

MICLVX : 0.25 : mg/L : LVX 유주균 MIC

MICDAP : 0.50 : mg/L : DAP 유주균 MIC

MICLZD : 2.0 : mg/L : LZD 유주균 MIC

MICCFZ : 1.0 : mg/L : CFZ 유주균 MIC (MSSA)

MBCVAN : 512 : mg/L : VAN 바이오필름 MBEC

MBCRIF : 1.0 : mg/L : RIF 바이오필름 MBEC

MBCLVX : 64 : mg/L : LVX 바이오필름 MBEC

MBCDAP : 32 : mg/L : DAP 바이오필름 MBEC

MBCLZD : 128 : mg/L : LZD 바이오필름 MBEC

MBCCFZ : 256 : mg/L : CFZ 바이오필름 MBEC

// ------------------------------- MAXIMUM KILL RATES (planktonic) -----------

EMXVAN : 0.45 : 1/h : VAN 최대 살균속도 (시간의존·완만)

EMXRIF : 0.80 : 1/h : RIF 최대 살균속도

EMXLVX : 1.20 : 1/h : LVX 최대 살균속도 (농도의존)

EMXDAP : 2.00 : 1/h : DAP 최대 살균속도 (급속 농도의존)

EMXLZD : 0.12 : 1/h : LZD 최대 살균속도 (정균제)

EMXCFZ : 0.80 : 1/h : CFZ 최대 살균속도

EMXLZDI : 0.85 : : LZD 최대 증식억제 분율 (정균 작용의 본체)

// ---- STATE TOLERANCE MULTIPLIERS (applied ON TOP of the biofilm PHI) ------

PHPVAN : 10.0 : : VAN 지속균 추가 내약 배수

PHPRIF : 25.0 : : RIF 지속균 추가 내약 배수

PHPLVX : 25.0 : : LVX 지속균 추가 내약 배수

PHPDAP : 8.0 : : DAP 지속균 추가 내약 배수

PHPLZD : 20.0 : : LZD 지속균 추가 내약 배수

PHPCFZ : 40.0 : : CFZ 지속균 추가 내약 배수 (증식 의존 최대)

PHSVAN : 3.0 : : VAN SCV 추가 내약 배수

PHSRIF : 2.0 : : RIF SCV 추가 내약 배수

PHSLVX : 4.0 : : LVX SCV 추가 내약 배수

PHSDAP : 3.0 : : DAP SCV 추가 내약 배수

PHSLZD : 1.5 : : LZD SCV 추가 내약 배수

PHSCFZ : 6.0 : : CFZ SCV 추가 내약 배수

PHIVAN : 100.0 : : VAN 세포내 추가 내약 배수

PHIRIF : 200.0 : : RIF 세포내 추가 내약 배수 (세포내는 비증식·SCV형)

PHILVX : 5.0 : : LVX 세포내 추가 내약 배수

PHIDAP : 50.0 : : DAP 세포내 추가 내약 배수 (파고솜에서 불활성)

PHILZD : 3.0 : : LZD 세포내 추가 내약 배수

PHICFZ : 60.0 : : CFZ 세포내 추가 내약 배수

CARVAN : 0.20 : : VAN 세포내 축적비

CARRIF : 6.00 : : RIF 세포내 축적비

CARLVX : 5.00 : : LVX 세포내 축적비

CARDAP : 0.50 : : DAP 세포내 축적비

CARLZD : 1.00 : : LZD 세포내 축적비

CARCFZ : 0.10 : : CFZ 세포내 축적비

// ------------------------------------------ BACTERIAL POPULATION ----------

MUP : 0.35 : 1/h : 유주균 최대 증식속도

MUB : 0.080: 1/h : 바이오필름 증식속도

MUS : 0.050: 1/h : SCV 증식속도

MUI : 0.005: 1/h : 세포내 균 증식속도

NMAX : 1e10 : CFU : 병소 전체 최대 수용력

NBCAP : 3e9 : CFU : 임플란트 표면 최대 정착 수용력

NPCAP : 1e9 : CFU : 비부착(관절액·농양) 균 최대 수용력

NICCAP : 2e8 : CFU : 세포내 저장고 최대 수용력 (숙주세포 수 제한)

KDNAT : 0.005: 1/h : 자연 사멸속도

KATT : 0.40 : 1/h : 부착속도 계수 (유주균 -> 바이오필름)

KDISP : 0.020: 1/h : 분산속도 계수 (agr 의존)

KPER0 : 0.0008:1/h : 기저 지속균 형성속도

APER : 6.0 : : 성숙·스트레스에 의한 지속균 형성 증폭

KWAKE : 0.0040:1/h : 지속균 재활성(wake-up) 속도

KSCV0 : 0.00050:1/h : SCV 형성속도

ASCV : 8.0 : : 국소 항생제 노출에 의한 SCV 유도 증폭

KREVS : 0.0040:1/h : SCV 정상형 복귀속도

KINT0 : 0.0060:1/h : 골모세포 내재화 속도

KRELIC : 0.0100:1/h : 세포내 균 방출속도 (숙주세포 사멸 포함)

KEXT : 2.0 : 1/h : 1 CFU 미만 소멸 강제 속도 (클론 단위 적용)

NEXT : 1.0 : CFU : 소멸 문턱 (1 세포)

// ------------------------------------------------- BIOFILM MATRIX ---------

KSEPS : 0.0035: 1/h : EPS 생성속도 (성숙 시상수 ~10일)

KDEPS : 0.0002: 1/h : EPS 소실속도 (유지된 임플란트의 기질은 오래 남음)

KEPSN : 3e7 : CFU : EPS 생성 반포화 표면균량

KMAT : 0.35 : : EPS 성숙 반포화 상수

KSAIP : 0.30 : 1/h : AIP 생성속도

KDAIP : 0.20 : 1/h : AIP 소실속도

KAIPN : 1e8 : CFU : AIP 생성 반포화 균량

KAIP : 0.40 : : agr 활성 반포화 AIP

// --------------------------------------------- RESISTANCE (mutant supply) --

MURPOB : 1e-8 : : rpoB 변이 빈도 (분열당)

