한 줄 요약. 케톤산증을 "인슐린이 없어 혈당이 오르고 케톤이 생긴다"로 놓지 않고, 배출 경로의 수가 다른 두 개의 결핍으로 놓았다. 포도당에는 출구가 둘(인슐린 매개 처리 + 신장 배출)이고 그 신장 출구의 컨덕턴스는 세포외액량 이다. 케토산에는 출구가 하나이고 그것은 포화한다. 우리가 재는 혈당은 상당 부분 용적(volume) 신호이고, 우리가 잘 재지 않는 케톤이 곧 질병이다.

One-line thesis. DKA is posed here not as "insulin is missing, so glucose rises and ketones appear", but as two deficits with different numbers of exits. Glucose has two (insulin-mediated disposal and a renal escape valve whose conductance is the extracellular volume); ketoacid has one, and it saturates. The glucose we measure is largely a volume signal; the ketone we usually do not measure is the disease.

아래에 인용된 숫자는 전부 dka_reference_check.py가 계산한 것이며, 그 전체 출력이 dka_reference_output.txt에 커밋되어 있습니다. 주장하지 않고 계산했습니다. Every number below was computed by dka_reference_check.py, whose complete output is committed. Nothing here is asserted.

Plasma [HCO₃⁻] is never integrated. At every derivative evaluation the model solves the physicochemical (Stewart) electroneutrality condition

[HCO3-](pH, PCO2) + [A-](pH) + [ketoanion] + [lactate] + [SIG]  =  SID_app
SID_app = Na + K + (Ca,Mg) - Cl - (unmetabolised infused organic anion)

for pH by bisection, then reads off the bicarbonate. This costs work per step and buys the following for free, with no book-keeping term anywhere in the code:

• making a ketoacid from a neutral triglyceride adds a strong anion → HCO₃⁻ falls 1:1 with the ketoanion;
• oxidising a ketoanion removes it → HCO₃⁻ is regenerated 1:1, so retained ketoanions are literally potential bicarbonate (437 mmol at presentation);
• excreting a ketoanion with Na⁺/K⁺ removes the anion and a strong cation → the anion gap closes but the bicarbonate does not recover: an organic acidosis is silently converted into a hyperchloraemic one;
• excreting it with NH₄⁺ removes only the anion → base is preserved, and NH₄Cl excretion (which also removes Cl⁻) is how the kidney repairs a hyperchloraemic acidosis;
• 0.9% saline (SID 0) diluting a plasma SID of ~49 is acidifying, automatically;
• acetate/gluconate/lactate in balanced crystalloids count as strong anions until metabolised, then vanish — transient dip, then base gain.

Secreted insulin passes the liver first (~50% first-pass extraction), so portal concentration is several-fold peripheral. The CPT-1 gate therefore sees PORTF·20·SEC more insulin than the adipocyte does; exogenous insulin has no such privilege. This single asymmetry is what makes HHS a different disease from DKA in the same equations.

No laboratory value is typed in anywhere. Every scenario integrates a prodrome from a healthy steady state with insulin withdrawn, and treatment starts from wherever that lands. The 24-hour prodrome of a 70 kg adult with type 1 diabetes and a moderate precipitating illness comes out as:

The classical deficit table is therefore reproduced, not entered.

Standard protocol (15 mL/kg then 250 mL/h of 0.9% saline, insulin 0.1 U/kg/h, KCl 40 mmol/L, dextrose titrated by a closed loop on the simulated glucose):

Glucose is at target 7.8 hours before the ketosis is gone — a factor of 3.7 in time. First-hour rates: glucose −75 mg/dL/h, BHB −2.76 mmol/L/h.

Fluid alone lowers glucose by 37 mg/dL in an hour while barely touching the ketone, because saline reopens the renal escape valve: GFR 52 → 64 → 83 mL/min over three hours, renal glucose clearance 1.50 → 1.74 L/h, urinary loss 39 mmol/h at a time when hepatic output is 105 mmol/h and total disposal 63. A large part of the early glucose fall is renal, not metabolic — which is exactly why the glucose is a poor guide to the disease.

