# Invariance-Based Safety Verification for Automated Anesthesia Delivery

## A Philosophy Engineering Approach to Preventing Perioperative Catastrophes

---

**Technical Whitepaper v1.0 — December 2025**

**Andrew H. Bond**  
San José State University  
Ethical Finite Machines  
andrew.bond@sjsu.edu

---

> *"The patient received a lethal overdose because the infusion pump was programmed in mg/kg/hr while the anesthesiologist calculated the dose in mcg/kg/min. The pharmacology was unchanged. The representation was inconsistent. A 10,000-fold error."*

---

## Executive Summary

This whitepaper presents a novel approach to automated anesthesia safety verification based on **representational invariance testing**—the principle that an anesthesia delivery system's dosing decisions should not depend on arbitrary choices in how drug concentrations, patient parameters, or physiological states are represented.

We apply the **ErisML/DEME framework** (Epistemic Representation Invariance & Safety ML / Democratically Governed Ethics Modules) to closed-loop anesthesia delivery, demonstrating how:

1. **The Bond Index (Bd)** can quantify the coherence of Target Controlled Infusion (TCI) systems across pharmacokinetic models, unit systems, and patient populations
2. **Declared transforms (G_declared)** map naturally to concentration unit conversions, pharmacokinetic model selection, patient parameter scaling, and monitoring modality substitution
3. **The Decomposition Theorem** separates implementation bugs (fixable via calibration) from fundamental specification conflicts (requiring clinical protocol redesign)
4. **Democratic governance profiles** allow multi-stakeholder requirements (anesthesiologists, surgeons, patients, hospitals, regulators) to be composed without contradiction

**Key finding**: An anesthesia delivery system with Bond Index Bd < 0.01 across standard transforms is **provably consistent** in its dosing decisions—it will not compute a safe dose under one pharmacokinetic model while computing a lethal dose under an equivalent model for the same patient.

**Market opportunity**: The global anesthesia equipment market exceeds $15B, with 300+ million surgeries performed annually worldwide. Post-Michael Jackson, post-Joan Rivers regulatory scrutiny demands rigorous safety verification—yet no current standard systematically tests for *representational consistency* across the dosing pipeline.

---

## Table of Contents

1. [Introduction: The Representational Failure Mode](#1-introduction-the-representational-failure-mode)
2. [Background: Anesthesia Delivery and Current Practice](#2-background-anesthesia-delivery-and-current-practice)
3. [The Invariance Framework for Anesthesia Systems](#3-the-invariance-framework-for-anesthesia-systems)
4. [Observables and Grounding (Ψ)](#4-observables-and-grounding-ψ)
5. [Declared Transforms (G_declared)](#5-declared-transforms-g_declared)
6. [The Bond Index for Anesthesia Delivery Systems](#6-the-bond-index-for-anesthesia-delivery-systems)
7. [Regime Transitions and Coherence Defects](#7-regime-transitions-and-coherence-defects)
8. [The Ψ-Incompleteness Challenge: Effect-Site Concentration](#8-the-ψ-incompleteness-challenge-effect-site-concentration)
9. [Multi-Stakeholder Governance](#9-multi-stakeholder-governance)
10. [Case Study: Propofol TCI for Colonoscopy Sedation](#10-case-study-propofol-tci-for-colonoscopy-sedation)
11. [Implementation Architecture](#11-implementation-architecture)
12. [Deployment Pathway](#12-deployment-pathway)
13. [Limitations and Future Work](#13-limitations-and-future-work)
14. [Conclusion](#14-conclusion)
15. [References](#15-references)

---

## 1. Introduction: The Representational Failure Mode

### 1.1 A Different Kind of Failure

Most anesthesia safety analysis focuses on **pharmacological failures**: drug allergies, unexpected interactions, equipment malfunction. These are important, and the medical field has developed sophisticated tools to address them (allergy screening, drug interaction databases, equipment certification).

But there is another failure mode that kills patients: **representational failures**—cases where the dosing system's *model* of the patient or drug becomes inconsistent across processing stages, not because monitors failed, but because the *way the system interprets data* contains hidden inconsistencies.

### 1.2 The Unit Conversion Problem in Medicine

Drug dosing involves multiple unit systems that clinicians must convert between:

```
Same dose expressed as:
  Propofol:    200 mg bolus
               20 mL of 1% solution (10 mg/mL)
               2.86 mg/kg for 70 kg patient
               0.048 mg/kg/min as infusion rate
               2.86 μg/kg/sec (different time unit!)
               
  Fentanyl:    100 mcg bolus
               0.1 mg (milligrams vs micrograms!)
               2 mL of 50 mcg/mL solution
               1.43 mcg/kg for 70 kg patient
```

**The Institute for Safe Medication Practices (ISMP)** reports that **unit conversion errors are among the most common causes of medication errors**, with 10-fold and 1000-fold errors occurring regularly.

### 1.3 The Dennis Quaid Case: A Famous Example

In 2007, actor Dennis Quaid's newborn twins received **1,000 times the intended dose of heparin** at Cedars-Sinai Medical Center. The vials of Heparin 10 units/mL and Hep-Lock 10,000 units/mL looked similar. The infusion pump calculated correctly—but with the wrong concentration input.

**The pharmacology was unchanged**—heparin acts the same regardless of how it's labeled. **The representation was wrong**—the system believed it was infusing a different concentration than it actually was.

### 1.4 The Pharmacokinetic Model Problem

Modern Target Controlled Infusion (TCI) systems use pharmacokinetic (PK) models to predict drug concentrations. Multiple validated models exist for the same drug:

```
Propofol PK models (all FDA-approved for different populations):
  
  Marsh model:    Cp = f(dose, weight, time)
                  Designed for: Adults
                  
  Schnider model: Cp = f(dose, weight, height, age, LBM, time)
                  Designed for: Adults, accounts for age/body composition
                  
  Paedfusor:      Cp = f(dose, weight, age, time)
                  Designed for: Pediatric patients
                  
  Kataria:        Cp = f(dose, weight, time)
                  Designed for: Children
```

**The same patient** with the same dose can have different predicted concentrations depending on which model is selected. If the system is inconsistent about which model it's using, dosing errors result.

### 1.5 The Philosophy Engineering Insight

For decades, questions like "Is this dose safe?" have been treated as matters of clinical judgment, experience, and protocol compliance. The **Philosophy Engineering** framework changes the question:

> We cannot test whether a dose is *optimal* in some absolute sense. But we **can** test whether a dosing system is **consistent**—whether it gives the same recommendation when the same clinical situation is described in different equivalent ways.

This is a *falsifiable* property. If we find a case where the system computes "2 mg/kg safe" under model A but "2 mg/kg lethal" under equivalent model B for the same patient, we have produced a **witness** to inconsistency. Witnesses enable debugging.

### 1.6 What This Whitepaper Offers

We present:

1. **A formal framework** for defining "equivalent representations" in anesthesia delivery (the transform suite G_declared)
2. **A quantitative metric** (the Bond Index Bd) that measures how consistently an anesthesia system treats equivalent clinical states
3. **A verification protocol** that can be applied to existing TCI systems without replacing them
4. **A governance mechanism** for composing safety requirements from multiple stakeholders
5. **A deployment roadmap** from simulation validation to FDA approval to clinical deployment

---

## 2. Background: Anesthesia Delivery and Current Practice

### 2.1 Why Anesthesia Safety Matters

General anesthesia requires maintaining a delicate balance—deep enough to prevent awareness and movement, light enough to avoid cardiorespiratory depression:

| Hazard | Mechanism | Timescale | Consequence |
|--------|-----------|-----------|-------------|
| **Hypoxia** | Respiratory depression, airway obstruction | Seconds–minutes | Brain damage, death |
| **Hypotension** | Cardiac depression, vasodilation | Seconds–minutes | Organ damage, death |
| **Awareness** | Inadequate anesthetic depth | Throughout surgery | Psychological trauma, PTSD |
| **Overdose** | Excessive drug administration | Minutes–hours | Prolonged recovery, death |
| **Aspiration** | Loss of airway reflexes | During induction/emergence | Pneumonia, death |
| **Malignant hyperthermia** | Genetic susceptibility + triggers | Minutes | Death if untreated |

**The stakes**: Over 300 million surgeries are performed globally each year. Anesthesia-related mortality has decreased dramatically (from 1:1,000 in 1960s to ~1:100,000 today) but still represents thousands of preventable deaths annually.

### 2.2 The Anesthesia Triad

General anesthesia typically involves three components:

| Component | Purpose | Primary Drugs | Monitoring |
|-----------|---------|---------------|------------|
| **Hypnosis** | Unconsciousness | Propofol, sevoflurane, desflurane | BIS, clinical signs |
| **Analgesia** | Pain suppression | Fentanyl, remifentanil, morphine | HR/BP response to stimulation |
| **Muscle relaxation** | Immobility | Rocuronium, vecuronium, cisatracurium | Train-of-four (TOF) |

Each component has different pharmacokinetics, different therapeutic windows, and different monitoring requirements.

### 2.3 Target Controlled Infusion (TCI)

Modern anesthesia delivery uses **Target Controlled Infusion**—computerized systems that:

1. Accept a target drug concentration (plasma or effect-site)
2. Use pharmacokinetic models to compute required infusion rate
3. Continuously adjust infusion to maintain target
4. Account for drug redistribution and elimination

```
┌─────────────────────────────────────────────────────────────────┐
│                    TCI SYSTEM ARCHITECTURE                      │
├─────────────────────────────────────────────────────────────────┤
│                                                                 │
│  INPUT: Target concentration (e.g., Ce = 4 μg/mL propofol)      │
│                         │                                       │
│                         ▼                                       │
│  ┌─────────────────────────────────────────────────────────┐   │
│  │              PHARMACOKINETIC MODEL                       │   │
│  │  • 3-compartment model (central, fast, slow)             │   │
│  │  • Patient covariates (weight, age, gender, etc.)        │   │
│  │  • Effect-site equilibration (ke0)                       │   │
│  └─────────────────────────────────────────────────────────┘   │
│                         │                                       │
│                         ▼                                       │
│  ┌─────────────────────────────────────────────────────────┐   │
│  │              INFUSION RATE CALCULATOR                    │   │
│  │  • Compute required rate to achieve/maintain target      │   │
│  │  • Account for redistribution, elimination               │   │
│  │  • Implement safety limits                               │   │
│  └─────────────────────────────────────────────────────────┘   │
│                         │                                       │
│                         ▼                                       │
│  OUTPUT: Infusion rate command (e.g., 150 mL/hr)                │
│                         │                                       │
│                         ▼                                       │
│  ┌─────────────────────────────────────────────────────────┐   │
│  │              INFUSION PUMP                               │   │
│  │  • Delivers drug at commanded rate                       │   │
│  │  • Monitors for occlusion, air-in-line                   │   │
│  └─────────────────────────────────────────────────────────┘   │
│                                                                 │
└─────────────────────────────────────────────────────────────────┘
```

### 2.4 Closed-Loop Anesthesia

Emerging systems close the loop by adding physiological feedback:

```
┌─────────────────────────────────────────────────────────────────┐
│                CLOSED-LOOP ANESTHESIA CONTROL                   │
├─────────────────────────────────────────────────────────────────┤
│                                                                 │
│  ┌───────────────┐                                              │
│  │    PATIENT    │◀──────────────────────────────────┐          │
│  └───────┬───────┘                                   │          │
│          │ (physiological response)                  │          │
│          ▼                                           │          │
│  ┌─────────────────────────────────────────────────────────┐   │
│  │              MONITORING                                  │   │
│  │  • BIS (anesthetic depth)                                │   │
│  │  • SpO2 (oxygenation)                                    │   │
│  │  • BP, HR (cardiovascular)                               │   │
│  │  • EtCO2 (ventilation)                                   │   │
│  └─────────────────────────────────────────────────────────┘   │
│          │                                                      │
│          ▼                                                      │
│  ┌─────────────────────────────────────────────────────────┐   │
│  │              CONTROLLER                                  │   │
│  │  • Compare measured vs. target (e.g., BIS = 50)          │   │
│  │  • PID or model-predictive control                       │   │
│  │  • Compute drug adjustment                               │   │
│  └─────────────────────────────────────────────────────────┘   │
│          │                                                      │
│          ▼                                                      │
│  ┌─────────────────────────────────────────────────────────┐   │
│  │              TCI SYSTEM                                  │   │
│  │  • Adjust target concentration                           │   │
│  │  • Execute infusion rate change                          │   │
│  └─────────────────────────────────────────────────────────┘   │
│          │                                                      │
│          └──────────────────────────────────────────────────────┘
│                                                                 │
└─────────────────────────────────────────────────────────────────┘
```

### 2.5 The Gap: Representational Consistency Testing

Current anesthesia device verification focuses on:

- **Drug delivery accuracy**: Does the pump deliver the commanded rate?
- **Alarm functionality**: Do alarms trigger at appropriate thresholds?
- **Electrical safety**: Does the device meet IEC 60601 standards?
- **Software verification**: Does the software meet IEC 62304 requirements?

What they do **not** systematically test:

- **Do different pharmacokinetic models yield consistent safety classifications for the same patient?**
- **Are unit conversions implemented consistently across the dosing pipeline?**
- **Does the system produce equivalent doses regardless of how patient parameters are entered?**
- **Is the controller stable across equivalent representations of physiological state?**

These are precisely the questions the Bond Index framework addresses.

---

## 3. The Invariance Framework for Anesthesia Systems

### 3.1 Core Definitions

**Definition 1 (Clinical State).** A clinical state σ is the complete specification of the patient's physiological status and drug exposure:

```
σ = (patient_params, vital_signs, drug_concentrations, surgical_phase, comorbidities)
```

where:
- `patient_params = (age, weight, height, gender, BSA, LBM, ...)`
- `vital_signs = (HR, BP, SpO2, RR, EtCO2, BIS, temperature, ...)`
- `drug_concentrations = {(drug_i, Cp_i, Ce_i, cumulative_dose_i)}`
- `surgical_phase = (induction, maintenance, emergence, recovery)`
- `comorbidities = (cardiac, respiratory, hepatic, renal, ...)`

**Definition 2 (Representation).** A representation r(σ) is a specific encoding of the clinical state in terms of:
- Unit system (mg vs. mcg, kg vs. lb, mL/hr vs. mg/kg/hr)
- Pharmacokinetic model (Marsh, Schnider, Minto, etc.)
- Body composition metric (TBW, IBW, LBM, ABW)
- Monitoring source (invasive vs. non-invasive BP, BIS vs. clinical signs)
- Time reference (clock time vs. time since induction)

**Definition 3 (Dosing Decision).** A dosing decision function D maps representations to drug delivery commands:

```
D: Representations → {infusion_rate, bolus_dose, hold, alarm, ⊥}
```

where ⊥ indicates insufficient information to determine safe action (should alert clinician).

**Definition 4 (Declared Transform).** A declared transform g ∈ G_declared is a mapping between representations that preserves the underlying clinical state:

```
g: r(σ) → r'(σ)    such that    σ is unchanged
```

### 3.2 The Consistency Requirement

**Axiom (Representational Invariance).** A consistent anesthesia delivery system must satisfy:

```
∀σ, ∀g ∈ G_declared:  D(r(σ)) = D(g(r(σ)))
```

In plain language: If two representations describe the same patient in the same clinical state, they must produce the same dosing decision.

### 3.3 Why This Matters for Anesthesia

Consider dosing propofol for a 70 kg adult:

```
Representation A (Marsh model, weight-based):
  Patient: 70 kg, 170 cm, 45 years, male
  Marsh model: V1 = 0.228 × 70 = 15.96 L
  Target Cp = 4 μg/mL
  → Bolus = 4 × 15.96 = 63.8 mg
  → Decision: BOLUS 64 mg

Representation B (Schnider model, includes age/LBM):
  Patient: 70 kg, 170 cm, 45 years, male
  LBM (James formula) = 57.8 kg
  Schnider model: V1 = 4.27 L (fixed, not weight-scaled)
  Target Ce = 4 μg/mL (effect-site, not plasma!)
  → Bolus calculation different due to ke0
  → Decision: BOLUS 89 mg (25% higher!)
```

Both models are validated. Both are used clinically. But they produce **different doses for the same patient at the same target**. If the system switches models without adjusting targets appropriately, dangerous inconsistencies result.

**In anesthesia, there is no margin for large errors**: A 2-fold overdose can cause apnea, cardiovascular collapse, and death.

