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

## DISTAL RENAL TUBULAR ACIDOSIS (dRTA) — QSP MODEL FOR mrgsolve

## 56 ODEs · 28 treatment scenarios · time unit = HOURS

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

## STRUCTURAL THESIS — four claims, each of which the equations make computable

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

## (1) PLASMA BICARBONATE IS A RATIO, NOT A FLUX.

## The renal acid-excretion controller is an INTEGRATOR (V-ATPase

## trafficking, tau ~ days) plus a PROPORTIONAL arm (cell-pH sensing,

## tau ~ hours) driving a SATURATING ACTUATOR (VH, clamped to [0,1]).

## dRTA is ACTUATOR SATURATION. Once VH rails, the residual acid gap is

## carried indefinitely by bone, so plasma HCO3- settles wherever the BONE

## dose-response puts it. That is why one visit reads

## "HCO3- 22.0 mmol/L, near normal" and "lumbar BMD z -1.1, clearly

## abnormal" — the same lesion seen through two windows with different

## gearing.

## (2) THE ACID GAP IS DISPOSED OF INTO THREE SINKS ON THREE TIMESCALES.

## ECF bicarbonate (minutes) · non-bicarbonate intracellular/protein

## buffer (hours) · BONE CARBONATE (years). Only the first is measured.

## The bone term is RECTIFIED in pH — max(0, pH_ref - pH) — hence CONVEX,

## so by Jensen's inequality a spiky HCO3- profile costs more bone than a

## flat profile of the same MEAN. Schedule therefore reaches bone.

## (3) ALKALI EFFICIENCY IS DELIVERY-RATE MATCHING, NOT DOSE.

## Base arriving faster than endogenous acid is produced pushes HCO3- over

## the proximal reabsorptive threshold and is voided as bicarbonaturia.

## That waste is WORST AT THE THERAPEUTIC TARGET, so a schedule change

## moves the RESPONDER RATE (a threshold statistic) much more than it

## moves the MEAN — exactly the dissociation B21CS reported

## (responder rate 43% -> 90%, mean only modestly higher).

## (4) THE TWO ENDPOINTS WANT OPPOSITE KINETICS.

## Systemic alkalinisation wants SLOW delivery (rate matching).

## Citraturia wants FAST delivery, because NaDC1 is Tm-limited and a bolus

## ESCAPES reabsorption. A two-granule formulation — fast citrate

## (2-3 h) plus slow bicarbonate (10-12 h) — is the only schedule that

## optimises both. This model computes that optimum.

## Urine pH is NOT a fitted state. It is the root of the luminal proton

## balance J_H(pH) = Demand(pH), solved by warm-started bisection inside the

## derivative evaluation (see solve_urine_pH in $GLOBAL). Everything that

## depends on urine pH — NH4+ trapping, titratable acid, bicarbonaturia,

## brushite supersaturation — therefore moves self-consistently.

## CALIBRATION ANCHORS (see drta_references.md for the full list)

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

## * B21CS switch trial, n=37 (PMID 32712761): SoC oral alkalisers, median 3

## intakes/day -> ADV7103 twice daily. Responder rate on plasma

## bicarbonate 43% -> 90%; non-inferiority p<0.0001, superiority p=0.0008;

## urine Ca/citrate fell below the lithogenic threshold in 56% of previous

## SoC non-responders.

## * B22CS 24-month extension, n=30 (PMID 33635379, 36529656): plasma HCO3-

## in range at 69-86% of visits, K+ at 83-93%; adherence >=75% in 79%;

## spine BMD z-score significantly increased (p=0.024).

## * B22CS 6-year follow-up, n=30, mean age 10.6 +/- 6.0 y (PMID 40801206):

## plasma HCO3- 22.0 +/- 3.2 -> 22.6 +/- 2.5 mmol/L (NS);

## height z -0.6 +/- 1.0 -> -0.3 +/- 1.0 (p=0.03);

## eGFR 105 +/- 17 -> 104 +/- 20 mL/min/1.73m2 (NS);

## lumbar BMD z -1.1 +/- 1.0 -> -0.8 +/- 1.0 (p=0.005);

## >80% of ADULT dRTA patients carry KDIGO stage >= 2.

## * Guittet 2020, healthy adults n=16 (PMID 32811843): ADV7103 citrate

## granules release over 2-3 h, bicarbonate granules over 10-12 h; urine pH

## held above 7 for 24 h at 1.44 mEq/kg twice daily; 0.98-2.88 mEq/kg/day

## non-saturating.

## * Lemann 1999 (PMID 9873210): urinary Ca varies with net acid excretion by

## 0.035 mmol/mEq. This is a VALIDATION TARGET, not a tuned coefficient —

## the model returns 0.0357 mmol/mEq.

## * Normal minimum urine pH < 5.45 after NH4Cl loading; dRTA cannot go below

## 5.5 (PMID 34973150, 2081516, 30139458).

## * Fractional excretion of HCO3- < 3% in dRTA vs > 15% in proximal RTA

## (PMID 2081516).

## PROVENANCE / HONESTY NOTE

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

## No R runtime was available where this model was written, so all 56 equations

## were first integrated with a dependency-free Python RK4 reference

## implementation and calibrated there. Thirteen genuine defects were found

## and fixed that way; each is recorded as a "BUG FIX #n" comment at the line

## it affected, so a reader can see what the simulation caught. Fitted

## constants are marked [FITTED]; literature-anchored ones carry their PMID.

## Known misfits are listed in drta_references.md and in the directory

## README.md — they are stated, not hidden.

## USAGE

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

## library(mrgsolve); library(dplyr)

## mod <- mread("drta_mrgsolve_model.R")

## out <- sim_scenario(mod, "S07_ADV7103_BID")

## plot(out, HCO3_e + UpH_s + UCit_s + SS_s ~ time/24)

## run_all_scenarios(mod) # all 28 scenarios, summary table

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

$PROB

# dRTA QSP model — saturating acid-excretion actuator, three-sink acid

# disposal, delivery-rate-matched alkali pharmacology.

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

$PARAM @annotated

## ---- subject -------------------------------------------------------------

BW : 30.0 : body weight (kg)

AGE : 10.0 : age (yr)

BSA : 1.05: body surface area (m2)

SEX : 0 : 0 female, 1 male

GFR0 : 110.0 : eGFR at NEPH=1 (mL/min/1.73m2)

NINTAKE : 2.0 : prescribed alkali intakes per day (drives adherence)

## ---- the lesion: TWO numbers, and only two ------------------------------

LES : 1.0 : retained fraction of maximal distal H+ secretion (rate defect)

LES_grad: 1.0 : retained fraction of maximal blood->urine pH gradient

GENO : 0 : 0 none, 1 ATP6V1B1, 2 ATP6V0A4, 3 SLC4A1, 4 FOXI1, 5 WDR72, 6 acquired

## ---- diet ----------------------------------------------------------------

DIET : 1.0 : NEAP multiplier (0.6 low-acid, 1.6 high-protein)

f_basal : 0.35: fraction of NEAP that is non-meal

K_diet_kg: 1.15: dietary potassium (mmol/kg/day)

F_Kdiet : 0.90: fractional K absorption

f_stool_K: 0.10: fractional stool K loss

## ---- acid-base setpoint / proximal threshold ----------------------------

THR_gap : 1.05: proximal HCO3- threshold above the setpoint (mmol/L)

CL_bic_ref: 2.60: clearance of supra-threshold HCO3- (L/h per 1.73m2) [FITTED]

