qsp/osteoarthritis/oa_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

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

## Osteoarthritis (OA) QSP Model — mrgsolve Implementation

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

## Disease: Primary knee osteoarthritis (KOA)

## Scope : Chondrocyte biology · ECM catabolism · Synovitis ·

## Subchondral bone remodelling · Pain signalling ·

## Drug PK/PD (NSAID · IA-CS · HA · Sprifermin · Tanezumab)

## Version : 1.0 (2026-06-23)

## Key References (parameter calibration):

## Thijssen E et al. Osteoarthritis Cartilage 2015 (MMP-13 dynamics)

## Hunter DJ et al. NEJM 2019 (OA pathophysiology review)

## Lohmander LS et al. Ann Rheum Dis 2005 (uCTX-II biomarker)

## Berenbaum F et al. Nat Rev Rheumatol 2013 (OA inflammation)

## Hochberg MC et al. Osteoarthritis Cartilage 2017 (JSW loss)

## Kloppenburg M et al. Ann Rheum Dis 2020 (OARSI guidelines)

## Dahlberg L et al. Lancet 2016 (sprifermin FORWARD RCT)

## Schnitzer TJ et al. NEJM 2019 (tanezumab Phase-3)

## Conaghan PG et al. Arthritis Rheum 2016 (celecoxib PK/PD)

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

library(mrgsolve)

library(dplyr)

library(ggplot2)

library(tidyr)

## ─────────────────────────────────────────────────────────────

## 1. MODEL CODE STRING

## ─────────────────────────────────────────────────────────────

oa_code <- '

$PROB

Osteoarthritis QSP Model

- 20 ODE compartments

- Drug PK: NSAID (celecoxib), IA corticosteroid, HA, Sprifermin, Tanezumab

- Disease PD: IL-1b, MMP-13, ADAMTS-5, Collagen II, Aggrecan,

Chondrocyte, Synovitis, Osteoclast, JSW, PGE2, Pain, uCTX-II, COMP

$PARAM

// ── Baseline/Normal States ──────────────────────────────────

ColII_0 = 100 // Collagen II baseline (% normal)

Agg_0 = 100 // Aggrecan baseline (% normal)

Chondro_0 = 100 // Chondrocyte count baseline (% normal)

JSW_0 = 5.5 // Joint space width baseline (mm), typical KOA at KL2

OC_0 = 1.0 // Osteoclast activity baseline (normalized)

OB_0 = 1.0 // Osteoblast activity baseline (normalized)

Syn_0 = 0.5 // Synovitis score baseline (0-3)

// ── Disease Progression Parameters ──────────────────────────

k_IL1b_syn = 0.10 // IL-1b synthesis rate (pg/mL/day) from synovitis

k_IL1b_deg = 0.35 // IL-1b elimination rate (/day), t1/2 ~2d

k_TNFa_syn = 0.08 // TNF-a synthesis (pg/mL/day)

k_TNFa_deg = 0.40 // TNF-a elimination (/day)

k_MMP13_syn = 0.15 // MMP-13 synthesis (ng/mL/day), IL1b/TNFa driven

k_MMP13_deg = 0.30 // MMP-13 elimination (/day)

k_ADAM5_syn = 0.12 // ADAMTS-5 synthesis (/day)

k_ADAM5_deg = 0.28 // ADAMTS-5 elimination (/day)

K_IL1_mmp = 2.0 // EC50 for IL-1b→MMP-13 (pg/mL)

Emax_IL1_mmp = 3.0 // Emax for IL-1b→MMP-13 fold induction

K_IL1_adam5 = 2.5 // EC50 for IL-1b→ADAMTS-5 (pg/mL)

k_ColII_syn = 0.008 // Collagen II synthesis (/day), Chondro-dependent

k_ColII_deg = 0.12 // Collagen II MMP-13-dependent degradation

K_mmp_col = 2.0 // EC50 for MMP-13→ColII deg (ng/mL)

k_Agg_syn = 0.010 // Aggrecan synthesis (/day)

k_Agg_deg = 0.15 // Aggrecan ADAMTS-5-dependent deg (/day per normalized unit)

K_adam5_agg = 1.5 // EC50 for ADAMTS-5→Aggrecan (normalized)

k_Chondro_prof= 0.003 // Chondrocyte proliferation rate (/day)

k_Chondro_apt= 0.006 // Chondrocyte apoptosis rate (/day) baseline + IL1b-driven

Emax_IL1_apt = 2.0 // Emax for IL-1b-driven apoptosis

K_IL1_apt = 5.0 // EC50 for IL-1b→apoptosis (pg/mL)

k_Syn_syn = 0.05 // Synovitis genesis (/day), ECM-loss driven

k_Syn_deg = 0.10 // Synovitis resolution (/day)