---

## 4. Observables and Grounding (Ψ)

### 4.1 The Observable Set for Anesthesia

Following the ErisML framework, we define the **grounding map Ψ** that specifies which clinical quantities the system has access to:

| Observable | Symbol | Sensors | Sample Rate | Accuracy |
|------------|--------|---------|-------------|----------|
| Heart rate | HR | ECG | 1–5 Hz (computed) | ±2 bpm |
| Blood pressure (non-invasive) | NIBP | Oscillometric cuff | 0.02–0.5 Hz | ±5 mmHg |
| Blood pressure (invasive) | ABP | Arterial line | 100–250 Hz | ±2 mmHg |
| Oxygen saturation | SpO2 | Pulse oximetry | 1 Hz | ±2% |
| End-tidal CO2 | EtCO2 | Capnography | 10–50 Hz | ±2 mmHg |
| Respiratory rate | RR | Capnography, impedance | 0.1–1 Hz | ±1 /min |
| Temperature | Temp | Thermistor | 0.1 Hz | ±0.1°C |
| Anesthetic depth | BIS | EEG-derived | 1 Hz | ±3 index points |
| Neuromuscular block | TOF | Train-of-four | 0.02 Hz (on demand) | ±5% |
| Drug infusion rate | Rate | Pump feedback | 1 Hz | ±2% |
| Cumulative drug dose | Dose | Pump integration | Continuous | ±1% |

### 4.2 Patient Parameters

Beyond real-time monitoring, anesthesia systems use patient-specific parameters:

| Parameter | Symbol | Source | Use |
|-----------|--------|--------|-----|
| Age | age | Patient record | PK model |
| Weight | TBW | Scale/reported | Dosing |
| Height | Ht | Measured/reported | LBM calculation |
| Gender | sex | Patient record | LBM calculation |
| Lean body mass | LBM | Calculated | PK model (some) |
| Body surface area | BSA | Calculated | Drug clearance |
| Serum creatinine | SCr | Lab | Renal function |
| Albumin | Alb | Lab | Protein binding |
| Hemoglobin | Hgb | Lab | Oxygen carrying |

### 4.3 Derived Quantities

Anesthesia systems compute critical derived quantities:

| Derived Observable | Formula/Method | Clinical Relevance |
|--------------------|----------------|-------------------|
| Predicted Cp (plasma) | PK model integration | Drug level in blood |
| Predicted Ce (effect-site) | PK/PD model with ke0 | Drug level at brain |
| Cardiac output (estimated) | Pulse contour, thoracic impedance | Drug distribution |
| Depth of anesthesia | BIS algorithm from EEG | Hypnosis assessment |
| MAC-equivalent | [volatile]/MAC_50 | Inhaled agent depth |
| Drug elimination | k10 × V1 × Cp | Clearance rate |

### 4.4 The Ψ-Incompleteness Challenge

Anesthesia faces **significant Ψ-incompleteness**:

| Unobservable | Why It Matters | Proxy |
|--------------|----------------|-------|
| **Brain drug concentration** | Determines actual effect | Predicted Ce from model |
| **Individual PK parameters** | Vary 2–10× between patients | Population models |
| **Pain level under anesthesia** | Can't ask unconscious patient | HR/BP response |
| **Awareness** | Patient may be paralyzed | BIS (imperfect) |
| **Surgical stimulation** | Unpredictable timing/intensity | Anticipation |
| **Drug interactions** | Complex, nonlinear | Known tables, caution |

**Critical insight**: The Bond Index framework explicitly acknowledges Ψ-incompleteness. When operating in regimes where key observables are unavailable or unreliable, the system should return ⊥ and alert the clinician—not guess.

---

## 5. Declared Transforms (G_declared)

### 5.1 Transform Categories for Anesthesia Systems

The transform suite G_declared defines which changes to the representation should **not** affect dosing decisions:

#### Category 1: Concentration Unit Transforms

| Transform | Example | Conversion |
|-----------|---------|------------|
| μg/mL ↔ ng/mL | Propofol | × 1000 |
| μg/mL ↔ mg/L | Equivalent | × 1 |
| mcg ↔ μg | Notation | Same unit |
| mg ↔ mcg | Critical! | × 1000 |
| % (w/v) ↔ mg/mL | Propofol 1% = 10 mg/mL | × 10 |

**This is the most dangerous transform category.** 10-fold and 1000-fold errors are common when confusing mg and mcg.

#### Category 2: Weight/Dose Unit Transforms

| Transform | Example | Conversion |
|-----------|---------|------------|
| kg ↔ lb | Weight | × 2.205 |
| mg/kg ↔ mg/lb | Dose per weight | × 2.205 |
| mg/kg/hr ↔ μg/kg/min | Infusion rate | × 16.67 |
| mg/kg/hr ↔ mg/kg/day | Different time base | × 24 |
| mL/hr ↔ mg/hr | Requires concentration! | × [conc] |

#### Category 3: Pharmacokinetic Model Transforms

| Transform | Example | Constraint |
|-----------|---------|------------|
| Marsh ↔ Schnider | Propofol in adults | Different targets! |
| Minto ↔ manufacturer | Remifentanil | Similar parameters |
| Pediatric ↔ adult model | Age-appropriate selection | Age-dependent |
| Cp targeting ↔ Ce targeting | Plasma vs. effect-site | ke0 differences |

**Critical**: PK model transforms are **not simple unit conversions**. Different models may require different target concentrations to achieve the same clinical effect. This must be accounted for.

#### Category 4: Body Composition Transforms

| Transform | Example | Formula |
|-----------|---------|---------|
| TBW ↔ IBW | Total vs. ideal weight | Height-based formulas |
| TBW ↔ LBM | Total vs. lean mass | James, Boer formulas |
| TBW ↔ ABW | Total vs. adjusted | ABW = IBW + 0.4(TBW-IBW) |
| BSA (DuBois) ↔ BSA (Mosteller) | Surface area formulas | Similar results |

**Clinical importance**: Lipophilic drugs (propofol) distribute into fat; hydrophilic drugs (muscle relaxants) do not. Wrong weight metric = wrong dose.

#### Category 5: Monitoring Source Transforms

| Transform | Example | Constraint |
|-----------|---------|------------|
| NIBP ↔ ABP | Non-invasive ↔ invasive BP | Bounded difference |
| SpO2 ↔ SaO2 | Pulse ox ↔ arterial blood gas | Known offset |
| BIS ↔ entropy | Different depth monitors | Similar scale |
| Clinical signs ↔ BIS | Expert assessment ↔ monitor | Correlation |

#### Category 6: Time System Transforms

| Transform | Example | Conversion |
|-----------|---------|------------|
| Clock time ↔ time since induction | Absolute ↔ relative | Subtract induction time |
| Minutes ↔ hours | Infusion duration | × 60 |
| Context-sensitive half-time | Different drugs, durations | Model-dependent |

### 5.2 The Transform Suite Document

```yaml
transform_id: UNIT_MG_TO_MCG
version: 1.0.0
category: concentration_units
description: "Convert milligrams to micrograms"
forward: "dose_mcg = dose_mg × 1000"
inverse: "dose_mg = dose_mcg / 1000"
semantic_equivalence: "Same mass of drug"
high_risk_warning: |
  THIS IS A HIGH-RISK TRANSFORM.
  10-fold and 1000-fold errors between mg and mcg are 
  among the most common causes of fatal medication errors.
  ISMP designates mcg/mg confusion as a "high-alert" issue.
validation:
  required_checks:
    - label_verification: true
    - double_check_protocol: true
    - smart_pump_DERS: true
```

### 5.3 Transforms That Are NOT Declared Equivalent

Some transformations **do** change the clinical situation:

| NOT Equivalent | Why |
|----------------|-----|
| Different patient | Different pharmacokinetics |
| Different drug | Different pharmacology |
| Different surgical phase | Different requirements |
| Healthy ↔ compromised | Different responses |
| Awakening | Patient is waking up |

---

## 6. The Bond Index for Anesthesia Delivery Systems

### 6.1 Definition

The **Bond Index (Bd)** quantifies how consistently an anesthesia delivery system treats equivalent representations:

```
Bd = D_op / τ
```

where:
- **D_op** is the observed coherence defect (measured inconsistency)
- **τ** is the human-calibrated threshold (the defect level clinicians consider "meaningful")

### 6.2 The Three Coherence Defects

#### Defect 1: Commutator (Ω_op)

**Question**: Does the order of transforms matter?

```
Ω_op(σ; g₁, g₂) = |D(g₂(g₁(r(σ)))) - D(g₁(g₂(r(σ))))|
```

**Anesthesia example**: Convert units, then apply PK model, vs. apply PK model, then convert units. Should yield same dose recommendation.

