Medical - Models for vitals

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Mathematical Functions

$$\mathrm{Clamp}(x,\ x_{min},\ x_{max}) := \begin{cases} x_{max} & \text{for } x \ge x_{max} \\ x_{min} & \text{for } x_{min} \ge x \\ x & \text{otherwise} \end{cases}$$

$$\mathrm{Map}(x,\ x_0,\ x_1,\ y_0,\ y_1) := \frac{y_1 - y_0}{x_1 - x_0} \cdot (x - x_0) + y_0$$

$$\mathrm{Lerp}(y_0,\ y_1,\ x) := \mathrm{Map}(x,\ 0,\ 1,\ y_0,\ y_1)$$

$$\mathrm{Lerp}^{-1}(y_0,\ y_1,\ x) := \mathrm{Map}(x,\ y_0,\ y_1,\ 0,\ 1)$$

$$\mathcal{U}(x_{min},\ x_{max}) \text{: RNG from uniform distribution}$$

$$\mathcal{N}(\bar{x},\ \sigma^2) \text{: RNG from normal distribution}$$

$$\mathrm{Hill}(x) := \frac{x}{1 + x}$$

[Kexanone]

ACE3 Medical

Vitals

Systolic pressure (SP):

$$SP = 9.4736842 \cdot CO \cdot SVR$$

• $SP\ [Pa]$: Systolic pressure
• $CO\ [m^3/s]$: Cardiac output
• $SVR\ [Pa \cdot s/m^3]$: Systemic vascular resistance

Diastolic pressure (DP):

$$DP = 6.3157894 \cdot CO \cdot SVR$$

• $DP\ [Pa]$: Diastolic pressure

Mean arterial pressure (MAP):

$$MAP = \frac{1}{3} SP + \frac{2}{3} DP = 7.368421 \cdot CO \cdot SVR$$

• $MAP\ [Pa]$: Mean arterial pressure

Pulse pressure (PP):

$$PP = SP - DP = 3.1578948 \cdot CO \cdot SVR$$

• $PP [Pa]$: Pulse pressure (Not calculated in ACE3)

Cardiac output (CO):

$$CO = HR \cdot SV$$

• $HR\ [s^{-1}]$: Heart rate
• $SV\ [m^3]$: Stroke volume

$$SV = SV_0 \cdot \mathrm{Lerp}^{-1}\Big(0.5,\ 1,\ \mathrm{Clamp}\big(BVR,\ 0.5,\ 1\big)\Big)$$

• $SV_0 = 9.5 \cdot 10^{-2}\ l$
• $BVR\ [\cdot]$: Blood volume ratio

Blood volume ratio (BVR):

$$BVR = \frac{BV}{BV_0}$$

• $BV_0 = 6.0\ l$

Blood volume (BV):

$$BV(t + \Delta t) = BV(t) - BR \cdot \Delta t + \Delta f(med.)$$

• $BV\ [m^3]$: Blood volume
• $BR\ [m^3/s]$: Bleeding rate
• $\Delta t\ [s]$: Time step ($\approx 1 s$)
• $\Delta f(med.)$: Some function to account for adjustments through medication

$$BR = WBP \cdot BC \cdot \max(CO,\ CO_{g})$$

• $WBP\ [\cdot]$: Wound bleeding percentage (How much of CO gets lost through wounds)
• $BC\ [\cdot]$: Bleeding coefficient (param, default = 1)
• $CO_{g} = 0.05\ l/min$ : Cardiac output due to gravity

Systemic vascular resistance (SVR):

$$SVR = SVR_0 + \Delta f(med.)$$

• $SVR_0 = 100\ mmHg \cdot min / l$

Heart rate (HR):

$$HR(t + \Delta t) = \begin{cases} \begin{cases} \mathrm{Clamp}\big(25,\ 35,\ \mathcal{N}(30,\ ?)\big) & \text{| CPR} \\ 0 & \text{| otherwise} \end{cases} & \bigg|\text{ cardiac arrest} \\ \\ \begin{cases} \min\big(HR_{tgt.},\ HR(t) + HRA(t) \cdot \Delta t \big) & |\ HRA \ge 0 \\ \max\big(HR_{tgt.},\ HR(t) + HRA(t) \cdot \Delta t \big) & |\ HRA < 0 \end{cases} & \bigg|\text{ otherwise} \end{cases}$$