Emax_ecm_syn = 1.5 // ECM loss → synovitis amplification

K_ecm_syn = 30.0 // EC50 for ECM loss → synovitis (% loss)

k_OC_form = 0.08 // Osteoclast formation rate (RANKL/OPG driven)

k_OC_deg = 0.15 // Osteoclast apoptosis rate (/day)

Emax_IL1_OC = 1.8 // IL-1b-driven RANKL → OC increase Emax

K_IL1_OC = 3.0 // EC50 (pg/mL)

k_OB_form = 0.06 // Osteoblast formation rate

k_OB_deg = 0.10 // Osteoblast elimination

k_JSW_loss = 0.0012 // JSW loss rate (mm/day), MMP-13 and ADAMTS-5 driven

k_JSW_repair = 0.0001 // Very slow JSW passive repair

k_PGE2_syn = 0.20 // PGE2 synthesis (ng/mL/day), COX-2/IL-1b driven

k_PGE2_deg = 0.60 // PGE2 elimination (/day), t1/2~28h

k_Pain_syn = 0.08 // Pain generation rate (VAS units/day)

k_Pain_deg = 0.12 // Pain resolution rate (/day)

Pain_baseline= 10.0 // Residual/structural pain floor (VAS)

Emax_PGE2_pain=60.0 // Emax PGE2→pain contribution (VAS units)

K_PGE2_pain = 0.5 // EC50 PGE2→pain (ng/mL)

Emax_Syn_pain= 20.0 // Synovitis → pain Emax

K_Syn_pain = 1.0 // EC50 synovitis

k_CTXII_syn = 0.05 // uCTX-II synthesis (nmol/mmolCr/day), ColII deg product

k_CTXII_deg = 0.25 // uCTX-II elimination (/day)

k_COMP_rel = 0.04 // COMP release rate (μg/mL/day)

k_COMP_deg = 0.20 // COMP elimination (/day)

// ── NSAID PK/PD (Celecoxib 200 mg) ──────────────────────────

ka_nsaid = 1.20 // Absorption rate constant (/h)

F_nsaid = 0.74 // Oral bioavailability (fraction)

Vd_nsaid = 455 // Volume of distribution (L), population mean

CL_nsaid = 27.7 // Clearance (L/h), t1/2≈11.5h

Kp_nsaid_jt = 0.40 // Plasma-to-joint partition (synovial fluid:plasma ~0.4)

IC50_cox2 = 0.042 // IC50 celecoxib for COX-2 (μg/mL)

Emax_cox2 = 0.95 // Max COX-2 inhibition fraction

hill_cox2 = 1.2 // Hill coefficient

// Naproxen alternative: ka=1.1/h, F=0.99, Vd=11L, CL=0.55L/h, IC50=0.28μg/mL

// ── IA Corticosteroid PK/PD (Triamcinolone acetonide 40 mg) ─

k_IACS_abs = 0.008 // Absorption from joint (slow, t1/2≈3.5d) (/h)

k_IACS_el = 0.25 // Plasma elimination (/h), t1/2~2.7h

Vd_IACS = 99 // Plasma Vd (L/70kg)

EC50_GR = 0.05 // EC50 for GR-mediated NF-kB inhibition (μg/mL joint)

Emax_GR_NF = 0.75 // Max NF-kB inhibition by GCS

// ── IA Hyaluronic Acid PK ────────────────────────────────────

k_HA_deg = 0.041 // HA degradation in joint (/h), t1/2≈17h

Emax_HA_pain = 0.35 // HA viscosupplement max pain reduction fraction

K_HA_pain = 0.8 // EC50 HA joint conc (mg/mL)

Emax_HA_IL1 = 0.25 // HA → IL-1b reduction Emax

// ── Sprifermin (FGF-18) PK/PD ───────────────────────────────

k_Sprif_deg = 0.044 // Elimination from joint (/h), t1/2≈16 min converted to /h = ln2/(16/60h)

Emax_FGF_col = 0.40 // Sprifermin max ColII synthesis increase fraction (FORWARD RCT)

K_FGF18 = 0.002 // EC50 (μg/mL joint)

// ── Tanezumab (anti-NGF mAb) PK/PD ──────────────────────────

ka_tanz = 0.0167 // SC absorption (/h), t1/2_abs≈41h → F=0.73

F_tanz = 0.73 // SC bioavailability

Vd_tanz = 7.8 // Vd (L/70kg) for IgG

CL_tanz = 0.0133 // CL (/h), t1/2≈23d

Emax_tanz_pain=45.0 // Max pain reduction (VAS units, tanezumab anti-NGF)