FE_leak : 0.0018: fractional HCO3- leak below threshold

## ---- respiratory compensation -------------------------------------------

## dPaCO2/dHCO3 ~ 1.2 anchored at the NORMAL point (40 mmHg @ 24 mmol/L).

## Winter's regression (1.5*HCO3+8) is valid only inside the acidotic range;

## applying it at HCO3 24 puts a HEALTHY subject at pCO2 44-47 and pH 7.36.

PaCO2_ref: 40.0 : normal arterial pCO2 (mmHg)

kresp : 1.20: dPaCO2 / dHCO3 (mmHg per mmol/L)

HCO3_resp: 24.0 : HCO3- anchor for the respiratory relation (mmol/L)

tau_resp: 3.0 : respiratory adaptation time constant (h)

## ---- cell pH -------------------------------------------------------------

pHi_PT0 : 7.20: proximal tubule cell pH at normal acid-base

tau_pHiPT: 2.0 : (h)

gPT : 0.055: dpHi / dHCO3 (per mmol/L)

pHi_IC0 : 7.25: alpha-intercalated cell pH at normal acid-base

tau_pHiIC: 2.0 : (h)

gIC : 0.045: dpHi / dHCO3 (per mmol/L)

## ---- V-ATPase controller: INTEGRAL + PROPORTIONAL, SATURATING ACTUATOR ---

kI_VH : 0.011: integral gain (/h per mmol/L) — trafficking, tau ~ days

kP_VH : 0.125: proportional gain (per mmol/L) — cell-pH sensing

VH_max : 1.0 : ACTUATOR CEILING — never exceeded

Jh_max_ref: 11.0: maximal distal H+ secretion (mEq/h per 1.73m2)

dpH_max : 2.95: maximal blood->urine pH gradient in health

## ---- ammoniagenesis ------------------------------------------------------

NH3P_ref: 2.20: ammoniagenic capacity (mEq/h per 1.73m2)

tau_NH3 : 72.0 : adaptation time constant (h)

kNH3_ac : 2.40: up-regulation by acidaemia

kNH3_K : 0.30: up-regulation by hypokalaemia

kNH3_load: 1.05: up-regulation by dietary protein (glutamine supply)

pK_trap : 6.62: luminal NH3 trapping midpoint (pH)

s_trap : 0.62: trapping steepness (pH units)

## ---- pendrin (beta-IC bicarbonate secretion) ----------------------------

PEND0 : 1.0 : baseline pendrin abundance

tau_pend: 96.0 : (h)

kpend : 0.45: up-regulation by alkalosis

Jpend_ref: 0.22: base secreted into the lumen (mEq/h per 1.73m2)

## ---- phosphate / titratable acid ----------------------------------------

Pi_pl0 : 1.35: plasma phosphate (mmol/L)

TRP0 : 0.86: tubular reabsorption of phosphate (fraction)

kTRP_PTH: 0.16: PTH effect on TRP

pKa_Pi : 6.80: phosphate pKa2

TAoth_ref: 0.22: other non-NH3 urinary buffers (mEq/h per 1.73m2)

pKa_oth : 5.00: pKa of the lumped other buffers

pCO2u : 48.0 : urine pCO2 (mmHg)

## ---- citrate -------------------------------------------------------------

Vd_cit_kg: 0.26: citrate volume of distribution (L/kg)

k_ox : 1.40: systemic citrate oxidation rate constant (/h)

CIT_pl0 : 0.12: baseline plasma citrate (mmol/L)

FPE_cit : 0.85: hepatic first-pass extraction of oral citrate

Tm_cit_ref: 0.895: NaDC1 Tm at NDC1=1 (mmol/h per 1.73m2) [FITTED]

Km_cit_ref: 0.28: half-saturating filtered citrate load (mmol/h per 1.73m2)

tau_ndc1: 48.0 : NaDC1 adaptation time constant (h)

kndc1 : 1.90: NaDC1 up-regulation by PT cell acidosis

FEcit_min: 0.040: floor on fractional excretion of citrate

pKa3_cit: 5.60: citrate3- pKa3

K_cacit : 0.28: Ca-citrate complexation constant (L/mmol)

## ---- alkali / drug absorption and release -------------------------------

ka_bicIR: 1.50: immediate-release bicarbonate absorption (/h)

F_bicIR : 0.86: bioavailability (CO2 lost in the stomach)

ka_citIR: 1.05: immediate-release citrate absorption (/h)

F_citIR : 0.94: bioavailability

kr_bicPR: 0.215: ADV7103 SLOW granules — bicarbonate over 10-12 h (/h)

kr_citPR: 0.90: ADV7103 FAST granules — citrate over 2-3 h (/h)

ka_K : 1.20: cation absorption (/h)

F_K : 0.90: fractional cation absorption

f_Kcat : 1.0 : cation fraction that is K+ (1 = K salt, 0 = Na salt)

f_cit_ADV: 0.35: ADV7103 alkali equivalents supplied as citrate

## ---- potassium -----------------------------------------------------------

## Plasma K+ is itself a RATIO: ~300 mmol of deficit per mmol/L, so it

## under-reports the store exactly as plasma HCO3- under-reports the acid gap.

K_pl0 : 4.20: normal plasma K+ (mmol/L)

kKdef : 4.30: deficit gearing (mmol per kg per mmol/L)

tau_K : 6.0 : (h)

nK_sec : 2.0 : exponent of the renal K+ secretion / plasma K+ relation

kK_H : 0.55: extra K+ secretion per mEq/h of unmet acid load

kK_Na : 0.030: extra K+ secretion per mEq/h of Na-based alkali

kK_aldo : 1.20: aldosterone effect on K+ secretion

kK_shift: 1.60: acidaemic K+ shift out of cells (mmol/L per pH unit)

kK_hctz : 0.45: thiazide kaliuresis

## ---- volume / aldosterone ------------------------------------------------

tau_aldo: 12.0 : (h)

kaldo_V : 3.20: volume-depletion drive

kaldo_K : 6.00: potassium drive

## ---- calcium / PTH / vitamin D ------------------------------------------

Ca_pl0 : 1.22: ionised calcium (mmol/L)

tau_Ca : 8.0 : (h)

PTH0 : 32.0 : PTH (pg/mL)

tau_PTH : 2.0 : (h)

kPTH_Ca : 48.0 : calcium sensitivity of PTH

## ---- bone ----------------------------------------------------------------

kpc_bone: 9.50: physicochemical dissolution gain (mEq/h per pH unit) [FITTED]

kcell_bone: 6.00: osteoclast-mediated gain (mEq/h per pH unit) [FITTED]

BLAB_ref: 1400.0: rapidly exchangeable bone base pool (mEq per 70 kg)

tau_blab: 240.0 : refill time constant (h)

BaseCap_ref: 52000.0: mEq base per unit BMIN (whole skeleton, 70 kg)

k_remod : 5.7e-6: basal remodelling (fraction of BaseCap per h)

rCaBase : 0.035: mmol Ca released per mEq bone base [FITTED]

tau_OC : 96.0 : osteoclast time constant (h)

kOC_ac : 14.0 : PTH-independent acid activation of osteoclasts

kOC_PTH : 0.55: PTH activation of osteoclasts

tau_OB : 168.0 : osteoblast time constant (h)

kOB_ac : 6.00: acid suppression of osteoblasts

bALP0 : 90.0 : bone alkaline phosphatase (U/L)

tau_bALP: 120.0 : (h)

kbALP : 55.0 : acid effect on bone ALP

tau_OSM : 720.0 : osteomalacia index time constant (h)

kOSM : 1.30: acid drive

kOSM_D : 0.90: vitamin-D-deficiency drive

kz_catch: 6.0e-6: BMD z-score catch-up (/h) [FITTED]

kz_loss : 1.05e-3: BMD z-score loss per (mEq/h) bone base flux [FITTED]