IC50 for suppression of lipolysis is 15 µU/mL; EC50 for glucose disposal is 60. Nothing else in the model distinguishes the two arms.

Across the 4-fold range 0.10 → 0.40 U/kg/h the antiketogenic arm moves only 0.649 → 0.833 (nearly saturated), so resolution is 17% faster for four times the insulin; the disposal arm moves 0.375 → 0.705 and is not saturated, so the glucose fall, the glucose nadir and the potassium nadir all scale with dose. Below the recommended range the lipolytic arm genuinely lapses (0.025–0.05 U/kg/h never clear the ketosis) — the other half of the same result. This is Kitabchi's 1976 finding derived from two half-maximal constants in the ratio 15 : 60, rather than assumed.

Identical insulin deficiency; only the capacity to drink varies. 36 h prodrome:

Nothing in the model knows the word "HHS". Glucose is set by how much of the filtered load the kidney is still permitted to throw away, and the ketone barely changes across the whole range — the two axes are independent.

Residual β-cell function restrains lipolysis (IC50 15) and, through portal privilege, the CPT-1 gate, long before it restrains glucose (EC50 60). That ordering is the whole difference between a ketoacidotic and a hyperosmolar presentation.

The same ketone burden at a glucose 133 mg/dL lower. A glucose-triggered diagnostic rule (>250 mg/dL) fires at 5.0 h without the drug and at 22.4 h with it, while the anion gap has already crossed 16 at 7.3 h. The diagnostic delay is 15 hours, and it is a property of the trigger, not of the disease.

The ledger: 102 mmol of ketoanion had already been excreted with Na⁺/K⁺ before arrival — over a 35 L bicarbonate space that is 2.9 mmol/L of base that cannot be recovered by oxidising anything, because the carbon carrying it has left the body. Against that, 625 mmol of chloride is infused during treatment. Removing the chloride load alone is worth +3.6 mmol/L of bicarbonate at 24 h.

And a result that inverts the intuition: impose AKI and the anion gap closes 5 hours EARLIER, because a ketoanion that cannot be filtered away has to be oxidised — and oxidising it regenerates the bicarbonate that excreting it loses. A failing kidney makes the gap look better and the base ledger worse.

At presentation the electroneutrality ledger reads:

strong ion difference             +48.79 mEq/L
weak acid (albumin, phosphate)    -13.45
ketoanion                         -16.55
lactate                            -4.43
other unmeasured strong anion      -6.82
--------------------------------  ------
                                   +7.53   = bicarbonate

The bicarbonate is 15% of the largest term in its own definition. Perturb only the fractional excretion of chloride by ±15%:

A 30% swing in one renal parameter moves the bicarbonate by 1.2 mmol/L and the pH by 0.086, while the gap and the ketone — which involve no cancellation — move by 1.4% and 0.7%. This is a property of the chemistry, not of the model. It is why the anion gap and the point-of-care ketone are the robust bedside variables, why "bicarbonate recovery" is a poor endpoint once chloride is being infused, and why this model's bicarbonate and pH predictions deserve less trust than its gap and ketone predictions. Stated as a warning label rather than buried.

Serum K spans 20% while the deficit spans 78%. Two patients here differ by 223 mmol of body potassium at serum values 0.19 mmol/L apart.

What the ADA "hold insulin if K < 3.3" rule buys, quantified:

The rule trades ~2.3 hours of extra ketoacidosis for 0.55 mmol/L of potassium. That is the trade, and it is why DKA cannot be treated with insulin alone even in a patient whose potassium looks high.

A 6-fold range of fluid rate moves peak brain volume by 0.2 percentage points, which is what the randomised trial found. Decomposing the swelling in a severe presentation: both mechanisms 0.47%, osmotic alone 0.00%, injury alone 0.94%. The osmotic rate-mismatch is the smaller contribution; the larger is an ischaemic insult that accrues before treatment starts, driven by hypocapnia and hypoperfusion — which is why the reported risk factors are a low PCO₂ and a high urea at presentation rather than anything done afterwards.

