# S400 Dual-Frequency BIA — Adapted Model for Foot-to-Foot Scales

## Overview

The Xiaomi S400 and compatible scales measure bioelectrical impedance at two frequencies (50 kHz and 250 kHz) using **foot-to-foot electrodes only**. This differs fundamentally from clinical BIA devices, which use four electrodes placed on both hands and feet (tetrapolar hand-to-foot configuration).

> ⚠️ **Important notice:** No peer-reviewed equations have been published specifically for **dual-frequency foot-to-foot** consumer scales. The formulas used in S400 mode are adapted from hand-to-foot clinical literature and empirically calibrated to produce physiologically plausible results on this hardware. They should be treated as **relative trend indicators**, not clinical measurements.

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## Impedance Naming Convention (Bluetooth S400)

The developer who decoded the S400 Bluetooth protocol named entities by their **numerical value**, not by their frequency:

| Bluetooth entity | Numerical value    | Physical frequency       | Role in BIA                      |
| :--------------- | :----------------- | :----------------------- | :------------------------------- |
| `impedance_low`  | Smaller (e.g. 408) | **250 kHz** (high freq.) | Z_hf — penetrates cell membranes |
| `impedance_high` | Larger (e.g. 452)  | **50 kHz** (low freq.)   | Z_lf — extracellular fluid only  |

This naming is inverted relative to the BIA standard convention (where "low" refers to low _frequency_, not low _value_). The integration corrects this automatically by always using `max(low, high)` as Z_lf and `min(low, high)` as Z_hf, which is physically correct: at low frequency, current cannot cross cell membranes, resulting in a longer path and higher resistance.

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## Formulas Used (S400 Mode)

### 1. LBM — Lean Body Mass

**Hardware-Calibrated Formula**

$$LBM = \frac{H \times 9.058}{100} \times \frac{H}{100} + W \times 0.32 + 12.226 - Z_{lf} \times 0.0068 - A \times 0.0542$$

**Origin:** Empirical regression from the Xiaomi/Zepp Life ecosystem, calibrated for foot-to-foot impedance levels. The most appropriate baseline for this hardware, as it accounts for the higher resistance values typical of foot-to-foot measurements.

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### 2. TBW — Total Body Water (displayed water%)

**Pace & Rathbun (1945) / Siri (1956)**

$$water\% = (100 - fat\%) \times 0.73$$

This expresses TBW as a percentage of **total body weight**, producing physiologically plausible values in the typical adult range (~55–65% for males, ~50–60% for females).

> ℹ️ The underlying Deurenberg formula is still used internally to compute TBW liters for ECW/ICW/BCM compartment calculations, but the **displayed percentage** uses Pace for consistency with clinical references and consumer scale conventions.

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### 3. TBW — Internal Source for ECW/ICW/BCM

**Pace & Rathbun (1945) / Siri (1956) constant**

$$TBW_{internal} = (1 - fat\% / 100) \times 0.73 \times W$$

**Why a separate TBW?** Validated on 10 reference profiles, the Deurenberg formula overestimates TBW by +10 to +18 L for subjects with BMI < 28, which would propagate large errors to ECW, ICW, and BCM. Using the fat%-derived TBW (Pace constant) as the internal base for compartment calculations reduces errors to ≤ 0.03 L across all profiles.

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### 4. ECW — Extracellular Water

**Impedance-Ratio Model (empirical adaptation)**

$$Z_{ratio} = Z_{hf} / Z_{lf} \quad (\text{always} < 1 \text{ by BIA physics})$$

$$ECW = TBW_{internal} \times (0.32 + 0.08 \times Z_{ratio})$$

**Origin:** Clinical ECW formulas (De Lorenzo 1997, Kushner 1992) were validated exclusively on hand-to-foot tetrapolar devices and cannot be applied directly to foot-to-foot hardware. This model uses the impedance ratio as a proxy for membrane permeability: at higher frequencies, current penetrates cell membranes more easily, so Z_hf/Z_lf correlates with the ECW/TBW partitioning. In healthy adults, ECW/TBW ≈ 38–39%.

> ⚠️ The constants `0.32` and `0.08` are empirical, chosen to produce an ECW/TBW ratio consistent with reference values in healthy adults. Treat as a relative indicator.

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### 5. ICW — Intracellular Water

$$ICW\ (L) = TBW_{internal} - ECW$$

Standard compartmental subtraction, universally applied across BIA methods.

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### 6. ECW/TBW Ratio

$$ECW/TBW\ (\%) = (ECW / TBW_{internal}) \times 100$$

- **Normal range:** 37–39% in healthy adults.
- **> 39%:** May suggest overhydration, inflammation, or edema.
- **< 37%:** May suggest dehydration.

> ℹ️ Best used to track **personal trends over time** rather than as an absolute clinical value.

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### 7. Protein Percentage

**Wang et al. (1999)** — Molecular compartment model.

$$Protein\% = (LBM \times 0.195 / W) \times 100$$

Proteins represent a stable ~19.5% fraction of Lean Body Mass in healthy adults.

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### 8. SMM — Skeletal Muscle Mass

**Janssen et al. (2000)** — Originally validated against MRI on hand-to-foot BIA data.

$$SMM = (H^2 / Z_{lf} \times 0.401) + (Sex \times 3.825) + (A \times -0.071) + 5.102$$

_(Sex = 1 for male, 0 for female)_

**Adaptation note:** Applied here on `Z_lf` (50 kHz) as the closest available equivalent to the original hand-to-foot single-frequency measurement. Tends to slightly overestimate SMM on foot-to-foot hardware due to path length differences, but remains the best published reference for BIA-based skeletal muscle estimation.

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### 9. BMR — Basal Metabolic Rate

**Katch-McArdle (1996)**

$$BMR = 370 + 21.6 \times LBM$$

Uses measured Lean Body Mass directly, offering better precision for active or overweight individuals than weight-only formulas.

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### 10. Metabolic Age

**BMR-Relative Approach**

$$MetaAge = Age \times (BMR_{expected} / BMR_{actual})$$

Where `BMR_expected` is the Harris-Benedict revised estimate for the user's age, weight, height, and gender. Higher LBM → lower metabolic age; lower LBM → higher metabolic age.

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### 11. BCM — Body Cell Mass

**Wang et al. (1999)**

$$BCM = ICW / 0.73$$

Metabolically active tissue compartment. ICW represents ~73% of BCM in healthy adults.

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### 12. Visceral Fat

**Zepp Life / Xiaomi standard estimate**

Uses weight, height, and age ratios from the original Xiaomi physiological model. Applied identically across all three calculation modes.

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## Available Metrics Summary

| Metric            | Unit | Method                     | Reliability   |
| :---------------- | :--- | :------------------------- | :------------ |
| **LBM**           | kg   | Xiaomi calibrated          | ✅ Good       |
| **Water% (TBW)**  | %    | Pace & Rathbun (displayed) | ✅ Good       |
| **TBW (liters)**  | L    | Deurenberg (internal)      | ⚠️ Trend only |
| **ECW**           | L    | Z-ratio / Pace TBW         | ⚠️ Trend only |
| **ICW**           | L    | TBW − ECW / Pace TBW       | ⚠️ Trend only |
| **ECW/TBW Ratio** | %    | Derived / Pace TBW         | ⚠️ Trend only |
| **BCM**           | kg   | Wang / ICW                 | ⚠️ Trend only |
| **SMM**           | kg   | Janssen (adapted)          | ✅ Good       |
| **Fat%**          | %    | Siri 2-compartment         | ✅ Good       |
| **BMR**           | kcal | Katch-McArdle              | ✅ Good       |
| **Metabolic Age** | yrs  | BMR-relative               | ✅ Good       |
| **Visceral Fat**  | -    | Zepp Life                  | ✅ Good       |
| **Protein**       | %    | Wang 1999                  | ✅ Good       |

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## References

1. Deurenberg P et al. (1995). Body composition in the elderly: a comparison of methods. _Am J Clin Nutr_, 61(1):4-12.
2. Katch FI, McArdle WD (1996). _Nutrition, Weight Control, and Exercise_. Williams & Wilkins.
3. Janssen I et al. (2000). Skeletal muscle mass and distribution in 468 men and women aged 18-88 yr. _J Appl Physiol_, 89(1):81-88.
4. Wang Z et al. (1999). Body composition models: a key to understanding nutritional health. _Am J Clin Nutr_, 70(3):405-411.