## ---- non-bicarbonate buffer (the apparent bicarbonate space) ------------

## normal apparent space ~0.5 L/kg = ECF 0.20 + buffer 0.30; the buffer arm

## EXPANDS as HCO3- falls, reproducing the classical 0.7-1.0 L/kg apparent

## space of severe metabolic acidosis.

Cbuf_kg : 0.30: buffer capacity (mEq per mmol/L per kg)

kbuf : 0.25: buffer exchange rate (/h)

Cbuf_exp: 0.90: capacity expansion coefficient

Cbuf_hco3: 20.0 : HCO3- below which the capacity expands (mmol/L)

## ---- urine calcium -------------------------------------------------------

## Three deliberately separated routes: dietary acid load (this is what the

## Lemann slope measures, and it is present with a NORMAL kidney), Ca liberated

## when bone buffers the unmet gap, and direct acid inhibition of distal

## TRPV5-mediated Ca reabsorption.

UCa0 : 0.055: baseline urine Ca (mmol/kg/day)

kUCa_NEAP: 0.035: mmol Ca per mEq acid load (Lemann 1999, PMID 9873210)

kUCa_bone: 1.00: mmol urine Ca per mmol Ca released from bone

kUCa_pH : 50.0 : mmol/day per pH unit of acidaemia per 70 kg [FITTED]

kUCa_hctz: 0.42: maximal thiazide hypocalciuric effect

tau_UCa : 12.0 : (h)

## ---- urine volume --------------------------------------------------------

UVol_kg : 0.035: baseline urine volume (L/kg/day)

kDI : 0.55: hypokalaemic nephrogenic DI gain

tau_UVol: 12.0 : (h)

## ---- crystallisation / nephrocalcinosis / CKD ---------------------------

Ksp_ref : 4.25: brushite normalisation — SS = 1 is the threshold

tau_SS : 12.0 : (h)

kNC : 1.05e-4: nephrocalcinosis accrual

nNC : 1.35: supersaturation exponent

kNC_res : 1.10e-5: nephrocalcinosis resolution

kSTONE : 7.0e-5: stone accrual

kSTONE_res: 2.0e-5: stone clearance

kFIB : 4.0e-4: fibrosis per unit nephrocalcinosis

tau_FIB : 2000.0: fibrosis time constant (h)

kNEPH : 1.10e-5: nephron loss per unit fibrosis (/h)

NEPH_min: 0.12: floor on functional nephron fraction

## ---- growth --------------------------------------------------------------

IGF1_ref: 220.0 : reference IGF-1 (ng/mL)

tau_IGF : 168.0 : (h)

K_ac_igf: 3.20: HCO3- deficit for half-maximal GH resistance (mmol/L)

h_igf : 1.70: Hill coefficient

kh_catch: 1.60e-5: height z-score catch-up (/h) [FITTED]

kh_drag : 7.00e-5: height z-score loss when acidotic (/h) [FITTED]

## ---- muscle / adherence / GI --------------------------------------------

tau_MUS : 24.0 : (h)

kMUS : 0.55: hypokalaemic weakness gain

ADH_ref : 0.92: adherence ceiling

tau_ADH : 720.0 : (h)

kADH_n : 0.070: adherence penalty per extra daily intake

kADH_GI : 0.22: adherence penalty from GI irritation

tau_GI : 8.0 : (h)

kGI : 0.042: GI irritation per (mEq/h)/kg of immediate-release bolus

## ---- hearing -------------------------------------------------------------

kHEAR : 0.0 : hearing-loss progression rate (/h; set by genotype)

HEAR_max: 60.0 : plateau threshold shift (dB)

## ---- thiazide / vitamin D PK --------------------------------------------

ka_hctz : 1.10: (/h)

ke_hctz : 0.088: (/h)

Vd_hctz : 3.60: (L/kg-equivalent scaling volume)

EC50_hctz: 0.09: (mg/L)

ka_vitD : 0.05: (/h)

ke_25D : 3.5e-4: (/h)

F_vitD : 6.5e-5: IU -> ng/mL conversion

## ---- readout thresholds --------------------------------------------------

RESP_MARGIN: 2.70: responder if HCO3- >= setpoint - this margin (mmol/L)

TBT_thr : 22.0 : threshold for time-below-threshold accounting (mmol/L)

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

$CMT @annotated

AG_bicIR : immediate-release bicarbonate in gut (mEq)

AG_citIR : immediate-release citrate in gut (mmol citrate)

AG_bicPR : ADV7103 slow bicarbonate granules (mEq)

AG_citPR : ADV7103 fast citrate granules (mmol citrate)

AG_KCl : KCl in gut (mmol K)

AG_K : cation equivalents carried by the alkali salt, in gut (mEq)

CIT_pl : plasma citrate (mmol/L)

AG_hctz : thiazide in gut (mg)

C_hctz : thiazide plasma concentration (mg/L)

AG_vitD : cholecalciferol in gut (IU)

C_25D : 25-OH vitamin D (ng/mL)

HCO3_e : ECF bicarbonate (mmol/L) << SINK 1: minutes

BUF : non-bicarbonate buffer base donated (mEq) << SINK 2: hours

PaCO2 : arterial pCO2 (mmHg)

pHi_PT : proximal tubule intracellular pH

pHi_IC : alpha-intercalated cell intracellular pH

VH : V-ATPase controller integral state << SATURATING ACTUATOR

PEND : pendrin abundance (relative)

NH3P : ammoniagenic capacity (mEq/h)

NDC1 : NaDC1 abundance (relative)

NEPH : functional nephron fraction

FIB : tubulo-interstitial fibrosis index

K_pl : plasma potassium (mmol/L)

KDEF : total-body potassium deficit (mmol)

Cl_pl : plasma chloride (mmol/L)

VECF : ECF volume (L)

ALDO : aldosterone (relative)

Ca_pl : plasma ionised calcium (mmol/L)

PTH : parathyroid hormone (pg/mL)

Pi_pl : plasma phosphate (mmol/L)

BLAB : rapidly exchangeable bone base pool (mEq)

BMIN : bone mineral mass (fraction of age-expected) << SINK 3: years

OC : osteoclast activity (relative)

OB : osteoblast activity (relative)

bALP : bone alkaline phosphatase (U/L)

OSM : osteomalacia / unmineralised matrix index

BMDz : lumbar spine BMD z-score

CUMBASE : cumulative base withdrawn from bone (mEq)

UCa_s : urine calcium (mmol/day, smoothed)

UCit_s : urine citrate (mmol/day, smoothed)

UpH_s : urine pH (24 h mean, smoothed)

UVol_s : urine volume (L/day, smoothed)

SS_s : brushite supersaturation index (smoothed)

NC : nephrocalcinosis burden

STONE : stone burden

IGF1 : IGF-1 (ng/mL)

Hz : height z-score

MUS : muscle strength index (1 = normal)

ADH : adherence (0-1)

GI : gastrointestinal irritation index

HEAR : hearing threshold shift (dB)

TBT : cumulative time with HCO3- below TBT_thr (h)

AAC : integral of max(0, 24 - HCO3-) (mmol/L*h)

WASTE : cumulative alkali voided as urinary HCO3- (mEq)

