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

## ANDROGENETIC ALOPECIA (male & female pattern hair loss) — QSP model

## 50 ODE compartments · 12 drug PK · 8 enzyme · 4 endocrine · 4 signalling

## · 5 systemic readouts · 17 follicle

## Reference subject: 5.07 cm2 (1-inch diameter) vertex target area,

## 1268 follicles. TIME UNIT: DAYS. Concentrations ng/mL.

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

## THE THESIS THIS MODEL IS BUILT AROUND

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

## Androgenetic alopecia is taught as hair LOSS. This model asserts that

## almost nothing is lost — the follicles are still there, they are

## smaller — and that every clinically important number follows from that

## one substitution once you notice that follicle size is only ever

## changed at ONE moment of the cycle: the exit from telogen.

## (1) NOTHING IS LOST; THINGS GET SMALLER.

## The reference patient (below) reaches the exact baseline of the

## pivotal finasteride trial — 876 non-vellus hairs in a 1-inch

## circle (Kaufman 1998, PMID 9777765, baseline 876) — while still

## owning 1006 of his original 1268 follicles. Of the 392-hair

## "deficit", 262 follicles are genuinely gone and 130 are alive and

## vellus. Every drug in this file competes for those 130.

## (2) THE RATCHET TURNS ONLY AT TELOGEN EXIT, SO THE DISEASE RUNS ON

## CYCLE TIME, NOT ON CALENDAR TIME.

## Cycle periods in the model (anagen shortened by androgen, ARD 0.95):

## terminal 832 d -> 2.2 cycles in 5 years

## intermediate 517 d -> 3.5

## small interm. 296 d -> 6.2

## vellus 167 d -> 10.9

## A terminal follicle gets ~2 chances to shrink in 5 years, which is

## why AGA takes decades; a vellus follicle gets ~11 chances to grow

## back, which is why treatment response is visible in ONE year.

## Both facts come from the same number.

## (3) THE DAILY SHED IS A RATIO, NOT A RATE.

## shed = N_total x f_telogen / T_telogen. Normal scalp:

## 100 000 x 0.090 / 100 d = 90 hairs/day

## the textbook "about 100 a day", derived rather than asserted. In

## the model's AGA patient the telogen fraction rises to 23.6% purely

## because anagen is shorter in the smaller classes — increased

## shedding in AGA is a CONSEQUENCE of miniaturisation, not a

## separate disease process, and the trichogram telogen percentage is

## a direct read-out of anagen duration.

## (4) THE FINASTERIDE DOSE-RESPONSE IS FLAT BECAUSE THE TARGET

## INTEGRATES. Finasteride's plasma half-life is 6 h; the enzyme it

## inactivates turns over with a half-life of 2 d, and the covalent

## enzyme-NADP-dihydrofinasteride adduct with a half-life of 30 d.

## Steady-state free enzyme is therefore k_deg/(k_deg + <k_on x C>),

## which depends on the AVERAGE concentration, not the peak. The

## model reproduces the whole published dose-response (Drake 1999,

## PMID 10495374; day 42 scalp / serum DHT, model vs observed):

## dose scalp DHT serum DHT

## 0.01 mg -18.3% (obs -14.9) -20.1% (obs n/a)

## 0.05 mg -43.1% (obs -61.6) -47.3% (obs -49.5)

## 0.2 mg -57.5% (obs -56.5) -63.1% (obs -68.6)

## 1 mg -63.6% (obs -64.1) -70.3% (obs -71.4)

## 5 mg -67.4% (obs -69.4) -74.9% (obs -72.2)

## Five times the dose buys 3.8 percentage points of scalp DHT and,

## downstream, FOUR HAIRS at one year (+105.2 vs +101.3 over placebo).

## (5) 5 mg BEATS 1 mg ON SCALP DHT BUT NOT ON SERUM DHT, AND THE MODEL

## SAYS WHY. Finasteride is ~2000x weaker on type I than type II in

## this parameterisation. At 1 mg free type I is 0.979; at 5 mg it is

## 0.903. Scalp skin — unlike blood — contains sebaceous type I, so

## only the scalp compartment can see that extra 7.6%. The published

## asymmetry (scalp -64.1 -> -69.4 while serum -71.4 -> -72.2) is

## recovered without a scalp-specific fitted parameter.

## (6) THE RECOVERABLE POOL IS THE RECENTLY MINIATURISED FOLLICLE, AND IT

## AGES OUT. Class 4 is split into A4a/C4a/T4a (recent vellus, can

## re-enlarge, PUW 0.75) and A4b/C4b/T4b (established vellus, PUW

## 0.05), with a one-way flux p_age between them. This single

## structure produces the plateau of the finasteride response, the

## failure of late starts, and the irreversibility of Norwood VI:

## start at TAHC 1150 -> 58.8% of the deficit recovered by 3 y

## start at TAHC 1000 -> 40.9%

## start at TAHC 876 -> 30.1%

## (7) HAIR WEIGHT MOVES TWICE AS FAR AS HAIR COUNT, BECAUSE MASS GOES AS

## d^2. Same simulation, two endpoints, at 12 months:

## finasteride 1 mg count +9.9% mass +18.6% ratio 1.88

## minoxidil 5% count +2.9% mass +5.9% ratio 2.05

## dutasteride count +12.3% mass +20.3% ratio 1.65

## Trials that count hairs are reading the smaller half of the effect.

