Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 1 Common Respiratory Care Equations % Accuracy: % Accuracy = Mean measured value reference value × 100 % Error: % Error = Mean measured value−Reference value Reference value × 100 %Body Humidity: %BH = (AH/43.8) × 100 Acute Alveolar Hyperventilation: ∆HCO− 3 = 0.2 × ∆PaCO2 Acute Alveolar Hypoventilation: ∆HCO− 3 = 0.10 × ∆PaCO2 Adjusting Oxygen Percentage: PaO2 desired FI O2 = (PaO 2 /PAO2 + PaCO2 × 1.25) × P ratio 1 B −PH2O Simpler but less accurate: PaO2 (initial) FI O2 (Initial) = PaO2 (Desired) FI O2 (New) Notes: Where PaO2 (initial) is the current arterial the current arterial oxygen tension (mm Hg), the FIO2 (initial) is the current setting for the fraction of inspired oxygen, the PaO2 (desired) is the arterial oxygen tension (mm Hg) that is desired, and the FIO2 (new) is the setting for the fraction of inspired oxygen. Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 2 Adjusting Ventilation: PaCO2 (1) × VT (1) = PaCO2 (2) × VT (2) Notes: Where PaCO2 (1) is the initial arterial carbon dioxide tension (mm Hg), VT (1) is the current tidal volume (L), PaCO2 (2) is the desired carbon dioxide tension (mm Hg), and VT (2) is the new tidal volume setting (L). Air-to-oxygen ratio: Liters air Liters O2 = (100−%O2 ) (%O2 −21) Notes: Where the flow rate values are the settings on the flow meters. The air-to-oxygen ration for each oxygen concentration is a fixed value. The “magic box” is a visual method of making this same calculation: 100 − %O2 21 %O2 100 %O2 − 21 Airway Pressure: Paw = Elastance (Vt ) + Resistance (flow) Airway Resistance: Raw = PIP−PPlateau Flow Notes: Where Raw (cm H2O/L/s) is the airways resistance, PIP is the peak inspiratory pressure (cm H2O), Pplateau is the plateau pressure (cm H2O). The presence of an artificial airway changes the airways resistance. This measurement if easily assessed on patients receiving ventilator support. A constant flow pattern (square) should be used for this measurement. Remember that ventilators normally report the flow rate in L/min and this value needs to be converted to L/s. Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 3 Alveolar Air Equation: PAO2 = FI O2 (PB − PH2O ) − [PACO2 × {FI O2 + (1 − FI O2 R )}] simplified form: PAO2 = PI O2 − (PACO2 × 1.25) Notes: Where PAO2 partial pressure of alveolar oxygen PACO2 partial pressure of alveolar carbon dioxide. The I refers to inspired. Thus, FIO2 is the fraction of inspired oxygen. Think of a fraction as changing the percentage of a value into a decimal. Divide the percentage by 100 to get the FIO2. Alveolar Carbon Dioxide Tension: PaCO2 = ̇ 2 ×0.863 VCO V̇A ̇ 2 is the CO2 Notes: Where PaCO2 is the partial pressure of alveolar carbon dioxide (mm Hg), VCO production (mL/Min), V̇A is alveolar ventilation (L/min), and .863 is the factor to convert temperature from STPD to BTPS. Alveolar Partial Pressure of Oxygen: PAO2 = FI O2 × (PB − 47) − (PACO2 ÷ 0.8) Alveolar Pressure: PA = ∆V CS Alveolar Ventilation: V̇A = (VT − VDSphys )ƒ Notes: Where V̇A is alveolar ventilation (L/min), VT - VDSphys is = VA , and ƒ is frequency (breaths/min). Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 4 Amount of Oxygen Carried by the Hemoglobin: 1.34( mL g g ) × Hb(dL) × SaO2 (%) Notes: Each gram of hemoglobin (Hb) carries 1.34 g mL g of oxygen. The average hemoglobin concentration is 15 dL. The SaO2 is the percentage of hemoglobin bound with oxygen. Anion Gap: Anion gap = [Na+ ] − ([CI− ] + [HCO3 − ]) Arm Muscle Area: AMA (males) = [MAC−(π×TSF)]2 𝐴𝑀𝐴 (𝑓𝑒𝑚𝑎𝑙𝑒𝑠) = 4π − 10 cm [𝑀𝐴𝐶 − (𝜋 × 𝑇𝑆𝐹)]2 − 6.5𝑐𝑚 4𝜋 Notes: AMA is the arm muscle area, MAC is the mid-arm circumference, and TSF is the triceps skinfold thickness (TSF) which is a measure of body fat. Arterial Partial Pressure of Carbon Dioxide: PaCO2 = 0.863×V̇CO2 V̇A Body Mass: weight (kg) BMI = height (m)2 or weight (lb) BMI = height (in)2 × 704.5 Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 5 Boyle’s Law: P×V=K <rule> Buoyancy: B = dw × V Notes: Where B is buoyancy, dw is the weight density of object and V is the object’s