K_tanz_NGF = 0.001 // EC50 for tanezumab plasma→NGF binding

// ── KL Grade / Disease Severity Modifier ─────────────────────

KL_grade = 2 // Kellgren-Lawrence grade at baseline (1-4)

// KL modifies disease load: KL_mult = 0.5 + 0.25*KL_grade

KL_mult = 1.0 // Runtime calculated from KL_grade

// Age effect

Age_yr = 62 // Patient age (years)

k_age_mod = 0.005 // Age-dependent disease rate modifier (/yr above 45)

// ── Input Flags ──────────────────────────────────────────────

NSAID_flag = 0 // 1 = NSAID dosing active

IACS_flag = 0 // 1 = IA corticosteroid given

HA_flag = 0 // 1 = IA HA viscosupplementation

Sprif_flag = 0 // 1 = Sprifermin dosing

Tanz_flag = 0 // 1 = Tanezumab dosing

$INIT

// ── Initial States ───────────────────────────────────────────

// Drug PK compartments (all zero at t=0)

A_NSAID_gut = 0

A_NSAID_plasma = 0

A_NSAID_joint = 0

A_IACS_joint = 0

A_IACS_plasma = 0

A_HA_joint = 0

A_Sprif_joint = 0

A_Tanz_depot = 0

A_Tanz_plasma = 0

// ── Disease State Initial Conditions ─────────────────────────

IL1b = 5.0 // pg/mL (elevated vs normal ~1 pg/mL in mild KOA)

TNFa = 3.0 // pg/mL

MMP13 = 8.0 // ng/mL (elevated in OA synovial fluid)

ADAM5 = 2.0 // normalized ADAMTS-5 activity

ColII = 70.0 // % normal (KL2 = moderate loss)

Aggrecan= 65.0 // % normal

Chondro = 75.0 // % normal chondrocyte count

Synovitis=1.0 // 0–3 scale (mild-moderate)

OC_act = 1.3 // osteoclast activity (slightly elevated)

OB_act = 1.1 // osteoblast activity

JSW = 3.8 // mm (mild joint space narrowing from 5.5 baseline)

PGE2_jt = 1.5 // ng/mL joint PGE2

VASPain = 45.0 // VAS pain 0-100 (moderate)

uCTXII = 0.40 // nmol/mmolCr urine CTX-II

COMP_s = 12.0 // μg/mL serum COMP

$ODE

// ─────────────────────────────────────────────────────────────

// Helper variables

// ─────────────────────────────────────────────────────────────

double age_mod = 1.0 + k_age_mod * (Age_yr - 45.0);

double KL_mod = 0.5 + 0.25 * KL_grade;

// NSAID plasma concentration (μg/mL)

double C_NSAID_plasma = A_NSAID_plasma / Vd_nsaid;

double C_NSAID_joint = A_NSAID_joint / (Vd_nsaid * 0.01); // ~1% of total in joint

// COX-2 inhibition fraction (Hill equation)

double COX2_inh = Emax_cox2 * pow(C_NSAID_joint, hill_cox2) /

(pow(IC50_cox2, hill_cox2) + pow(C_NSAID_joint, hill_cox2));

double COX2_eff = 1.0 - COX2_inh * NSAID_flag; // residual COX-2 activity

// IA corticosteroid joint concentration (μg/mL)

double C_IACS = A_IACS_joint;

// GR-mediated NF-kB inhibition

double GR_inh = (Emax_GR_NF * C_IACS / (EC50_GR + C_IACS)) * IACS_flag;

double NF_kB_eff = 1.0 - GR_inh; // multiplier on NF-kB driven processes

// HA joint concentration (mg/mL)

double C_HA = A_HA_joint;

double HA_pain_inh = Emax_HA_pain * C_HA / (K_HA_pain + C_HA) * HA_flag;

double HA_IL1_inh = Emax_HA_IL1 * C_HA / (K_HA_pain + C_HA) * HA_flag;

// Sprifermin joint concentration (μg/mL)

double C_Sprif = A_Sprif_joint;

double FGF18_eff = Emax_FGF_col * C_Sprif / (K_FGF18 + C_Sprif) * Sprif_flag;

// Tanezumab plasma concentration (μg/mL)

double C_Tanz = A_Tanz_plasma / Vd_tanz;

// Tanezumab sequesters NGF → proportional pain reduction

double Tanz_pain_red = Emax_tanz_pain * C_Tanz / (K_tanz_NGF + C_Tanz) * Tanz_flag;

// ECM loss (0–100, where 0=normal, 100=complete loss)

double ECM_loss = 100.0 - 0.5*(ColII + Aggrecan);

double ECM_loss_pos = (ECM_loss < 0) ? 0.0 : ECM_loss;