## (8) MINOXIDIL'S "DREAD SHED" IS THE MECHANISM WORKING, NOT A SIDE

## EFFECT. Minoxidil sulfate shortens telogen (T_T 100 -> 49 d at

## full effect). Telogen exit REQUIRES release of the club hair, so

## an abrupt shortening must produce a transient rise in shedding

## before it produces any hair. No term in the model represents

## shedding as toxicity, yet the responder's shed goes

## 2.38 -> 3.88 hairs/day per target area (x1.63) at day 10,

## back below placebo by day 140, +25 hairs by 12 months.

## (9) MINOXIDIL IS A PRODRUG, SO THE TRIAL MEAN IS A MIXTURE. Follicular

## SULT1A1 activity splits the population; the same 5% solution gives

## +39.8 hairs vs placebo at 12 months in a responder and +18.7 in a

## non-responder. A trial reporting the mean is reporting neither.

## (10) STOPPING FINASTERIDE RESTORES DHT IN TEN DAYS AND LOSES THE HAIR

## OVER YEARS. After withdrawal at 2 y: scalp DHT 0.364 -> 0.980 by

## day 10 (enzyme resynthesis, t1/2 2 d), while the hair count is

## still +110 at day 20, +101 at 1 y and +79 at 3 y post-stop. The

## two clocks differ by two orders of magnitude.

## (11) THE SAME ARITHMETIC EXCUSES IMPERFECT ADHERENCE. Taking 4 doses in

## 7 leaves scalp DHT at 0.389 instead of 0.364 and costs 10 hairs at

## one year (+91.4 vs +101.3 over placebo) — 90% of the benefit from

## 57% of the tablets, because the enzyme, not the plasma, is the

## thing being titrated.

## (12) TOPICAL FINASTERIDE SEPARATES THE TWO COMPARTMENTS. With separate

## scalp and systemic enzyme pools, 0.25% topical gives scalp DHT

## -47.6% and serum DHT -27.4% (observed ~-25%), keeping 78% of the

## hair benefit for 42% of the systemic androgen signal.

## (13) IN FEMALE PATTERN HAIR LOSS EVERY ANTI-ANDROGEN SCALES WITH THE

## ANDROGENIC SHARE OF THE DRIVE, AND MINOXIDIL DOES NOT. ARIND

## carries androgen-independent inhibitory drive. Holding presenting

## severity fixed (age 33.5 y, 899 hairs, 216 vellus) and varying only

## the SPLIT between GS and ARIND, with the drug model untouched:

## androgenic share finasteride spironolactone minoxidil 5%

## 34% +53.2 +136.6 +67.2

## 22% +35.2 +99.9 +67.1

## 11% +15.7 +53.4 +67.1

## Minoxidil is identical to three significant figures; every

## anti-androgen is proportional. The negative post-menopausal

## finasteride trial (Price 2000, PMID 11050579) is therefore read as

## a low-androgenic-share POPULATION, not as an inert drug — the same

## model gives +101.3 in the man.

## CALIBRATION SUMMARY (targets, and how the model does)

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

## Kaufman 1998 (PMID 9777765) baseline 876 hairs / 5.1 cm2 ...... 876 (by construction)

## Kaufman 1998 finasteride 1 mg vs placebo, 1 year: +107 ......... +101.3

## Kaufman 1998 finasteride 1 mg vs placebo, 2 years: +138 ........ +139.2

## Drake 1999 (PMID 10495374) five-point DHT dose-response ........ see (4) above

## Gubelin Harcha 2014 (PMID 24411083) dutasteride > finasteride

## at 24 weeks .................................................. +80.8 vs +63.4

## Clark 2004 (PMID 15126539) dutasteride serum DHT ~-94% ......... -96.1%

## normal daily shed ~100 hairs/day ............................... 90

## normal telogen fraction 9-14% .................................. 9.0%

## topical finasteride serum DHT ~-25% (Piraccini 2022, 34634163) .. -27.4%

## WHERE THE MODEL DISAGREES WITH THE LITERATURE (reported, not tuned away)

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

## * Dutasteride SCALP DHT: model -90.3% at day 42, literature range

## -51% to -79%. The model's scalp compartment has only a 6%

## 5AR-independent floor (W_ALT); a larger floor would fit dutasteride

## but would then break the finasteride 1 mg point, which is the better

## measured of the two. Reported as an over-prediction.

## * Drake's 0.05 mg SCALP point (-61.6%) is not fitted (model -43.1%).

## That point is non-monotonic in the source data — 0.05 mg is reported

## as suppressing scalp DHT MORE than 0.2 mg (-56.5%) — which no

## mass-action model can produce. The 0.05 mg SERUM point is fitted

## well (-47.3 vs -49.5).

## * Setipiprant: with PGD_EFF = 0.18 the model predicts +46 hairs vs

## placebo at 12 months, which the phase 2a programme did not see.

## Rather than delete the arm, the model is used the other way round:

## to be consistent with a null trial the PGD2 arm can carry at most

## ~5% of the inhibitory drive (see the PGD_EFF scan in the README).

## * Placebo decline: model -14.7 hairs at 1 y vs ~-21 published; the

## model's untreated arm is linear where the trial's accelerates.

## IMPLEMENTATION NOTE

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

## Every equation here was first written and run in an independent

## Python/scipy implementation (fixed-step RK4, 50 states). That

## cross-implementation found four real defects: (i) a dropped

## up-transition flux (class 3 -> class 2) that silently destroyed

## follicles and made finasteride look WORSE than placebo; (ii) a

## 1000-fold concentration unit error (V in L against k_on in per-ng/mL)

## that put free type-II enzyme at zero for every dose; (iii) a

## testosterone feedback loop with DHT weighted at 0.65, which produced a

## 47% rise in testosterone on finasteride instead of the observed ~10%;

## and (iv) an RK4 step of 0.05 d that went unstable at 5 mg finasteride

## (k_on x C_max = 224/d) and returned NaN for the entire arm. All four

## are fixed here. Numbers quoted above are from the corrected runs.

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

library(mrgsolve)

library(dplyr)

aga_code <- '

$PARAM @annotated

// ---------------------------------------------------------------- finasteride

FIN_F : 0.65 : finasteride oral bioavailability

FIN_KA : 15.0 : finasteride absorption rate (1/d)

FIN_V : 76000 : finasteride V/F (mL)

FIN_KE : 2.7726 : finasteride elimination (1/d, t1/2 6 h)

FIN_KWASH : 1.5 : topical finasteride wash-off from scalp depot (1/d)

FIN_KFT : 0.62 : topical finasteride scalp depot -> scalp tissue (1/d)

FIN_SCL : 20.0 : mg of scalp depot -> ng/mL of scalp tissue scaling

FIN_KDIF : 4.0 : plasma <-> scalp tissue exchange (1/d)

FIN_KOUTT : 4.0 : scalp tissue finasteride clearance (1/d)

TOPFIN_F : 0.002 : systemic bioavailability of topical finasteride

// ---------------------------------------------------------------- dutasteride

DUT_F : 0.60 : dutasteride bioavailability

DUT_KA : 5.0 : dutasteride absorption (1/d)

DUT_V1 : 100000 : dutasteride central volume (mL)

DUT_V2 : 300000 : dutasteride peripheral volume (mL)

DUT_Q : 20000 : dutasteride intercompartmental clearance (mL/d)

DUT_CL : 7500 : dutasteride clearance (mL/d)

// ---------------------------------------------------------------- minoxidil

MX_KFOL : 0.55 : scalp depot -> follicle (1/d)

MX_KSYS : 0.020 : scalp depot -> systemic (1/d) [=> 1.1% absorbed]

MX_KWASH : 1.20 : scalp depot wash-off (1/d)

MX_KFOUT : 2.0 : follicular minoxidil elimination (1/d)

MX_KSULT : 3.0 : SULT1A1 sulfotransferase scalar (1/d)

MX_KSOUT : 1.5 : minoxidil sulfate elimination (1/d)

MX_VO : 180000 : minoxidil systemic volume (mL)

MX_KEO : 4.0774 : minoxidil systemic elimination (1/d, t1/2 4 h)

MX_FORAL : 0.90 : oral minoxidil bioavailability

MX_FOLOR : 5.5 : systemic -> follicular minoxidil transfer

MX_EC50 : 12.0 : minoxidil sulfate EC50 (follicular units)

MX_HILL : 1.5 : minoxidil Hill coefficient

MX_GTT : 1.05 : telogen shortening gain

MX_GTA : 0.40 : anagen prolongation gain

SULT : 2.0 : follicular SULT1A1 activity (responder 2.0 / non 0.25)

// ---------------------------------------------------------------- other drugs

SPI_V : 90000 : canrenone volume (mL)

SPI_KE : 0.9902 : canrenone elimination (1/d)

SPI_F : 0.70 : spironolactone bioavailability

SPI_IC50 : 90.0 : canrenone AR-antagonism IC50 (ng/mL)

// ------------------------------------------------------- 5-alpha-reductase

KDEG_E : 0.3466 : 5AR protein turnover (1/d, t1/2 2 d)

KON_FIN2 : 5.24 : finasteride capture of type II (per ng/mL per d)

KON_FIN1 : 0.00262: finasteride capture of type I (2000x weaker)

KON_DUT2 : 0.50 : dutasteride capture of type II

KON_DUT1 : 0.40 : dutasteride capture of type I

KDISS1 : 0.0231 : adduct dissociation type I (1/d, t1/2 30 d)

KDISS2 : 0.0231 : adduct dissociation type II (1/d, t1/2 30 d)

// ----------------------------------------------------------------- HPG axis

KLH : 4.0 : LH production (1/d)

KDLH : 4.0 : LH elimination (1/d)

FB_A_T : 0.70 : testosterone weight in gonadal negative feedback

FB_A_D : 0.30 : DHT weight in gonadal negative feedback

FB_H : 2.0 : feedback Hill coefficient

KSYN_T : 4.0 : testosterone synthesis scalar (1/d)

KEL_T : 4.0 : testosterone elimination (1/d)

W_T2_SER : 0.765 : fraction of serum DHT from type II

W_T1_SER : 0.235 : fraction of serum DHT from type I

KEL_DHTS : 4.0 : serum DHT equilibration (1/d)

W_LOC : 0.52 : scalp DHT from local synthesis

W_SER : 0.42 : scalp DHT taken up from serum

W_ALT : 0.06 : scalp DHT from 5AR-independent routes

W_T1_LOC : 0.30 : type I share of local scalp synthesis

W_T2_LOC : 0.70 : type II share of local scalp synthesis

KEL_DHTC : 4.0 : scalp DHT equilibration (1/d)

AROM : 1.0 : aromatase scalar (raise for female arm)

LOC5AR : 1.0 : follicular 5AR capacity (0.30 in the female arm)

// ------------------------------------------------------- AR signalling in DP

AR_EC50 : 0.55 : scalp DHT EC50 for AR signal (normalised)

AR_H : 2.0 : AR Hill coefficient

K_ARN : 0.0990 : nuclear AR signal turnover (1/d, t1/2 7 d)

K_DKK : 0.1386 : DKK-1 turnover (1/d, t1/2 5 d)

K_WNT : 0.1386 : Wnt tone turnover (1/d)

K_PGD : 0.2310 : PGD2 turnover (1/d, t1/2 3 d)

PGD_GAIN : 0.45 : AR-driven rise in PGD2

PGD_EFF : 0.18 : weight of the PGD2 arm in total inhibitory drive

SETI : 0.0 : setipiprant / CRTH2 blockade (0-1)

ARIND : 0.05 : androgen-INDEPENDENT inhibitory drive

GS : 0.90 : genetic susceptibility (0-1.3)

// --------------------------------------------------------- follicle kinetics

NTOT : 1268 : follicles in the 5.07 cm2 target area

TA1 : 1000 : anagen duration, terminal (d)

TA2 : 560 : anagen duration, intermediate (d)

TA3 : 250 : anagen duration, small intermediate (d)

TA4 : 70 : anagen duration, vellus (d)

TCAT : 17 : catagen duration (d)

TTEL : 100 : telogen duration (d)

D1 : 78 : shaft diameter, terminal (um)

D2 : 60 : shaft diameter, intermediate (um)

D3 : 43 : shaft diameter, small intermediate (um)

D4 : 24 : shaft diameter, vellus (um)

G1 : 0.36 : linear growth rate, terminal (mm/d)

G2 : 0.31 : linear growth rate, intermediate

G3 : 0.24 : linear growth rate, small intermediate

G4 : 0.13 : linear growth rate, vellus

PDMAX : 0.17 : maximum per-cycle probability of shrinking one class

PUMAX : 2.20 : restitution scalar (per-cycle, x (1-ARD))

PUMX : 0.40 : minoxidil-driven restitution (androgen-independent)

PUW2 : 1.00 : restitution weight, intermediate -> terminal

PUW3 : 0.90 : restitution weight, small interm -> intermediate

PUW4A : 0.75 : restitution weight, RECENT vellus -> small interm

PUW4B : 0.05 : restitution weight, ESTABLISHED vellus

PLOSS : 0.055 : per-cycle probability of follicle death (vellus only)

PAGE : 0.15 : per-cycle recent vellus -> established vellus

FA_AR : 0.30 : fractional anagen shortening at full androgen drive

// -------------------------------------------------------- systemic readouts

K_PSA : 0.0277 : PSA turnover (1/d, t1/2 25 d)

K_SEB : 0.0693 : sebum turnover (1/d)

K_MAP : 0.3466 : MAP turnover (1/d)

MAP0 : 92 : baseline mean arterial pressure (mmHg)

MAP_G : 0.65 : MAP fall per ng/mL of systemic minoxidil

K_HTR : 0.0231 : hypertrichosis index turnover (1/d)

HTR_G : 8.0 : hypertrichosis gain

$CMT @annotated

FING : finasteride gut depot (ng)

FINC : finasteride plasma (ng/mL)

FINSK : topical finasteride scalp depot (mg)

FINT : finasteride scalp tissue (ng/mL)

DUTG : dutasteride gut depot (ng)

DUTC : dutasteride central (ng/mL)

DUTP : dutasteride peripheral (ng/mL)

MXSK : topical minoxidil scalp depot (mg)

MXDF : follicular minoxidil (units)

MXSF : follicular minoxidil SULFATE (units)

MXDC : minoxidil systemic (ng/mL)

SPIC : canrenone plasma (ng/mL)

E1 : free 5AR type I, systemic (fraction)

EI1 : inhibited 5AR type I, systemic

E2 : free 5AR type II, systemic (fraction)

EI2 : inhibited 5AR type II, systemic

E1S : free 5AR type I, SCALP (fraction)

EI1S : inhibited 5AR type I, scalp

E2S : free 5AR type II, SCALP (fraction)

EI2S : inhibited 5AR type II, scalp

LH : luteinising hormone (normalised)

TST : serum testosterone (normalised)

DHTS : serum DHT (normalised)

DHTC : scalp / follicular DHT (normalised)

ARN : nuclear AR signal in dermal papilla

DKK1 : DKK-1 / TGF-beta paracrine inhibitor

PGD2 : prostaglandin D2 (normalised)

WNT : Wnt / beta-catenin growth tone

PSA : serum PSA (normalised)

EST : serum estradiol (normalised)

SEB : sebum output (normalised)

MAP : mean arterial pressure (mmHg)

HTR : hypertrichosis index

A1 : anagen, terminal

A2 : anagen, intermediate

A3 : anagen, small intermediate

A4A : anagen, vellus RECENT

A4B : anagen, vellus ESTABLISHED

C1 : catagen, terminal

C2 : catagen, intermediate

C3 : catagen, small intermediate

C4A : catagen, vellus recent

C4B : catagen, vellus established

T1 : telogen, terminal

T2 : telogen, intermediate

T3 : telogen, small intermediate

T4A : telogen, vellus recent

T4B : telogen, vellus established

LOST : follicles lost (fibrous streamers)

CSHED : cumulative shed hairs (target area)

$MAIN

// initial condition: an intact 18-year-old scalp, all terminal, at

// cycle-phase steady state. The presenting patient is produced by a

// burn-in (see the scenario section at the bottom of the file).

double cyc0 = TA1 + TCAT + TTEL;

A1_0 = NTOT * TA1 / cyc0;

C1_0 = NTOT * TCAT / cyc0;

T1_0 = NTOT * TTEL / cyc0;

E1_0 = 1.0; E2_0 = 1.0; E1S_0 = 1.0; E2S_0 = 1.0;

LH_0 = 1.0; TST_0 = 1.0; DHTS_0 = 1.0; DHTC_0 = 1.0;

PGD2_0 = 1.0; WNT_0 = 1.0; PSA_0 = 1.0; EST_0 = 1.0; SEB_0 = 1.0;

MAP_0 = MAP0;

$ODE

// ------------------------------------------------------------------- guards

double fc = FINC > 0 ? FINC : 0.0;

double ft = FINT > 0 ? FINT : 0.0;

double dc = DUTC > 0 ? DUTC : 0.0;

double msk = MXSK > 0 ? MXSK : 0.0;

double mdf = MXDF > 0 ? MXDF : 0.0;

double msf = MXSF > 0 ? MXSF : 0.0;

double mdc = MXDC > 0 ? MXDC : 0.0;

double spc = SPIC > 0 ? SPIC : 0.0;

double dhc = DHTC > 1e-9 ? DHTC : 1e-9;

// ----------------------------------------------------------------------- PK

dxdt_FING = -FIN_KA*FING;

dxdt_FINC = FIN_KA*FING/FIN_V - FIN_KE*FINC;

dxdt_FINSK = -(FIN_KFT + FIN_KWASH)*FINSK;

dxdt_FINT = FIN_KFT*FINSK*FIN_SCL + FIN_KDIF*fc - FIN_KOUTT*FINT;

dxdt_DUTG = -DUT_KA*DUTG;

dxdt_DUTC = (DUT_KA*DUTG - DUT_CL*DUTC - DUT_Q*DUTC + DUT_Q*DUTP)/DUT_V1;

dxdt_DUTP = (DUT_Q*DUTC - DUT_Q*DUTP)/DUT_V2;

dxdt_MXSK = -(MX_KFOL + MX_KSYS + MX_KWASH)*MXSK;

dxdt_MXDF = MX_KFOL*msk + MX_FOLOR*mdc - MX_KFOUT*MXDF - MX_KSULT*SULT*MXDF;

dxdt_MXSF = MX_KSULT*SULT*mdf - MX_KSOUT*MXSF;

dxdt_MXDC = MX_KSYS*msk*1e6/MX_VO - MX_KEO*MXDC;

dxdt_SPIC = -SPI_KE*SPIC;

// -------------------------------------- 5AR: mechanism-based, quasi-irrevers.

// Free enzyme obeys E_ss = k_deg / (k_deg + <k_on x C>) : the TARGET, not the

// plasma, is what the dose titrates. This one line is thesis (4) and (11).

double kon1 = KON_FIN1*fc + KON_DUT1*dc; // systemic pool

double kon2 = KON_FIN2*fc + KON_DUT2*dc;

double kon1s = KON_FIN1*ft + KON_DUT1*dc; // scalp pool (topical-aware)

double kon2s = KON_FIN2*ft + KON_DUT2*dc;

dxdt_E1 = KDEG_E - KDEG_E*E1 - kon1 *E1 + KDISS1*EI1;

dxdt_EI1 = kon1 *E1 - KDISS1*EI1 - KDEG_E*EI1;

dxdt_E2 = KDEG_E - KDEG_E*E2 - kon2 *E2 + KDISS2*EI2;

dxdt_EI2 = kon2 *E2 - KDISS2*EI2 - KDEG_E*EI2;

dxdt_E1S = KDEG_E - KDEG_E*E1S - kon1s*E1S + KDISS1*EI1S;

dxdt_EI1S = kon1s*E1S - KDISS1*EI1S - KDEG_E*EI1S;

dxdt_E2S = KDEG_E - KDEG_E*E2S - kon2s*E2S + KDISS2*EI2S;

dxdt_EI2S = kon2s*E2S - KDISS2*EI2S - KDEG_E*EI2S;