volume. Carbon Dioxide Production: V̇CO2 = FE̅ CO2 × V̇E Notes: Where the V̇CO2 (mL/min) is the CO2 production, the FE̅ CO2 is the average value of expired carbon dioxide, and the V̇E (mL/min) is the expired minute ventilation. a) There is no need to measure FI CO2 because the amount of inhaled carbon dioxide is very small. TheFE̅ CO2 is measured by collecting expired gas in a large reservoir such as a Douglas bag. Newer capnometers use microprocessors to calculate the average (mixed) carbon dioxide value from individual breaths. Cardiac Index: cardiac output CI = body surface area Cardiac Output: CO = Stoke volume × Heart rate Notes: In clinical practice, the CO is measured by Fick equation (see later in document), the thermodilution through a pulmonary artery catheter (Swan-Ganz), and is estimated by transesophageal echocardiography. Celsius to Kelvin: Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 6 °K = °C + 273 Cerebral Perfusion Pressure (CPP): CPP = ICP − MAP Notes: Where ICP is intracranial pressure and MAP is mean arterial blood pressure. Charles’ Law: V T =K Combined Gas Law: P1 ×V1 T1 = P2 ×V2 T2 Notes: Where P is pressure, V is volume, T is Temperature. Remember that if water vapor is present, subtract this from the pressure and the temperature needs to be converted to °K. Compliance: tidal volume Compliance = plateau pressure−PEEP Corrected Pressure: PC = Fgas × (PT − PH2 O ) Current: Current (A) = Voltage (V) Resistance (Ω) Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 7 Cylinder Factor: (see also Gas Cylinders) Cubic feet (full cylinder)×28.3 Cylinder Factor (L/psig) = Pressure (full cylinder)in psig Dalton’s Law: Partial Pressure = Fractional Concentration × Total Pressure Dead Space to Tidal Volume Ratio: VD VT = PaCO2 −PE ̅ CO2 PaCO2 V Notes: Where VD is the dead space tot tidal volume ration (%), PaCO2 is the partial pressure of T arterial carbon dioxide (mmHg), andPE̅ CO2 is the partial pressure of mixed expired carbon dioxide (mm Hg). The measurement for PE̅ CO2 is a mixed expired carbon dioxide value. This value can be measured by collecting expired gas in a large reservoir such as a Douglas bag. Newer capnometers use microprocessors to calculate the average (mixed) carbon dioxide value from individual breaths. Dead Space Ventilation: V̇D = VD × ƒ Notes: Where V̇D is dead space ventilation (L/min), VD is dead space volume (L), and ƒ if frequency (breaths/min). Density: d= m V Notes: Where d is the density, m is the object’s mass, and V is the volume the mass occupies. In clinical practice, weight is used for the mass which provides a weight density dw and for gases, mass is measured by gmw and the volume is expressed in liters. Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 8 or weight dw = volume Notes: Where dw is the weight density. Desired Fractional Inspired Oxygen: Desired FI O2 = [PaO2 (desired)×FI O2 (known)] PaO2 (known) Desire Respiratory Rate: Desired ƒ = Known PaCO2 ×Known ƒ Desired PaCO2 Desired Tidal Volume: Desired VT = Known PaCO2 ×Known VT Desired PaCO2 deWeir Equation: EE = [3.941 (V̇O2 ) + 1.106 (V̇CO2 )] × 1.44 − [2.17 (urinary nitrogen)] Diffusion Capacity of the Lung for Carbon Monoxide (DLCO): DLCO = VE (FI CO − FE CO) ÷ (PACO) Diffusion Capacity of the Lung for Carbon Monoxide by a Single Breath: DLCOSB = [60(VA ) ÷ t(PB − 47)] × in (FACO0 ÷ FACOt ) Dilution: Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 9 V1 C1 = V2 C2 Drug Dosage: Drug Solute (in g) %(as decimal) = Total solution (in ml) Duration of Flow of a Cylinder: (see also Gas Cylinder) Duration of Flow (min) = Pressure (psig)×Cylinder Factor Flow (L/min) Duration of Gas: (see also Gas Cylinder) Duration of gas (min) = Amount of gas in cylinder (L) Flow (L/min) Dynamic Compliance: VT PIP − PEEP Efficiency of Ventilation for Single Breath: VT = VA + VD Effective Total Lung Capacity: VA = VC × (FIHe ÷ FEHe) Ejection Fraction: SV EF = EDV Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 10 Notes: Where EF is ejection fraction (%), SV is stroke volume (mL), and EDV is end-diastolic volume (mL). Enghoff-Modified Bohr Equation: [PaCO2 − PE̅ CO2 ]/PaCO2 = VD /VT Equation of Motion: Pvent + Pmus = (E × V) + (R × V̇) Notes: Where Pvent is the pressure (cm H2O) used by the ventilator, Pmus is the pressure (cm H2O) the respiratory muscles create, E is elastance, R is resistance, and V̇ is the flow rate (L/min). Estimated Tidal Volume: VT = (TI × VL̇/min) 60 Fahrenheit to Kelvin: °C = 5/9 ×(°F − 32) then °K = °C − 273 Fick Equation: Q̇t = V̇O2 ÷ [C(a − v̅)O2 × 10] Notes: The basic formula for Fick's first law of diffusion addresses the rate of diffusion across a membrane. This diffusion rate is affected by the membrane's area, diffusion constant, and the concentration gradients on both sides of the membrane. Fick’s First Law of Diffusion: V̇gas = [(A × D × S) ÷ T](P1 − P2 ) Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 11 Notes: Where V̇gas is gas transport across the membrane, A is the cross-sectional area for diffusion, D is the diffusion coefficient of the gas, S is the solubility of the gas, T is the thickness of the membrane, andP1 − P2 is the pressure gradient across the membrame. When applying this equation to the lungs, the concentration gradient in Fick's diffusion equation is calculated by using Henry's law. The volume of gas is affected by the solubility (S) and partial pressure (P). Flow Resistance: R= (P1 −P2 ) V̇ Flow Rate: V V̇ = t Notes: Where V̇ is flow rate (L/min or L/s), V is volume (L), and t is time (min or s). or P V̇ = R Notes: Where V̇ is flow rate (L/s), P is pressure (cm H2O), and R is resistance. Fractional Hemoglobin Saturation: Fractional O2 Hb = [HHb+O O2 Hb 2 Hb+COHb+MetHb] Functional Residual Capacity (FRC): FRC = (vol He ÷ Fi He) × [(Fi He − Fƒ He) ÷ Fƒ He] FRC (or TLC) by Nitrogen Washout Technique: FRC = (VE × FE N2 ) ÷ (0.78 − FA N2 ) Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 12 Functional Hemoglobin Saturation: O Hb 2 Functional O2 Hb = [HHb+O 2 Hb] Gas Cylinders Duration of flow = Cylinder pressure × Cylinder factor x 1 flow rate Notes: Where units of measure are duration of flow (min), cylinder pressure (psig), cylinder factors (L/psig), and flow rate (L/min). The cylinder factor for an H cylinder of oxygen is 3.14 L/psig. The cylinder factor of an E cylinder of oxygen is 0.28 L/psig. Use a cylinder pressure of 200 psig lower than the actual cylinder pressure for patient safety. It is dangerous for patients when tanks empty earlier than expected. Most full cylinders contain 2200 psig. Gas in Liquid O2 Cylinder: Amount of gas in cylinder = Liquid O2 weight (b)×860 2.5lb/L Gas Mixing: VF CF = V1 C1 + V2 C2 Gay-Lussac’s Law: P T =K Graham’s Law: Dgas = 1 √gmw Notes: Where Dgas is the diffusion rate of the gas, and gmw is the gram molecular weight. Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 13 Gram Equivalent Weight Value: gEq = Gram atomic (formula)weight Valence Harris-Benedict Equation: men: BEE = 66 + (13.7)(weight) + (5)(height) − (6.8)(age) women: BEE = 65 + (9.6)(weight) + (1.8)(height) − (4.7)(age) Heat Gain/Loss by Radiation: E t = ekA(T2 − T1 ) Hemoglobin Saturation: SAO2 = [HbO2 ÷ Total HB] × 100 Henderson-Hasselbach Equation: pH = pK + log[(HCO− 3 )/(H2 CO3 )] simplified form: [H + ] = 24×PaCO2 (mm Hg) HCO3 − (mEq/L) Notes: Where HCO− 3 is bicarbonate and H2 CO3 is carbonic acid. In clinical practice The carbonic acid is not measured directly. This value is estimated by measuring the amount of dissolved carbon dioxide. The ratio is base ( HCO− 3 ) to acid ( H2 CO3 or CO2 ) needs to be 20:1 for a pH = 7.40. Henry’s Law: V = α × Pgas Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 14 Notes: Where V is the volume of the dissolved gas, is the solubility coefficient and Pgas is the partial pressure of the gas above the liquid. At higher temperatures, the gases are less soluble. The higher kinetic energy makes the gas escape from the solution. The solubility of oxygen at body temperature (37°C) is 0.003 mL/ mmHg/dL. Therefore: Dissolved oxygen = PaO2 × 0.003 mL/ mmHg/dL. Ideal Body Weight: men: IBW (lbs) = 106 + 6[height (in) − 60] women: IBW (lbs) = 105 + 5[height (in) − 60] Inspiratory Resistance: Raw = ∆P/∆F = (Ppeak −Pplat ) flow Karvonen Formula: [(HR max − HR rest ) × (50% to 70%)] + HR rest Notes: Where HR max is the patient’s maximal heart rate and HR rest is the resting heart rate. Laminar Flow: P = K1 × V̇ Laplace’s Law: As applied to a drop P= 2ST r As applied to a bubble P= 4ST r Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 15 As applied to a cylinder P= ST r Notes: Where P is the pressure inside the liquid bubble, ST is the surface tension, and r is the radius of the bubble. Law of Continuity: (A1 × v1 ) + (A2 × v2 ) + (An × vn ) = k Left Ventricular Stroke Work: LVSW = (MAP × SV) × 0.0136 Liquid Pressure: PL = h × dw Notes: Where PL is the liquid’s static pressure, h is the height of the liquid and dw is the weight density. Lung Compliance: ∆V (liters) CL = ∆P pl (cm H2 O) Notes: Where CL is the compliance (L/cm H2O) for the lung. ∆V (L) is the change in volume from the beginning to the end of inspiration. ∆Ppl (cm H2O) is the change in pressure from the beginning to the end of inspiration. Mean Airway Pressure: 1 inspiratory time Paw ̅̅̅̅ = 2 [PIP × (total respiratory cycle)] or Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 16 1 inspiratory time Paw ̅̅̅̅ = 2 (PIP − PEEP) × ( total cycle time ) + PEEP Notes: Where Paw ̅̅̅̅ is the mean airway pressure, PIP is the peak inspiratory pressure (cm H2O). Mean Arterial Blood Pressure: MAP = SBP+(2×DBP) 3 Notes: Where MAP is mean arterial pressure (mm Hg), SBP is systolic blood pressure (mm Hg), and DBO is diastolic blood pressure (mm Hg). Mean Arterial Pressure: Volume Mean Arterial Pressure = Capacity <rule> Mechanical Dead Space to Add: PaCO2(1) (VT −VD −VDmech(1) ) VDmech(2) = VT − VDphys − ( PaCO2(2) ) Milligram Equivalent Weight: gEq mEq = 1000 Minute Ventilation: V̇E = VT × 𝑓 Notes: Where V̇E is minute ventilation (L/min), VT is tidal volume (L), and f is breaths/min. Modified Fick Equation: Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 17 kcal REE ( ) = Cardiac Output × Hemoglobin × (SaO2 − Sv̅O2 ) × 95.18 day Modified deWeir Equation: Notes: Where REE (kcal/day) is the resting energy expenditure, consumption, and (mL/min) is the O2 is the CO2 production. This equation is used to predict the patient’s caloric needs at rest. Nitrogen Balance: Ohm’s Law: Notes: The current is measured in amperes (A), voltage is measured in volts (V), and resistance is measures in Ohms (Ω). Oxygen Consumption: Oxygen Consumption by Ventilatory Muscles: (Arterial) Oxygen Content: Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 18 Notes: Where CaO2 is the arterial oxygen content (mL/dL), Hb is the total hemoglobin (g/dL), and SaO2 is the percentage of hemoglobin saturated with oxygen (%). Oxygen Cost of Breathing: Oxygen Delivery: Notes: Where DO2 is oxygen delivery (mL/min), CaO2 is the arterial oxygen content (mL/dL). Oxygen Flow: Oxygen Percentage: Notes: Where the air flow rate is the reading from the air flowmeter (L/min), the O2 flow rate is the reading from the oxygen flowmeter (L/min), and the total flow rate is the sum of the readings from the air and oxygen flowmeters. Pack Years: Partial Pressure of Alveolar Nitrogen: Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 19 Partial Pressure of Inspired Oxygen: Passive Inhalation: Percent Gas Exhaled: Percent Leak: Percent leak = [ (VT insp − VT exh) VT insp Percentage Shunt: pH: Physically Dissolved Oxygen: × 100] Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 20 Physiologic Dead Space: Physiologic Shunt: Notes: Where is the ratio of shunted blood to the total cardiac output, the is the oxygen content (mL/dL) measured from a pulmonary capillary (end-capillary) blood sample, the is the oxygen content (mL/dL) from an arterial blood sample, and the the oxygen content (mL/dL) from a mixed-venous blood sample. Poiseuille’s Law: Notes: Where length, is the driving pressure gradient, n is the viscosity of the fluid, l is the tube’s is the fluid’s flow rate, and r is the tube’s radius. Predicted PaO2 with Age: Predicted Postoperative