// ─────────────────────────────────────────────────────────────

// DRUG PK ODEs

// ─────────────────────────────────────────────────────────────

// NSAID

dxdt_A_NSAID_gut = -ka_nsaid * A_NSAID_gut;

dxdt_A_NSAID_plasma = F_nsaid * ka_nsaid * A_NSAID_gut * NSAID_flag

- (CL_nsaid / Vd_nsaid) * A_NSAID_plasma

- Kp_nsaid_jt * (CL_nsaid / Vd_nsaid) * A_NSAID_plasma;

dxdt_A_NSAID_joint = Kp_nsaid_jt * (CL_nsaid / Vd_nsaid) * A_NSAID_plasma

- (CL_nsaid / Vd_nsaid) * A_NSAID_joint;

// IA Corticosteroid

dxdt_A_IACS_joint = -k_IACS_abs * A_IACS_joint;

dxdt_A_IACS_plasma = k_IACS_abs * A_IACS_joint - k_IACS_el * A_IACS_plasma;

// IA Hyaluronic Acid

dxdt_A_HA_joint = -k_HA_deg * A_HA_joint;

// Sprifermin (IA FGF-18)

dxdt_A_Sprif_joint = -k_Sprif_deg * A_Sprif_joint;

// Tanezumab (SC mAb)

dxdt_A_Tanz_depot = -ka_tanz * A_Tanz_depot;

dxdt_A_Tanz_plasma = F_tanz * ka_tanz * A_Tanz_depot - (CL_tanz / Vd_tanz) * A_Tanz_plasma;

// ─────────────────────────────────────────────────────────────

// DISEASE PD ODEs

// ─────────────────────────────────────────────────────────────

// IL-1β (pg/mL) — master catabolic cytokine

// Synthesis driven by synovitis, age/KL severity; inhibited by GCS and HA

double IL1b_pos = (IL1b < 0.01) ? 0.01 : IL1b;

double IL1b_syn_rate = k_IL1b_syn * Synovitis * age_mod * KL_mod

* NF_kB_eff * (1.0 - HA_IL1_inh);

dxdt_IL1b = IL1b_syn_rate - k_IL1b_deg * IL1b;

// TNF-α (pg/mL) — co-driver with IL-1b

double TNFa_syn_rate = k_TNFa_syn * Synovitis * NF_kB_eff;

dxdt_TNFa = TNFa_syn_rate - k_TNFa_deg * TNFa;

// MMP-13 (ng/mL) — dominant collagenase in OA cartilage

// Induced by IL-1b, TNFa; inhibited by TIMP/GCS

double IL1b_TNFa_stim = (IL1b / (K_IL1_mmp + IL1b)) * Emax_IL1_mmp

+ TNFa / (1.0 + TNFa);

double MMP13_syn_rate = k_MMP13_syn * IL1b_TNFa_stim * NF_kB_eff;

dxdt_MMP13 = MMP13_syn_rate - k_MMP13_deg * MMP13;

// ADAMTS-5 (normalized activity) — dominant aggrecanase

double ADAM5_syn_rate = k_ADAM5_syn * (IL1b / (K_IL1_adam5 + IL1b))

* NF_kB_eff;

dxdt_ADAM5 = ADAM5_syn_rate - k_ADAM5_deg * ADAM5;

// Collagen Type II (% normal, 0–100)

// Synthesis: chondrocyte-dependent, FGF18 anabolic boost

// Degradation: MMP-13 dependent (Hill), floor at 0

double ColII_pos = (ColII < 0.5) ? 0.5 : ColII;

double col_syn = k_ColII_syn * (Chondro / Chondro_0) * (1.0 + FGF18_eff);

double col_deg = k_ColII_deg * MMP13 / (K_mmp_col + MMP13) * ColII_pos * age_mod;

dxdt_ColII = col_syn - col_deg;

// Aggrecan (% normal, 0–100)

double Agg_pos = (Aggrecan < 0.5) ? 0.5 : Aggrecan;

double agg_syn = k_Agg_syn * (Chondro / Chondro_0) * (1.0 + 0.5*FGF18_eff);

double agg_deg = k_Agg_deg * ADAM5 / (K_adam5_agg + ADAM5) * Agg_pos * age_mod;

dxdt_Aggrecan = agg_syn - agg_deg;

// Chondrocyte population (% normal)

double Chondro_pos = (Chondro < 1.0) ? 1.0 : Chondro;

double chondro_prolif = k_Chondro_prof * Chondro_pos;

double chondro_apopt = (k_Chondro_apt + k_Chondro_apt * Emax_IL1_apt *

IL1b / (K_IL1_apt + IL1b)) * Chondro_pos * age_mod;

dxdt_Chondro = chondro_prolif - chondro_apopt;