MUGYRA : 3e-9 : : gyrA 변이 빈도 (분열당)

FITRPOB : 0.15 : : rpoB 변이체 적합도 비용

FITGYRA : 0.10 : : gyrA 변이체 적합도 비용

// --------------------------------------------------- INNATE IMMUNITY ------

KIMM : 5e6 : CFU/h : 최대 식균 제거 용량 (완전 동원 시)

KMIMM : 1e6 : CFU : 식균 제거 포화 상수 (용량 공유)

FBFACT : 0.12 : : 이물 관련 국소 면역결핍 계수

FRUST : 0.030: : 좌절 식균작용 잔여 효율 (바이오필름)

FICIMM : 0.020: : 세포내 균에 대한 면역 효율

KPMN : 500 : cells/uL : 면역 동원 반포화 PMN 밀도

PMN0 : 50 : cells/uL : 정상 관절액 호중구 밀도

MONO0 : 150 : cells/uL : 정상 관절액 단핵구 밀도

KINPMN : 7.5 : cells/uL/h : PMN 유입 기저속도

KOUTPMN : 0.15 : 1/h : PMN 소실속도

EMXCHEM : 1200 : : 주화성 최대 동원 배수

KCHEM : 0.35 : : 주화성 반포화 신호

KTOXP : 0.35 : 1/h : 백혈구독소에 의한 PMN 사멸 최대속도

KINMON : 1.5 : cells/uL/h : 단핵구 유입 기저속도

KOUTMON : 0.010: 1/h : 단핵구 소실속도

EMXMON : 18 : : 단핵구 동원 최대 배수

KBACS : 1e7 : CFU : 세균 PAMP 신호 반포화 균량

KSMDSC : 0.030: 1/h : MDSC 동원 속도 (바이오필름 특이)

KDMDSC : 0.020: 1/h : MDSC 소실속도

KEPSM : 0.30 : : MDSC 동원 반포화 EPS

EMDSC : 0.70 : : MDSC 최대 면역억제 분율

KMD : 0.50 : : MDSC 억제 반포화

KSMAC : 0.10 : 1/h : 대식세포 활성화 속도

KDMAC : 0.050: 1/h : 대식세포 비활성화 속도

// ------------------------------------------------------- CYTOKINES --------

IL1B0 : 2.0 : pg/mL : IL-1beta 기저

TNFA0 : 5.0 : pg/mL : TNF-alpha 기저

IL60 : 3.0 : pg/mL : IL-6 기저

IL100 : 4.0 : pg/mL : IL-10 기저

KDIL1 : 0.70 : 1/h : IL-1beta 소실

KDTNF : 0.50 : 1/h : TNF-alpha 소실

KDIL6 : 0.25 : 1/h : IL-6 소실

KDIL10 : 0.30 : 1/h : IL-10 소실

EIL1 : 60 : : IL-1beta 최대 유도 배수

ETNF : 40 : : TNF-alpha 최대 유도 배수

EIL6 : 120 : : IL-6 최대 유도 배수

EIL10 : 30 : : IL-10 최대 유도 배수

// -------------------------------------------------- BIOMARKERS ------------

CRP0 : 3.0 : mg/L : CRP 기저

KDCRP : 0.0365:1/h : CRP 소실 (t1/2 19 h)

ECRP : 40 : : CRP 최대 유도 배수

KCRP : 60 : pg/mL : CRP 유도 반포화 IL-6

ESR0 : 10 : mm/h : ESR 기저

KDESR : 0.00413:1/h : ESR 소실 (t1/2 7일)

EESR : 8.0 : : ESR 최대 유도 배수

ADEF0 : 0.30 : S/CO : alpha-defensin 기저

KDADEF : 0.10 : 1/h : alpha-defensin 소실

EADEF : 12 : : alpha-defensin 최대 유도 배수

KADEF : 4000 : cells/uL : alpha-defensin 유도 반포화 PMN

// ----------------------------------------------------- BONE / OSTEOLYSIS --

KSRANK : 0.10 : 1/h : RANKL 생성 기저

KDRANK : 0.10 : 1/h : RANKL 소실

ERANK : 9.0 : : 사이토카인에 의한 RANKL 최대 유도

KSOPG : 0.10 : 1/h : OPG 생성 기저

KDOPG : 0.10 : 1/h : OPG 소실

KINOBL : 0.40 : 1/h : 골모세포 유입

KOUTOBL : 0.0040:1/h : 골모세포 소실

KAPOP : 0.0040:1/h : 골모세포 아포토시스 최대속도

KINOCL : 1.20 : 1/h : 파골세포 유입 최대

KRR : 1.00 : : RANKL/OPG 반포화

KOUTOCL : 0.0060:1/h : 파골세포 소실

KFORM : 0.020: %/h : 골 형성 속도 계수

KRES : 0.020: %/h : 골 흡수 속도 계수

KHOM : 0.00050:1/h : 골량 항상성 복귀

KLOOSE : 0.0030:1/h : 계면 이완 진행 계수

KREPAIR : 0.00050:1/h : 계면 회복 계수

KSSEQ : 0.0020:1/h : 부골(sequestrum) 형성 계수

KCSEQ : 0.00030:1/h : 부골 자연 소실

KSEQA : 0.30 : : 부골에 의한 약물 접근 차단 반포화

SEQMAX : 3.0 : : 부골 최대 축적량

// ---------------------------------------------------------- TOXICITY ------

SCR0 : 0.90 : mg/dL : 혈청 크레아티닌 기저

KELCR : 0.030: 1/h : 크레아티닌 소실

ENEPH : 0.55 : : VAN 최대 신독성 분율

AUCTHR : 600 : mg*h/L : VAN AUC24 신독성 문턱

KNEPH : 250 : mg*h/L : VAN 신독성 반포화

PLT0 : 250 : 1e9/L : 혈소판 기저

KOUTPLT : 0.00417:1/h : 혈소판 소실 (수명 10일)