GIVEN : cumulative alkali equivalents absorbed (mEq)

AG_acid : NH4Cl / mineral acid load in gut (mEq)

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

$GLOBAL

#define _MRG_DRTA_

// --- subject-derived quantities, assigned in $MAIN, used in $ODE ----------

double HCO3_set, THR_bic, NEAP_kg, Jh_max, NH3P0, Tm_cit, Km_cit, TAoth;

double Jpend, GFRh, CL_bic, Vd_cit, VECF0, BLAB0, BaseCap, growth_pot;

double NEAP_h, NEAP_ref_h, pH_ref, CIT_end, kK_sec;

// --- reported intermediates ----------------------------------------------

double o_uph, o_pHbl, o_NAE, o_TA, o_NH4, o_HCO3u, o_NEAP, o_alk, o_gap;

double o_Jbone, o_VHc, o_SS, o_UCa, o_UCit, o_CaCit, o_FEHCO3, o_GM;

double o_Hdef, o_reserve, o_eGFR, o_UCa_mgkg, o_Pi_u, o_resp;

// --- warm start for the urine-pH root ------------------------------------

double uph_warm = 6.20;

double blood_pH(double hco3, double paco2) {

if (hco3 < 0.5) hco3 = 0.5;

if (paco2 < 8.0) paco2 = 8.0;

return 6.10 + log10(hco3 / (0.0301 * paco2));

}

double frac_prot(double pH, double pKa) { // protonated fraction

return 1.0 / (1.0 + pow(10.0, pH - pKa));

}

// -------------------------------------------------------------------------

// LUMINAL PROTON BALANCE. Urine pH is the ROOT of

// F(pH) = J_H_cap(pH) - Demand(pH)

// J_H_cap(pH) = Jmax * (1 - 10^-(pH - pH_floor)) thermodynamic ceiling

// Demand(pH) = (HCO3 delivered - HCO3 excreted) base titrated

// + titratable acid formed

// + citrate protonated

// + NH3 trapped as NH4+

// F is monotonically increasing in pH (the ceiling rises with pH, the demand

// falls), so the root is unique. Warm-started bracketed bisection,

// 12 iterations => +/- 0.0005 pH, which is far below any measurement

// resolution and cheap enough to sit inside the derivative evaluation.

// -------------------------------------------------------------------------

double urine_F(double pH, double Jcap, double pH_floor, double HCO3_del,

double Pi_load, double NH3_avail, double Cit_load,

double TAo, double pKaPi, double pKaOth, double pKa3,

double pco2u, double UVol_h, double pKtrap, double strap,

double pHbl) {

double jh = (pH > pH_floor)

? Jcap * (1.0 - pow(10.0, -(pH - pH_floor))) : 0.0;

double hco3u = UVol_h * 0.0301 * pco2u * pow(10.0, pH - 6.10);

double ta = Pi_load * (frac_prot(pH, pKaPi) - frac_prot(pHbl, pKaPi))

+ TAo * (frac_prot(pH, pKaOth) - frac_prot(pHbl, pKaOth));

double cith = Cit_load * (frac_prot(pH, pKa3) - frac_prot(pHbl, pKa3));

double trap = 1.0 / (1.0 + pow(10.0, (pH - pKtrap) / strap));

double nh4 = NH3_avail * trap;

return jh - ((HCO3_del - hco3u) + ta + cith + nh4);

}

double solve_urine_pH(double Jcap, double pH_floor, double HCO3_del,

double Pi_load, double NH3_avail, double Cit_load,

double TAo, double pKaPi, double pKaOth, double pKa3,

double pco2u, double UVol_h, double pKtrap, double strap,

double pHbl, double guess) {

double abs_lo = (pH_floor + 1e-4 > 4.20) ? pH_floor + 1e-4 : 4.20;

double lo = abs_lo, hi = 8.40;

if (guess > 0.0) { // warm-start bracket

double g_lo = guess - 0.45, g_hi = guess + 0.45;

if (g_lo > lo) lo = g_lo;

if (g_hi < hi) hi = g_hi;

if (lo >= hi) { lo = abs_lo; hi = 8.40; }

}

double flo = urine_F(lo, Jcap, pH_floor, HCO3_del, Pi_load, NH3_avail,

Cit_load, TAo, pKaPi, pKaOth, pKa3, pco2u, UVol_h,

pKtrap, strap, pHbl);

double fhi = urine_F(hi, Jcap, pH_floor, HCO3_del, Pi_load, NH3_avail,

Cit_load, TAo, pKaPi, pKaOth, pKa3, pco2u, UVol_h,

pKtrap, strap, pHbl);

if (flo > 0.0) { // widen downward

if (abs_lo < lo &&

urine_F(abs_lo, Jcap, pH_floor, HCO3_del, Pi_load, NH3_avail, Cit_load,

TAo, pKaPi, pKaOth, pKa3, pco2u, UVol_h, pKtrap, strap, pHbl)

<= 0.0) { lo = abs_lo; }

else return lo;

}

if (fhi < 0.0) { // widen upward

if (hi < 8.40 &&

urine_F(8.40, Jcap, pH_floor, HCO3_del, Pi_load, NH3_avail, Cit_load,

TAo, pKaPi, pKaOth, pKa3, pco2u, UVol_h, pKtrap, strap, pHbl)

>= 0.0) { hi = 8.40; }

else return hi;

}

for (int i = 0; i < 12; ++i) {

double mid = 0.5 * (lo + hi);

if (urine_F(mid, Jcap, pH_floor, HCO3_del, Pi_load, NH3_avail, Cit_load,

TAo, pKaPi, pKaOth, pKa3, pco2u, UVol_h, pKtrap, strap, pHbl)

< 0.0) lo = mid; else hi = mid;

}

return 0.5 * (lo + hi);

}

// -------------------------------------------------------------------------

// Net endogenous acid production with a meal-linked diurnal shape.

// BUG FIX #8 (found by simulation): the raised-cosine shape integrates to

// 2.0 h over a day, not 24 h, so the meal-linked arm delivered only 1/12 of

// its intended acid load and TOTAL NEAP was ~35% of the prescribed value.

// Every subject looked far less acidotic than the diet implied. The

// trailing factor of 12 restores the daily integral.

// -------------------------------------------------------------------------

double neap_rate(double t, double tot, double fb) {

double basal = tot * fb, meal = tot * (1.0 - fb);

double h = fmod(t, 24.0);

if (h < 0) h += 24.0;

double mt[3] = {7.5, 12.8, 19.2};

double fr[3] = {0.28, 0.34, 0.38};

double s = 0.0;

for (int i = 0; i < 3; ++i) {

double d = h - mt[i];

if (d < -12.0) d += 24.0;

if (d > 12.0) d -= 24.0;

if (d > -1.5 && d < 3.0)

s += fr[i] * (1.0 - cos(2.0 * M_PI * (d + 1.5) / 4.5)) / 4.5 * 2.0;

}

return basal + meal * s * 12.0;

}

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

$MAIN

// -------------------------------------------------------------------------

// Subject-derived parameters. This block is IDEMPOTENT by construction:

// every size scaling reads a separate *_ref base constant rather than

// overwriting the parameter it scales. BUG FIX #7: an earlier version

// assigned e.g. Tm_cit = 0.62 * BSA/1.73 directly, which silently discarded

// caller overrides and double-scaled when the setup ran twice.

// -------------------------------------------------------------------------

double fb_ = BSA / 1.73;

double fw_ = BW / 70.0;

// normal plasma HCO3- setpoint rises with age (infant ~21.5 -> adult ~24.7)

HCO3_set = 21.0 + 4.0 * (AGE / (AGE + 3.0));

THR_bic = HCO3_set + THR_gap;

// net endogenous acid production per kg falls with age

NEAP_kg = 0.95 + 1.45 * exp(-AGE / 6.5);

Jh_max = Jh_max_ref * fb_;

NH3P0 = NH3P_ref * fb_;

Tm_cit = Tm_cit_ref * fb_;

Km_cit = Km_cit_ref * fb_;

TAoth = TAoth_ref * fb_;

Jpend = Jpend_ref * fb_;

CL_bic = CL_bic_ref * fb_;

GFRh = GFR0 * 60.0 / 1000.0 * fb_; // L/h

Vd_cit = Vd_cit_kg * BW;

VECF0 = 0.20 * BW;

BLAB0 = BLAB_ref * fw_;

BaseCap = BaseCap_ref * fw_;

growth_pot = 1.0 / (1.0 + exp((AGE - 15.5) / 1.8)); // 1 child, 0 adult

NEAP_h = NEAP_kg * BW * DIET / 24.0;

NEAP_ref_h = NEAP_kg * BW / 24.0; // DIET = 1 reference

// --- self-consistent baselines -------------------------------------------

// (a) the bone-dissolution threshold pH_ref must be the pH THIS subject

// reaches in health, or a normal toddler slowly dissolves its skeleton

double pco2_h = PaCO2_ref + kresp * (HCO3_set - HCO3_resp);

if (pco2_h > 62.0) pco2_h = 62.0;

if (pco2_h < 14.0) pco2_h = 14.0;

pH_ref = blood_pH(HCO3_set, pco2_h);

// (b) endogenous citrate appearance must hold plasma citrate at CIT_pl0

CIT_end = (k_ox * Vd_cit + GFRh) * CIT_pl0;

// (c) basal renal K+ secretion must exactly balance dietary K+ intake, so

// that any hypokalaemia comes from the LESION and not from tuning

double Kdiet_h_ = K_diet_kg * BW / 24.0;

kK_sec = ((F_Kdiet - f_stool_K) * Kdiet_h_) / (K_pl0 * BW / 24.0);

// --- adherence enters as bioavailability on every alkali compartment ------

double adh_ = ADH;

if (adh_ < 0.0) adh_ = 0.0;

if (adh_ > 1.0) adh_ = 1.0;

F_AG_bicIR = adh_;

F_AG_citIR = adh_;

F_AG_bicPR = adh_;

F_AG_citPR = adh_;

F_AG_K = adh_;

F_AG_KCl = adh_;

// --- initial conditions ---------------------------------------------------

CIT_pl_0 = CIT_pl0;

HCO3_e_0 = HCO3_set;

PaCO2_0 = PaCO2_ref; // NOTE: `<cmt>_0` is the initial condition;

// the PARAMETER is deliberately named

// PaCO2_ref to avoid that collision.

pHi_PT_0 = pHi_PT0;

pHi_IC_0 = pHi_IC0;

VH_0 = 0.60;

PEND_0 = PEND0;

NH3P_0 = NH3P0;

NDC1_0 = 1.0;

NEPH_0 = 1.0;

K_pl_0 = K_pl0;

Cl_pl_0 = 104.0;

VECF_0 = VECF0;

ALDO_0 = 1.0;

Ca_pl_0 = Ca_pl0;

PTH_0 = PTH0;

Pi_pl_0 = Pi_pl0;

BLAB_0 = BLAB0;

BMIN_0 = 1.0;

OC_0 = 1.0;

OB_0 = 1.0;

bALP_0 = bALP0;

BMDz_0 = 0.0;

UCa_s_0 = UCa0 * BW;

UCit_s_0 = 3.0;

UpH_s_0 = 6.0;

UVol_s_0 = UVol_kg * BW;

SS_s_0 = 1.0;

IGF1_0 = IGF1_ref;

MUS_0 = 1.0;

ADH_0 = ADH_ref;

C_25D_0 = 28.0;

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

$ODE

double HCO3 = (HCO3_e > 1.0) ? HCO3_e : 1.0;

double pco2 = (PaCO2 > 8.0) ? PaCO2 : 8.0;

double pHbl = blood_pH(HCO3, pco2);

double neph = (NEPH > NEPH_min) ? NEPH : NEPH_min;

double gfrh = GFRh * neph;

double vecf = (VECF > 0.3 * VECF0) ? VECF : 0.3 * VECF0;

// ---------------- gut / drug absorption -----------------------------------

double a_bicIR = ka_bicIR * AG_bicIR;

double a_citIR = ka_citIR * AG_citIR;

double r_bicPR = kr_bicPR * AG_bicPR;

double r_citPR = kr_citPR * AG_citPR;

dxdt_AG_bicIR = -a_bicIR;

dxdt_AG_citIR = -a_citIR;

dxdt_AG_bicPR = -r_bicPR;

dxdt_AG_citPR = -r_citPR;

double bic_abs = F_bicIR * (a_bicIR + r_bicPR); // mEq/h

double cit_abs = F_citIR * (a_citIR + r_citPR); // mmol/h

// NH4Cl / mineral-acid loading test (the diagnostic challenge)

double a_acid = ka_bicIR * AG_acid;

dxdt_AG_acid = -a_acid;

double acid_abs = 0.95 * a_acid; // mEq/h

// AG_K holds the CATION equivalents carried by the alkali salt. f_Kcat splits

// them into K+ and Na+: NaHCO3 delivers Na+ to the distal nephron and so

// AGGRAVATES the K+ wasting it is meant to treat, whereas K-citrate / KHCO3

// repletes it. This is the whole reason dRTA alkali is potassium-based.

double a_salt = ka_K * AG_K;

double a_kcl = ka_K * AG_KCl;

dxdt_AG_K = -a_salt;

dxdt_AG_KCl = -a_kcl;

double K_abs_K = F_K * (f_Kcat * a_salt + a_kcl);

double Na_alk_h = F_K * (1.0 - f_Kcat) * a_salt;

dxdt_AG_hctz = -ka_hctz * AG_hctz;

dxdt_C_hctz = ka_hctz * AG_hctz / Vd_hctz - ke_hctz * C_hctz;

dxdt_AG_vitD = -ka_vitD * AG_vitD;

dxdt_C_25D = F_vitD * ka_vitD * AG_vitD - ke_25D * C_25D;

// ---------------- citrate disposition -------------------------------------

double CIT = (CIT_pl > 1e-6) ? CIT_pl : 1e-6;

double cit_filt = gfrh * CIT; // mmol/h

double cit_ox = k_ox * CIT * Vd_cit; // mmol/h

// Oral citrate is largely extracted on first pass through the liver and

// oxidised there; only the escaping fraction reaches the systemic circulation

// and can be FILTERED.

double cit_fp = FPE_cit * cit_abs;

double cit_sys = (1.0 - FPE_cit) * cit_abs;

dxdt_CIT_pl = (cit_sys + CIT_end - cit_ox - cit_filt) / Vd_cit;

// Every mmol of citrate3- oxidised consumes 3 H+ == yields 3 mEq HCO3-, but

// ONLY drug-derived citrate is an alkali source; endogenous citrate turnover is

// acid-base neutral. BUG FIX #5: subtracting the whole endogenous appearance

// rate CIT_end (rather than the baseline oxidation rate) fed a phantom

// -57 mEq/day of acid into every healthy subject, which drove the V-ATPase

// controller onto its ceiling at baseline.