// Synovitis score (0–3)

// Driven by ECM damage (FN fragments, DAMP release), amplified by macrophage

// Resolved by anti-inflammatory mediators; GCS reduces

double Syn_drive = k_Syn_syn * (1.0 + Emax_ecm_syn * ECM_loss_pos /

(K_ecm_syn + ECM_loss_pos)) * age_mod;

double Syn_res = k_Syn_deg * Synovitis * NF_kB_eff;

dxdt_Synovitis = Syn_drive - Syn_res;

// Osteoclast activity (normalized)

// RANKL/OPG imbalance driven by IL-1b, TNFa

double OC_drive = k_OC_form * (1.0 + Emax_IL1_OC * IL1b / (K_IL1_OC + IL1b));

dxdt_OC_act = OC_drive - k_OC_deg * OC_act;

// Osteoblast activity (normalized)

// Slightly activated by TGF-b from OC; GCS suppresses new bone

double OB_drive = k_OB_form * (1.0 + 0.2 * OC_act) * NF_kB_eff;

dxdt_OB_act = OB_drive - k_OB_deg * OB_act;

// Joint Space Width (mm)

// Loss driven by ECM degradation and OC activity; very slow passive

double JSW_pos = (JSW < 0.5) ? 0.5 : JSW;

double jsw_loss = k_JSW_loss * (MMP13 / (K_mmp_col + MMP13) +

ADAM5 / (K_adam5_agg + ADAM5)) * JSW_pos * age_mod;

double jsw_gain = k_JSW_repair * (1.0 + FGF18_eff);

dxdt_JSW = jsw_gain - jsw_loss;

// PGE2 in joint (ng/mL)

// Driven by COX-2 activity (which is inhibited by NSAID)

double PGE2_syn = k_PGE2_syn * IL1b / (2.0 + IL1b) * COX2_eff;

dxdt_PGE2_jt = PGE2_syn - k_PGE2_deg * PGE2_jt;

// VAS Pain Score (0–100)

// PGE2 (inflammatory pain), synovitis, structural pain, central sensitization

// Reduced by NSAID (via PGE2), HA (viscosupplementation), tanezumab (NGF)

double PGE2_pain = Emax_PGE2_pain * PGE2_jt / (K_PGE2_pain + PGE2_jt);

double Syn_pain = Emax_Syn_pain * Synovitis / (K_Syn_pain + Synovitis);

double Struct_pain = Pain_baseline * (1.0 - JSW / JSW_0); // structural pain

double Pain_target = PGE2_pain + Syn_pain + Struct_pain

- HA_pain_inh * 20.0

- Tanz_pain_red;

double Pain_target_clamp = (Pain_target < 0.0) ? 0.0 :

(Pain_target > 100.0) ? 100.0 : Pain_target;

dxdt_VASPain = k_Pain_syn * (Pain_target_clamp - VASPain);

// uCTX-II (nmol/mmolCr) — urinary type II collagen degradation biomarker

// Reflects active collagen catabolism; released when MMP-13 cleaves ColII

double CTX2_syn = k_CTXII_syn * col_deg * 10.0; // scaled to nM range

dxdt_uCTXII = CTX2_syn - k_CTXII_deg * uCTXII;

// Serum COMP (μg/mL) — reflects cartilage stress/damage

double COMP_drive = k_COMP_rel * (1.0 + Synovitis + MMP13 / 5.0);

dxdt_COMP_s = COMP_drive - k_COMP_deg * COMP_s;

$TABLE

// ── Derived Outputs ─────────────────────────────────────────

double COX2_pct_inh = COX2_inh * NSAID_flag * 100.0;

// KOOS (0–100 scale, higher = better)

// Components: pain, symptoms, ADL, sport, QoL

// Approximate from VAS pain and JSW

double KOOS_pain = 100.0 - VASPain * 0.60;

double KOOS_func = 50.0 + (JSW / JSW_0) * 30.0 - (Synovitis / 3.0) * 10.0;

double KOOS_est = 0.6 * KOOS_pain + 0.4 * KOOS_func;

KOOS_est = (KOOS_est < 0) ? 0 : (KOOS_est > 100) ? 100 : KOOS_est;

// WOMAC pain subscale (0–20, lower = better)

double WOMAC_pain_sub = VASPain / 5.0;

// Cartilage volume estimate (% baseline)

double CartVol_pct = 0.5 * (ColII / ColII_0) + 0.5 * (Aggrecan / Agg_0);

// ECM degradation score

double ECM_degradation = 100.0 - CartVol_pct * 100.0;

// NSAID plasma concentration

double C_nsaid_pl = A_NSAID_plasma / Vd_nsaid;

double C_nsaid_jt = A_NSAID_joint / (Vd_nsaid * 0.01);

double C_IACS_jt = A_IACS_joint;

double C_HA_jt = A_HA_joint;

double C_Sprif_jt = A_Sprif_joint;

double C_Tanz_pl = A_Tanz_plasma / Vd_tanz;

// GR inhibition percentage

double GR_inhibition_pct = ((Emax_GR_NF * C_IACS_jt / (EC50_GR + C_IACS_jt))

* IACS_flag) * 100.0;

$CAPTURE

COX2_pct_inh C_nsaid_pl C_nsaid_jt C_IACS_jt C_HA_jt C_Sprif_jt C_Tanz_pl

GR_inhibition_pct KOOS_est WOMAC_pain_sub CartVol_pct ECM_degradation

IL1b TNFa MMP13 ADAM5 ColII Aggrecan Chondro Synovitis OC_act OB_act

JSW PGE2_jt VASPain uCTXII COMP_s

## ─────────────────────────────────────────────────────────────

## 2. COMPILE MODEL

## ─────────────────────────────────────────────────────────────

mod <- mcode("oa_qsp", oa_code)

## ─────────────────────────────────────────────────────────────

## 3. DOSING REGIMENS

## ─────────────────────────────────────────────────────────────

# Simulation period: 2 years (730 days)

sim_end <- 730

# NSAID: Celecoxib 200 mg BID (every 12h), continuous

# Dose = 200 mg; Vd = 455 L → 200 mg dose input

nsaid_regimen <- ev(amt = 200, ii = 12, addl = sim_end * 2, cmt = "A_NSAID_gut")

# IA Corticosteroid: Triamcinolone acetonide 40 mg Q3 months (4 injections/year)

# 40 mg dose into IACS joint compartment

iacs_regimen <- ev(time = c(0, 91, 182, 273, 365, 456, 547, 638),

amt = 40, cmt = "A_IACS_joint")

# IA Hyaluronic Acid: Synvisc 16 mg × 3 weekly (2 courses, at 0 and 6m)

ha_regimen <- ev(time = c(0, 7, 14, 182, 189, 196),

amt = 16, cmt = "A_HA_joint")

# Sprifermin (FGF-18): 30 μg IA every 12 weeks × 5 (FORWARD RCT inspired)

# Dose in μg directly into joint

sprif_regimen <- ev(time = c(0, 84, 168, 252, 336, 420, 504, 588, 672),

amt = 0.030, cmt = "A_Sprif_joint")

# Tanezumab: 2.5 mg SC every 8 weeks (anti-NGF mAb, Phase-3 inspired)

tanz_regimen <- ev(time = c(0, 56, 112, 168, 224, 280, 336, 392, 448,

504, 560, 616, 672, 728),

amt = 2.5, cmt = "A_Tanz_depot")

## ─────────────────────────────────────────────────────────────

## 4. TREATMENT SCENARIOS

## ─────────────────────────────────────────────────────────────

run_scenario <- function(mod, regimen, params_override = list(), label = "Scenario") {

p_mod <- do.call(param, c(list(mod), params_override))

out <- mrgsim(p_mod, ev = regimen, end = sim_end, delta = 1,

carry_out = "evid,amt,cmt")

as_tibble(out) %>% mutate(Scenario = label)

}

# Scenario 1: Disease progression, no treatment

scen1 <- run_scenario(mod, ev(),

params_override = list(NSAID_flag=0, IACS_flag=0, HA_flag=0, Sprif_flag=0, Tanz_flag=0),

label = "1. No Treatment (Natural History)")

# Scenario 2: NSAID continuous (Celecoxib 200 mg BID)

scen2 <- run_scenario(mod, nsaid_regimen,

params_override = list(NSAID_flag=1, IACS_flag=0, HA_flag=0, Sprif_flag=0, Tanz_flag=0),

label = "2. Celecoxib 200mg BID")

# Scenario 3: IA Corticosteroid (Triamcinolone 40mg Q3mo)

scen3 <- run_scenario(mod, iacs_regimen,

params_override = list(NSAID_flag=0, IACS_flag=1, HA_flag=0, Sprif_flag=0, Tanz_flag=0),

label = "3. IA Triamcinolone 40mg Q3mo")

# Scenario 4: IA Hyaluronic Acid (viscosupplementation)

scen4 <- run_scenario(mod, ha_regimen,

params_override = list(NSAID_flag=0, IACS_flag=0, HA_flag=1, Sprif_flag=0, Tanz_flag=0),

label = "4. IA Hyaluronic Acid 3x")