EPLT : 0.60 : : LZD 최대 혈소판 생성 억제

KPLT : 9.0 : mg/L : LZD 혈소판독성 반포화 농도

ALT0 : 25 : U/L : ALT 기저

KDALT : 0.0144: 1/h : ALT 소실 (t1/2 48 h)

EALT : 4.0 : : RIF 최대 ALT 유도

KALT : 6.0 : mg/L : RIF 간독성 반포화 농도

// ------------------------------------------------ SURGERY / STRATEGY ------

TSURG1 : 1e9 : h : 1차 수술 시각 (기본: 시행 안 함)

LOGK1 : 0.0 : log10 : 1차 수술 균 제거량 (DAIR 2.5 / 1단계 4.5 / 2단계 6)

EXCH1 : 0 : : 1차 수술 임플란트 교환 여부 (1 = 교환, EPS 초기화)

TSURG2 : 1e9 : h : 2차 수술 시각 (2단계 재치환)

LOGK2 : 0.0 : log10 : 2차 수술 균 제거량

EXCH2 : 0 : : 2차 수술 임플란트 교환 여부

TSDUR : 1.0 : h : 수술 제거 적용 구간 길이

SPCF0 : 0.0 : mg : 스페이서 빠른 방출 풀 초기량

SPCS0 : 0.0 : mg : 스페이서 느린 방출 풀 초기량

// -------------------------------------------------------- ENDPOINTS -------

PSEED : 0.35 : : 생존 1 CFU 당 재발 확률 (임플란트 유지 시)

PSEEDX : 0.10 : : 생존 1 CFU 당 재발 확률 (임플란트 교환 시)

$CMT @annotated

VANC : VAN 중심구획 (mg)

VANP : VAN 말초구획 (mg)

VANB : VAN 골 유리농도 (mg/L)

RIFG : RIF 위장관 흡수구획 (mg)

RIFC : RIF 중심구획 (mg)

RIFB : RIF 골 유리농도 (mg/L)

ENZR : CYP3A4 유도 상태 (배수)

LVXG : LVX 위장관 흡수구획 (mg)

LVXC : LVX 중심구획 (mg)

LVXB : LVX 골 유리농도 (mg/L)

DAPC : DAP 중심구획 (mg)

DAPB : DAP 골 유리농도 (mg/L)

LZDG : LZD 위장관 흡수구획 (mg)

LZDC : LZD 중심구획 (mg)

LZDB : LZD 골 유리농도 (mg/L)

CFZC : CFZ 중심구획 (mg)

CFZB : CFZ 골 유리농도 (mg/L)

SPCF : 스페이서 빠른 방출 풀 (mg)

SPCS : 스페이서 느린 방출 풀 (mg)

LOCJ : 관절액 국소 항생제 농도 (mg/L)

NP : 유주균 (CFU)

NB : 바이오필름 정착균 (CFU)

NPER : 지속균 persisters (CFU)

NSCV : 소집락변이체 SCV (CFU)

NIC : 세포내 균 (CFU)

RP : rpoB 변이 유주균 (CFU)

RB : rpoB 변이 바이오필름균 (CFU)

QP : gyrA 변이 유주균 (CFU)

QB : gyrA 변이 바이오필름균 (CFU)

EPS : 바이오필름 기질 성숙도 (0-1)

AIP : agr 자가유도 펩티드 (정규화)

PMN : 관절액 호중구 (cells/uL)

MONO : 관절액 단핵구 (cells/uL)

MDSC : 골수유래 억제세포 (정규화)

MAC : 활성 대식세포 (정규화)

IL1B : IL-1beta (pg/mL)

TNFA : TNF-alpha (pg/mL)

IL6 : IL-6 (pg/mL)

IL10 : IL-10 (pg/mL)

CRP : 혈청 CRP (mg/L)

ESR : 혈청 ESR (mm/h)

ADEF : 관절액 alpha-defensin (S/CO)

RANKL : RANKL (정규화)

OPG : OPG (정규화)

OBL : 골모세포 (정규화 100)

OCL : 파골세포 (정규화 100)

BONEV : 임플란트 주위 골량 (% 기저)

LOOSEN : 계면 이완 지수 (0-100)

SEQ : 부골 / 무혈관 격막 (정규화)

SCR : 혈청 크레아티닌 (mg/dL)

PLT : 혈소판 (1e9/L)

ALT : 혈청 ALT (U/L)

AUCF : VAN AUC24 이동추정 (mg*h/L)

$MAIN

VANB_0 = 0.0; RIFB_0 = 0.0; LVXB_0 = 0.0;

DAPB_0 = 0.0; LZDB_0 = 0.0; CFZB_0 = 0.0;

ENZR_0 = 1.0;

SPCF_0 = SPCF0;

SPCS_0 = SPCS0;

LOCJ_0 = 0.0;

// Inoculum. NOTE: this is driven by the PARAMETER INOC0, not by init().

// $MAIN runs on every record, so writing a literal here would silently

// override anything passed to mrgsolve::init() -- change INOC0 instead.