• $HR_{tgt.}\ [s^{-1}]$: Target heart rate
• $HRA\ [s^{-2}]$: Heart rate acceleration

$$HRA(t) = \begin{cases} -\mathrm{round}\Big(\frac{HR(t)}{10}\Big) & \text{| class 4 hemorrhage} \\ \frac{1}{2} \cdot \mathrm{round}\big(HR_{tgt.} - HR(t)\big) & \text{| otherwise} \end{cases}$$

$$HR_{tgt.} = \max\Big(0,\ \max\big(f(MAP,\ BV),\ f(PL)\big) + \Delta f(SpO_2,\ \Delta OD) + \Delta f(med.)\Big)$$

$$f(MAP,\ BV) = \begin{cases} HR(t) \cdot \frac{107 \cdot BVR}{\max(45,\ MAP)} & \text{| at least class 3 hemorrhage} \\ HR_0 & \text{| otherwise} \end{cases}$$

$$f(PL) = HR_0 + \begin{cases} 50 \cdot PL & |\ PL > 0.2 \\ 0 & \text{| otherwise} \end{cases}$$

• $PL\ [\cdot]$: Pain level
• $HR_0 = 80\ min^{-1}$

$$\Delta f(SpO_2,\ \Delta OD) = 2 \cdot \big(97 - SpO_2\big) - 1000 \cdot \Delta OD$$

• $SpO_2$ [%]: Peripheral oxygen saturation
• $\Delta OD\ [?]$: Additional oxygen demand

Peripheral oxygen saturation ($SpO_2$):

$$SpO_2(t + \Delta t) = \mathrm{Clamp}\bigg(0,\ 100,\ SpO_2(t) + \frac{dSpO_2}{dt}(t) \cdot \Delta t \bigg)$$

• $\frac{dSpO_2}{dt}$ [%/s]: Rate of change of $SpO_2$

$$\frac{dSpO_2}{dt}(t) = OD_0 + \Delta OD + 0.00368 \cdot HR(t) \cdot OS$$

• $OD_0 = -0.25\ ?$ : Base oxygen demand
• $OS\ [\cdot]$: Oxygen saturation

$$OS = \bigg(\frac{p_{O_2}}{p_{O_2, 0}}\bigg)^{1-3 \cdot p_{O_2}}$$

• $p_{O_2}\ [Pa]$: Partial pressure of oxygen (capped at $p_{O_2, 0}$)
• $p_{O_2, 0} = 0.255\ bar$

Oxygen demand (OD):

$$\Delta OD = \begin{cases} \big((1 - AER) \cdot 0.1 \big) - \big((1 - ANR) \cdot 0.05 \big) & \text{| advanced fatigue} \\ 0 & \text{| otherwise} \end{cases}$$

• $AER\ [\cdot]$: Aerobic reserve
• $ANR\ [\cdot]$: Anaerobic reserve

State Machine

[Kexanone]

[WIP] Anvil Medical Proposal V1

Vitals

Mean arterial pressure (MAP):

$$MAP = CO \cdot SVR$$

• $MAP\ [\mathrm{kPa}]$: Mean arterial pressure
• $CO\ [\mathrm{ml/s}]$: Cardiac output
• $SVR\ [\mathrm{kPa \cdot s/ml}]$: Systemic vascular resistance

