{
  "name": "HemodynamicConstraint",
  "main": {
    "name": "HemodynamicConstraint",
    "namespace": "sofa::component::constraintset",
    "module": "SofaTestPlugin",
    "include": "constraint/HemodynamicConstraint.h",
    "doc": "Surf Constraint",
    "inherits": [
      "BaseOpenSurfaceConstraint"
    ],
    "templates": [
      "sofa::defaulttype::Vec3Types"
    ],
    "data_fields": [
      {
        "name": "d_Par",
        "type": "Real",
        "xmlname": "Par",
        "help": "Current state at time t"
      },
      {
        "name": "d_ParInf",
        "type": "Real",
        "xmlname": "ParInf",
        "help": "Asymptotic arterial pressure when flow is null"
      },
      {
        "name": "d_Kar",
        "type": "Real",
        "xmlname": "Kar",
        "help": "Valve param"
      },
      {
        "name": "d_Rc",
        "type": "Real",
        "xmlname": "Rc",
        "help": "Windkessel param (Rc)"
      },
      {
        "name": "d_C",
        "type": "Real",
        "xmlname": "C",
        "help": "Windkessel param (C)"
      },
      {
        "name": "d_Rp",
        "type": "Real",
        "xmlname": "Rp",
        "help": "Windkessel param (Rp)"
      },
      {
        "name": "d_Pat",
        "type": "Real",
        "xmlname": "Pat",
        "help": "Input pressure at time t"
      },
      {
        "name": "d_nextPat",
        "type": "Real",
        "xmlname": "nextPat",
        "help": "Input pressure at time t+dt"
      },
      {
        "name": "d_Kat",
        "type": "Real",
        "xmlname": "Kat",
        "help": "Hemodynamic coefficient between pressure and flow"
      }
    ],
    "links": [],
    "methods": [
      {
        "name": "init",
        "return_type": "void",
        "params": [],
        "is_virtual": false,
        "is_pure_virtual": false,
        "is_static": false,
        "access": "public"
      },
      {
        "name": "getConstraintResolution",
        "return_type": "void",
        "params": [
          {
            "name": "cParams",
            "type": "const ConstraintParams *"
          },
          {
            "name": "resTab",
            "type": "int &"
          },
          {
            "name": "offset",
            "type": "unsigned int &"
          }
        ],
        "is_virtual": true,
        "is_pure_virtual": false,
        "is_static": false,
        "access": "public"
      }
    ]
  },
  "desc": {
    "description": "The `HemodynamicConstraint` is a specialized constraint component in the SOFA framework, primarily designed for simulating hemodynamics and fluid dynamics within cardiovascular systems. It inherits from `BaseOpenSurfaceConstraint`, indicating its role in managing open surface constraints related to blood flow and pressure interactions.\n\nThis component interacts with other components through methods such as `init()` for initialization and `getConstraintResolution()` which is overridden to provide specific constraint resolution for hemodynamic calculations. The latter integrates parameters like arterial pressures (`d_Par`), asymptotic pressures (`d_ParInf`), valve coefficients (`d_Kar`), Windkessel model parameters (`d_Rc`, `d_C`, `d_Rp`), and input pressure values at different times (`d_Pat`, `d_nextPat`). These parameters are crucial for modeling the dynamics of blood flow and pressures in cardiovascular simulations.\n\nPractical usage involves setting these data fields appropriately to reflect the specific hemodynamic conditions being simulated. The `m_ejection_container` struct maintains state variables that track hemodynamic states at different time points, enabling accurate modeling of pressure and flow interactions over time."
  },
  "maths": {
    "maths": "### Mathematical and Physical Description of the Hemodynamic Constraint Component\n\nThe `HemodynamicConstraint` component is designed to simulate hemodynamics in cardiovascular systems, focusing on pressure interactions and blood flow dynamics. It leverages a Windkessel model for the arterial system, which simplifies the complex arterial network into a lumped parameter system consisting of resistances and capacitances.\n\n#### Core Equations and Parameters\n\nThe following parameters are critical to this simulation:\n- **Arterial Pressure (\\(P_{ar}\\))**: This is the pressure in the artery at a given time. It varies based on cardiac output, arterial compliance, and resistance.\n- **Asymptotic Arterial Pressure (\\(P_{arInf}\\))**: The baseline or resting pressure of the artery when flow is zero.\n- **Windkessel Parameters**:\n  - \\(R_p\\): Peripheral resistance representing the impedance to blood flow in the arterial system.\n  - \\(C\\): Arterial compliance, a measure of the ability of the arteries to expand and store volume during systole.\n  - \\(R_c\\): Characteristic resistance related to vessel size and geometry.\n\n#### Key Equations\n1. **Recurrence Relation for Arterial Pressure**:\n   The arterial pressure at time step \\(t+dt\\) is given by the recurrence relation, accounting for compliance and resistance dynamics:\n   \n   $$\nP_{ar}(t + dt) = P_{arInf} + \\frac{a_1 Q_{ar} - R_c Q_{arPrev} + P_{ar}(t) - P_{arInf}}{a_2}\n$$\n\n   where,\n   \n   $$\na_1 = dt \\left(\\frac{R_p + R_c}{RC}\\right) + R_c, \\quad a_2 = 1 + \\frac{dt}{RC},\n$$\n   and \\(Q_{ar}\\) and \\(Q_{arPrev}\\) are the current and previous ejection flows respectively.