# Scenario 5: Sprifermin (DMOAD, cartilage structural modification)

scen5 <- run_scenario(mod, sprif_regimen,

params_override = list(NSAID_flag=0, IACS_flag=0, HA_flag=0, Sprif_flag=1, Tanz_flag=0),

label = "5. Sprifermin 30μg IA Q12w")

# Scenario 6: Tanezumab (anti-NGF, pain-focused)

scen6 <- run_scenario(mod, tanz_regimen,

params_override = list(NSAID_flag=0, IACS_flag=0, HA_flag=0, Sprif_flag=0, Tanz_flag=1),

label = "6. Tanezumab 2.5mg SC Q8w")

# Scenario 7: Combination — NSAID + IA Corticosteroid

combo_regimen7 <- ev(nsaid_regimen, iacs_regimen)

scen7 <- run_scenario(mod, combo_regimen7,

params_override = list(NSAID_flag=1, IACS_flag=1, HA_flag=0, Sprif_flag=0, Tanz_flag=0),

label = "7. Celecoxib + IA Triamcinolone")

# Scenario 8: Sprifermin + NSAID (structural + symptomatic)

combo_regimen8 <- ev(nsaid_regimen, sprif_regimen)

scen8 <- run_scenario(mod, combo_regimen8,

params_override = list(NSAID_flag=1, IACS_flag=0, HA_flag=0, Sprif_flag=1, Tanz_flag=0),

label = "8. Sprifermin + Celecoxib")

# Combine all scenarios

all_scen <- bind_rows(scen1, scen2, scen3, scen4, scen5, scen6, scen7, scen8)

## ─────────────────────────────────────────────────────────────

## 5. SENSITIVITY ANALYSIS — Parameter Uncertainty

## ─────────────────────────────────────────────────────────────

# Vary key parameters ±30% for no-treatment scenario

sens_params <- c("k_MMP13_syn", "k_ColII_deg", "k_Agg_deg", "k_JSW_loss", "k_Pain_syn")

sens_results <- list()

for (p in sens_params) {

for (mult in c(0.7, 1.0, 1.3)) {

pmod <- param(mod, setNames(list(param(mod)[[p]] * mult), p))

out <- mrgsim(pmod, ev = ev(),

param = list(NSAID_flag=0, IACS_flag=0, HA_flag=0, Sprif_flag=0, Tanz_flag=0),

end = sim_end, delta = 7) %>%

as_tibble() %>%

mutate(SensParam = p, SensMult = as.character(mult))

sens_results[[paste0(p, "_", mult)]] <- out

}

sens_df <- bind_rows(sens_results)

## ─────────────────────────────────────────────────────────────

## 6. PLOTS

## ─────────────────────────────────────────────────────────────

theme_oa <- theme_bw(base_size = 11) +

theme(legend.position = "bottom",

legend.text = element_text(size = 8),

strip.background = element_rect(fill = "#E3F2FD"),

plot.title = element_text(face = "bold", size = 12))

scen_colors <- c("1. No Treatment (Natural History)" = "#B71C1C",

"2. Celecoxib 200mg BID" = "#1565C0",

"3. IA Triamcinolone 40mg Q3mo" = "#2E7D32",

"4. IA Hyaluronic Acid 3x" = "#6A1B9A",

"5. Sprifermin 30μg IA Q12w" = "#E65100",

"6. Tanezumab 2.5mg SC Q8w" = "#00838F",

"7. Celecoxib + IA Triamcinolone" = "#F9A825",

"8. Sprifermin + Celecoxib" = "#AD1457")

# Plot 1: VAS Pain over time

p1 <- ggplot(all_scen, aes(time, VASPain, color = Scenario)) +

geom_line(linewidth = 0.9) +

scale_color_manual(values = scen_colors) +

labs(title = "OA QSP Model: VAS Pain Score over 2 Years",

x = "Time (days)", y = "VAS Pain (0–100)",

color = "Treatment Scenario") +

scale_y_continuous(limits = c(0, 100)) +

theme_oa

print(p1)

# Plot 2: Joint Space Width (structural outcome)

p2 <- ggplot(all_scen, aes(time, JSW, color = Scenario)) +

geom_line(linewidth = 0.9) +

scale_color_manual(values = scen_colors) +

labs(title = "Joint Space Width (mm) — Structural Outcome",

x = "Time (days)", y = "JSW (mm)") +

geom_hline(yintercept = 2.0, linetype = "dashed", color = "gray50",

linewidth = 0.5) +

annotate("text", x = 700, y = 2.1, label = "TKR threshold ~2mm",

size = 3, color = "gray50") +

theme_oa

print(p2)