NP_0 = INOC0;

NB_0 = 0.0;

NPER_0 = 0.0;

NSCV_0 = 0.0;

NIC_0 = 0.0;

RP_0 = 0.0; RB_0 = 0.0; QP_0 = 0.0; QB_0 = 0.0;

EPS_0 = 0.0;

AIP_0 = 0.0;

PMN_0 = PMN0;

MONO_0 = MONO0;

MDSC_0 = 0.02;

MAC_0 = 0.10;

IL1B_0 = IL1B0;

TNFA_0 = TNFA0;

IL6_0 = IL60;

IL10_0 = IL100;

CRP_0 = CRP0;

ESR_0 = ESR0;

ADEF_0 = ADEF0;

RANKL_0 = 1.0;

OPG_0 = 1.0;

OBL_0 = 100.0;

OCL_0 = 100.0;

BONEV_0 = 100.0;

LOOSEN_0 = 0.0;

SEQ_0 = 0.0;

SCR_0 = SCR0;

PLT_0 = PLT0;

ALT_0 = ALT0;

AUCF_0 = 0.0;

$ODE

// Guard: the stiff 10-log bacterial collapse can leave states a hair below

// zero, and pow(negative, 3) then poisons the extinction term.

if(NP < 0) NP = 0.0; if(NB < 0) NB = 0.0;

if(NPER < 0) NPER = 0.0; if(NSCV < 0) NSCV = 0.0;

if(NIC < 0) NIC = 0.0; if(RP < 0) RP = 0.0;

if(RB < 0) RB = 0.0; if(QP < 0) QP = 0.0;

if(QB < 0) QB = 0.0; if(EPS < 0) EPS = 0.0;

if(AIP < 0) AIP = 0.0; if(SEQ < 0) SEQ = 0.0;

// =====================================================================

// 0. Surgical burden-reduction pulses (Pillar 3)

// A pulse of height ln(10)*LOGK/TSDUR applied for TSDUR hours multiplies

// every bacterial state by exactly 10^-LOGK. No event plumbing needed and

// the arithmetic is exact.

// =====================================================================

double SR1 = 0.0, SR2 = 0.0, WIPE = 0.0;

if(SOLVERTIME >= TSURG1 && SOLVERTIME < (TSURG1 + TSDUR)) {

SR1 = 2.302585093*LOGK1/TSDUR;

if(EXCH1 > 0.5) WIPE = 1.0;

}

if(SOLVERTIME >= TSURG2 && SOLVERTIME < (TSURG2 + TSDUR)) {

SR2 = 2.302585093*LOGK2/TSDUR;

if(EXCH2 > 0.5) WIPE = 1.0;

}

double SURG = SR1 + SR2;

// Debridement also strips matrix; an implant exchange strips it completely.

// A RETAINED implant keeps its adherent matrix: debridement removes pus and

// necrotic soft tissue, not the biofilm bonded to metal. Only an exchange

// wipes the matrix. This asymmetry is what makes DAIR a race against EPS.

double EPSCUT = SURG*(WIPE > 0.5 ? 1.0 : 0.05);

double SEQCUT = SURG*(WIPE > 0.5 ? 1.0 : 0.10);

// =====================================================================

// 1. Pharmacokinetics

// =====================================================================

double CVAN = VANC/VCVAN;

double CVANP= VANP/VPVAN;

double CRIF = RIFC/VCRIF;

double CLVX = LVXC/VCLVX;

double CDAP = DAPC/VCDAP;

double CLZD = LZDC/VCLZD;

double CCFZ = CFZC/VCCFZ;

// Rifampicin autoinduction and its induction of linezolid clearance

double CLRIF = CLRIF0*ENZR;

double CLLZD = CLLZD0*(1.0 + RIFLZD*(ENZR - 1.0));

dxdt_VANC = -CLVAN*CVAN - QVAN*CVAN + QVAN*CVANP;

dxdt_VANP = QVAN*CVAN - QVAN*CVANP;

dxdt_RIFG = -KARIF*RIFG;

dxdt_RIFC = KARIF*FRIF*RIFG - CLRIF*CRIF;

dxdt_ENZR = KENZ*((1.0 + EIND*CRIF/(3.0 + CRIF)) - ENZR);

dxdt_LVXG = -KALVX*LVXG;

dxdt_LVXC = KALVX*FLVX*LVXG - CLLVX*CLVX;

dxdt_DAPC = -CLDAP*CDAP;

dxdt_LZDG = -KALZD*LZDG;

dxdt_LZDC = KALZD*FLZD*LZDG - CLLZD*CLZD;

dxdt_CFZC = -CLCFZ*CCFZ;

// Free BONE concentrations. FB = (AUC_bone/AUC_plasma) x fu (Pillar 2)

double FBVAN = PENVAN*FUVAN;

double FBRIF = PENRIF*FURIF;

double FBLVX = PENLVX*FULVX;

double FBDAP = PENDAP*FUDAP;

double FBLZD = PENLZD*FULZD;

double FBCFZ = PENCFZ*FUCFZ;

dxdt_VANB = KEQVAN*(FBVAN*CVAN - VANB);

dxdt_RIFB = KEQRIF*(FBRIF*CRIF - RIFB);

dxdt_LVXB = KEQLVX*(FBLVX*CLVX - LVXB);

dxdt_DAPB = KEQDAP*(FBDAP*CDAP - DAPB);

dxdt_LZDB = KEQLZD*(FBLZD*CLZD - LZDB);

dxdt_CFZB = KEQCFZ*(FBCFZ*CCFZ - CFZB);

// Local elution from antibiotic-loaded cement (two-pool, burst + matrix)

dxdt_SPCF = -KFAST*SPCF;

dxdt_SPCS = -KSLOW*SPCS;

dxdt_LOCJ = (KFAST*SPCF + KSLOW*SPCS)/VJOINT - KCLJ*LOCJ;

// Rolling 24 h vancomycin AUC estimate (for nephrotoxicity)

dxdt_AUCF = (24.0*CVAN - AUCF)/24.0;

// =====================================================================

// 2. Biofilm state variables and drug access

// =====================================================================

double EPSM = EPS/(EPS + KMAT); // maturity 0-1

double AGRA = AIP/(AIP + KAIP); // agr activity 0-1

double ACCESS= 1.0/(1.0 + SEQ/KSEQA); // sequestrum shielding

// Effective drug concentrations seen by each niche

double VANEP = (VANB + FLOCP*LOCJ);

double VANEB = (VANB + FLOCB*LOCJ)*ACCESS;

double RIFEB = RIFB*ACCESS;

double LVXEB = LVXB*ACCESS;

double DAPEB = DAPB*ACCESS;

double LZDEB = LZDB*ACCESS;

double CFZEB = CFZB*ACCESS;