// ------------------------------------------------------------------ HPG axis

double fb = 1.0/(1.0 + pow(FB_A_T*TST + FB_A_D*DHTS, FB_H));

double fb0 = 1.0/(1.0 + pow(FB_A_T + FB_A_D, FB_H));

dxdt_LH = KLH*(fb/fb0) - KDLH*LH;

dxdt_TST = KSYN_T*LH - KEL_T*TST;

double ser = (W_T2_SER*E2 + W_T1_SER*E1)*TST;

dxdt_DHTS = KEL_DHTS*(ser - DHTS);

double loc = LOC5AR*(W_T1_LOC*E1S + W_T2_LOC*E2S)*TST;

dxdt_DHTC = KEL_DHTC*(W_LOC*loc + W_SER*DHTS + W_ALT - DHTC);

// --------------------------------------------- AR signalling, dermal papilla

double spi = 1.0/(1.0 + spc/SPI_IC50);

double h_now = pow(dhc, AR_H)/(pow(AR_EC50,AR_H) + pow(dhc,AR_H));

double h_ref = 1.0/(pow(AR_EC50,AR_H) + 1.0);

double arraw = GS*spi*h_now/h_ref;

dxdt_ARN = K_ARN*(arraw - ARN);

dxdt_DKK1 = K_DKK*(ARN - DKK1);

dxdt_PGD2 = K_PGD*(1.0 + PGD_GAIN*ARN - PGD2);

double msfh = pow(msf, MX_HILL);

double mx = msfh/(pow(MX_EC50,MX_HILL) + msfh);

dxdt_WNT = K_WNT*((1.0 - 0.6*DKK1 + 0.5*mx) - WNT);

double pgds = (1.0 - SETI)*((PGD2 > 1.0 ? PGD2 - 1.0 : 0.0)/PGD_GAIN);

double ARD = (1.0 - PGD_EFF)*DKK1 + PGD_EFF*pgds + ARIND;

if (ARD < 0.0) ARD = 0.0;

if (ARD > 1.5) ARD = 1.5;

double ARDc = ARD > 1.0 ? 1.0 : ARD;

// --------------------------------------------------------- the follicle cycle

// Everything above enters here and NOWHERE else. Class transitions occur only

// in the telogen->anagen flux, which is why the disease runs on cycle time.

double TTe = TTEL/(1.0 + MX_GTT*mx);

double fAn = (1.0 - FA_AR*ARDc)*(1.0 + MX_GTA*mx);

double TAe1 = TA1*fAn, TAe2 = TA2*fAn, TAe3 = TA3*fAn, TAe4 = TA4*fAn;

double pdn = PDMAX*ARD; if (pdn > 0.85) pdn = 0.85;

double pu0 = PUMAX*(1.0 - ARDc) + PUMX*mx;

double pu2 = pu0*PUW2, pu3 = pu0*PUW3, pu4a = pu0*PUW4A, pu4b = pu0*PUW4B;

double cap = 0.90 - pdn; if (cap < 0.0) cap = 0.0;

if (pu2 > cap) pu2 = cap;

if (pu3 > cap) pu3 = cap;

if (pu4a > cap) pu4a = cap;

if (pu4b > cap) pu4b = cap;

double plo = PLOSS*ARD; if (plo > 0.85) plo = 0.85;

double f1 = T1/TTe, f2 = T2/TTe, f3 = T3/TTe, f4a = T4A/TTe, f4b = T4B/TTe;

dxdt_A1 = f1*(1.0 - pdn) + f2*pu2 - A1/TAe1;

dxdt_A2 = f2*(1.0 - pdn - pu2) + f1*pdn + f3*pu3 - A2/TAe2;

dxdt_A3 = f3*(1.0 - pdn - pu3) + f2*pdn + f4a*pu4a + f4b*pu4b - A3/TAe3;

dxdt_A4A = f4a*(1.0 - pu4a - plo - PAGE) + f3*pdn - A4A/TAe4;

dxdt_A4B = f4b*(1.0 - pu4b - plo) + f4a*PAGE - A4B/TAe4;

dxdt_C1 = A1/TAe1 - C1/TCAT;

dxdt_C2 = A2/TAe2 - C2/TCAT;

dxdt_C3 = A3/TAe3 - C3/TCAT;

dxdt_C4A = A4A/TAe4 - C4A/TCAT;

dxdt_C4B = A4B/TAe4 - C4B/TCAT;

dxdt_T1 = C1/TCAT - f1;

dxdt_T2 = C2/TCAT - f2;

dxdt_T3 = C3/TCAT - f3;

dxdt_T4A = C4A/TCAT - f4a;

dxdt_T4B = C4B/TCAT - f4b;

dxdt_LOST = (f4a + f4b)*plo;

dxdt_CSHED = f1 + f2 + f3 + f4a + f4b;

// -------------------------------------------------------- systemic readouts

dxdt_PSA = K_PSA*((0.42 + 0.58*(0.25*E1 + 0.75*E2)) - PSA);

dxdt_EST = 4.0*(AROM*TST*(1.0 + 0.04*(1.0 - DHTS)) - EST);

dxdt_SEB = K_SEB*((0.25 + 0.75*E1S) - SEB);

dxdt_MAP = K_MAP*((MAP0 - MAP_G*mdc) - MAP);

dxdt_HTR = K_HTR*(HTR_G*mdc - HTR);