Total ketone falls monotonically, but the acetoacetate concentration rises for the first two hours and its fraction goes from 10.4% to 21.7% as hepatic redox normalises. At 4 h the total ketone is down 61% while the nitroprusside-visible pool (AcAc + acetone) is down 14%. Acetone, with a 23 h half-life, is still 0.26 mmol/L at 24 h — days of ketotic breath and positive strips after the disease has gone.

Severe presentation (pH 7.069, HCO₃ 6.8, gap 30.5, BHB 16.1):

Bicarbonate buys +0.233 pH units at 1 h, +0.029 at 12 h and +0.011 at 24 h. The acid was never the reservoir — the ketoanion was, and insulin sets the rate at which it is oxidised back into bicarbonate. The costs are computed too: a higher PCO₂ (the respiratory drive is switched off by the very pH it corrected) and 200 mmol of extra sodium in a patient whose brain is already adapting to a falling osmolality.

Plasma insulin has a 5-minute half-life; the CPT-1 gate reopens within an hour of stopping, and ketogenesis restarts from a fat store that treatment never emptied. The gap reopens to 15.4 within six hours.

Mild-to-moderate presentation, 0.3 U/kg s.c. load then 0.2 U/kg every 2 h:

The plasma trough falls to 13 µU/mL — below the 15 µU/mL lipolysis IC50, because plasma insulin has a 5-minute half-life and no bolus regimen can avoid a trough. But the effect site only falls to 26 µU/mL (1.73× the IC50), because it is a 21-minute low-pass filter on that profile. That filter is the whole reason an intermittent subcutaneous protocol can suppress ketogenesis at all — and it says exactly where it breaks: lengthen the interval, or slow absorption (shock, oedema, regular insulin at kₐ 0.5 instead of 1.5 /h), and it is the effect-site trough, not the peak, that crosses the constant.

Alcoholic ketoacidosis comes out with BHB:AcAc 18.4 (against 8.6 in DKA) at a glucose of 184 mg/dL: the redox term, not the insulin term, is doing the work — which is why a nitroprusside strip, blind to BHB, is at its most misleading in exactly this patient.

The last row is not a bug and is worth reading: in an untreated type 1 patient plasma insulin is essentially zero, so the insulin IC50s and the portal ratio have no leverage over the presentation. They have all their leverage over the response to treatment (R3) and over the HHS phenotype (R5). The parameters that set what walks through the door and the parameters that set what happens next are almost disjoint.

• electroneutrality residual, max over 24 h: 1.9 × 10⁻¹² mEq/L
• anion gap reconstructed from its parts at 6 h: 14.573 vs 14.573
• sodium: ECF content 1551 → 1819 mmol = infused 491 − urinary 223 (residual 0)
• potassium: total body +6.7 mmol vs (infused 133.6 − urinary 126.9) = +6.7; closure residual 3.8 × 10⁻¹² mmol (0.000% of throughput)
• ketone: retained 437 → 12 mmol, 30 mmol excreted, the remaining 395 mmol oxidised — regenerating bicarbonate 1:1, which is where the recovered base came from.

42 ODEs. Volume and electrolytes (VECF, VICF, NAE, CLE, KE, KI, PHOSE, ORGA) · substrates and ketones (GLU, GLYCO, FFA, LAC, ACAC, BHB, ACET) · nitrogen (UREA, CREA) · respiratory (PCO₂) · insulin (INSSC, INSP, INSEF, INSES) · counter-regulatory (GCG, CORT, EPI, IR, REDOX) · renal (GFRR, ALDO, NH4C) · β-cell (BETAF) · illness (ILL) · brain (OSMB, VBR, INJ) · CNS (MENT) · six cumulative ledgers (UKET, UKETN, UGLU, UKCUM, UVOL, CLIN) plus two mass-balance audit trackers.