Pulse pressure (PP):

$$PP =\ ?$$

• $PP\ [\mathrm{kPa}]$: Pulse pressure

Systolic pressure (SP):

$$SP = MAP + \frac{2}{3}PP$$

• $SP\ [\mathrm{kPa}]$: Systolic pressure

Diastolic pressure (DP):

$$DP = MAP - \frac{1}{3}PP$$

• $DP\ [\mathrm{kPa}]$: Diastolic pressure

Cardiac output (CO):

$$CO = HR \cdot SV$$

• $HR\ [\mathrm{Hz}]$: Heart rate
• $SV\ [\mathrm{ml}]$: Stroke volume

$$SV = SV_0 \cdot\ ?$$

• $SV_0 = 95\ \mathrm{ml}$

Blood volume (BV):

$$\frac{dBV}{dt}(t) = \frac{dBT}{dt}(t) - WBP(t) \cdot \max\big(CO(t),\ CO_{g}\big)$$

• $BV\ [\mathrm{ml}]$: Blood volume
• $\frac{dBT}{dt}\ [\mathrm{ml/s}]$: Blood transfusion rate
• $WBP\ [\cdot]$: Wound bleeding percentage (How much of CO gets lost through wounds). Certain type of bleedings can be reduced through "alpha action" of drugs (see KAM)
• $CO_{g} = 0.05\ l/min$ : Cardiac output due to gravity

Systemic vascular resistance (SVR):

$$SVR = \frac{8 \cdot L_{BV}}{\pi} \cdot \frac{\eta_B}{r_{BV}^4} \qquad\text{(based on Hagen–Poiseuille)}$$

• $L_{BV}\ [\mathrm{mm}]$: = Length of blood vessel
• $\eta_B\ [\mathrm{kPa \cdot s}]$: Dynamic viscosity of blood
• $r_{BV}\ [\mathrm{mm}]$: Radius of blood vessel

$$\eta_B = \eta_{B,0} + \Delta f(med.)$$

$$r_{BV} = r_{BV,0} + \Delta f(med.)$$

• $\Delta f(med.)$: Some function to account for adjustments through medication

Heart rate (HR):

$$\frac{dHR}{dt}(t) = k_{HR} \cdot \big(HR_{tgt.}(t) - HR(t) \big)$$

• $HR_{tgt.}\ [\mathrm{Hz}]$: Target heart rate
• $k_{HR}\ [\mathrm{s^{-1}}]$: HR rate constant (parameter for tweaking how fast adjustment to new target takes place)

$$HR_{tgt.}(t) = \mathrm{AceHill}\big(\mathrm{\underline{x}};\ \underline{\underline{\mathrm{K}_{HR}}},\ 0,\ HR_0,\ HR_{max}\big)$$

• $HR_{max} = \frac{10}{3}\ \mathrm{Hz}$
• $HR_0 = \frac{4}{3}\ \mathrm{Hz}$

Peripheral oxygen saturation ($SpO_2$):

$$OS = \mathrm{Hill}\bigg(\Big(\frac{p_{O_2}}{3.5\ \mathrm{kPa}}\Big)^{2.85} \bigg)$$

• $OS\ [\cdot]$: Oxygen saturation
• $p_{O_2}\ [\mathrm{kPa}]$: Partial pressure of oxygen

Medication

Pharmacokinetics

IV Bolus

$$\mathrm{[A]}(t) = \frac{k_a}{k_a-k_d} \cdot \Big(\exp\big(-k_d \cdot t \big) - \exp\big(-k_a \cdot t \big)\Big) \cdot \mathrm{[P]}_0$$

• $\mathrm{[A]}\ [\mathrm{nM}]$: Concentration of drug in active state
• $\mathrm{[P]}_0\ [\mathrm{nM}]$: Initial concentration of drug in precursor state
• $k_a\ [\mathrm{s^{-1}}]$: Activation rate constant
• $k_d\ \mathrm{[s^{-1}}]$: Deactivation rate constant

IV Infusion (with steady-state approximation of precursor)

$$\mathrm{[A]}(t) = \frac{v_{IV,0}}{k_d} \cdot \begin{dcases} \Big( 1 + \exp\big(-k_d \cdot t \big) \Big) & \Big|\ t \le \Delta t_{IV} \\ \exp\big(-k_d \cdot t \big) & \big|\ t > \Delta t_{IV} \\ \end{dcases}$$

• $v_{IV}\ [\mathrm{nM/s}]$: Infusion rate
• $\Delta t_{IV}\ [\mathrm{s}]$: Infusion duration

Additional notes

• Exponential tails can be dealt with by subtracting the curve by a constant (see approaches to truncating LJ)

Pharmacodynamics

$$Q = \mathrm{AceHill}\big(\mathrm{\underline{x}};\ \mathrm{\underline{\underline{K}}},\ Q_{min},\ Q_0,\ Q_{max}\big) := (Q_{max} - Q_{min}) \cdot \ \mathrm{Hill}\Bigg(C_0 \cdot \frac{1 + \sum_i \mathcal{X}^+_i}{1 + \sum_i \mathcal{X}^-_i} \Bigg) + Q_{min}$$

$$C_0 = \frac{Q_0 + Q_{min}}{Q_{max} - Q_0 - 2 \cdot Q_{min}}$$

$$\mathcal{X}^+_i = \begin{dcases} \Big(\frac{\mathcal{J}^+_i}{\mathcal{J}^-_i} \cdot \frac{\mathrm{[X]}_i}{K_{ii}}\Big)^{\gamma_i} & \Big|\ K_{ii} > 0 \\ 0 & \text{| otherwise} \end{dcases}$$

$$\mathcal{X}^-_i = \begin{dcases} \Big(\frac{\mathcal{J}^+_i}{\mathcal{J}^-_i} \cdot \frac{\mathrm{[X]}_i}{-K_{ii}}\Big)^{\gamma_i} & \Big|\ K_{ii} < 0 \\ 0 & \text{| otherwise} \end{dcases}$$

$$\mathcal{J}^+_i = 1 + \sum_{j \neq i} \begin{dcases} \frac{\mathrm{[X]}_j}{K_{ij}} & \Big|\ K_{ij} > 0 \\ 0 & \text{| otherwise} \end{dcases}$$

$$\mathcal{J}^-_i = 1 - \sum_{j \neq i} \begin{dcases} \frac{\mathrm{[X]}_j}{K_{ij}} & \Big|\ K_{ij} < 0 \\ 0 & \text{| otherwise} \end{dcases}$$

• $Q$: Some quantity $\big(Q \in [Q_{min}, Q_{max}]\big)$
• $Q_0$: Quantity's value in absence of drugs $\big(Q_0 \in (Q_{min}, Q_{max})\big)$
• $\mathcal{X}_i\ [\cdot]$: Term for drug $\mathrm{X}_i$ with either positive $(\mathcal{X}^+_i)$ or negative effect $(\mathcal{X}^-_i)$ on $Q$
• $\mathcal{J}_i\ [\cdot]$: Synergistic $(\mathcal{J}^+_i)$ and antagonistic interactions $(\mathcal{J}^-_i)$ for drug $\mathrm{X}_i$
• $\mathrm{\underline{x}}\ [\mathrm{nM}]$: Vector containing drug concentrations $\big(x_i = \mathrm{[X]}_i\big)$
• $\mathrm{[X]}_i\ [\mathrm{nM}]$: Concentration of drug $\mathrm{X}_i$
• $\mathrm{\underline{\underline{K}}}\ [\mathrm{nM}]$: Matrix containing dissociation constants on diagonal and interaction constants on off-diagonals
• $K_{ii}\ [\mathrm{nM}]$: Dissociation constant of drug $\mathrm{X}_i$. Smaller values lead to a steeper slope in $Q$ vs $\mathrm{[X]}_i$
• $K_{ij}\ [\mathrm{nM}]$: Interaction constant for drug $\mathrm{X}_j$ to enhance (if positive) or inhibit (if negative) the effect of drug $\mathrm{X}_i$
• $\gamma_i\ [\cdot]$: Hill coefficient of drug $\mathrm{X}_i$