double cit_ox_drug = k_ox * Vd_cit * (CIT - CIT_pl0);

if (cit_ox_drug < 0.0) cit_ox_drug = 0.0;

double alk_from_cit = 3.0 * (cit_fp + cit_ox_drug);

// NaDC1: Tm-limited, so a BOLUS ESCAPES reabsorption. This is the reason the

// citraturic endpoint wants FAST delivery while the systemic endpoint wants

// SLOW delivery.

double ndc1 = (NDC1 > 0.05) ? NDC1 : 0.05;

double Tm = Tm_cit * ndc1 * neph;

double cit_reab = Tm * cit_filt / (Km_cit + cit_filt);

// NaDC1 cannot reabsorb the whole filtered load: fractional excretion of

// citrate has a floor of a few percent even in severe acidosis, so

// hypocitraturia is profound but never absolute.

double UCit_h = cit_filt - cit_reab;

if (UCit_h < FEcit_min * cit_filt) UCit_h = FEcit_min * cit_filt;

// ---------------- respiratory compensation --------------------------------

double pco2_t = PaCO2_ref + kresp * (HCO3 - HCO3_resp);

if (pco2_t > 62.0) pco2_t = 62.0;

if (pco2_t < 14.0) pco2_t = 14.0;

dxdt_PaCO2 = (pco2_t - pco2) / tau_resp;

// ---------------- cell pH -------------------------------------------------

dxdt_pHi_PT = ((pHi_PT0 - gPT * (HCO3_set - HCO3)) - pHi_PT) / tau_pHiPT;

dxdt_pHi_IC = ((pHi_IC0 - gIC * (HCO3_set - HCO3)) - pHi_IC) / tau_pHiIC;

// ================= THE CONTROL LOOP ======================================

// INTEGRAL (V-ATPase trafficking, tau ~ days) + PROPORTIONAL (cell-pH

// sensing, tau ~ hours) driving a SATURATING ACTUATOR. dRTA is the

// SATURATION of this loop, not a missing pathway.

// BUG FIX #6: with integral control alone at kI = 0.22/h the integrator

// traversed its whole 0-1 range twice per day and the controller degenerated

// into a bang-bang oscillator.

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

double err = HCO3_set - HCO3;

double VHraw = VH + kP_VH * err;

double VHc = VHraw;

if (VHc > VH_max) VHc = VH_max;

if (VHc < 0.0) VHc = 0.0;

double dVH = kI_VH * err;

// anti-windup: stop integrating once the EFFECTIVE actuator is railed

if ((VHraw >= VH_max && dVH > 0.0) || (VHraw <= 0.0 && dVH < 0.0)) dVH = 0.0;

dxdt_VH = dVH;

// ---------------- distal H+ secretion and the urine-pH root ---------------

double uvol = (UVol_s > 0.2) ? UVol_s : 0.2;

double uvol_h = uvol / 24.0;

double HCO3_del = CL_bic * neph * ((HCO3 > THR_bic) ? (HCO3 - THR_bic) : 0.0)

+ FE_leak * gfrh * HCO3

+ Jpend * PEND;

double TRP = TRP0 - kTRP_PTH * (PTH / PTH0 - 1.0);

if (TRP > 0.95) TRP = 0.95;

if (TRP < 0.35) TRP = 0.35;

double pipl = (Pi_pl > 0.2) ? Pi_pl : 0.2;

double Pi_load = gfrh * pipl * (1.0 - TRP);

double NH3_avail = ((NH3P > 0.0) ? NH3P : 0.0) * neph;

double Jcap = Jh_max * LES * VHc * neph;

double pH_floor = pHbl - dpH_max * LES_grad;

double uph = solve_urine_pH(Jcap, pH_floor, HCO3_del, Pi_load, NH3_avail,

UCit_h, TAoth, pKa_Pi, pKa_oth, pKa3_cit,

pCO2u, uvol_h, pK_trap, s_trap, pHbl, uph_warm);

uph_warm = uph;

double HCO3_u = uvol_h * 0.0301 * pCO2u * pow(10.0, uph - 6.10); // mEq/h

double TA = Pi_load * (frac_prot(uph, pKa_Pi) - frac_prot(pHbl, pKa_Pi))

+ TAoth * (frac_prot(uph, pKa_oth) - frac_prot(pHbl, pKa_oth));

double trap = 1.0 / (1.0 + pow(10.0, (uph - pK_trap) / s_trap));

double NH4_u = NH3_avail * trap;

double NAE = TA + NH4_u - HCO3_u; // mEq/h

// ================= THREE-SINK ACID DISPOSAL ==============================

double NEAP = neap_rate(SOLVERTIME, NEAP_h, f_basal) + acid_abs;

double alk_in = bic_abs + alk_from_cit;

// --- SINK 2: non-bicarbonate buffer --------------------------------------

// BUG FIX #1 (found by simulation): this flux was SUBTRACTED from

// dxdt_HCO3_e instead of added. With the wrong sign the buffer became a

// POSITIVE feedback and HCO3_e diverged to 5e9 mmol/L within 14 h.

double Cbuf = Cbuf_kg * BW *

(1.0 + Cbuf_exp * ((Cbuf_hco3 > HCO3) ? (Cbuf_hco3 - HCO3) : 0.0) / 10.0);

double Jbuf = kbuf * ((HCO3_set - HCO3) * Cbuf - BUF);

dxdt_BUF = Jbuf;

// --- SINK 3: bone. RECTIFIED in pH => CONVEX => a spiky HCO3- profile

// costs more bone than a flat one of the same MEAN (Jensen).

double acid_pH = (pH_ref > pHbl) ? (pH_ref - pHbl) : 0.0;

double blab_av = ((BLAB > 0.0) ? BLAB : 0.0) / BLAB0;

double bmin_ = (BMIN > 0.2) ? BMIN : 0.2;

double oc_ = (OC > 0.0) ? OC : 0.0;

double Jbone = (kpc_bone * bmin_ * blab_av + kcell_bone * oc_)

* acid_pH * (BW / 70.0);

dxdt_BLAB = (BLAB0 - BLAB) / tau_blab - Jbone;

dxdt_CUMBASE = Jbone;

dxdt_BMIN = k_remod * (OB - OC) - Jbone / BaseCap;

dxdt_HCO3_e = (alk_in + NAE - NEAP + Jbuf + Jbone) / vecf;

dxdt_GIVEN = alk_in;

dxdt_WASTE = HCO3_u;

// ---------------- renal adaptation ----------------------------------------

// BUG FIX #10: ammoniagenic capacity was driven ONLY by the plasma HCO3-

// error -- but that error is small precisely BECAUSE the kidney is

// compensating, so the ammonia arm could never be recruited and the V-ATPase

// controller railed at 1.0 on a merely high-protein diet. Dietary protein

// supplies the glutamine, so NEAP itself must drive it.

double errp = (err > 0.0) ? err : 0.0;

double ac_n = errp / 6.0; if (ac_n > 1.0) ac_n = 1.0;

double kdef_n = (K_pl0 - K_pl) / 1.5; if (kdef_n < 0.0) kdef_n = 0.0;

double NH3_t = NH3P0 * (1.0 + kNH3_load * (NEAP_h / NEAP_ref_h - 1.0)

+ kNH3_ac * ac_n + kNH3_K * kdef_n);

dxdt_NH3P = (NH3_t - NH3P) / tau_NH3;

double alk_n = (err < 0.0) ? (-err) : 0.0;

dxdt_PEND = (PEND0 * (1.0 + kpend * alk_n / 3.0) - PEND) / tau_pend;

double pti = (pHi_PT0 > pHi_PT) ? (pHi_PT0 - pHi_PT) : 0.0;

dxdt_NDC1 = ((1.0 + kndc1 * pti) - NDC1) / tau_ndc1;

// ---------------- potassium ----------------------------------------------

// Hdef == the H+ the collecting duct FAILED to secrete, expressed as the unmet

// acid load. Zero in health by construction, so no parameter has to be

// re-tuned to keep a normal subject in potassium balance.

double Hdef = NEAP_h - NAE - alk_in;

if (Hdef < 0.0) Hdef = 0.0;

double Kdiet_h = K_diet_kg * BW / 24.0;

double K_in = F_Kdiet * Kdiet_h + K_abs_K;

// BUG FIX #12: renal K+ secretion was LINEAR in plasma K+ with a weak

// aldosterone arm, so plasma K+ was almost undefended -- tripling K+ intake

// with K-citrate drove it to 6.6 mmol/L and Na-based alkali drove it to 1.4.