# Plot 3: Collagen II and Aggrecan

p3 <- all_scen %>%

pivot_longer(c(ColII, Aggrecan), names_to = "ECM", values_to = "Value") %>%

ggplot(aes(time, Value, color = Scenario)) +

geom_line(linewidth = 0.8) +

facet_wrap(~ECM, scales = "fixed",

labeller = labeller(ECM = c(ColII = "Collagen II (%)", Aggrecan = "Aggrecan (%)"))) +

scale_color_manual(values = scen_colors) +

labs(title = "Cartilage ECM Components over Time",

x = "Time (days)", y = "% Baseline") +

scale_y_continuous(limits = c(0, 110)) +

theme_oa

print(p3)

# Plot 4: Inflammatory Biomarkers

p4 <- all_scen %>%

pivot_longer(c(IL1b, TNFa, MMP13), names_to = "Marker", values_to = "Value") %>%

ggplot(aes(time, Value, color = Scenario)) +

geom_line(linewidth = 0.8) +

facet_wrap(~Marker, scales = "free_y",

labeller = labeller(Marker = c(IL1b = "IL-1β (pg/mL)",

TNFa = "TNF-α (pg/mL)",

MMP13 = "MMP-13 (ng/mL)"))) +

scale_color_manual(values = scen_colors) +

labs(title = "Inflammatory Mediators over Time",

x = "Time (days)", y = "Concentration") +

theme_oa

print(p4)

# Plot 5: Circulating Biomarkers (uCTX-II, COMP)

p5 <- all_scen %>%

pivot_longer(c(uCTXII, COMP_s), names_to = "Biomarker", values_to = "Value") %>%

ggplot(aes(time, Value, color = Scenario)) +

geom_line(linewidth = 0.8) +

facet_wrap(~Biomarker, scales = "free_y",

labeller = labeller(Biomarker = c(uCTXII = "uCTX-II (nmol/mmolCr)",

COMP_s = "Serum COMP (μg/mL)"))) +

scale_color_manual(values = scen_colors) +

labs(title = "Circulating OA Biomarkers",

x = "Time (days)", y = "Concentration") +

theme_oa

print(p5)

# Plot 6: KOOS Estimated Score

p6 <- ggplot(all_scen, aes(time, KOOS_est, color = Scenario)) +

geom_line(linewidth = 0.9) +

scale_color_manual(values = scen_colors) +

labs(title = "Estimated KOOS Score (0–100, higher = better)",

x = "Time (days)", y = "KOOS (estimated)") +

scale_y_continuous(limits = c(0, 100)) +

theme_oa

print(p6)

# Plot 7: Sensitivity analysis — JSW at 2 years

sens_end <- sens_df %>%

filter(time == max(time)) %>%

group_by(SensParam, SensMult) %>%

summarise(JSW_end = mean(JSW), Pain_end = mean(VASPain), .groups = "drop")

p7 <- ggplot(sens_end, aes(SensParam, JSW_end, fill = SensMult)) +

geom_col(position = "dodge") +

scale_fill_manual(values = c("0.7" = "#4CAF50", "1.0" = "#2196F3", "1.3" = "#F44336"),

labels = c("−30%", "Baseline", "+30%")) +

labs(title = "Sensitivity Analysis: JSW at 2 Years",

x = "Parameter", y = "JSW (mm)", fill = "Parameter Change") +

coord_flip() +

theme_oa

print(p7)

# Summary table at 1 year (day 365)

summary_1yr <- all_scen %>%

filter(abs(time - 365) < 1) %>%

group_by(Scenario) %>%

summarise(

VAS_Pain = round(mean(VASPain), 1),

JSW_mm = round(mean(JSW), 2),

KOOS = round(mean(KOOS_est), 1),

ColII_pct = round(mean(ColII), 1),

Aggrecan_pct = round(mean(Aggrecan), 1),

MMP13_ngmL = round(mean(MMP13), 2),

uCTXII = round(mean(uCTXII), 3),

COMP_ug_mL = round(mean(COMP_s), 2),

.groups = "drop"

)

cat("\n═══════════ OA QSP: 1-Year Outcome Summary ═══════════\n")

print(summary_1yr, n = 10)

cat("\n═══════════ Parameter Reference ═══════════\n")

cat("KL Grade at baseline:", param(mod)$KL_grade, "\n")

cat("Patient Age:", param(mod)$Age_yr, "years\n")

cat("JSW_0 (normal reference):", param(mod)$JSW_0, "mm\n")

cat("Simulation period: 730 days (2 years)\n")

FilesExpand file tree

oa_mrgsolve_model.R

Latest commit

History

oa_mrgsolve_model.R

File metadata and controls