// Maturity-scaled biofilm tolerance multipliers (Pillar 4)

double PBVAN = PHIMAT(MBCVAN/MICVAN, EPSM);

double PBRIF = PHIMAT(MBCRIF/MICRIF, EPSM);

double PBLVX = PHIMAT(MBCLVX/MICLVX, EPSM);

double PBDAP = PHIMAT(MBCDAP/MICDAP, EPSM);

double PBLZD = PHIMAT(MBCLZD/MICLZD, EPSM);

double PBCFZ = PHIMAT(MBCCFZ/MICCFZ, EPSM);

double MS = (MSSA > 0.5) ? 1.0 : 0.0;

// =====================================================================

// 3. Antibiotic kill by bacterial state

// Every term is EMAX * C / (EC50_state + C) with

// EC50_state = MIC x (biofilm PHI) x (state PHI).

// =====================================================================

// --- planktonic (rifampicin-susceptible, FQ-susceptible) ---

double KLP = EKILL(EMXVAN, VANEP, MICVAN)

+ EKILL(EMXRIF, RIFB , MICRIF)

+ EKILL(EMXLVX, LVXB , MICLVX)

+ EKILL(EMXDAP, DAPB , MICDAP)

+ EKILL(EMXLZD, LZDB , MICLZD)

+ MS*EKILL(EMXCFZ, CFZB, MICCFZ);

double INHP = EMXLZDI*LZDB/(MICLZD + LZDB);

// --- biofilm-embedded ---

double KLB = EKILL(EMXVAN, VANEB, MICVAN*PBVAN)

+ EKILL(EMXRIF, RIFEB, MICRIF*PBRIF)

+ EKILL(EMXLVX, LVXEB, MICLVX*PBLVX)

+ EKILL(EMXDAP, DAPEB, MICDAP*PBDAP)

+ EKILL(EMXLZD, LZDEB, MICLZD*PBLZD)

+ MS*EKILL(EMXCFZ, CFZEB, MICCFZ*PBCFZ);

double INHB = EMXLZDI*LZDEB/(MICLZD*PBLZD + LZDEB);

// --- persisters ---

double KLPER = EKILL(EMXVAN, VANEB, MICVAN*PBVAN*PHPVAN)

+ EKILL(EMXRIF, RIFEB, MICRIF*PBRIF*PHPRIF)

+ EKILL(EMXLVX, LVXEB, MICLVX*PBLVX*PHPLVX)

+ EKILL(EMXDAP, DAPEB, MICDAP*PBDAP*PHPDAP)

+ EKILL(EMXLZD, LZDEB, MICLZD*PBLZD*PHPLZD)

+ MS*EKILL(EMXCFZ, CFZEB, MICCFZ*PBCFZ*PHPCFZ);

// --- small colony variants ---

double KLSCV = EKILL(EMXVAN, VANEB, MICVAN*PBVAN*PHSVAN)

+ EKILL(EMXRIF, RIFEB, MICRIF*PBRIF*PHSRIF)

+ EKILL(EMXLVX, LVXEB, MICLVX*PBLVX*PHSLVX)

+ EKILL(EMXDAP, DAPEB, MICDAP*PBDAP*PHSDAP)

+ EKILL(EMXLZD, LZDEB, MICLZD*PBLZD*PHSLZD)

+ MS*EKILL(EMXCFZ, CFZEB, MICCFZ*PBCFZ*PHSCFZ);

// --- intracellular: cellular accumulation ratio x free bone concentration,

// shielded by the same avascular sequestrum (the canalicular reservoir

// sits inside dead cortical bone) ---

double VANIC = CARVAN*VANB*ACCESS; double RIFIC = CARRIF*RIFB*ACCESS;

double LVXIC = CARLVX*LVXB*ACCESS; double DAPIC = CARDAP*DAPB*ACCESS;

double LZDIC = CARLZD*LZDB*ACCESS; double CFZIC = CARCFZ*CFZB*ACCESS;

double KLIC = EKILL(EMXVAN, VANIC, MICVAN*PBVAN*PHIVAN)

+ EKILL(EMXRIF, RIFIC, MICRIF*PBRIF*PHIRIF)

+ EKILL(EMXLVX, LVXIC, MICLVX*PBLVX*PHILVX)

+ EKILL(EMXDAP, DAPIC, MICDAP*PBDAP*PHIDAP)

+ EKILL(EMXLZD, LZDIC, MICLZD*PBLZD*PHILZD)

+ MS*EKILL(EMXCFZ, CFZIC, MICCFZ*PBCFZ*PHICFZ);

// --- rpoB mutants: identical EXCEPT rifampicin does nothing ---

double KLRP = EKILL(EMXVAN, VANEP, MICVAN)

+ EKILL(EMXLVX, LVXB , MICLVX)

+ EKILL(EMXDAP, DAPB , MICDAP)

+ EKILL(EMXLZD, LZDB , MICLZD)

+ MS*EKILL(EMXCFZ, CFZB, MICCFZ);

double KLRB = EKILL(EMXVAN, VANEB, MICVAN*PBVAN)

+ EKILL(EMXLVX, LVXEB, MICLVX*PBLVX)

+ EKILL(EMXDAP, DAPEB, MICDAP*PBDAP)

+ EKILL(EMXLZD, LZDEB, MICLZD*PBLZD)

+ MS*EKILL(EMXCFZ, CFZEB, MICCFZ*PBCFZ);

// --- gyrA mutants: identical EXCEPT levofloxacin does nothing ---

double KLQP = EKILL(EMXVAN, VANEP, MICVAN)

+ EKILL(EMXRIF, RIFB , MICRIF)