// Real K+ homeostasis has very high loop gain.

double Ksec = kK_sec * K_pl0 * pow(K_pl / K_pl0, nK_sec) * BW / 24.0

* (1.0 + kK_aldo * (ALDO - 1.0)) * neph

+ kK_H * Hdef

+ kK_Na * Na_alk_h

+ kK_hctz * C_hctz / (EC50_hctz + C_hctz) * BW / 24.0 * 0.05;

dxdt_KDEF = Ksec + f_stool_K * Kdiet_h - K_in;

double Kpl_t = K_pl0 - KDEF / (kKdef * BW) + kK_shift * (7.40 - pHbl);

if (Kpl_t > 7.5) Kpl_t = 7.5;

if (Kpl_t < 1.4) Kpl_t = 1.4;

dxdt_K_pl = (Kpl_t - K_pl) / tau_K;

double vdep = (VECF0 - vecf) / VECF0; if (vdep < 0.0) vdep = 0.0;

double kexc = (K_pl - K_pl0); if (kexc < 0.0) kexc = 0.0;

dxdt_ALDO = ((1.0 + kaldo_V * vdep + kaldo_K * kexc) - ALDO) / tau_aldo;

dxdt_VECF = (VECF0 * (1.0 - 0.05 * errp / 6.0) - vecf) / 24.0;

dxdt_Cl_pl = ((104.0 + (HCO3_set - HCO3) * 0.85) - Cl_pl) / 6.0;

// ---------------- calcium / PTH / phosphate ------------------------------

double Ca_bone = rCaBase * Jbone; // mmol/h

double UCa_basal = UCa0 * BW / 24.0;

double hz_ = C_hctz / (EC50_hctz + C_hctz);

double UCa_h = (UCa_basal

+ kUCa_NEAP * (NEAP - alk_in - NEAP_ref_h)

+ kUCa_bone * Ca_bone

+ kUCa_pH * acid_pH * (BW / 70.0) / 24.0)

* (1.0 - kUCa_hctz * hz_);

if (UCa_h < 0.0) UCa_h = 0.0;

// serum Ca stays normal because the bone Ca is spilled into the urine --

// dRTA is hypercalciuria WITHOUT hypercalcaemia.

double ca_ex = UCa_h - UCa_basal; if (ca_ex < 0.0) ca_ex = 0.0;

double Ca_t = Ca_pl0 + 0.35 * (Ca_bone - ca_ex);

if (Ca_t > 1.55) Ca_t = 1.55;

if (Ca_t < 0.85) Ca_t = 0.85;

dxdt_Ca_pl = (Ca_t - Ca_pl) / tau_Ca;

double dlow = (25.0 - C_25D) / 25.0; if (dlow < 0.0) dlow = 0.0;

double PTH_t = PTH0 * (1.0 + kPTH_Ca * (Ca_pl0 - Ca_pl) + 0.35 * dlow);

if (PTH_t < 3.0) PTH_t = 3.0;

dxdt_PTH = (PTH_t - PTH) / tau_PTH;

dxdt_Pi_pl = ((Pi_pl0 - 0.10 * (PTH / PTH0 - 1.0)) - Pi_pl) / 12.0;

// ---------------- bone cells --------------------------------------------

double OC_t = 1.0 + kOC_ac * acid_pH + kOC_PTH * (PTH / PTH0 - 1.0);

if (OC_t < 0.2) OC_t = 0.2;

dxdt_OC = (OC_t - OC) / tau_OC;

double OB_t = 1.0 - kOB_ac * acid_pH;

if (OB_t < 0.15) OB_t = 0.15;

dxdt_OB = (OB_t - OB) / tau_OB;

dxdt_bALP = (bALP0 * (1.0 + kbALP * acid_pH * 0.5) * (1.0 + 0.5 * growth_pot)

- bALP) / tau_bALP;

double dlow2 = (20.0 - C_25D) / 20.0; if (dlow2 < 0.0) dlow2 = 0.0;

dxdt_OSM = ((kOSM * acid_pH * 12.0 + kOSM_D * dlow2) - OSM) / tau_OSM;

dxdt_BMDz = kz_catch * (0.0 - BMDz) * (0.35 + 0.65 * growth_pot)

- kz_loss * Jbone / (BW / 70.0);

// ---------------- urine --------------------------------------------------

double kdi_n = (3.8 - K_pl) / 1.0; if (kdi_n < 0.0) kdi_n = 0.0;

dxdt_UVol_s = (UVol_kg * BW * (1.0 + kDI * kdi_n) - UVol_s) / tau_UVol;

dxdt_UCa_s = (UCa_h * 24.0 - UCa_s) / tau_UCa;

dxdt_UCit_s = (UCit_h * 24.0 - UCit_s) / tau_UCa;

dxdt_UpH_s = (uph - UpH_s) / 12.0;

// brushite (CaHPO4) supersaturation: free Ca2+ x HPO4(2-) / Ksp. All three

// arms of the therapeutic trade-off appear here: Ca up, citrate down, pH up.

double Ca_c = UCa_h * 24.0 / uvol;

double Cit_c = UCit_h * 24.0 / uvol;

double Pi_u_c = Pi_load * 24.0 / uvol;

double Ca_free = Ca_c / (1.0 + K_cacit * Cit_c);

double HPO4 = Pi_u_c / (1.0 + pow(10.0, pKa_Pi - uph));

double SS = Ca_free * HPO4 / Ksp_ref;

dxdt_SS_s = (SS - SS_s) / tau_SS;

double drive = (SS > 1.0) ? pow(SS - 1.0, nNC) : 0.0;

dxdt_NC = kNC * drive - kNC_res * NC;

dxdt_STONE = kSTONE * drive - kSTONE_res * STONE;

double ncp = (NC > 0.0) ? NC : 0.0;

double FIB_t = 1.0 - exp(-kFIB * ncp * 1000.0);

dxdt_FIB = (FIB_t - FIB) / tau_FIB;

// BUG FIX #2: nephron loss was driven by the target-minus-state GAP, which

// vanishes at steady state and made established nephrocalcinosis harmless.

double fibp = (FIB > 0.0) ? FIB : 0.0;

dxdt_NEPH = -kNEPH * fibp * neph;