The load-bearing parameters are four half-maximal insulin concentrations taken from dose-response literature rather than fitted — lipolysis 15, potassium shift 25, hepatic output 30, peripheral disposal 60 µU/mL — and the ordering of those four numbers generates R1, R3, R5 and R14.

Closed-loop protocols. The dextrose switch, the insulin taper and the ADA potassium rule are feedback on the simulated patient, not events on a clock, so the protocol is an output of the model as much as an input to it. That is what makes R9's "hours insulin held" a computed quantity.

Allometry. Flux, clearance and volume parameters scale with body weight; rate constants, concentrations, IC50s and stoichiometries do not. The 30 kg child in R15 is a 30 kg child, not a 70 kg adult with a relabelled weight.

1. 중탄산염의 조건수. Documented quantitatively in R8. The gap and the ketone are the trustworthy outputs; the bicarbonate and pH are not, and the model says so with numbers rather than a disclaimer.
2. 뇌부종의 분해는 추론이다. The 0.00%/0.94% split between osmotic and ischaemic mechanisms in R10 reproduces the reported risk factors and the negative fluid-rate trial, but the partition itself cannot be measured in patients. It should be read as "the osmotic story is insufficient", not as a calibrated ratio.
3. 문맥:말초 인슐린 비(PORTF)가 DKA/HHS 구분 전체를 지고 있다. It is taken from fasting first-pass extraction studies; whether the same ratio holds during a hyperglycaemic crisis with hepatic congestion is unknown. This is the model's single largest structural bet.
4. 케톤 산화 용량의 포화/비포화 분할 (VMAX_KOX vs KLIN_KOX) is constrained by steady-state concentrations, not measured. The non-saturable arm is what keeps the system bounded, so this is not a cosmetic choice.
5. 0.05 U/kg/h에서 케톤산증이 40시간 내에 해소되지 않는다. Reported as-is in R3. Whether real patients on half the recommended dose fail this badly is not established; the model is extrapolating below the range its constants were measured in.
6. 선행 질환은 단일 스칼라(ILL, decaying with τ 30 h). There is no infection dynamics, no source control, no antibiotic effect. A patient whose sepsis is worsening cannot be represented.
7. 응고·심전기생리·약물유전학 없음. Venous thromboembolism and arrhythmia appear on the mechanistic map as outcomes but are not dynamical states; the potassium panel reports a concentration, not a rhythm.
8. 첫 시간 혈당 강하가 다소 빠르다 (−75 mg/dL/h against the ADA's stated expectation of 50–75). The model sits at the upper edge of the guideline band. The cause is identifiable — renal clearance at a presenting GFR of 52 mL/min — and was left alone rather than tuned, because R2 depends on that flux being honest.

# the mechanistic map
dot -Tsvg dka_qsp_model.dot -o dka_qsp_model.svg
dot -Tpng -Gdpi=150 dka_qsp_model.dot -o dka_qsp_model.png

# every number in this README, from an independent implementation
python3 dka_reference_check.py            # ~6 min, no dependencies at all
python3 dka_reference_check.py --quick    # just the presentation table

# the mrgsolve model
source("dka_mrgsolve_model.R")
pr <- present();  print(t(presentation_row(pr$sim)))
print(resolution_table(sc1_standard()))
print(lapply(sc10_phenotypes(), presentation_row))

# the dashboard
shiny::runApp("dka_shiny_app.R")

교육 및 연구 목적의 QSP 모델입니다. 개별 환자 데이터에 적합·검증되지 않았으며 실제 임상 의사결정, 처방, 규제 제출에 사용해서는 안 됩니다. 파라미터는 공개 문헌에 근거한 근사치이고, 위에 기록한 한계는 축소하지 않고 그대로 남겨 두었습니다.

Educational and research use only. This model has not been fitted or validated against individual patient data and must not be used to guide treatment. The limitations above are recorded at full strength rather than minimised.