$TABLE

double N1 = A1 + C1 + T1;

double N2 = A2 + C2 + T2;

double N3 = A3 + C3 + T3;

double N4A = A4A + C4A + T4A;

double N4B = A4B + C4B + T4B;

double NTO = N1 + N2 + N3 + N4A + N4B;

// TARGET-AREA HAIR COUNT: non-vellus hairs in the 1-inch circle.

double TAHC = N1 + N2 + N3;

double VELL = N4A + N4B;

// HAIR MASS INDEX: mass goes as diameter SQUARED times linear growth rate.

double HMI = 100.0*(N1*D1*D1*G1 + N2*D2*D2*G2 + N3*D3*D3*G3

+ (N4A+N4B)*D4*D4*G4)/(NTOT*D1*D1*G1);

double MDIA = NTO > 1e-9 ? (N1*D1 + N2*D2 + N3*D3 + (N4A+N4B)*D4)/NTO : 0.0;

double TELPC = NTO > 1e-9 ? 100.0*(T1+T2+T3+T4A+T4B)/NTO : 0.0;

// daily shed, target area and whole-scalp equivalent (x 100000/NTOT)

double TTe_o = TTEL/(1.0 + MX_GTT*(pow(MXSF>0?MXSF:0,MX_HILL)

/(pow(MX_EC50,MX_HILL)+pow(MXSF>0?MXSF:0,MX_HILL))));

double SHEDTA = (T1+T2+T3+T4A+T4B)/TTe_o;

double SHEDSC = SHEDTA*100000.0/NTOT;

// sexual adverse-effect index and modelled incidence (%)

double SEXIDX = 1.0 - (0.35*E1 + 0.65*E2);

double SEXPCT = 2.1 + 2.6*SEXIDX;

double PSAADJ = 2.0*PSA; // the "double the PSA" clinical rule

$CAPTURE TAHC HMI VELL MDIA TELPC SHEDTA SHEDSC N1 N2 N3 N4A N4B

SEXIDX SEXPCT PSAADJ

mod <- mcode("aga", aga_code)

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

## DOSING HELPERS

## Finasteride and dutasteride enter FING / DUTG as ng of ABSORBED drug.

## Topical minoxidil enters MXSK in mg, twice daily.

## Topical finasteride enters BOTH FINSK (mg) and FING (its 0.2% systemic

## share), twice daily.

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

ev_fin <- function(mg = 1, days = 1825, F = 0.65)

ev(amt = mg*1e6*F, cmt = "FING", ii = 1, addl = days - 1)

ev_dut <- function(mg = 0.5, days = 1825, F = 0.60)

ev(amt = mg*1e6*F, cmt = "DUTG", ii = 1, addl = days - 1)

ev_mxt <- function(mg = 50, days = 1825) # 1 mL of 5% BID

ev(amt = mg, cmt = "MXSK", ii = 0.5, addl = 2*days - 1)

ev_mxo <- function(mg = 5, days = 1825, F = 0.90, V = 180000)

ev(amt = mg*1e6*F/V, cmt = "MXDC", ii = 1, addl = days - 1)

ev_spi <- function(mg = 100, days = 1825, F = 0.70, V = 90000)

ev(amt = mg*1e6*F/V, cmt = "SPIC", ii = 1, addl = days - 1)

ev_topfin <- function(mg = 2.5, days = 1825, Fsys = 0.002) # 0.25%, BID

c(ev(amt = mg, cmt = "FINSK", ii = 0.5, addl = 2*days - 1),

ev(amt = mg*1e6*Fsys, cmt = "FING", ii = 0.5, addl = 2*days - 1))

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

## THE PRESENTING PATIENT IS NOT ASSUMED — IT IS GROWN

## Start from an intact 18-year-old scalp and run the untreated model

## until the target-area hair count falls to the published trial baseline

## of 876 hairs / 5.07 cm2. With GS = 0.90 this takes 47.8 years.

## Severity is therefore a CLOCK SPEED, not a state:

## GS 0.70 -> never reaches 876 within a lifetime

## GS 0.90 -> age 47.8 y

## GS 1.00 -> age 33.6 y

## GS 1.25 -> age 30.7 y

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

burn_in <- function(mod, gs = 0.90, target = 876, maxy = 60) {

out <- mod %>% param(GS = gs) %>%

mrgsim(end = maxy*365, delta = 30, atol = 1e-10, rtol = 1e-8) %>% as_tibble()

k <- which(out$TAHC <= target)[1]

if (is.na(k)) k <- nrow(out)

list(state = as.numeric(out[k, cmt(mod)]), age = 18 + out$time[k]/365,

tahc = out$TAHC[k], out = out)

}

run_arm <- function(mod, y0, events = NULL, end = 1825, ...) {

m <- mod %>% init(setNames(as.list(y0$state), cmt(mod)))

extra <- list(...)

if (length(extra)) m <- do.call(param, c(list(m), extra))

if (is.null(events)) m %>% mrgsim(end = end, delta = 5, atol = 1e-10, rtol = 1e-8)

else m %>% mrgsim(data = events, end = end, delta = 5,

atol = 1e-10, rtol = 1e-8)