// ---------------- growth / muscle / adherence ---------------------------

double acid_idx = (HCO3_set > HCO3) ? (HCO3_set - HCO3) : 0.0;

double GM = 1.0 / (1.0 + pow(acid_idx / K_ac_igf, h_igf));

dxdt_IGF1 = (IGF1_ref * GM - IGF1) / tau_IGF;

dxdt_Hz = kh_catch * (0.0 - Hz) * growth_pot * GM

- kh_drag * (1.0 - GM) * growth_pot;

double kw = (3.5 - K_pl) / 1.5; if (kw < 0.0) kw = 0.0;

double MUS_t = 1.0 - kMUS * pow(kw, 1.5);

if (MUS_t < 0.05) MUS_t = 0.05;

dxdt_MUS = (MUS_t - MUS) / tau_MUS;

// BUG FIX #13: the GI-irritation drive was normalised by BW/70 and then divided

// by 25, so ANY immediate-release bolus saturated it at the min(1, GI) cap and

// gastrointestinal tolerability became a binary switch rather than a

// dose-graded penalty. The drive is now the bolus absorption rate PER KG,

// which is what a patient actually feels. Note the PROLONGED-RELEASE granules

// (r_bicPR, r_citPR) do not enter: that is the tolerability advantage.

dxdt_GI = kGI * (a_bicIR + 3.0 * a_citIR) / BW - GI / tau_GI;

double gi_c = (GI < 1.0) ? GI : 1.0;

double nx = NINTAKE - 2.0; if (nx < 0.0) nx = 0.0;

double ADH_t = ADH_ref * exp(-kADH_n * nx) * (1.0 - kADH_GI * gi_c);

dxdt_ADH = (ADH_t - ADH) / tau_ADH;

dxdt_HEAR = kHEAR * ((HEAR_max > HEAR) ? (HEAR_max - HEAR) : 0.0);

// ---------------- exposure integrals ------------------------------------

dxdt_TBT = (HCO3 < TBT_thr) ? 1.0 : 0.0;

dxdt_AAC = (24.0 > HCO3) ? (24.0 - HCO3) : 0.0;

// ---------------- reported intermediates --------------------------------

o_uph = uph; o_pHbl = pHbl;

o_NAE = NAE * 24.0; o_TA = TA * 24.0; o_NH4 = NH4_u * 24.0;

o_HCO3u = HCO3_u * 24.0; o_NEAP = NEAP * 24.0; o_alk = alk_in * 24.0;

o_gap = (NEAP - alk_in - NAE) * 24.0;

o_Jbone = Jbone * 24.0; o_VHc = VHc; o_SS = SS; o_GM = GM;

o_UCa = UCa_h * 24.0; o_UCit = UCit_h * 24.0;

o_UCa_mgkg = UCa_h * 24.0 * 40.08 / BW;

o_CaCit = (UCa_h * 24.0 * 40.08) / ((UCit_h * 24.0 * 192.12 > 0.02)

? UCit_h * 24.0 * 192.12 : 0.02);

o_FEHCO3 = 100.0 * HCO3_u / ((gfrh * HCO3 > 1e-9) ? gfrh * HCO3 : 1e-9);

o_Hdef = Hdef * 24.0;

o_reserve = (NEAP_h > 1e-9) ? (Jh_max * LES / NEAP_h) : 0.0;

o_eGFR = GFR0 * neph;

o_Pi_u = Pi_load * 24.0;

o_resp = (HCO3 >= HCO3_set - RESP_MARGIN) ? 1.0 : 0.0;

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

$TABLE

double pH_blood = o_pHbl;

double urine_pH = o_uph;

double NAE_day = o_NAE;

double TA_day = o_TA;

double NH4_day = o_NH4;

double HCO3u_day = o_HCO3u;

double NEAP_day = o_NEAP;

double alkali_day = o_alk;

double acid_gap = o_gap;

double bone_day = o_Jbone;

double VH_eff = o_VHc;

double SS_inst = o_SS;

double UCa_day = o_UCa;

double UCa_mgkg = o_UCa_mgkg;

double UCit_day = o_UCit;

double CaCit = o_CaCit;

double FE_HCO3 = o_FEHCO3;

double Hdef_day = o_Hdef;

double reserve = o_reserve;

double eGFR = o_eGFR;

double Pi_urine = o_Pi_u;

double GrowthMult = o_GM;

double responder = o_resp;

double HCO3_target = HCO3_set;

double waste_frac = (GIVEN > 1.0) ? (WASTE / GIVEN) : 0.0;

$CAPTURE

pH_blood urine_pH NAE_day TA_day NH4_day HCO3u_day NEAP_day alkali_day

acid_gap bone_day VH_eff SS_inst UCa_day UCa_mgkg UCit_day CaCit FE_HCO3

Hdef_day reserve eGFR Pi_urine GrowthMult responder HCO3_target waste_frac

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

## R-SIDE HELPERS AND THE 28 SCENARIOS

## (mrgsolve ignores everything after $ENV that is not a recognised block;

## the code below is sourced by the accompanying driver, and is also valid

## standalone R.)

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

$ENV

## ---- compartment indices, for building events ---------------------------

CMT <- c(AG_bicIR = 1, AG_citIR = 2, AG_bicPR = 3, AG_citPR = 4,

AG_KCl = 5, AG_K = 6, AG_hctz = 8, AG_vitD = 10, AG_acid = 56)

## ---- canonical subjects -------------------------------------------------

SUBJ <- list(

infant = list(BW = 8.0, AGE = 0.75, BSA = 0.42, GFR0 = 85),

toddler= list(BW = 12.0, AGE = 2.0, BSA = 0.54, GFR0 = 105),

child = list(BW = 30.0, AGE = 10.0, BSA = 1.05, GFR0 = 115),

teen = list(BW = 52.0, AGE = 15.0, BSA = 1.55, GFR0 = 110),

adult = list(BW = 70.0, AGE = 35.0, BSA = 1.80, GFR0 = 105)

)

## ---- lesion severities --------------------------------------------------

## LES = retained H+-pump Vmax (rate defect); LES_grad = retained maximal

## blood->urine pH gradient (gradient defect). Both are dimensionless

## fractions of normal, and they are the ONLY way the lesion enters.

LESION <- list(

none = list(LES = 1.00, LES_grad = 1.00),

incomplete = list(LES = 0.34, LES_grad = 0.70), # normal HCO3, fails acid load

complete = list(LES = 0.18, LES_grad = 0.58),

severe = list(LES = 0.12, LES_grad = 0.52), # infantile ATP6V0A4-like

gradonly = list(LES = 1.00, LES_grad = 0.58), # pure gradient defect

rateonly = list(LES = 0.18, LES_grad = 1.00) # pure rate defect

)

## ---- regimen builder ----------------------------------------------------

## kind: "none" | "bicIR" | "citIR" | "ADV" | "mixed"

## cation: "K" (potassium salt) | "Na" (sodium salt) | "NaK"

make_regimen <- function(kind = "none", mEq_kg_day = 0, times = c(7, 13, 19),

cation = "K", days = 365, BW = 30, f_cit = NULL,

KCl_mmol_day = 0, hctz_mg = 0, vitD_IU = 0,

start_day = 0) {

ev <- NULL

add <- function(cmt, amt, ii_times) {

for (ct in ii_times)

ev <<- rbind(ev, data.frame(

time = (start_day:(days - 1)) * 24 + ct, cmt = cmt, amt = amt,

evid = 1))

}

if (kind != "none" && mEq_kg_day > 0) {

per <- mEq_kg_day * BW / length(times)

fc <- if (!is.null(f_cit)) f_cit else

switch(kind, bicIR = 0, citIR = 1, ADV = 0.35, mixed = 0.5, 0)

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