+ EKILL(EMXDAP, DAPB , MICDAP)

+ EKILL(EMXLZD, LZDB , MICLZD)

+ MS*EKILL(EMXCFZ, CFZB, MICCFZ);

double KLQB = EKILL(EMXVAN, VANEB, MICVAN*PBVAN)

+ EKILL(EMXRIF, RIFEB, MICRIF*PBRIF)

+ EKILL(EMXDAP, DAPEB, MICDAP*PBDAP)

+ EKILL(EMXLZD, LZDEB, MICLZD*PBLZD)

+ MS*EKILL(EMXCFZ, CFZEB, MICCFZ*PBCFZ);

// =====================================================================

// 4. Innate immunity (Pillar 1)

// =====================================================================

double NTOT = NP + NB + NPER + NSCV + NIC + RP + RB + QP + QB;

double NSURF = NB + NPER + RB + QB; // surface-attached pool

double CROWD = 1.0 - NTOT/NMAX; if(CROWD < 0.0) CROWD = 0.0;

double PMNF = PMN/(PMN + KPMN);

double MDSUP = 1.0 - EMDSC*MDSC/(KMD + MDSC);

double FB = (IMPL > 0.5) ? FBFACT : 1.0;

double IMMCAP= KIMM*PMNF*FB*IMSUP*MDSUP;

// Phagocyte capacity is a SHARED resource, allocated across the accessible

// burden in proportion to how reachable each state is. Writing one

// Michaelis-Menten sink per compartment (the obvious but wrong way) makes a

// rare mutant clone face the entire immune system as if it were alone, which

// silently sterilises exactly the subpopulation the model exists to track.

double NPHAG = (NP + RP + QP)

+ FRUST*(NB + RB + QB + NSCV)

+ FICIMM*NIC;

double IMMTOT= IMMCAP*NPHAG/(KMIMM + NPHAG);

double SHR = IMMTOT/(NPHAG + 1e-30);

double IMMP = SHR*NP;

double IMMB = SHR*FRUST*NB; // frustrated phagocytosis

double IMMS = SHR*FRUST*NSCV;

double IMMI = SHR*FICIMM*NIC;

double IMMRP = SHR*RP;

double IMMRB = SHR*FRUST*RB;

double IMMQP = SHR*QP;

double IMMQB = SHR*FRUST*QB;

// =====================================================================

// 5. Bacterial state transitions

// =====================================================================

double SFREE = (IMPL > 0.5) ? (1.0 - NSURF/NBCAP) : 0.0;

if(SFREE < 0.0) SFREE = 0.0;

double ATT = KATT*SFREE;

double DISP = KDISP*AGRA;

double STRESS= KLB/(KLB + 0.05);

double KPER = KPER0*(1.0 + APER*(EPSM + STRESS));

double KSCV = KSCV0*(1.0 + ASCV*LOCJ/(LOCJ + 20.0));

double KINT = KINT0*(OBL/100.0);

// Smooth sub-single-cell extinction. Applied PER CLONE, not per phenotypic

// state: a cell can move between planktonic / sessile / persister / SCV /

// intracellular, so "fewer than one organism left" is a property of the

// lineage. Applying it per compartment annihilates the rare rpoB clone the

// instant it dips below 1 CFU in any one state and destroys Pillar 3.

double NSUSC = NP + NB + NPER + NSCV + NIC;

double NRPOB = RP + RB;

double NGYRA = QP + QB;

double EXS = KEXT/(1.0 + pow(NSUSC/NEXT, 3.0));

double EXR = KEXT/(1.0 + pow(NRPOB/NEXT, 3.0));

double EXQ = KEXT/(1.0 + pow(NGYRA/NEXT, 3.0));

// Carrying capacities: the free (joint fluid / abscess) pool and the

// intracellular reservoir are each bounded by their own niche, not only by

// the lesion total. Without this the dispersed planktonic pool overruns the

// biofilm and the infection becomes spuriously vancomycin-curable.

double NFREE = NP + RP + QP;

double CROWDP= CROWD*(1.0 - NFREE/NPCAP); if(CROWDP < 0.0) CROWDP = 0.0;

double CROWDI= CROWD*(1.0 - NIC/NICCAP); if(CROWDI < 0.0) CROWDI = 0.0;

// Growth fluxes (needed for mutant supply: mutants arise at DIVISION)

double GFP = MUP*(1.0 - INHP)*CROWDP*NP;

double GFB = MUB*(1.0 - INHB)*CROWD*NB;

double GFS = MUS*CROWD*NSCV;

double GFI = MUI*CROWDI*NIC;

double MUTR = MURPOB*(GFP + GFB); // rpoB mutant supply (Pillar 3)

double MUTQ = MUGYRA*(GFP + GFB); // gyrA mutant supply

dxdt_NP = GFP - MUTR - MUTQ

- (KDNAT + KLP + EXS + SURG)*NP

- IMMP - ATT*NP + DISP*NB - KINT*NP + KRELIC*NIC;

dxdt_NB = GFB

+ ATT*NP - DISP*NB

- (KDNAT + KLB + EXS + SURG)*NB

- IMMB - KPER*NB + KWAKE*NPER - KSCV*NB + KREVS*NSCV;

dxdt_NPER = KPER*NB - KWAKE*NPER - (KLPER + EXS + SURG)*NPER;

dxdt_NSCV = GFS + KSCV*NB - KREVS*NSCV

- (KDNAT + KLSCV + EXS + SURG)*NSCV - IMMS;

dxdt_NIC = GFI + KINT*NP - KRELIC*NIC

- (KLIC + EXS + 0.5*SURG)*NIC - IMMI;

dxdt_RP = MUTR + MUP*(1.0 - FITRPOB)*(1.0 - INHP)*CROWDP*RP

- (KDNAT + KLRP + EXR + SURG)*RP - IMMRP - ATT*RP + DISP*RB;

dxdt_RB = MUB*(1.0 - FITRPOB)*(1.0 - INHB)*CROWD*RB

+ ATT*RP - DISP*RB

- (KDNAT + KLRB + EXR + SURG)*RB - IMMRB;

dxdt_QP = MUTQ + MUP*(1.0 - FITGYRA)*(1.0 - INHP)*CROWDP*QP

- (KDNAT + KLQP + EXQ + SURG)*QP - IMMQP - ATT*QP + DISP*QB;

dxdt_QB = MUB*(1.0 - FITGYRA)*(1.0 - INHB)*CROWD*QB

+ ATT*QP - DISP*QB

- (KDNAT + KLQB + EXQ + SURG)*QB - IMMQB;