#### Defect 2: Mixed (μ)

**Question**: Does the same transform behave differently in different contexts?

**Anesthesia example**: mg↔mcg conversion for bolus vs. infusion. The conversion is the same, but rounding/truncation may differ.

#### Defect 3: Permutation (π₃)

**Question**: Do three-way compositions have hidden interactions?

**Anesthesia example**: Convert units → apply PK model → adjust for body composition. All 6 orderings should yield consistent results.

### 6.3 Deployment Tiers

| Bd Range | Tier | Interpretation | Action |
|----------|------|----------------|--------|
| < 0.01 | **Negligible** | Excellent coherence | Approve for clinical use |
| 0.01 – 0.1 | **Low** | Minor inconsistencies | Deploy with enhanced monitoring |
| 0.1 – 1.0 | **Moderate** | Significant inconsistencies | Remediate before deployment |
| 1 – 10 | **High** | Severe inconsistencies | Do not deploy |
| > 10 | **Severe** | Fundamental incoherence | Complete redesign |

### 6.4 Calibration Protocol for Medical Devices

The threshold τ is determined empirically:

1. **Recruit raters**: Anesthesiologists, pharmacologists, patient safety officers (n ≥ 30)
2. **Generate test pairs**: Clinical scenarios with known transform relationships
3. **Collect judgments**: "Should these produce the same dose recommendation?"
4. **Fit threshold**: Find defect level where 95% agree the difference matters
5. **Set τ**: Conservative estimate—for anesthesia, human life is at stake

For anesthesia delivery systems, typical calibration yields **τ ≈ 0.05** (clinicians expect < 5% deviation in doses across equivalent representations).

### 6.5 Application to Specific Anesthesia Functions

| Function | Key Transforms | Target Bd |
|----------|----------------|-----------|
| **Bolus calculation** | Units, weight | < 0.01 |
| **Infusion rate** | Units, PK model | < 0.01 |
| **TCI targeting** | PK model, Cp vs. Ce | < 0.05 |
| **Closed-loop control** | Monitor source, model | < 0.01 |
| **Drug interaction** | Multiple drugs | < 0.1 |
| **Alarm thresholds** | Units, monitor | < 0.001 |

---

## 7. Regime Transitions and Coherence Defects

### 7.1 The Regime Transition Problem

Anesthesia proceeds through distinct phases with different goals:

```
┌──────────────────────────────────────────────────────────────────┐
│                    ANESTHESIA PHASES                             │
├──────────────────────────────────────────────────────────────────┤
│                                                                  │
│  PHASE 1: PRE-INDUCTION                                          │
│  ──────────────────────                                          │
│  • Patient assessment complete                                   │
│  • Monitors attached, baseline recorded                          │
│  • Anxiolysis if needed (midazolam)                              │
│  • Preoxygenation                                                │
│  • Key risk: Aspiration during induction                         │
│                                                                  │
│              ↓ (induction drugs given)                           │
│                                                                  │
│  PHASE 2: INDUCTION                                              │
│  ───────────────────                                             │
│  • Rapid transition from awake to unconscious                    │
│  • Loss of airway reflexes                                       │
│  • Potential for apnea, hypotension                              │
│  • Airway management (intubation, LMA)                           │
│  • Key risk: Failed airway, cardiovascular collapse              │
│                                                                  │
│              ↓ (airway secured, stable)                          │
│                                                                  │
│  PHASE 3: MAINTENANCE                                            │
│  ────────────────────                                            │
│  • Steady-state anesthesia                                       │
│  • Surgical stimulation varies                                   │
│  • Titrate to response (BIS, hemodynamics)                       │
│  • Ongoing: analgesia, muscle relaxation                         │
│  • Key risk: Awareness, hemodynamic instability                  │
│                                                                  │
│              ↓ (surgery ending)                                  │
│                                                                  │
│  PHASE 4: EMERGENCE                                              │
│  ───────────────────                                             │
│  • Decrease/stop anesthetic agents                               │
│  • Allow patient to awaken                                       │
│  • Return of airway reflexes                                     │
│  • Extubation when appropriate                                   │
│  • Key risk: Laryngospasm, vomiting, pain                        │
│                                                                  │
│              ↓ (extubated, responding)                           │
│                                                                  │
│  PHASE 5: RECOVERY (PACU)                                        │
│  ────────────────────────                                        │
│  • Continued monitoring                                          │
│  • Pain management                                               │
│  • Watch for delayed complications                               │
│  • Discharge criteria met                                        │
│  • Key risk: Respiratory depression, PONV                        │
│                                                                  │
└──────────────────────────────────────────────────────────────────┘
```

### 7.2 Phase-Specific Parameters

Different phases require different control strategies:

| Parameter | Induction | Maintenance | Emergence |
|-----------|-----------|-------------|-----------|
| **BIS target** | 40–60 (rapid) | 40–60 (stable) | 60–80 (rising) |
| **Propofol Ce** | 4–6 μg/mL | 2–4 μg/mL | 1–2 μg/mL |
| **Response to hypotension** | Vasopressor | Reduce anesthetic | Reduce anesthetic |
| **Opioid dosing** | Bolus | Infusion | Reduce/stop |
| **Relaxant** | Intubating dose | PRN redose | Reverse |

### 7.3 Coherence Across Phase Transitions

A key test: **Does the system give consistent doses for patients near phase boundaries?**

**Example**: Transition from induction to maintenance:
- Induction mode: Target Ce = 5 μg/mL
- Maintenance mode: Target Ce = 3 μg/mL
- Transition: Which target applies? When exactly does it switch?

**Incoherent behavior** (witness): At the transition, the system briefly uses induction parameters with maintenance mode, computing a dose appropriate for neither phase.

### 7.4 Testing Phase Boundary Coherence

The Bond Index framework tests phase boundary coherence by:

1. **Generating boundary states**: σ where phase assignment is ambiguous
2. **Applying transforms**: PK models, unit systems
3. **Checking consistency**: Same dose regardless of representation?

High defect rates at phase boundaries indicate:
- Asynchronous parameter updates
- Discontinuous control law
- Timing vulnerabilities

---

## 8. The Ψ-Incompleteness Challenge: Effect-Site Concentration

### 8.1 The Fundamental Limitation

The most important variable in anesthesia—**drug concentration at the effect site (brain)**—cannot be directly measured:

```
Observable: Plasma concentration Cp (from TCI model)
Desired: Effect-site concentration Ce (at brain)
Actual brain concentration: Unknown (would require invasive sampling)

Relationship: Ce lags Cp due to blood-brain barrier transit
             Ce(t) = Ce(t-1) + ke0 × (Cp(t) - Ce(t-1)) × dt
             
Problem: ke0 varies 2–3× between patients
         Cp itself is only predicted, not measured
```

### 8.2 The ErisML/DEME Response

The framework explicitly handles Ψ-incompleteness:

1. **Acknowledge uncertainty**: Predicted Ce has significant uncertainty bounds
2. **Conservative default**: When uncertainty is high, dose conservatively
3. **Return ⊥ when appropriate**: If monitors fail or data is inconsistent, alert clinician
4. **Test across PK models**: Bond Index should hold even with different PK assumptions

### 8.3 Model-Invariant Safety

A key principle: **Core safety decisions should be invariant to PK model selection within clinically validated options**

```
PK Model A (Marsh): Predicted Ce = 3.2 μg/mL → BIS = 52
PK Model B (Schnider): Predicted Ce = 4.1 μg/mL → BIS = 52

Actual patient: BIS = 52 (observed)

Coherent safety response:
  - Both models predict "adequate anesthesia" given observed BIS
  - Dose adjustment based on BIS, not predicted Ce
  - Safety judgment: ADEQUATE_DEPTH (same regardless of model)
```

**What varies**: The specific predicted concentration
**What should NOT vary**: The safety classification for the observable physiological state

### 8.4 The BIS-Guided Approach

Because Ce cannot be measured, closed-loop systems use BIS (or equivalent) as the primary feedback:

```yaml
bispectral_index_control:
  target: 50  # Units: BIS index (0-100)
  acceptable_range: [40, 60]
  
  response_to_deviation:
    BIS > 60:
      action: increase_propofol_target
      magnitude: "+0.5 μg/mL Ce"
      
    BIS < 40:
      action: decrease_propofol_target
      magnitude: "-0.5 μg/mL Ce"
      
    BIS in [40, 60]:
      action: maintain_current
      
  override_conditions:
    hypotension: prioritize_hemodynamics
    desaturation: prioritize_oxygenation
```

This approach is inherently more robust to PK model uncertainty because it responds to measured effect, not predicted concentration.

---

## 9. Multi-Stakeholder Governance

### 9.1 The Multi-Stakeholder Challenge

Anesthesia delivery involves multiple stakeholders with different priorities:

| Stakeholder | Primary Concerns | Requirements |
|-------------|------------------|--------------|
| **Anesthesiologist** | Patient safety, smooth case | Clinical control, override capability |
| **Surgeon** | Operative conditions, OR time | Immobility, hemodynamic stability |
| **Patient** | Survival, no awareness, comfort | Safety, minimal side effects |
| **Hospital** | Liability, efficiency | Standards compliance, documentation |
| **Nursing** | Workflow, patient handoff | Clear protocols, monitoring |
| **FDA/Regulator** | Public safety | Device safety, efficacy |
| **Insurer** | Claims cost | Risk documentation |

### 9.2 DEME Governance Profiles for Anesthesia

The DEME framework allows clinical requirements to be composed into **governance profiles**:

```yaml
profile_id: "general_anesthesia_adult_asa1_v2.1"
patient_classification:
  asa_class: 1  # Healthy
  age_range: [18, 65]
  excluded_conditions: [cardiac, hepatic, renal, morbid_obesity]

stakeholders:
  - id: patient_safety
    weight: 0.40
    priorities:
      - prevent_hypoxia
      - prevent_hypotension
      - prevent_awareness
      - minimize_drug_exposure
    hard_vetoes:
      - spo2_min: 90  # percent
      - map_min: 60   # mmHg
      - bis_max: 60   # during surgery
      - propofol_rate_max: 200  # mcg/kg/min
      
  - id: anesthesiologist_control
    weight: 0.25
    priorities:
      - clinical_override_capability
      - smooth_induction
      - stable_maintenance
      - controlled_emergence
    constraints:
      - manual_override: always_available
      - alarm_visibility: high_priority
      
  - id: surgical_requirements
    weight: 0.20
    priorities:
      - adequate_depth
      - muscle_relaxation
      - hemodynamic_stability
    constraints:
      - tof_target: 0  # complete paralysis for abdominal
      - bis_range: [40, 50]  # deep for sensitive surgery
      
  - id: institutional_compliance
    weight: 0.10
    priorities:
      - documentation
      - protocol_adherence
      - liability_protection
    constraints:
      - timestamp_all_events: true
      - record_all_doses: true
      
  - id: efficiency
    weight: 0.05
    priorities:
      - minimize_time_to_readiness
      - predictable_emergence
    constraints:
      - induction_time_max: 5  # minutes
      - emergence_time_target: 10  # minutes after last dose

aggregation:
  method: weighted_sum_with_vetoes
  veto_behavior: any_stakeholder_veto_honored
  conflict_resolution: safety_always_priority
```

### 9.3 The "Never Events" as Hard Vetoes

Certain outcomes are unacceptable regardless of other priorities:

```yaml
never_events:  # Hard vetoes - absolute constraints
  - patient_death_from_overdose
  - patient_death_from_hypoxia
  - intraoperative_awareness_with_recall
  - medication_10x_error
  - wrong_patient
  - wrong_drug
  
response_to_never_event_risk:
  detection: any_indication_of_above
  action: immediate_halt_and_alert
  override: requires_explicit_clinician_confirmation
```

### 9.4 Consistency Checking

Before enabling closed-loop control, the governance profile is checked for:

1. **Veto consistency**: Do hard constraints from different stakeholders conflict?
2. **Priority consistency**: Are clinical and efficiency goals compatible?
3. **Protocol completeness**: Are all phases and transitions covered?

**Example conflict**: Surgeon wants deep anesthesia (BIS 35); safety limits BIS > 40. Resolution: Safety constraint takes precedence; document disagreement.

---

## 10. Case Study: Propofol TCI for Colonoscopy Sedation

### 10.1 Scenario Description

**Procedure**: Outpatient colonoscopy with moderate sedation

**Patient**: 
- 55-year-old male
- 85 kg, 175 cm (BMI 27.8)
- ASA Class 2 (controlled hypertension)
- No allergies

**Anesthesia plan**:
- Propofol TCI (Schnider model)
- Target: BIS 70–80 (moderate sedation, not general anesthesia)
- Spontaneous ventilation maintained
- Supplemental O2 via nasal cannula

### 10.2 Observable Set (Ψ)

| Observable | Monitor | Value (baseline) | Rate |
|------------|---------|------------------|------|
| Heart rate | ECG | 72 bpm | Continuous |
| Blood pressure | NIBP | 138/82 mmHg | q3 min |
| SpO2 | Pulse oximetry | 98% | Continuous |
| RR | Capnography | 14 /min | Continuous |
| EtCO2 | Capnography | 38 mmHg | Continuous |
| Sedation level | BIS | 97 (awake) | Continuous |
| Responsiveness | Clinical | Awake | Intermittent |

### 10.3 Transform Suite

For this case study, we apply 14 transforms:

| ID | Transform | Category |
|----|-----------|----------|
| T1 | mg ↔ mcg (×1000) | Concentration units |
| T2 | μg/mL ↔ ng/mL | Concentration units |
| T3 | mg/kg/hr ↔ mcg/kg/min | Infusion rate |
| T4 | mL/hr ↔ mg/hr (concentration-dependent) | Volume/mass |
| T5 | kg ↔ lb | Weight units |
| T6 | Marsh ↔ Schnider model | PK model |
| T7 | Cp target ↔ Ce target | Targeting mode |
| T8 | TBW ↔ LBM dosing | Body composition |
| T9 | TBW ↔ ABW dosing | Body composition |
| T10 | BIS ↔ clinical assessment | Monitoring source |
| T11 | NIBP ↔ ABP | Blood pressure source |
| T12 | Clock time ↔ time since induction | Time reference |
| T13 | Simulation ↔ clinical-equivalent | Sim validation |
| T14 | Manual ↔ closed-loop | Control mode |

### 10.4 Dosing Logic Under Test

The TCI system implements:

```
PROPOFOL TCI (SCHNIDER MODEL):

INDUCTION:
  Target Ce = 2.5 μg/mL (moderate sedation)
  Initial rate = high (achieve target quickly)
  Monitor BIS every 10 seconds
  
MAINTENANCE:
  IF BIS > 80 (too light):
    Increase Ce target by 0.3 μg/mL
  IF BIS < 65 (too deep):
    Decrease Ce target by 0.3 μg/mL
  IF BIS ∈ [65, 80]:
    Maintain current target
    
SAFETY LIMITS:
  Maximum Ce target: 4.0 μg/mL
  Maximum rate: 200 mcg/kg/min
  IF SpO2 < 92%: PAUSE infusion, alert
  IF MAP < 60 mmHg: REDUCE target, alert
  IF RR < 8: PAUSE infusion, alert
  
EMERGENCE:
  Stop infusion when procedure complete
  Monitor until BIS > 90 and responsive
```

### 10.5 Bond Index Evaluation

**Test protocol**:
1. Generate 500 representative clinical states across sedation depth
2. Apply each of 14 transforms at 5 intensity levels
3. Compute dosing decision before and after transform
4. Calculate coherence defects

**Results**:

```
═══════════════════════════════════════════════════════════════════
              BOND INDEX EVALUATION RESULTS
═══════════════════════════════════════════════════════════════════

System:        Propofol TCI Sedation System v3.1
Transform suite: G_declared_propofol_sedation_v1.0 (14 transforms)
Test cases:    500 states × 14 transforms × 5 intensities = 35,000

───────────────────────────────────────────────────────────────────
                      BOND INDEX
───────────────────────────────────────────────────────────────────
  Bd_mean = 0.0082   [0.0067, 0.0098] 95% CI
  Bd_p95  = 0.041
  Bd_max  = 0.28

  TIER: NEGLIGIBLE
  DECISION: ✅ Meets clinical safety threshold

───────────────────────────────────────────────────────────────────
                  DEFECT BREAKDOWN
───────────────────────────────────────────────────────────────────
  Ω_op (commutator):     0.0052  █████
  μ (mixed):             0.0023  ██
  π₃ (permutation):      0.0007  █

───────────────────────────────────────────────────────────────────
                TRANSFORM SENSITIVITY
───────────────────────────────────────────────────────────────────
  T1  (mg↔mcg):          0.000   (perfect - hard-coded validation)
  T2  (μg/mL↔ng/mL):     0.000   (perfect)
  T3  (mg/kg/hr↔mcg/kg/min): 0.002   
  T4  (mL/hr↔mg/hr):     0.008   █
  T5  (kg↔lb):           0.000   (perfect)
  T6  (Marsh↔Schnider):  0.089   █████████  ← HIGHEST
  T7  (Cp↔Ce target):    0.062   ██████  ← Second highest
  T8  (TBW↔LBM):         0.034   ███
  T9  (TBW↔ABW):         0.021   ██
  T10 (BIS↔clinical):    0.041   ████
  T11 (NIBP↔ABP):        0.003   
  T12 (clock↔elapsed):   0.000   (perfect)
  T13 (sim↔clinical):    0.015   ██
  T14 (manual↔closed):   0.012   █

───────────────────────────────────────────────────────────────────
                   WORST WITNESS
───────────────────────────────────────────────────────────────────
  Transform: T6 (Marsh ↔ Schnider PK model)
  Patient: 55 yo male, 85 kg, 175 cm
  
  Schnider model (system default):
    Target Ce = 2.5 μg/mL
    Predicted Ce at t=5 min: 2.48 μg/mL
    Infusion rate: 142 mL/hr (1% propofol = 1420 mg/hr)
    
  Marsh model (alternative):
    Target Cp = 2.5 μg/mL (same numeric value)
    Predicted Cp at t=5 min: 2.52 μg/mL
    Infusion rate: 198 mL/hr (1980 mg/hr) ← 39% higher!
    
  Defect: 0.28 (>25% dose difference)
  
  ROOT CAUSE: Marsh uses Cp targeting; Schnider uses Ce targeting
              Same numeric target = different clinical depth
              Schnider Ce 2.5 ≈ Marsh Cp 3.5 for equivalent effect
  
  CLINICAL IMPACT: Marsh dose would produce deeper sedation
                   than intended with Schnider target
  
  RECOMMENDATION: 
    - PK model switch must include target adjustment
    - Display current model prominently
    - Require confirmation when switching models

───────────────────────────────────────────────────────────────────
                PHASE BOUNDARY ANALYSIS
───────────────────────────────────────────────────────────────────
  Induction start:       Bd = 0.008  
  Induction → Maintenance: Bd = 0.015  █
  Maintenance stable:    Bd = 0.005  
  Desaturation event:    Bd = 0.003  (safety override consistent)
  Emergence:             Bd = 0.012  █

───────────────────────────────────────────────────────────────────
                HIGH-RISK TRANSFORM VALIDATION
───────────────────────────────────────────────────────────────────
  mg ↔ mcg (T1):         PASSED (hard-coded, double-checked)
  10-fold error check:   BLOCKED by Drug Error Reduction System
  1000-fold error check: BLOCKED by rate limits
  
  Unit safety: ✅ VALIDATED

═══════════════════════════════════════════════════════════════════
```

### 10.6 Decomposition Analysis

Applying the Decomposition Theorem:

```
Total defect: Ω = 0.0082 (mean)

Gauge-removable (Ω_gauge): 0.0031 (38%)
  - Fixable via:
    - PK model-specific target recommendations
    - Body composition calculator improvements
    - Better sim-to-clinical calibration
  
Intrinsic (Ω_intrinsic): 0.0051 (62%)
  - Fundamental:
    - Different PK models ARE different (validated on different populations)
    - BIS vs. clinical assessment are genuinely different measures
    - Cp vs. Ce targeting reflect different pharmacological approaches
  - Requires clinical decision (not software fix)
```

**Interpretation**: A significant portion (62%) of the observed variation reflects **legitimate clinical choices** between validated approaches. The Bond Index helps identify where consistency is achievable vs. where clinical judgment must choose between valid alternatives.

### 10.7 Remediation Plan

| Issue | Root Cause | Remediation | Expected Improvement |
|-------|------------|-------------|----------------------|
| Marsh↔Schnider | Different target meanings | Auto-adjust target on model switch | 0.089 → 0.02 |
| Cp↔Ce targeting | Different ke0 effects | Default Ce; warn on Cp mode | 0.062 → 0.02 |
| BIS↔clinical | Different measurement | Calibration lookup table | 0.041 → 0.02 |
| TBW↔LBM | Obesity handling | BMI-based auto-selection | 0.034 → 0.01 |

**Post-remediation target**: Bd < 0.03 (accounting for irreducible clinical variation)

---

## 11. Implementation Architecture

### 11.1 Integration with Existing Medical Devices

The Bond Index framework integrates **non-invasively** with existing anesthesia systems:

```
┌─────────────────────────────────────────────────────────────────┐
│                ANESTHESIA WORKSTATION                           │
├─────────────────────────────────────────────────────────────────┤
│                                                                 │
│  ┌───────────┐   ┌───────────┐   ┌───────────┐   ┌──────────┐  │
│  │  Patient  │──▶│ Monitors  │──▶│  TCI/ADS  │──▶│  Pumps   │  │
│  │           │   │           │   │           │   │          │  │
│  └───────────┘   └─────┬─────┘   └─────┬─────┘   └────┬─────┘  │
│                        │               │              │         │
│                        ▼               ▼              ▼         │
│                 ┌──────────────────────────────────────────┐    │
│                 │      ANESTHESIA INFORMATION SYSTEM       │    │
│                 │              (AIMS)                      │    │
│                 └──────────────────────────────────────────┘    │
│                                      │                          │
└──────────────────────────────────────┼──────────────────────────┘
                                       │
                                       ▼
┌─────────────────────────────────────────────────────────────────┐
│              BOND INDEX VERIFICATION LAYER                      │
│                   (Non-invasive, offline)                       │
├─────────────────────────────────────────────────────────────────┤
│                                                                 │
│  ┌─────────────────────────────────────────────────────────┐   │
│  │              DATA ACQUISITION                            │   │
│  │  • AIMS data export (HL7, FHIR)                          │   │
│  │  • Simulator interface (PKPD models)                     │   │
│  │  • Device log files                                      │   │
│  └─────────────────────────────────────────────────────────┘   │
│                             │                                   │
│                             ▼                                   │
│  ┌─────────────────────────────────────────────────────────┐   │
│  │              TRANSFORM ENGINE                            │   │
│  │  • Unit conversions (mg↔mcg, etc.)                       │   │
│  │  • PK model alternatives                                 │   │
│  │  • Body composition adjustments                          │   │
│  │  • Monitoring source substitution                        │   │
│  └─────────────────────────────────────────────────────────┘   │
│                             │                                   │
│                             ▼                                   │
│  ┌─────────────────────────────────────────────────────────┐   │
│  │              DOSING LOGIC EVALUATOR                      │   │
│  │  • Mirror of TCI algorithm                               │   │
│  │  • Evaluate original and transformed                     │   │
│  │  • Compare dose recommendations                          │   │
│  └─────────────────────────────────────────────────────────┘   │
│                             │                                   │
│                             ▼                                   │
│  ┌─────────────────────────────────────────────────────────┐   │
│  │              BOND INDEX CALCULATOR                       │   │
│  │  • Compute Ω_op, μ, π₃                                   │   │
│  │  • Phase-specific analysis                               │   │
│  │  • High-risk transform flagging                          │   │
│  │  • Generate witnesses                                    │   │
│  └─────────────────────────────────────────────────────────┘   │
│                             │                                   │
│                             ▼                                   │
│  ┌─────────────────────────────────────────────────────────┐   │
│  │              REPORTING & FDA SUBMISSION                  │   │
│  │  • 510(k) evidence package                               │   │
│  │  • Post-market surveillance                              │   │
│  │  • Adverse event correlation                             │   │
│  │  • Continuous improvement                                │   │
│  └─────────────────────────────────────────────────────────┘   │
│                                                                 │
└─────────────────────────────────────────────────────────────────┘
```

### 11.2 Deployment Modes

| Mode | Description | Timing | Use Case |
|------|-------------|--------|----------|
| **Pre-market verification** | Full test suite before submission | Months pre-FDA | Regulatory evidence |
| **Design validation** | Test during development | Development | Design control |
| **Manufacturing QC** | Test each production unit | Manufacturing | Quality assurance |
| **Post-market monitoring** | Analyze field data | Continuous | Surveillance |
| **Adverse event investigation** | Deep dive after incident | As needed | Root cause |

### 11.3 Integration with Drug Libraries (DERS)

Modern infusion pumps include Drug Error Reduction Systems (DERS):

```yaml
drug_library_integration:
  propofol:
    concentrations_allowed: [10, 20]  # mg/mL (1%, 2%)
    dose_unit: mcg/kg/min
    soft_limit_high: 150  # mcg/kg/min - warning
    hard_limit_high: 250  # mcg/kg/min - blocked
    soft_limit_low: 25    # mcg/kg/min - warning
    
    bond_index_validation:
      unit_transform_test: PASSED
      concentration_transform_test: PASSED
      weight_transform_test: PASSED
      
  fentanyl:
    concentrations_allowed: [10, 50]  # mcg/mL
    dose_unit: mcg/kg/hr
    hard_limit_high: 5    # mcg/kg/hr
    
    high_risk_flag: true
    reason: "mcg vs mg confusion frequent"
    additional_validation:
      - label_verification_required: true
      - barcode_scan: recommended
```

### 11.4 Regulatory Isolation

For FDA submission, strict isolation is maintained:

| Requirement | Implementation |
|-------------|----------------|
| **No patient contact** | Verification runs offline on recorded/simulated data |
| **IEC 62304 compliant** | Software development lifecycle documented |
| **Validated transforms** | Each transform clinically validated |
| **Audit trail** | All test results logged, tamper-evident |
| **Design controls** | Per 21 CFR 820.30 |

---

## 12. Deployment Pathway

### 12.1 Phase 1: Simulation Validation (Years 1-2)

**Objective**: Demonstrate Bond Index framework on PKPD simulation

**Activities**:
- Implement G_declared transforms for propofol, remifentanil
- Validate on standard PKPD simulators (STANPUMP, Tivatrainer)
- Partner with academic anesthesia department
- Test across patient populations (pediatric, adult, elderly, obese)
- Publish in Anesthesia & Analgesia or British Journal of Anaesthesia

**Deliverables**:
- Validated transform suite for IV anesthetics
- Simulation toolbox
- Technical paper demonstrating concept

**Resources**: $300K, 3 FTE, 2 years

### 12.2 Phase 2: Retrospective Clinical Validation (Years 2-3)

**Objective**: Validate on real clinical data (de-identified)

**Activities**:
- Partner with hospital AIMS (Anesthesia Information Management System)
- Analyze retrospective cases (n > 10,000)
- Correlate Bd with adverse events (hypoxia, hypotension, awareness)
- Validate that low-Bd systems have fewer complications
- IRB approval for retrospective analysis

**Deliverables**:
- Clinical correlation evidence
- Retrospective study publication
- Preliminary safety signal

**Resources**: $500K, 4 FTE, 1.5 years

### 12.3 Phase 3: Prospective Clinical Study (Years 3-5)

**Objective**: Prospective validation in controlled setting

**Activities**:
- Design prospective observational study
- IRB approval, informed consent
- Enroll patients for TCI procedures (n > 500)
- Real-time Bd monitoring (non-interventional)
- Correlate Bd with outcomes

**Study design**:
- Observational, non-interventional (Bd doesn't change care)
- Primary endpoint: Correlation of Bd with hemodynamic events
- Secondary: Correlation with BIS deviations, recovery time

**Deliverables**:
- Prospective clinical evidence
- Clinical study publication
- Foundation for 510(k)

**Resources**: $1.5M, 6 FTE, 2 years

### 12.4 Phase 4: FDA 510(k) Submission (Years 5-7)

**Objective**: Regulatory clearance for clinical use

**Predicate device**: Existing TCI systems (Alaris, Orchestra, Base Primea)

**Submission contents**:
- Device description
- Substantial equivalence argument
- Performance testing (Bond Index verification)
- Software documentation (IEC 62304)
- Electrical safety (IEC 60601)
- Biocompatibility (if patient contact)
- Clinical evidence

**510(k) strategy**:
- Position as quality/safety enhancement to existing TCI
- Not a new clinical indication
- Build on predicate substantial equivalence

**Deliverables**:
- 510(k) clearance
- FDA listing
- Commercial authorization

**Resources**: $2M, 8 FTE, 2 years

### 12.5 Phase 5: Clinical Adoption (Years 7+)

**Objective**: Widespread hospital adoption

**Activities**:
- Partner with TCI/pump manufacturers (BD, Fresenius, B. Braun)
- Integration with existing drug libraries
- Training programs for anesthesiologists
- Post-market surveillance
- Continuous improvement

**Market approach**:
- License to device manufacturers
- Hospital-direct for standalone verification
- Integrate with AIMS vendors

**Market potential**: $100M+ annual revenue at maturity

---

## 13. Limitations and Future Work

### 13.1 Current Limitations

| Limitation | Description | Mitigation |
|------------|-------------|------------|
| **Patient variability** | PKPD varies 2–10× between patients | Feedback control (BIS) reduces dependence on model |
| **Ψ-incompleteness** | Cannot measure brain concentration | Use validated PK models, adjust based on effect |
| **Regulatory burden** | FDA approval takes 5+ years | Partner with established device manufacturer |
| **Clinician skepticism** | "Automation can't replace judgment" | Position as safety tool, not replacement |
| **Drug interactions** | Complex, nonlinear | Conservative assumptions, clinician override |

### 13.2 What We Do NOT Claim

- **Completeness**: The Bond Index verifies consistency for declared transforms only. Novel dosing errors may exist.
- **Correctness**: We verify that doses are consistent, not that they are optimal for every patient.
- **Elimination of adverse events**: Even consistent systems can harm patients if underlying models are wrong.
- **Clinician replacement**: The system augments, not replaces, clinical judgment.

### 13.3 Future Work

1. **Inhaled anesthetics**: Extend to MAC-based dosing (sevoflurane, desflurane)
2. **Regional anesthesia**: Extend to local anesthetic dosing
3. **Pediatrics**: Specialized transforms for children
4. **Obesity**: Improved body composition handling
5. **AI/ML integration**: Verify consistency of ML-based dosing systems

---

## 14. Conclusion

Automated anesthesia delivery presents a compelling application domain for invariance-based safety verification:

1. **Hard constraints exist**: SpO2 > 90%, MAP > 60 mmHg, BIS in range
2. **Transforms are well-defined**: Unit conversions, PK models, body composition metrics
3. **Stakes are extreme**: 300+ million surgeries annually; errors cause death
4. **Monitoring is mature**: BIS, SpO2, capnography provide rich feedback
5. **Regulatory pathway exists**: FDA 510(k) with predicate devices
6. **Market is substantial**: $15B+ anesthesia equipment market

### The Dennis Quaid Lesson

The Quaid twins received 1,000× the intended heparin dose because of a concentration representation error. The pump calculated correctly. The input was wrong. **Representational consistency testing could have caught this.**

### The PK Model Challenge

Different pharmacokinetic models—all FDA-approved, all clinically validated—produce different dose recommendations for the same patient at the same target. This is not a bug; it's inherent in population-based pharmacology. The Bond Index framework helps identify where this variation matters and ensures that **within** a chosen approach, the system is internally consistent.

### The Path Forward

The anesthesia community has the monitoring technology, the PKPD knowledge, and the regulatory framework to adopt rigorous consistency verification. What has been missing is a formal approach to asking: "Does our dosing system give consistent recommendations across equivalent representations?"

The ErisML/DEME Bond Index framework provides that approach.

> *"The obstacle to anesthesia safety is not that we cannot deliver drugs accurately. It is that we might not verify our dosing systems are representationally consistent. The Bond Index makes that verification possible."*

---

## 15. References

1. Absalom, A. R., & Struys, M. M. R. F. (2007). *An Overview of TCI & TIVA* (2nd ed.). Academia Press.

2. Schnider, T. W., et al. (1998). "The influence of method of administration and covariates on the pharmacokinetics of propofol in adult volunteers." *Anesthesiology*, 88(5), 1170-1182.

3. Marsh, B., et al. (1991). "Pharmacokinetic model driven infusion of propofol in children." *British Journal of Anaesthesia*, 67(1), 41-48.

4. Minto, C. F., et al. (1997). "Influence of age and gender on the pharmacokinetics and pharmacodynamics of remifentanil." *Anesthesiology*, 86(1), 10-23.

5. Institute for Safe Medication Practices (ISMP). (2020). *ISMP List of High-Alert Medications*.

6. FDA. (2014). *Infusion Pump Improvement Initiative*. Center for Devices and Radiological Health.

7. IEC 60601-1:2005. *Medical electrical equipment – Part 1: General requirements for basic safety and essential performance*.

8. IEC 62304:2006. *Medical device software – Software life cycle processes*.

9. 21 CFR Part 820. *Quality System Regulation*.

10. Hemmerling, T. M., & Le, N. (2011). "Brief review: Neuromuscular monitoring: an update for the clinician." *Canadian Journal of Anesthesia*, 58(1), 80-92.

11. Liu, N., et al. (2011). "Closed-loop coadministration of propofol and remifentanil guided by bispectral index: a randomized multicenter study." *Anesthesia & Analgesia*, 112(3), 546-557.

12. Bond, A. H. (2025). "A Categorical Framework for Verifying Representational Consistency in Machine Learning Systems." *IEEE Transactions on Artificial Intelligence* (under review).

13. Bond, A. H. (2025). "The Grand Unified AI Safety Stack (GUASS) v12.0." Technical whitepaper.

14. Struys, M. M. R. F., et al. (2016). "The history of target-controlled infusion." *Anesthesia & Analgesia*, 122(1), 56-69.

15. Egan, T. D. (2019). "Are opioids necessary for general anesthesia?" *British Journal of Anaesthesia*, 123(6), e465-e467.

---

## Appendix A: Transform Suite Template

```yaml
# G_declared for anesthesia delivery systems
# Version: 1.0.0
# Domain: Propofol TCI sedation
# Date: 2025-12-27

metadata:
  domain: medical_devices
  subdomain: anesthesia_delivery
  drug: propofol
  author: ErisML Team
  regulatory_status: investigational
  hash: sha256:f1g2h3i4...

transforms:
  - id: UNIT_MG_TO_MCG
    category: concentration_units
    description: "Convert milligrams to micrograms"
    forward: "dose_mcg = dose_mg × 1000"
    inverse: "dose_mg = dose_mcg / 1000"
    semantic_equivalence: "Same mass of drug"
    high_risk: true
    ismp_alert: true
    validation:
      hard_coded_check: true
      ders_integration: required
      
  - id: RATE_MG_KG_HR_TO_MCG_KG_MIN
    category: infusion_rate
    description: "Convert mg/kg/hr to mcg/kg/min"
    forward: "rate_mcg_kg_min = rate_mg_kg_hr × 1000 / 60"
    inverse: "rate_mg_kg_hr = rate_mcg_kg_min × 60 / 1000"
    conversion_factor: 16.667
    semantic_equivalence: "Same drug delivery rate"
    
  - id: PK_MARSH_TO_SCHNIDER
    category: pharmacokinetic_model
    description: "Switch between Marsh and Schnider propofol models"
    parameters:
      marsh_input: [weight]
      schnider_input: [weight, height, age, gender]
    warning: |
      Target values are NOT equivalent between models.
      Schnider Ce ≈ Marsh Cp with ke0 adjustment.
      Recommend clinical review when switching.
    semantic_equivalence: "Different models, may require target adjustment"
    target_adjustment_table:
      schnider_ce_2.0: marsh_cp_2.8
      schnider_ce_3.0: marsh_cp_4.2
      schnider_ce_4.0: marsh_cp_5.6
      
  - id: WEIGHT_TBW_TO_LBM
    category: body_composition
    description: "Convert total body weight to lean body mass"
    forward_male: "LBM = 1.1 × TBW - 128 × (TBW/Ht)²"
    forward_female: "LBM = 1.07 × TBW - 148 × (TBW/Ht)²"
    method: "James formula"
    semantic_equivalence: "Same patient, different mass metric"
    clinical_guidance: |
      Use LBM for hydrophilic drugs (remifentanil, rocuronium)
      Use TBW or adjusted for lipophilic drugs (propofol)
      
  # ... additional transforms
```

---

## Appendix B: Dosing Logic Specification Template

```yaml
# TCI Dosing System Specification
# For Bond Index verification

system_id: "Propofol_TCI_Sedation_v3.1"
regulatory_class: Class II (510(k))
verification_date: 2025-12-27

drug:
  name: propofol
  concentrations: [10, 20]  # mg/mL
  therapeutic_index: moderate
  high_alert: true
  
patient_parameters:
  required: [weight, age]
  optional: [height, gender]
  derived: [lbm, bsa]
  
pharmacokinetic_model:
  default: schnider
  alternatives: [marsh, paedfusor]
  target_mode: effect_site  # Ce
  
dosing_limits:
  bolus_max: 2.5  # mg/kg
  rate_max: 200   # mcg/kg/min
  rate_min: 25    # mcg/kg/min
  cumulative_warning: 10  # mg/kg total
  
safety_interlocks:
  - condition: "SpO2 < 92"
    action: pause_infusion
    alert: high_priority
    
  - condition: "MAP < 60"
    action: reduce_target_20_percent
    alert: high_priority
    
  - condition: "BIS < 40"
    action: reduce_target_10_percent
    alert: medium_priority
    
  - condition: "no_pulse_ox_signal > 30s"
    action: alert_only
    alert: high_priority
    
control_law:
  type: proportional_on_error
  feedback: bis
  target: 70  # for sedation
  gain: 0.3  # μg/mL per BIS point
  update_interval: 10  # seconds
  rate_limit: 0.5  # μg/mL per minute
  
phases:
  - id: induction
    target: 2.5  # μg/mL Ce
    duration: until_bis_below_80
    
  - id: maintenance
    target: titrate_to_bis_70
    range: [1.5, 4.0]  # μg/mL Ce
    
  - id: emergence
    target: 0
    monitor_until: bis_above_90
```

---

## Appendix C: Glossary

| Term | Definition |
|------|------------|
| **ABW** | Adjusted Body Weight (for obese patients) |
| **AIMS** | Anesthesia Information Management System |
| **ASA class** | American Society of Anesthesiologists physical status classification |
| **BIS** | Bispectral Index (EEG-derived depth of anesthesia monitor) |
| **Ce** | Effect-site concentration (predicted drug level at brain) |
| **Cp** | Plasma concentration |
| **DERS** | Drug Error Reduction System (smart pump feature) |
| **EtCO2** | End-tidal carbon dioxide |
| **IBW** | Ideal Body Weight |
| **ke0** | Effect-site equilibration rate constant |
| **LBM** | Lean Body Mass |
| **MAC** | Minimum Alveolar Concentration (inhaled anesthetic potency) |
| **MAP** | Mean Arterial Pressure |
| **NIBP** | Non-Invasive Blood Pressure |
| **PK/PD** | Pharmacokinetic/Pharmacodynamic |
| **SpO2** | Peripheral oxygen saturation (pulse oximetry) |
| **TBW** | Total Body Weight |
| **TCI** | Target Controlled Infusion |
| **TOF** | Train of Four (neuromuscular monitoring) |
| **Ψ (Psi)** | Observable set — clinical quantities available to the system |

---

## Appendix D: FDA 510(k) Submission Outline

| Section | Content | Bond Index Evidence |
|---------|---------|---------------------|
| **1. Device description** | TCI with Bond Index verification | System architecture |
| **2. Indications for use** | Same as predicate | — |
| **3. Substantial equivalence** | Predicate: existing TCI systems | Enhanced safety feature |
| **4. Performance testing** | Standard + Bond Index | Full Bd evaluation results |
| **5. Software documentation** | IEC 62304 Level B | Transform validation |
| **6. Electrical safety** | IEC 60601-1 | Standard testing |
| **7. Biocompatibility** | N/A (no patient contact) | — |
| **8. Sterility** | N/A | — |
| **9. Labeling** | Warnings, instructions | Transform limitations |
| **10. Clinical evidence** | Retrospective + prospective studies | Bd correlation with outcomes |

---

**Document version**: 1.0.0  
**Last updated**: December 2025  
**License**: AGI-HPC Responsible AI License v1.0

---

<p align="center">
  <em>"The Bond Index is the deliverable. Everything else is infrastructure."</em>
</p>