\n\n2. **Phase Transition Logic**:\n   - **Filling Phase**: When ventricular pressure (\\(P_v\\)) is less than atrial pressure (\\(P_a\\)), blood fills the ventricle, and flow is driven by \\(K_{at}\\), an atrioventricular valve coefficient.\n   \n     $$\nP_v(t + dt) = P_{a}(t + dt) = \\frac{dt K_{at} P_a - dfree}{K_{at} + W}\n$$\n\n   - **Isometric Contraction Phase**: During this phase, the ventricle contracts isometrically without significant ejection flow.\n     \n     $$\nP_v(t + dt) = \\frac{-dfree}{W}\n$$\n\n   - **Ejection Phase**: When \\(P_v > P_{ar}\\), blood is ejected into the artery, and flow follows from a balance of forces.\n    \n     $$\n\\lambda(line) = (A - B dfree) / (B W + a_2 K_{at})\n$$\n\n3. **Parameter Definitions**:\n   - \\(K_{ar}\\): Aortic valve coefficient, governing the flow from the left ventricle into the artery.\n   - \\(Q_t\\): Time-varying ejection or inflow rate, used in the recurrence calculations to update pressures and flows.\n\n#### Numerical Integration Scheme\nThe component uses a forward Euler method for numerical integration of the differential equations over small time steps \\(dt\\). This is evident from the recurrence relations where new state variables (like arterial pressure) are calculated using previous values and known parameters.\n\n#### Initialization and State Updates\n- The `init()` function initializes the system states based on initial conditions and sets up data structures to hold these values over time.\n- The `getConstraintResolution()` function updates constraints by recalculating pressures, checking phase transitions, and adjusting lambda (the Lagrange multiplier) based on current ventricular pressure and ejection status.\n\n#### Practical Application\nThe practical usage of this component involves setting the parameters according to a specific cardiovascular system model. For example:\n- Calibrating \\(R_p\\), \\(C\\), and \\(R_c\\) based on empirical data or theoretical models of arterial behavior.\n- Specifying initial pressures (like \\(P_{arInf}\\)) to reflect baseline conditions.\n\nThese parameters are updated every time step in the simulation, reflecting changes in flow rates, pressure distributions, and the overall hemodynamic state."
  },
  "summary": {
    "abstract": "The `HemodynamicConstraint` component simulates hemodynamics in cardiovascular systems by managing open surface constraints related to blood flow and pressure interactions, using Windkessel parameters and phase transition logic.",
    "sheet": "# HemodynamicConstraint\n\n## Overview\nThe `HemodynamicConstraint` is a specialized constraint component designed for simulating hemodynamics in cardiovascular systems. It manages open surface constraints related to blood flow and pressure interactions by leveraging the Windkessel model, which simplifies arterial dynamics into lumped parameters.\n\n## Mathematical Model\n### Core Equations and Parameters\nThe following equations govern the simulation of arterial pressures and flows:\n1. **Recurrence Relation for Arterial Pressure**:\n   The arterial pressure at time step \\(t + dt\\) is given by:\n   \n   $P_{ar}(t + dt) = P_{arInf} + \\frac{a_1 Q_{ar} - R_c Q_{arPrev} + P_{ar}(t) - P_{arInf}}{a_2}$\n\n   where,\n   \n   $a_1 = dt \\left(\\frac{R_p + R_c}{RC}\\right) + R_c, \\quad a_2 = 1 + \\frac{dt}{RC},$\n\n   and \\(Q_{ar}\\) and \\(Q_{arPrev}\\) are the current and previous ejection flows respectively.\n\n2. **Phase Transition Logic**:\n   - **Filling Phase**: When ventricular pressure (\\(P_v\\)) is less than atrial pressure (\\(P_a\\)), blood fills the ventricle, and flow is driven by \\(K_{at}\\), an atrioventricular valve coefficient.\n     \n     $P_v(t + dt) = P_{a}(t + dt) = \\frac{dt K_{at} P_a - dfree}{K_{at} + W}$\n\n   - **Isometric Contraction Phase**: During this phase, the ventricle contracts isometrically without significant ejection flow.\n     \n     $P_v(t + dt) = \\frac{-dfree}{W}$\n\n   - **Ejection Phase**: When \\(P_v > P_{ar}\\), blood is ejected into the artery, and flow follows from a balance of forces.\n     \n     $\\lambda(line) = (A - B dfree) / (B W + a_2 K_{at})$\n\n3. **Parameter Definitions**:\n   - \\(K_{ar}\\): Aortic valve coefficient, governing the flow from the left ventricle into the artery.\n   - \\(Q_t\\): Time-varying ejection or inflow rate, used in the recurrence calculations to update pressures and flows.\n\n## Parameters and Data\nThe significant data fields exposed by this component are:\n- **d_Par**: Current state at time t (Real)\n- **d_ParInf**: Asymptotic arterial pressure when flow is null (Real)\n- **d_Kar**: Valve param (Real)\n- **d_Rc**: Windkessel param (Rc) (Real)\n- **d_C**: Windkessel param (C) (Real)\n- **d_Rp**: Windkessel param (Rp) (Real)\n- **d_Pat**: Input pressure at time t (Real)\n- **d_nextPat**: Input pressure at time t+dt (Real)\n- **d_Kat**: Hemodynamic coefficient between pressure and flow (Real)"
  }
}