// =====================================================================

// 6. Biofilm matrix and quorum sensing

// =====================================================================

dxdt_EPS = KSEPS*(NSURF/(NSURF + KEPSN))*(1.0 - EPS) - KDEPS*EPS - EPSCUT*EPS;

dxdt_AIP = KSAIP*(NP + NB)/((NP + NB) + KAIPN) - KDAIP*AIP;

// =====================================================================

// 7. Innate immune cells, MDSC and cytokines

// =====================================================================

double BACSIG = NTOT/(NTOT + KBACS);

double ACTSIG = BACSIG*MDSUP;

double TOXL = AGRA*BACSIG;

dxdt_PMN = KINPMN*(1.0 + EMXCHEM*ACTSIG/(KCHEM + ACTSIG))

- KOUTPMN*PMN - KTOXP*TOXL*PMN;

dxdt_MONO = KINMON*(1.0 + EMXMON*BACSIG/(KCHEM + BACSIG))

- KOUTMON*MONO;

dxdt_MDSC = KSMDSC*EPS/(EPS + KEPSM) - KDMDSC*MDSC;

dxdt_MAC = KSMAC*BACSIG - KDMAC*MAC;

dxdt_IL1B = KDIL1*IL1B0*(1.0 + EIL1*ACTSIG) - KDIL1*IL1B;

dxdt_TNFA = KDTNF*TNFA0*(1.0 + ETNF*ACTSIG) - KDTNF*TNFA;

dxdt_IL6 = KDIL6*IL60 *(1.0 + EIL6*ACTSIG) - KDIL6*IL6;

dxdt_IL10 = KDIL10*IL100*(1.0 + EIL10*MDSC/(KMD + MDSC)) - KDIL10*IL10;

// =====================================================================

// 8. Systemic biomarkers (2018 ICM criteria read-outs)

// =====================================================================

dxdt_CRP = KDCRP*CRP0*(1.0 + ECRP*(IL6 - IL60)/(KCRP + (IL6 - IL60))) - KDCRP*CRP;

dxdt_ESR = KDESR*ESR0*(1.0 + EESR*(IL6 - IL60)/(KCRP + (IL6 - IL60))) - KDESR*ESR;

dxdt_ADEF = KDADEF*ADEF0*(1.0 + EADEF*(PMN - PMN0)/(KADEF + (PMN - PMN0))) - KDADEF*ADEF;

// =====================================================================

// 9. Bone: RANKL/OPG, osteoclasts, osteolysis, loosening, sequestrum

// =====================================================================

double CYTB = (TNFA - TNFA0)/(50.0 + (TNFA - TNFA0))

+ (IL1B - IL1B0)/(30.0 + (IL1B - IL1B0))

+ (IL6 - IL60 )/(80.0 + (IL6 - IL60 ));

if(CYTB < 0.0) CYTB = 0.0;

double APOS = 0.6*AGRA*BACSIG + NIC/(NIC + 1e6);

dxdt_RANKL = KSRANK*(1.0 + ERANK*CYTB/3.0) - KDRANK*RANKL;

dxdt_OPG = KSOPG*(OBL/100.0) - KDOPG*OPG;

dxdt_OBL = KINOBL - KOUTOBL*OBL - KAPOP*APOS*OBL;

double RR = RANKL/(OPG + 1e-6);

dxdt_OCL = KINOCL*RR/(KRR + RR) - KOUTOCL*OCL;

dxdt_BONEV = KFORM*(OBL/100.0) - KRES*(OCL/100.0) + KHOM*(100.0 - BONEV);

double BLOSS = 100.0 - BONEV; if(BLOSS < 0.0) BLOSS = 0.0;

dxdt_LOOSEN= KLOOSE*BLOSS*(1.0 - LOOSEN/100.0) - KREPAIR*LOOSEN;

dxdt_SEQ = KSSEQ*(KRES*(OCL/100.0))*BACSIG*10.0*(1.0 - SEQ/SEQMAX)

- KCSEQ*SEQ - SEQCUT*SEQ;

// =====================================================================

// 10. Drug toxicity

// =====================================================================

double EXAUC = AUCF - AUCTHR; if(EXAUC < 0.0) EXAUC = 0.0;

double NEPH = ENEPH*EXAUC/(KNEPH + EXAUC);

dxdt_SCR = KELCR*SCR0 - KELCR*(1.0 - NEPH)*SCR;

dxdt_PLT = KOUTPLT*PLT0*(1.0 - EPLT*CLZD/(KPLT + CLZD)) - KOUTPLT*PLT;

dxdt_ALT = KDALT*ALT0*(1.0 + EALT*CRIF/(KALT + CRIF)) - KDALT*ALT;

$TABLE

double NTOTAL = NP + NB + NPER + NSCV + NIC + RP + RB + QP + QB;

if(NTOTAL < 0.0) NTOTAL = 0.0;

double NSUSC = NP + NB + NPER + NSCV + NIC;

double NRES = RP + RB + QP + QB;

capture LOGNTOT = log10(NTOTAL + 1e-12);

capture LOGNP = log10(NP + 1e-12);

capture LOGNB = log10(NB + 1e-12);

capture LOGNPER = log10(NPER+ 1e-12);

capture LOGNSCV = log10(NSCV+ 1e-12);

capture LOGNIC = log10(NIC + 1e-12);

capture LOGNRES = log10(NRES+ 1e-12);

capture RESFRAC = NRES/(NTOTAL + 1e-12);

// --- Pillar 2 read-out: achievable free bone concentration over MBEC -------

capture CPVAN = VANC/VCVAN;

capture CPRIF = RIFC/VCRIF;

capture CPLVX = LVXC/VCLVX;

capture CPDAP = DAPC/VCDAP;

capture CPLZD = LZDC/VCLZD;

capture CPCFZ = CFZC/VCCFZ;

capture RATVAN = (VANB + FLOCB*LOCJ)/MBCVAN;

capture RATRIF = RIFB/MBCRIF;

capture RATLVX = LVXB/MBCLVX;

capture RATDAP = DAPB/MBCDAP;

capture RATLZD = LZDB/MBCLZD;

capture RATCFZ = CFZB/MBCCFZ;

capture RATMAX = RATVAN;

if(RATRIF > RATMAX) RATMAX = RATRIF;

if(RATLVX > RATMAX) RATMAX = RATLVX;

if(RATDAP > RATMAX) RATMAX = RATDAP;

if(RATLZD > RATMAX) RATMAX = RATLZD;

if(MSSA > 0.5 && RATCFZ > RATMAX) RATMAX = RATCFZ;

// --- Pillar 3 read-out: mutant supply --------------------------------------

capture PRPOB = 1.0 - exp(-MURPOB*NTOTAL); // P(pre-existing rpoB mutant)

capture PGYRA = 1.0 - exp(-MUGYRA*NTOTAL);

capture EMUTR = MURPOB*NTOTAL; // expected rpoB mutants present

// --- Pillar 4 read-out: biofilm maturity ------------------------------------

capture EPSMAT = EPS/(EPS + KMAT);

// --- clinical / diagnostic read-outs ---------------------------------------

capture SYNWBC = PMN + MONO;

capture PMNPCT = 100.0*PMN/(PMN + MONO + 1e-9);

// 2018 ICM minor-criteria score (CRP 2, D-dimer 2, synovial WBC 3,

// PMN% 2, alpha-defensin 3, LE 3): >=6 infected, 2-5 inconclusive

double SC = 0.0;

if(CRP > 10.0) SC += 2.0;

if(SYNWBC > 3000.0)SC += 3.0;

if(PMNPCT > 80.0) SC += 2.0;

if(ADEF > 1.0) SC += 3.0;

capture ICMSC = SC;

// Cure probability: PSEED is the chance that ONE surviving CFU regrows.

double PS = (EXCH1 > 0.5 || EXCH2 > 0.5) ? PSEEDX : PSEED;

capture PCURE = exp(-PS*NTOTAL);

capture PFAIL = 1.0 - exp(-PS*NTOTAL);

capture CRCL = 100.0*(SCR0/SCR); // crude creatinine-clearance proxy

capture AKIFL = (SCR > 1.5*SCR0) ? 1.0 : 0.0;

capture TCPFL = (PLT < 100.0) ? 1.0 : 0.0;

capture HEPFL = (ALT > 3.0*ALT0) ? 1.0 : 0.0;

pji_mod <- mrgsolve::mcode_cache("pji", pji_code)

pji_mod <- mrgsolve::update(pji_mod, end = 4320, delta = 4,

atol = 1e-8, rtol = 1e-6, maxsteps = 500000)

## =============================================================================

## DOSING BUILDERS

## Standard adult regimens; amt in mg, time in hours.

## =============================================================================

DAY <- 24

ev_van <- function(start = 0, weeks = 6, amt = 1000, ii = 12)

ev(time = start, cmt = "VANC", amt = amt, ii = ii, addl = ceiling(weeks*7*24/ii) - 1)

ev_rif <- function(start = 0, weeks = 12, amt = 450, ii = 12)

ev(time = start, cmt = "RIFG", amt = amt, ii = ii, addl = ceiling(weeks*7*24/ii) - 1)

ev_lvx <- function(start = 0, weeks = 12, amt = 750, ii = 24)

ev(time = start, cmt = "LVXG", amt = amt, ii = ii, addl = ceiling(weeks*7*24/ii) - 1)

ev_dap <- function(start = 0, weeks = 6, amt = 560, ii = 24)

ev(time = start, cmt = "DAPC", amt = amt, ii = ii, addl = ceiling(weeks*7*24/ii) - 1)

ev_lzd <- function(start = 0, weeks = 6, amt = 600, ii = 12)

ev(time = start, cmt = "LZDG", amt = amt, ii = ii, addl = ceiling(weeks*7*24/ii) - 1)

ev_cfz <- function(start = 0, weeks = 6, amt = 2000, ii = 8)

ev(time = start, cmt = "CFZC", amt = amt, ii = ii, addl = ceiling(weeks*7*24/ii) - 1)

## =============================================================================

## SCENARIOS

## All start from a 100 CFU intra-operative contamination of a hip/knee

## prosthesis at t = 0 and are simulated for 180 days (4320 h).

## Diagnosis is assumed at day 21 (acute post-operative PJI) unless stated.

## =============================================================================

DX <- 21*DAY # diagnosis / surgery time for the acute presentations

DXC <- 90*DAY # delayed (chronic) presentation

scen <- list(

## 1. Natural history, no treatment ---------------------------------------

"01_무치료 자연경과" = list(

par = list(),

ev = ev(time = 0, cmt = "NP", amt = 0)),

## 2. Antibiotics only, implant retained, NO surgery -----------------------

## ("medical management" — the arm that shows Pillar 2 alone is not enough)

"02_수술없이 반코마이신 단독" = list(

par = list(),

ev = ev_van(DX, 6)),

## 3. DAIR + vancomycin monotherapy 6 weeks --------------------------------

"03_DAIR + 반코마이신 6주" = list(