}

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

## SCENARIO LIBRARY — 18 arms

## Model outputs quoted below are from the verified Python reference

## implementation of the identical equations; run this file to reproduce.

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

if (interactive()) {

P0 <- burn_in(mod) # age 47.8 y, TAHC 876, 262 lost

## ---- 1 natural history (placebo) -14.7 hairs at 1 y, -71.5 at 5 y

s01 <- run_arm(mod, P0)

## ---- 2 finasteride 1 mg +86.5 at 1 y (+101.3 vs placebo; obs +107)

s02 <- run_arm(mod, P0, ev_fin(1))

## ---- 3 finasteride 0.2 mg +94.2 vs placebo — 1/5 the dose, 7 hairs

s03 <- run_arm(mod, P0, ev_fin(0.2))

## ---- 4 finasteride 5 mg +105.2 vs placebo — 5x the dose, 4 hairs

s04 <- run_arm(mod, P0, ev_fin(5))

## ---- 5 dutasteride 0.5 mg +122.7 vs placebo at 1 y; scalp DHT -90%

s05 <- run_arm(mod, P0, ev_dut(0.5))

## ---- 6 topical minoxidil 5% BID, SULT1A1 RESPONDER +39.8 vs placebo

s06 <- run_arm(mod, P0, ev_mxt(50), SULT = 2.0)

## ---- 7 topical minoxidil 5% BID, NON-RESPONDER +18.7 vs placebo

s07 <- run_arm(mod, P0, ev_mxt(50), SULT = 0.25)

## ---- 8 oral minoxidil 5 mg +36.7 vs placebo, MAP -3.6 mmHg

s08 <- run_arm(mod, P0, ev_mxo(5))

## ---- 9 oral minoxidil 1 mg +7.5 vs placebo, MAP -0.7 mmHg

s09 <- run_arm(mod, P0, ev_mxo(1))

## ---- 10 finasteride 1 mg + topical minoxidil 5% +116.2 vs placebo

s10 <- run_arm(mod, P0, c(ev_fin(1), ev_mxt(50)))

## ---- 11 dutasteride + minoxidil +132.8 vs placebo

s11 <- run_arm(mod, P0, c(ev_dut(0.5), ev_mxt(50)))

## ---- 12 topical finasteride 0.25% BID

## scalp DHT -47.6%, serum DHT -27.4%, +78.6 vs placebo

s12 <- run_arm(mod, P0, ev_topfin(2.5))

## ---- 13 withdrawal after 2 years

## scalp DHT 0.364 -> 0.980 in 10 days; hair +110 -> +79 over 3 y

s13 <- run_arm(mod, P0, ev_fin(1, days = 730))

## ---- 14 imperfect adherence, 4 doses in 7

## +91.4 vs placebo at 1 y — 90% of the benefit, 57% of the tablets

ev14 <- do.call(rbind, lapply(seq(0, 1824), function(d)

if ((d %% 7) < 4) as.data.frame(ev(amt = 1e6*0.65, cmt = "FING", time = d))

else NULL))

s14 <- run_arm(mod, P0, ev14)

## ---- 15 late start: the same drug begun two years later

s15 <- run_arm(mod, P0, ev(amt = 1e6*0.65, cmt = "FING", time = 730,

ii = 1, addl = 1094))

## ---- 16 setipiprant (CRTH2 blockade). With PGD_EFF = 0.18 the model

## predicts +46 vs placebo, which the phase 2a did NOT see; a null

## trial requires PGD_EFF <= ~0.05 (see README).

s16 <- run_arm(mod, P0, NULL, SETI = 1)

## ---- 17 FEMALE PATTERN HAIR LOSS, post-menopausal

## androgen-independent drive dominant; finasteride nearly null

FP <- mod %>% param(GS = 0.45, ARIND = 0.62, LOC5AR = 0.30, AROM = 1.6,

W_LOC = 0.30, W_SER = 0.30, W_ALT = 0.40)

P0f <- burn_in(FP, gs = 0.45, target = 900, maxy = 45)

s17a <- run_arm(FP, P0f, ev_fin(1), end = 730)

s17b <- run_arm(FP, P0f, ev_mxt(50), end = 730)

s17c <- run_arm(FP, P0f, ev_spi(100), end = 730)

## ---- 18 severity ladder: how long each susceptibility takes to get there

for (g in c(0.70, 0.80, 0.90, 1.00, 1.10, 1.25))

message(sprintf("GS %.2f -> presenting age %.1f y", g, burn_in(mod, g)$age))

}

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

## READ-OUT NOTES

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

## TAHC non-vellus hairs per 5.07 cm2 — the endpoint of every pivotal

## trial, and the one that under-reads the effect (see HMI).

## HMI hair mass index, 100 = an intact terminal scalp. Because mass

## goes as d^2, HMI moves ~1.9x as far as TAHC.

## SHEDSC whole-scalp-equivalent daily shed. At baseline the model gives

## 90 hairs/day from 100 000 x 0.090 / 100 d — the textbook number

## is an identity between three parameters, not an observation.

## TELPC trichogram telogen percentage = T_T/(T_A+T_C+T_T); it rises in

## AGA only because T_A falls, so it is an anagen-duration assay.

## PSAADJ 2 x PSA — the clinical correction after any 5AR inhibitor.

## SEXPCT modelled sexual adverse-effect incidence, anchored on

## 3.8% vs 2.1% placebo for finasteride 1 mg.

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

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