Value: Notes: Patients need at least a 40% predicted postoperative value to tolerate surgery. is Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 21 Pressure: Pressure Support: Pressure Support Ventilation to overcome WOBI: Pressure-Time Index: Pulmonary Function Impairment: Pulmonary Function Improvement: Notes: Where “Pre” is the pulmonary function variable before a bronchodilator and “Post” is the pulmonary function variable after using a bronchodilator. The calculation is used to determine how much the pulmonary function variable improved after taking a bronchodilator. Pulmonary Vascular Resistance: Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 22 Notes: Where PVR is pulmonary vascular resistance, MPAP is mean pulmonary artery pressure and MLAP is mean left atrial pressure. Cardiac output is the same as pulmonary blood flow. You must look at the pressure that it takes to move the blood through the pulmonary circulation. Multiply by 80 to convert from to Rapid, Shallow Breathing Index: Notes: Where RSBI is the rapid, shallow breathing index (breaths/min/L), ƒ is the respiratory rate (breaths/min), and VT is the tidal volume (L). Relative Humidity: Respiratory Exchange Ratio: Notes: Where R is the respiratory exchange ratio, is the O2 consumption (mL/min) and is the CO2 carbon dioxide production (mL/min). Respiratory Quotient: Notes: Where RQ refers to the respiratory quotient, RQ refers to the ratio of tissues. This ratio at complete rest is treated as an RQ measurement. Respiratory Rate: to in the Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 23 Resting Energy Expenditure (REE): men: women: Required Frequency: Reynold’s Number: Right Ventricular Stroke Work: RV: Spontaneous Tidal Volume: Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 24 Starling Equilibrium: Notes: Where is the flow of fluid between the capillaries and interstitial space, is the capillary permeability coefficient for fluids and electrolytes, is the capillary hydrostatic pressure, is the interstial fluid hydrostatic pressure, is the capillary permeability coefficient for proteins, is the capillary oncotic pressure, and is the interstitial fluid oncotic pressure. On the arterial side: fluid, oxygen, and nutrients are pushed toward cells and into the interstitial fluid. On the venous side: fluid, carbon dioxide, and cellular waste products are pulled into the capillaries. Excess fluid in the interstitial space is absorbed into the lymphatic system. Static Compliance: Stroke Index: Stroke Volume: Systemic Vascular Resistance: Notes: Where SVR is systemic vascular resistance. Thoracic Gas Volume: Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 25 Tidal Volume: Time Constant: Notes: Where C is compliance (L/cm H2O) and R is resistance (cm H2O)/L/s) TLC: Total Compliance: Total Cycle Time: Total Oxygen Content: Total PEEP: Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 26 Tracheal Pressure: Transairway Pressure: Transitional Flow: Transpulmonary Pressure: Notes: Where is transpulmonary pressure, is transairway pressure and is transalveolar pressure Transpulmonary Pressure Gradient: Notes: Where is transpulmonary pressure, Transrespiratory Pressure Gradient: Transthoracic Pressure Gradient: is alveolar pressure and is pleural pressure. Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 27 Turbulent Flow: Vascular Resistance: Ventilatory Equivalent for Carbon Dioxide: 𝑉̇𝐸 𝑉̇ 𝐶𝑂2 Notes: Where 𝑉̇𝐸 is the expired minute ventilation(mL/min) and 𝑉̇ 𝐶𝑂2 is the O2 consumption (mL/min). Ventilatory Equivalent for Oxygen: 𝑉̇𝐸 𝑉̇ 𝑂2 Notes: Where 𝑉̇𝐸 is the expired minute ventilation(mL/min) and 𝑉̇ 𝑂2 is the O2 consumption (mL/min). Ventilatory Reserve: Formulas Egan’s Fundamentals of Respiratory Care, 10th Edition 28 Notes: Where ventilator capacity (L/min) is measured by the maximum voluntary ventilation (MVV) or by multiplying the FEV1 by 40, and the (L/min) is the highest minute ventilation that occurs at the end of a cardiopulmonary exercise test. Volume: Volume Lost: Work of Breathing: Work of Breathing Provided by Ventilator: