**math**is an essential part of respiratory therapy, serving as the foundation for various diagnostic and therapeutic procedures.

Students entering the field of respiratory care are often surprised to learn that their success will depend, in part, on their ability to master a range of **mathematical equations and calculations**.

However, this isn’t a reason to be intimidated.

The **calculations performed by respiratory therapists** are quite straightforward as long as you learn the formulas, which are listed and explained below.

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## Respiratory Therapy Formulas and Calculations

**Respiratory therapy**involves various formulas and calculations that are essential for diagnosing and treating patients effectively. This includes the following:

### Minute Ventilation (VE)

**Minute ventilation (VE)** measures the total volume of air entering or leaving the lungs in one minute. It is a crucial parameter for assessing the adequacy of ventilation in both spontaneously breathing and mechanically ventilated patients.

Formula:

VE = Respiratory Rate x Tidal Volume

**Example:**

A patient has the following bedside spirometry results:

- Respiratory Rate = 12
- Tidal Volume = 450 mL
- Dead Space = 147 mL
- Vital Capacity = 1.2 L

Based on this data, what is the patient’s minute ventilation?

**Calculation:**

VE = Respiratory Rate x Tidal Volume

VE = 12 x 450

VE = 5,400 mL/min

Divide by 1,000 to convert mL to L.

**Answer:**

VE = 5.4 L/min

### Alveolar Minute Ventilation (VA)

**Alveolar minute ventilation (VA)** calculates the volume of air that reaches the alveoli and participates in gas exchange per minute. It’s important for evaluating how well the lungs are ventilating the areas where gas exchange occurs.

Formula:

VA = Respiratory Rate x (Tidal Volume – Deadspace)

**Example:**

A patient has the following bedside spirometry results:

- Rate = 12
- Tidal Volume = 450 mL
- Dead Space = 147 mL
- Vital Capacity = 1.2 L

Based on this data, what is the alveolar minute ventilation?

**Calculation:**

VA = Respiratory Rate x (Tidal Volume – Deadspace)

VA = 12 x (450 – 147)

VA = 3,636 mL/min

Divide by 1,000 to convert mL to L.

**Answer:**

VA = 3.6 L/min

### Airway Resistance (Raw)

**Airway resistance (Raw)** quantifies the resistance to airflow in the respiratory airways. It helps in diagnosing and managing conditions that alter the airways’ diameter, such as asthma or chronic obstructive pulmonary disease (COPD).

Formula:

Raw = (PIP – Plateau pressure) / Flow

**Example:**

An adult patient receiving mechanical ventilation has a PIP of 30 cmH2O and a plateau pressure of 10 cmH2O, and the set flow rate is 60 L/min. What is the airway resistance?

**Calculation:**

Raw = (PIP – Plateau pressure) / Flow

Raw = (30 – 10) / 1

**Answer:**

Raw = 20 cmH2O/L/sec

### Mean Airway Pressure (Paw)

**Mean airway pressure (Paw)** represents the average pressure within the airways over the entire respiratory cycle. It is significant in mechanical ventilation management and influences oxygenation and ventilation.

Formula:

Paw = ((Inspiratory Time x Frequency) / 60) x (PIP – PEEP) + PEEP

**Example:**

The following data was obtained on a 63-year-old female patient who is receiving mechanical ventilatory support:

- Rate = 12/min
- Tidal Volume = 450 mL
- Inspiratory Time = 1.3 seconds
- PIP = 25
- PEEP = 5

What is the mean airway pressure?

**Calculation:**

Paw = ((Inspiratory Time x Frequency) / 60) x (PIP – PEEP) + PEEP

Paw = ((1.3 x 12) / 60) x (25 – 5) + 5

Paw = ((15.6) / 60) x (25 – 5) + 5

Paw = 0.26 x (25 – 5) + 5

Paw = 5.2 + 5

**Answer:**

Paw = 10.2 cmH2O

### Work of Breathing (WOB)

**Work of breathing (WOB)** measures the effort required for breathing, including both the patient’s and the ventilator’s effort if the patient is mechanically ventilated. It’s vital for assessing respiratory muscle load and fatigue.

Formula:

WOB = Change in Pressure x Change in Volume

**Example:**

An adult patient is intubated and receiving mechanical ventilation. Given the following data, calculate the work of breathing:

- ∆P = 6.9 cmH2O
- ∆V = 0.8 L

**Calculation:**

WOB = Change in Pressure x Change in Volume

WOB = 6.9 x 0.8

**Answer:**

WOB = 5.5 cmH2O/L

### Alveolar-Arterial Oxygen Tension Gradient (P(A-a)O2)

**Alveolar-arterial oxygen tension gradient (P(A-a)O2)** calculates the difference between the oxygen pressure in the alveoli and the arterial oxygen pressure, indicating how efficiently oxygen moves from the lungs to the blood.

Formula:

P(A-a)O2 = PAO2 – PaO2

**Example:**

The following data was obtained on an adult patient:

- PaO2 = 87 mmHg
- PAO2 = 107 mmHg

What is the alveolar-arterial oxygen tension gradient?

**Calculation:**

P(A-a)O2 = PAO2 – PaO2

P(A-a)O2 = 107 – 87

**Answer:**

P(A-a)O2 = 20 mmHg

### Alveolar Oxygen Tension (PAO2)

**Alveolar oxygen tension (PAO2)** estimates the partial pressure of oxygen in the alveoli, which is crucial for understanding the lung’s ability to oxygenate the blood.

Formula:

PAO2 = (PB – PH2O) x FiO2 – (PaCO2 / 0.8)

**Example:**

The following data was obtained on an adult patient:

- FiO2 = 40%
- PaCO2 = 35 mmHg
- PB = 760 mmHg
- PH2O = 47 mmHg

What is the PAO2?

**Calculation:**

PAO2 = (PB – PH2O) x FiO2 – (PaCO2 / 0.8)

PAO2 = (760 – 47) x 0.40 – (35 / 0.8)

PAO2 = (713 x 0.40) – 43.75

**Answer:**

PAO2 = 241.5 mmHg

### Arterial/Alveolar Oxygen Tension (a/A) Ratio

The **arterial/alveolar oxygen tension (a/A) ratio** compares the partial pressure of oxygen in arterial blood (PaO2) to that in the alveoli (PAO2), assessing the efficiency of gas exchange in the lungs.

Formula:

(a/A) Ratio = PaO2/PAO2

**Example:**

A patient has a PaO2 of 108 mmHg and a PAO2 of 300 mmHg. What is the arterial/alveolar oxygen tension ratio?

**Calculation:**

(a/A) Ratio = PaO2/PAO2

(a/A) Ratio = 108 / 300

(a/A) Ratio = 0.36 mmHg

Multiply by 100 to convert to a percentage.

**Answer:**

(a/A) Ratio = 36%

### Arterial Oxygen Content (CaO2)

**Arterial oxygen content (CaO2)** measures the total amount of oxygen in the arterial blood, including the oxygen bound to hemoglobin and the oxygen dissolved in the blood plasma. It provides insight into the blood’s oxygen-carrying capacity.

Formula:

CaO2 = (Hb x 1.34 x SaO2) + (PaO2 x 0.003)

**Example:**

An adult patient has a PaO2 of 95 mmHg, an oxygen saturation of 96%, and a hemoglobin of 13 g/dL. What is the arterial oxygen content?

**Calculation:**

CaO2 = (Hb x 1.34 x SaO2) + (PaO2 x 0.003)

CaO2 = (13 x 1.34 x 0.96) + (95 x 0.003)

CaO2 = 16.72 + 0.285

**Answer:**

CaO2 = 17 vol%

### End-Capillary Oxygen Content (CcO2)

**End-capillary oxygen content (CcO2)** calculates the oxygen content in capillary blood at the end of the pulmonary capillaries. It reflects the maximum oxygen level that blood can carry after passing through the lungs’ gas exchange areas.

Formula:

CcO2 = (Hb x 1.34 x SaO2) + (PAO2 x 0.003)

**Example:**

An adult patient has a PAO2 of 88 mmHg, an oxygen saturation of 94%, and a hemoglobin of 16 g/dL. What is the end-capillary oxygen content?

**Calculation:**

CcO2 = (Hb x 1.34 x SaO2) + (PAO2 x 0.003)

CcO2 = (16 x 1.34 x 0.94) + (88 x 0.003)

CcO2 = 20.15 + 0.264

**Answer:**

CcO2 = 20.4 vol%

### Mixed Venous Oxygen Content (CvO2)

**Mixed venous oxygen content (CvO2)** quantifies the amount of oxygen in the venous blood returning to the lungs from the body, providing insight into how much oxygen is being consumed by the body’s tissues.

Formula:

CvO2 = (Hb x 1.34 x SvO2) + (PvO2 x 0.003)

**Example:**

An adult patient has a PvO2 of 38 mmHg, a mixed venous saturation of 74%, and a hemoglobin of 14 g/dL. What is the mixed venous oxygen content?

**Calculation:**

CvO2 = (Hb x 1.34 x SvO2) + (PvO2 x 0.003)

CvO2 = (14 x 1.34 x 0.74) + (38 x 0.003)

CvO2 = 13.88 + 0.114

**Answer:**

CvO2 = 14 vol%

### Shunt Equation (QS/QT)

The **shunt equation (QS/QT)** calculates the proportion of cardiac output that does not participate in gas exchange due to bypassing the ventilated areas of the lung. This is a critical measure in assessing lung function and gas exchange efficiency.

Formula:

QS/QT = (CcO2 – CaO2) / (CcO2 – CvO2)

**Example:**

The following data was provided for an adult patient:

- CcO2 = 20.8 vol%
- CaO2 = 19.3 vol%
- CvO2 = 13.9 vol%

Calculate the physiologic shunt percentage for this patient.

**Calculation:**

QS/QT = (CcO2 – CaO2) / (CcO2 – CvO2)

QS/QT = (20.8 – 19.3) / (20.8 – 13.9)

QS/QT = 1.5 / 6.9

QS/QT = 0.22

Multiply by 100 to convert to a percentage.

**Answer:**

QS/QT = 22%

### Modified Shunt Equation (QS/QT)

An adaptation of the traditional shunt equation, the **modified shunt equation (QS/QT)** incorporates adjustments for practical clinical measurements, helping to refine the assessment of blood flow that bypasses the lungs’ gas exchange areas.

Formula:

QS/QT = ((PAO2 – PaO2) x 0.003) / ((CaO2 – CvO2) + (PAO2 – PaO2) x 0.003)

**Example:**

The following data was provided for an adult patient:

- PaO2 = 155 mmHg
- PAO2 = 650 mmHg
- CaO2 = 19.9 vol%
- CvO2 = 13.2 vol%

Calculate the shunt percentage for this patient.

**Calculation:**

QS/QT = ((PAO2 – PaO2) x 0.003) / ((CaO2 – CvO2) + (PAO2 – PaO2) x 0.003)

QS/QT = ((650 – 155) x 0.003)) / ((19.9 – 13.2) + (650 – 155) x 0.003)

QS/QT = 1.49 / 6.7 + 495 x 0.003

QS/QT = 1.49 / 8.19

QS/QT = 0.18

Multiply by 100 to convert to a percentage.

**Answer:**

QS/QT = 18%

### Arterial-Mixed Venous Oxygen Content Difference (C(a-v)O2)

**Arterial-mixed venous oxygen content difference (C(a-v)O2)** measures the difference between the oxygen content in arterial blood and mixed venous blood, indicating the amount of oxygen extracted by the body’s tissues.

Formula:

C(a-v)O2 = CaO2 – CvO2

**Example:**

An adult patient in the ICU has a CaO2 of 19.2 vol% and a CvO2 of 14.7 vol%. Calculate the arterial-mixed venous oxygen content difference.

**Calculation:**

C(a-v)O2 = CaO2 – CvO2

C(a-v)O2 = 19.2 – 14.7

**Answer:**

C(a-v)O2 = 4.5 vol%

### Oxygen-to-Air Entrainment Ratio (O2:Air)

**Oxygen-to-air entrainment ratio (O2:Air)** calculates the ratio of oxygen to air required to achieve a specific oxygen concentration in a delivered gas mixture, useful in setting up oxygen therapy devices.

Formula:

O2:Air = 1 : (100 – FiO2) / (FiO2 – 2)

**Example:**

What is the air-to-oxygen entrainment ratio of 60%?

**Calculation:**

O2:Air = 1 : (100 – FiO2) / (FiO2 – 2)

O2:Air = 1 : (100 – 60) / (60 – 2)

O2:Air = 1 : (40 / 58)

**Answer:**

O2:Air = 1 : 0.7

### Arterial Oxygen Saturation Estimation (SaO2)

**Arterial oxygen saturation estimation (SaO2)** estimates the saturation of oxygen in arterial blood, providing a quick assessment of a patient’s oxygenation status without the need for blood gas analysis.

Formula:

SaO2 = PaO2 + 30

**Example:**

An adult patient has a PaO2 of 63 mmHg. What is the estimated SaO2?

**Calculation:**

SaO2 = PaO2 + 30

Estimated SaO2 = 63 + 30

**Answer:**

Estimated SaO2 = 93 mmHg

### PaO2/FiO2 Ratio (P/F Ratio)

The **PaO2/FiO2 ratio (P/F ratio)** compares arterial oxygen partial pressure (PaO2) to fractional inspired oxygen (FiO2) and is a key indicator of lung function, particularly in assessing the severity of acute respiratory distress syndrome (ARDS).

Formula:

P/F Ratio = PaO2 / FiO2

**Example:**

An adult patient who is receiving oxygen at an FiO2 of 40% has a PaO2 of 88 mmHg. What is the PaO2/FiO2 ratio?

**Calculation:**

P/F Ratio = PaO2 / FiO2

P/F Ratio = 88 / 0.4

**Answer:**

P/F Ratio = 220 mmHg

### Oxygenation Index (OI)

The **oxygenation index (OI)** assesses the effectiveness of oxygenation and ventilatory support by calculating the product of mean airway pressure, fractional inspired oxygen, and arterial oxygen tension.

Formula:

OI = ((Paw x FiO2) / PaO2) x 100

**Example:**

The following data was obtained on an adult patient:

- FiO2 = 40%
- PaO2 = 80 mmHg
- Mean Airway Pressure = 9.8 cmH2O

What is the oxygenation index?

**Calculation:**

OI = ((Paw x FiO2) / PaO2) x 100

OI = ((9.8 x 0.4) / 80) x 100

OI = (3.92 / 80) x 100

OI = 0.049 x 100

**Answer:**

OI = 4.9

### Oxygen Consumption (VO2)

**Oxygen consumption (VO2)** represents the amount of oxygen consumed by the body per minute, an essential measure of metabolic rate and the body’s efficiency in using oxygen.

Formula:

VO2 = Cardiac Output x C(a-v)O2

**Example:**

What is the total oxygen consumption of an adult patient with the following data:

- Cardiac Output = 6.2 L/min
- C(a-v)O2 = 5 vol%

**Calculation:**

VO2 = Cardiac Output x C(a-v)O2

VO2 = 6.2 x 0.05

VO2 = 0.31 L/min

Multiply by 1,000 to convert L to mL.

**Answer:**

VO2 = 310 mL/min

### Oxygen Extraction Ratio (O2ER)

The **oxygen extraction ratio (O2ER)** calculates the proportion of oxygen extracted from the blood by the tissues during one circulation of the body, indicative of the balance between oxygen delivery and tissue oxygen demand.

Formula:

O2ER = (CaO2 – CvO2) / CaO2

**Example:**

An adult patient has an arterial oxygen content of 18 vol% and a mixed venous oxygen content of 13 vol%. What is the oxygen extraction ratio?

**Calculation:**

O2ER = (CaO2 – CvO2) / CaO2

O2ER = (18 – 13) / 18

O2ER = 0.2778

**Answer:**

O2ER = 27.8 vol%

### FiO2 Estimation for Nasal Cannula

The **FiO2 estimation for a nasal cannula** helps determine the fraction of inspired oxygen (FiO2) delivered by a nasal cannula based on the flow rate of oxygen, providing a guide for adjusting supplemental oxygen therapy.

Formula:

FiO2 = 20 + (4 x Liter Flow)

**Example:**

An adult patient receiving oxygen therapy via nasal cannula at 4 L/min. What is the estimated FiO2?

**Calculation:**

FiO2 = 20 + (4 x Liter Flow)

FiO2 = 20 + (4 x 4)

FiO2 = 20 + 16

**Answer:**

FiO2 = 36%

### Oxygen Cylinder Duration

Oxygen cylinder duration calculates the duration a portable oxygen tank will last given a specific flow rate and cylinder pressure, essential for managing oxygen therapy in ambulatory patients.

Formula:

Duration = (Gauge Pressure x Tank Factor) / Liter Flow

**Example:**

A patient is receiving oxygen via nasal cannula at 2 L/min from a size E tank with 2,200 psig. How long will the tank deliver oxygen?

**Cylinder Tank Factors:**

- D Cylinder = 0.16
- E Cylinder = 0.28
- G Cylinder = 2.41
- H Cylinder = 3.14
- M Cylinder = 1.56

**Calculation:**

Duration = (Gauge Pressure x Tank Factor) / Liter Flow

Duration = (2,200 x 0.28) / 2

Duration = 616 / 2

Duration = 308 minutes

Divide by 60 to convert minutes to hours.

**Answer:**

Duration = 5 hours and 8 minutes

### Liquid Oxygen System Duration

The **liquid oxygen system duration** formula determines how long a liquid oxygen system will supply oxygen at a set flow rate, important for planning oxygen therapy in various settings.

Formula:

Duration = (344 x Liquid Weight) / Flow

**Example:**

A liquid oxygen system with a weight of 3 lbs is being used and the patient is receiving oxygen via nasal cannula with a flow of 2 L/min. How long will the liquid oxygen system last?

**Calculation:**

Duration = (344 x Liquid Weight) / Flow

Duration = (344 x 3) / 2

Duration = 1,032 / 2

Duration = 516 minutes

Divide by 60 to convert minutes to hours.

**Answer:**

Duration = 8 hours and 36 minutes

### Cardiac Index (CI)

**Cardiac index (CI)** measures the cardiac output (blood flow from the heart) relative to the body surface area, providing a normalized assessment of heart function.

Formula:

CI = Cardiac Output / Body Surface Area

**Example:**

A 59-year-old female patient has a cardiac output of 5 L/min and a body surface area of 2.7 m2. What is the cardiac index?

**Calculation:**

CI = Cardiac Output / Body Surface Area

CI = 5 / 2.7

**Answer:**

CI = 1.85 L/min/m2

### Cardiac Output (QT)

**Cardiac output (QT)** calculates the total volume of blood the heart pumps per minute, a critical measure of cardiovascular health and function.

Formula:

QT = Heart Rate x Stroke Volume

**Example:**

A 57-year-old male patient has a heart rate of 94 beats/min and a stroke volume of 44 mL/beat. What is the cardiac outlook?

**Calculation:**

QT = Heart Rate x Stroke Volume

QT = 94 x 44

QT = 4,136 mL/min

Divide by 1,000 to convert mL to L.

**Answer:**

QT = 4.1 L/min

### Cardiac Output (CO) Fick’s Method

Utilizing the **Fick principle**, this method measures **cardiac output** based on oxygen consumption and the difference in oxygen content between arterial and venous blood, offering a direct assessment of heart efficiency.

Formula:

CO = (O2 Consumption / CaO2 – CvO2)

**Example:**

The following data was obtained on an adult patient:

- Body Surface Area = 1.7 m2
- CaO2 = 21 vol%
- CvO2 = 16 vol%

Calculate the cardiac output using Fick’s method.

**Calculation:**

First, calculate O2 Consumption:

O2 Consumption = Body Surface Area x 130

O2 Consumption = 130 x 1.7

O2 Consumption = 221

Then, calculate Cardiac Output using Fick’s Method.

CO = (O2 Consumption / CaO2 – CvO2)

CO = (221 / 21 – 16)

CO = (221 / 0.05)

CO = 4,420 mL/min

Divide by 1,000 to convert mL to L.

**Answer:**

CO = 4.42 L/min

### Cerebral Perfusion Pressure (CPP)

**Cerebral perfusion pressure (CPP)** is the net pressure gradient causing blood flow to the brain, crucial for ensuring adequate oxygenation and functioning of brain tissue.

Formula:

CPP = Mean Arterial Pressure – Intracranial Pressure

**Example:**

An adult patient has a mean arterial pressure of 88 mmHg and an intracranial pressure of 15 mmHg. Calculate the cerebral perfusion pressure.

**Calculation:**

CPP = Mean Arterial Pressure – Intracranial Pressure

CPP = 88 – 15

**Answer:**

CPP = 73 mmHg

### Mean Arterial Pressure (MAP)

**Mean arterial pressure (MAP)** calculates the average arterial pressure throughout one cardiac cycle, vital for assessing the overall blood pressure in the body’s arterial system.

Formula:

MAP = (Systolic BP + (2 x Diastolic BP)) / 3

**Example:**

An adult patient has a blood pressure measurement of 130/90 mmHg. What is the mean arterial pressure?

**Calculation:**

MAP = (Systolic BP + (2 x Diastolic BP)) / 3

MAP = (130 + (2 x 90)) / 3

MAP = 310 / 3

**Answer:**

MAP = 103.3 mmHg

### Stroke Volume (SV)

**Stroke volume (SV)** is the amount of blood pumped by the left ventricle of the heart in one contraction, indicating heart pumping efficiency.

Formula:

SV = Cardiac Output / Heart Rate

**Example:**

A 58-year-old female patient has a heart rate of 92/min and a cardiac output of 6 L/min. What is her stroke volume?

**Calculation:**

SV = Cardiac Output / Heart Rate

SV = 6 / 92

SV = 0.065 L

Multiply by 1,000 to convert to mL.

**Answer:**

SV = 65 mL

### Maximum Heart Rate (HRmax)

**Maximum heart rate (HRmax)** estimates the highest heart rate an individual can achieve without severe problems through exercise stress, used in exercise prescription and health assessment.

Formula:

HRmax = 220 – Age

**Example:**

What is the maximum heart rate of a 44-year-old female patient?

**Calculation:**

HRmax = 220 – Age

HRmax = 220 – 44

**Answer:**

HRmax = 176 beats/min

### Heart Rate on an EKG Strip (HR)

The heart rate (HR) on an EKG strip method calculates heart rate based on the number of large squares between R waves on an EKG strip, offering a quick way to estimate heart rate without electronic monitors.

Formula:

HR = 300 / # of large boxes between R waves

**Example:**

An adult patient has the following EKG results:

What is the patient’s heart rate?

**Calculation:**

HR = 300 / # of large boxes between R waves

There are 3 large boxes between the R waves on the EKG strip.

HR = 300 / 3

**Answer:**

HR = 100 beats/min

### Respiratory Quotient (RQ)

**Respiratory quotient (RQ)** is the ratio of carbon dioxide produced to oxygen consumed while food is being metabolized. It’s useful for determining the predominant type of metabolic fuel the body uses.

Formula:

RQ = VCO2 / VO2

**Example:**

A 45-year-old male patient is undergoing a metabolic study. The measurements taken indicate that he is consuming oxygen at a rate of 250 mL/min and producing carbon dioxide at a rate of 200 mL/min. Calculate the respiratory quotient (RQ) for this patient.

**Calculation:**

RQ = VCO2 / VO2

RQ = 200 / 250

**Answer:**

RQ = 0.8

### Systemic Vascular Resistance (SVR)

**Systemic vascular resistance (SVR)** calculates the resistance to blood flow offered by all of the systemic vasculature, excluding the pulmonary circulation. It’s important for understanding blood pressure regulation and cardiac function.

Formula:

SVR = (MAP – CVP) x (80 / Cardiac Output)

**Example:**

An adult patient has the following measurements:

- Cardiac Output = 4.0 L/min
- Central Venous Pressure = 9 mmHg
- Mean Arterial Pressure = 75 mmHg

What is the systemic vascular resistance (SVR)?

**Calculation:**

SVR = (MAP – CVP) x (80 / Cardiac Output)

SVR = (75 – 9) x (80 / 4)

SVR = 66 x 20

**Answer:**

SVR = 1,320 dynes/sec/cm5

### Pulmonary Vascular Resistance (PVR)

**Pulmonary vascular resistance (PVR)** is similar to SVR but specific to the lungs. It measures the resistance to blood flow in the pulmonary circulation, which is helpful in diagnosing and managing pulmonary hypertension.

Formula:

PVR = (MPAP – PCWP) x (80 / Cardiac Output)

**Example:**

An adult patient has the following measurements:

- Cardiac Output = 5.0 L/min
- Mean Pulmonary Artery Pressure = 23 mmHg
- Pulmonary Capillary Wedge Pressure = 7 mmHg

What is the pulmonary vascular resistance (PVR)?

**Calculation:**

PVR = (MPAP – PCWP) x (80 / Cardiac Output)

PVR = (23 – 7) x (80 / 5)

PVR = 16 x 16

**Answer:**

PVR = 256 dynes/sec/cm5

### Static Compliance (Cst)

**Static compliance (Cst)** measures the change in volume for any given change in pressure in the respiratory system when no air is flowing, reflecting lung and chest wall elasticity.

Formula:

Cst = Tidal Volume / (Plateau Pressure – PEEP)

**Example:**

An adult patient who is receiving mechanical ventilation has a tidal volume of 450 mL, peak pressure of 30 cmH2O, plateau pressure of 22 cmH2O, and a PEEP of 5. What is the static compliance?

**Calculation:**

Cst = Tidal Volume / (Plateau Pressure – PEEP)

Cst = 450 / (22 – 5)

**Answer:**

Cst = 26.5 mL/cmH2O

### Dynamic Compliance (Cdyn)

Unlike static compliance, **dynamic compliance (Cdyn)** measures lung and chest wall compliance during active inhalation and exhalation, indicating the effort required for breathing.

Formula:

Cdyn = Tidal Volume / (Peak Pressure – PEEP)

**Example:**

An adult patient who is receiving mechanical ventilation has a tidal volume of 450 mL, peak pressure of 30 cmH2O, plateau pressure of 22 cmH2O, and a PEEP of 5. What is the dynamic compliance?

**Calculation:**

Cdyn = Tidal Volume / (Peak Pressure – PEEP)

Cdyn = 450 / (30 – 5)

**Answer:**

Cdyn = 18 mL/cmH2O

### Deadspace to Tidal Volume Ratio (VD/VT)

The **deadspace to tidal volume ratio (VD/VT)** assesses the proportion of each breath that does not participate in gas exchange, important for evaluating ventilation efficiency.

Formula:

(VD/VT) = (PaCO2 – PECO2) / PaCO2

**Example:**

An adult patient has a PaCO2 of 44 mmHg and a PECO2 of 34 mmHg. What is the deadspace to tidal volume ratio?

**Calculation:**

(VD/VT) = (PaCO2 – PECO2) / PaCO2

(VD/VT) = (44 – 34) / 44

(VD/VT) = 10 / 44

**Answer:**

(VD/VT) = 23%

### Children Dosage Estimation

The **child dosage estimation** provides a method for estimating the appropriate medication dose for children compared to adults based on their age, ensuring safe and effective treatment.

Formula:

Child Dose = (Age / Age + 12) x Adult Dose

**Example:**

An adult male is receiving a medication dose of 44 mg. What is the appropriate dose for a 9-year-old boy?

**Calculation:**

Child Dose = (Age / Age + 12) x Adult Dose

Child Dose = (9 / (9 + 12)) x 44

Child Dose = 0.429 x 44

**Answer:**

Child Dose = 18.9 mg

### Infant Dosage Estimation

The **infant dosage estimation** provides a method for estimating the appropriate medication dose for infants compared to adults based on their body weight.

Formula:

Infant Dose = (Body Weight in lbs / 150) x Adult Dose

**Example:**

An adult male is receiving a medication dose of 37 mg. What is the appropriate dose for a 10-month-old infant?

**Calculation:**

Infant Dose = (Body Weight in lbs / 150) x Adult Dose

Infant Dose = (10 / 150) x 37

Infant Dose = 0.067 x 37

**Answer:**

Infant Dose = 2.5 mg

### Infant and Children Dosage Estimation (Fried’s Rule)

**Fried’s rule** is another method to calculate medication doses for infants and children, using the age in months to tailor drug dosages more precisely.

Formula:

Infant or Child Dose = (Age in Months / 150) x Adult Dose

**Example:**

An adult female is receiving a medication dose of 44 mg. What is the appropriate dose for a 9-month-old infant?

**Calculation:**

Infant or Child Dose = (Age in Months / 150) x Adult Dose

Infant Dose = (9 / 150) x 44

Infant Dose = 0.06 x 44

**Answer:**

Infant Dose = 2.6 mg

### Anion Gap

**Anion gap** calculates the difference between the measured cations and the measured anions in serum, plasma, or urine. It helps in identifying the causes of metabolic acidosis, a condition where the body produces too much acid or the kidneys are not removing enough acid from the body.

Formula:

Anion Gap = Na+ – (Cl- + HCO3-)

**Example:**

A patient has the following data:

- Na+ = 144 mEq/L
- Cl- = 104 mEq/L
- HCO3 = 24 mEq/L

Calculate the anion gap.

**Calculation:**

Anion Gap = Na+ – (Cl- + HCO3-)

Anion Gap = 144 – (104 + 24)

Anion Gap = 144 – 128

**Answer:**

Anion Gap = 16 mEq/L

### Body Surface Area (BSA)

**Body surface area (BSA)** estimates the total surface area of the human body. It’s used in medical practices to calculate dosages for medications and the expected physiological outputs that vary with body size, such as renal clearance.

Formula:

BSA = ((4 x Body Weight) + 7) / (Body Weight + 90)

**Example:**

What is the body surface area of an adult female patient who weighs 153 lbs?

**Calculation:**

First, convert lbs. to kg.

153 lbs / 2.2 = 69.5 kg

Then, plug the numbers into the formula.

BSA = ((4 x Body Weight) + 7) / (Body Weight + 90)

BSA = ((4 x 69.5) + 7) / (69.5 + 90)

BSA = 285 / 159.5

**Answer:**

BSA = 1.79 m2

### Elastance

**Elastance** measures the stiffness of the lung and chest wall system, indicating the effort required to expand the lungs during inhalation. It is the reciprocal of compliance, highlighting the relationship between pressure change and volume change in the respiratory system.

Formula:

Elastance = Change in Pressure / Change in Volume

**Example:**

An adult patient is intubated and receiving mechanical ventilation. Given the following data, calculate the elastance:

- ∆P = 6 cmH2O
- ∆V = 0.7 L

**Calculation:**

Elastance = Change in Pressure / Change in Volume

Elastance = 6 / 0.7

**Answer:**

Elastance = 8.6 cmH2O/L

### Smoking Use Calculation (Pack Years)

The **smoking pack years** formula quantifies the amount of smoking exposure an individual has, combining the number of packs of cigarettes smoked per day with the number of years the person has smoked. It’s used to assess risk factors in conditions like lung cancer and chronic obstructive pulmonary disease (COPD).

Formula:

Pack Years = (Packs Smoked per Day) x (Number of Years Smoked)

**Example:**

A 54-year-old male patient has been smoking 2 packs of cigarettes per day for 27 years. What is his smoking history in pack years?

**Calculation:**

Pack Years = (Packs Smoked per Day) x (Number of Years Smoked)

Pack Years = 2 x 27

**Answer:**

Pack Years = 54

### Suction Catheter Size Estimation

The **suction catheter size estimation** helps determine the appropriate size of a suction catheter for tracheal suctioning based on the internal diameter of the endotracheal or tracheostomy tube. This ensures effective clearance of secretions without harming the airway.

Formula:

Catheter Size = (Internal Diameter / 2) x 3

**Example:**

A 62-year-old male patient with retained secretions is intubated with a size 8 endotracheal tube. What size catheter would you recommend for suctioning?

**Calculation:**

Catheter Size = (Internal Diameter / 2) x 3

Catheter Size = (8 / 2) x 3

**Answer:**

Catheter Size = 12 Fr

### Endotracheal Tube Size Estimation in Children

The **endotracheal tube size estimation in children** helps determine the appropriate size of an endotracheal tube for pediatric patients based on age, ensuring a balance between effective ventilation and minimizing airway injury.

Formula:

Tube Size = (Age + 16) / 4

**Example:**

Intubation is required for a 4-year-old child in the emergency department. What size tube would you recommend?

**Calculation:**

Tube Size = (Age + 16) / 4

Tube Size = 20 / 4

**Answer:**

Tube Size = 5.0 mm

### Celsius to Fahrenheit Temperature Conversion

The **Celsius to Fahrenheit conversion** formula is used to convert temperature measurements from Celsius to Fahrenheit, enabling healthcare professionals to easily switch between temperature scales, which is essential in clinical settings where patient body temperature needs to be accurately monitored and communicated.

Formula:

˚F = (˚C x 1.8) + 32

**Example:**

A temperature of 30˚C is what temperature in Fahrenheit?

**Calculation:**

˚F = (˚C x 1.8) + 32

˚F = (30 x 1.8) + 32

**Answer:**

˚F = 86

### Fahrenheit to Celsius Temperature Conversion

The **Fahrenheit to Celsius conversion** formula converts Fahrenheit temperatures to Celsius. It’s vital in medical practice to maintain consistency in temperature recording and interpretation across various healthcare systems.

Formula:

˚C = (˚F – 32) / 1.8

**Example:**

A temperature of 69˚F is what temperature in Celsius?

**Calculation:**

˚C = (˚F – 32) / 1.8

˚C = (69 – 32) / 1.8

**Answer:**

˚C = 20.6

### Celsius to Kelvin Temperature Conversion

The **Celsius to Kelvin conversion** formula is used to convert temperature measurements from Celsius to Kelvin. While not commonly used in direct patient care, it’s relevant in scientific research and studies related to respiratory therapy and medicine.

Formula:

K = ˚C + 273

**Example:**

A temperature of 33˚C is what temperature in Kelvins?

**Calculation:**

K = ˚C + 273

K = 33 + 273

**Answer:**

K = 306

### Helium/Oxygen Conversion (He/O2)

The **helium/oxygen conversion (He/O2)** formula calculates the flow rates for helium/oxygen mixtures used in respiratory therapies. Such mixtures can reduce airway resistance and improve ventilation in patients with obstructive lung diseases.

Formula:

Actual Flow = Given Flow Rate x Factor

**Example:**

A patient is receiving a 70%Helium/30%Oxygen mixture on a flow rate of 8 L/min. What is the actual flow rate?

**Calculation:**

To complete this calculation, you must remember the Heliox factors:

- 80/20% Mixture = 1.8
- 70/30% Mixture = 1.6

Actual Flow = Given Flow Rate x Factor

Actual Flow = 8 x 1.6

**Answer:**

Actual Flow = 12.8 L/min

### Boyle’s Law

**Boyle’s law** describes how the pressure and volume of a gas are inversely related, assuming a constant temperature and amount of gas. It’s fundamental in understanding how changes in thoracic pressure can affect lung volumes.

Formula:

P1 x V1 = P2 x V2

**Example:**

During a pulmonary function test, a patient inhales deeply, expanding their lung volume from 2 liters to 6 liters. Given an initial lung pressure of 120 mmHg at 2 liters, calculate the new lung pressure at 6 liters, assuming constant temperature and gas amount.

**Calculation:**

P1 x V1 = P2 x V2

P2 = (P1 x V1) / V2

P2 = (120 x 2) / 6

**Answer:**

P2 = 40 mmHg

### Charles’s Law

**Charles’s law** states that the volume of a gas increases with increasing temperature, assuming pressure remains constant, relevant in the study of gas behaviors under different conditions.

Formula:

V1 / T1 = V2 / T2

**Example:**

A sealed container holding a gas is heated from a temperature of 20°C to 80°C. The initial volume of the gas is 2 liters. Assuming the pressure and the amount of gas remain constant throughout the experiment, calculate the final volume of the gas using Charles’s Law.

**Calculation:**

First, convert the temperatures from Celsius to Kelvin by adding 273.

Then, solve for V2 using Charles’s Law.

V1 / T1 = V2 / T2

V2 = V1 x (T2 / T1)

V2 = 2 x (353 / 293)

**Answer:**

V2 = 2.4 L

### Gay-Lussac’s Law

**Gay-Lussac’s law** describes the direct relationship between the pressure of a gas and its temperature, keeping its volume constant. In respiratory therapy, it helps explain the effects of heated gases on pressure within closed systems, such as in certain mechanical ventilation scenarios.

Formula:

P1 / T1 = P2 / T2

**Example:**

A patient is undergoing a respiratory function test where the pressure inside the lungs is initially measured at 100 mmHg at a room temperature of 25°C. The patient then performs an exercise that increases the temperature inside the lungs to 37°C, which is the body’s normal internal temperature. Assuming the volume of the lungs remains constant during the exercise, calculate the new pressure inside the lungs using Gay-Lussac’s Law.

**Calculation:**

First, convert the temperatures from Celsius to Kelvin by adding 273.

Then, solve for P2 using Gay-Lussac’s Law.

P1 / T1 = P2 / T2

P2 = (P1 / T1) x T2

P2 = (100 / 298) x 310

**Answer:**

P2 = 104 mmHg

### LaPlace’s Law

**LaPlace’s law** describes the relationship between the pressure inside a spherical structure (like alveoli), its surface tension, and its radius. This principle is important in understanding the mechanics of alveolar inflation and deflation, particularly the role of surfactants in reducing surface tension and stabilizing alveoli.

Formula:

P = (2 x Surface Tension) / Radius

**Example:**

A patient with a pulmonary condition has a spherical alveolus with a radius of 0.2 cm. The surface tension of the liquid lining the alveolus is estimated to be 0.0375 cmHg/cm. Using LaPlace’s Law, calculate the pressure inside this alveolus required to prevent it from collapsing.

**Calculation:**

P = (2 x Surface Tension) / Radius

P = (2 x 0.0375) / 0.2

**Answer:**

P = 0.375 cmHg

### Total Lung Capacity (TLC)

**Total lung capacity (TLC)** is the maximum amount of air the lungs can hold. It’s the sum of vital capacity and residual volume, and it provides valuable information about lung health and function, particularly in diagnosing restrictive or obstructive lung diseases.

Formulas:

TLC = IRV + VT + ERV + RV

TLC = VC + RV

TLC = IC + FRC

**Example:**

Pulmonary function testing (PFT) was performed on an adult patient with the following results:

- Tidal Volume = 600 mL
- Inspiratory Reserve Volume = 3,000 mL
- Expiratory Reserve Volume = 1,300 mL
- Residual Volume = 1,100 mL
- Vital Capacity = 4,900 mL

What is the total lung capacity (TLC)?

**Calculation:**

TLC = VC + RV

TLC = 4,900 + 1,100

**Answer:**

TLC = 6,000 mL

### Vital Capacity (VC)

**Vital capacity (VC)** represents the total volume of air that can be exhaled after a maximal inhalation. It is a key measurement in respiratory function tests, helping to assess the strength of thoracic muscles and the elasticity of the lungs.

Formulas:

VC = IRV + VT + ERV

VC = IC + ERV

VC = TLC – RV

**Example:**

Pulmonary function testing (PFT) was performed on an adult patient with the following results:

- Tidal Volume = 600 mL
- Inspiratory Reserve Volume = 3,000 mL
- Expiratory Reserve Volume = 1,300 mL
- Residual Volume = 1,100 mL

What is the vital capacity (VC)?

**Calculation:**

VC = IRV + VT + ERV

VC = 3,000 + 600 + 1,300

**Answer:**

VC = 4,900 mL

### Inspiratory Capacity (IC)

**Inspiratory capacity (IC)** is the maximum amount of air that can be inhaled from the end of a normal exhalation. This measure is used to evaluate lung function and detect restrictive lung diseases.

Formulas:

IC = IRV + VT

IC = TLC – FRC

IC = VC – ERV

**Example:**

Pulmonary function testing (PFT) was performed on an adult patient with the following results:

- Tidal Volume = 600 mL
- Inspiratory Reserve Volume = 3,000 mL
- Expiratory Reserve Volume = 1,300 mL
- Residual Volume = 1,100 mL

What is the inspiratory capacity (IC)?

**Calculation:**

IC = IRV + VT

IC = 3,000 + 600

**Answer:**

IC = 3,600 mL

### Functional Residual Capacity (FRC)

Functional residual capacity (FRC) is the volume of air remaining in the lungs after a normal exhalation. It’s important to understand how diseases or conditions affect lung volume.

Formulas:

FRC = ERV + RV

FRC = TLC – IC

**Example:**

Pulmonary function testing (PFT) was performed on an adult patient with the following results:

- Tidal Volume = 600 mL
- Inspiratory Reserve Volume = 3,000 mL
- Expiratory Reserve Volume = 1,300 mL
- Residual Volume = 1,100 mL

What is the functional residual capacity (FRC)?

**Calculation:**

FRC = ERV + RV

FRC = 1,300 + 1,100

**Answer:**

FRC = 2,400 mL

### Time Constant (t)

The time constant (t) is a concept in respiratory physiology that describes how quickly the lungs can fill or empty. It is the product of lung compliance and airway resistance, indicating the efficiency of ventilation.

Formula:

t = Compliance x Resistance

**Example:**

A 63-year-old male patient who is receiving ventilatory support has a compliance of 0.09 L/cmH2O and a total resistance of 4 cmH2O/L/sec. Calculate the time constant:

**Calculation:**

t = Compliance x Resistance

t = 0.09 x 4

**Answer:**

t = 0.36 seconds

### Ideal Body Weight (IBW)

**Ideal body weight (IBW)** estimates the optimal weight for calculating medical dosages and the settings for mechanical ventilation based on height, providing a standardized measure to guide clinical decisions.

Formula:

IBW = 50 kg + (2 x Number of Inches over 5 feet)

**Example:**

A 5’7″ female patient is receiving positive pressure ventilatory support. Calculate her ideal body weight.

**Calculation:**

IBW = 50 kg + (2 x Number of Inches over 5 feet)

IBW = 50 + (2 x 7)

IBW = 50 + 14

**Answer:**

IBW = 64 kg

### Tidal Volume (VT)

**Tidal volume (VT)** is the amount of air that is inhaled or exhaled during a normal breath. It’s a fundamental measurement in both spontaneous breathing and mechanical ventilation, reflecting the volume of air moved into or out of the lungs with each breath.

Formula:

VT = Flow Rate x Inspiratory Time

**Example:**

An adult patient who is intubated and receiving ventilatory support has a flow rate of 7 L/min and an inspiratory time of 0.7 seconds. What is the tidal volume?

**Calculation:**

First, you must convert the flow rate from L/min to mL/sec.

7 L/min = 7,000 mL/60 seconds

7,000 / 60 = 116.7 mL/sec

Then, plug the numbers into the formula.

VT = Flow Rate x Inspiratory Time

VT = 116.7 x 0.7

**Answer:**

VT = 8.2 mL

### Exhaled Tidal Volume (VT)

**Exhaled tidal volume (VT)** refers to the volume of air exhaled during a normal breath. It’s important for monitoring ventilation in mechanically ventilated patients to ensure that the set tidal volume closely matches the volume actually received by the lungs.

Formula:

VT = Minute Ventilation / Frequency

**Example:**

An adult patient is receiving mechanical ventilation with the following data:

- Rate = 12/min
- Minute Ventilation = 7.2 L/min

What is the exhaled tidal volume?

**Calculation:**

VT = Minute Ventilation / Frequency

VT = 7.2 / 12

VT = 0.6 L

Multiply by 1,000 to convert L to mL.

**Answer:**

VT = 600 mL

### Corrected Tidal Volume (VT)

**Corrected tidal volume (VT)** accounts for the compression of gas within the ventilator circuit, especially in humidified systems. The corrected tidal volume provides a more accurate representation of the actual volume of air that reaches the patient’s lungs.

Formula:

VT = Expired Tidal Volume – Tube Volume

**Example:**

An adult patient is receiving mechanical ventilatory support with the following data:

- Expired tidal volume = 600 mL
- PIP = 30 cmH2O
- PEEP = 5 cmH2O
- Tubing compression factor = 3 mL/cmH2O

What is the corrected tidal volume?

**Calculation:**

First, calculate the tube volume.

Tube Volume = Pressure Change x 3 mL/cmH2O

Tube Volume = (30 – 5) x 3

Tube Volume = 75 mL

Then, plug the numbers into the formula.

VT = Expired Tidal Volume – Tube Volume

VT = 600 – 75

**Answer:**

VT = 525 mL

### Pressure Support Ventilator Setting (PSV)

The **pressure support ventilator setting (PSV)** is an assisted breathing mode where the ventilator provides a preset level of pressure support to augment the patient’s spontaneous breaths. This setting reduces the work of breathing by helping to overcome the resistance of the breathing circuit and the airways.

Formula:

PSV = ((Peak Pressure – Plateau Pressure) / Set Flow) x Peak Flow

**Example:**

An adult patient is receiving mechanical ventilation in the SIMV mode with the following settings:

- VT = 400 mL
- Rate = 12 breaths/min
- PIP = 40 cmH2O
- Pplat = 20 cmH2O
- Inspiratory flow = 60 L/min or (1 L/sec)

The patient is breathing spontaneously with a spontaneous rate of 12 breaths/min and a spontaneous peak inspiratory flow of 30 L/min (0.5 L/sec). What level of PSV is needed to overcome the imposed work of breathing?

**Calculation:**

PSV = ((Peak Pressure – Plateau Pressure) / Set Flow) x Peak Flow

PSV = ((40 – 20) / 1) x 0.5

**Answer:**

PSV = 10 cmH2O

### Rapid Shallow Breathing Index (RSBI)

**Rapid shallow breathing index (RSBI)** is calculated by dividing the respiratory rate by the tidal volume (in liters), providing a measure to predict the success of weaning a patient from mechanical ventilation. A lower RSBI indicates a higher likelihood of weaning success.

Formula:

RSBI = Rate / Tidal Volume

**Example:**

An adult patient on the ventilator has a rate of 12/min and a tidal volume of 500 mL. What is the rapid shallow breathing index (RSBI)?

**Calculation:**

Divide by 1,000 to convert mL to L.

RSBI = Rate / Tidal Volume

RSBI = 12 / 0.5

**Answer:**

RSBI = 24 breaths/min/L

### Minimum Flow Rate in Mechanical Ventilation

**Minimum flow rate in mechanical ventilation** is the lowest flow rate that can be set on a ventilator to ensure adequate gas exchange and patient comfort during mechanical ventilation. It must be adjusted based on the patient’s ventilatory demands and the clinical situation.

Formula:

Flow Rate = Minute Ventilation x I:E Ratio Sum of Parts

**Example:**

An adult patient is receiving mechanical ventilation with the following settings:

- Tidal Volume = 550 mL
- Rate = 12/min
- I:E ratio = 1:3

What is the required minimum flow rate?

**Calculation:**

First, calculate the Minute Ventilation.

VE = Rate x Tidal Volume

VE = 12 x 550 mL

VE = 6,600 mL

Divide by 1,000 to convert mL to L.

VE = 6.6 L

Then, use the formula to calculate the Minimum Flow Rate.

Flow Rate = Minute Ventilation x I:E Ratio Sum of Parts

Flow Rate = 6.6 x (3 + 1)

**Answer:**

Flow Rate = 26.4 L/min

### Winters’ Formula

Winters’ formula is used to predict the expected compensatory response (in terms of PaCO2) to a primary metabolic acid-base disturbance. It helps clinicians assess whether the respiratory compensation is appropriate or if there is a mixed acid-base disorder.

Formula:

Expected PaCO2 = (1.5 x HCO3-) + 8 ± 2

**Example:**

An adult patient has the following arterial blood gas results:

- pH = 7.34
- PaCO2 = 28 mmHg
- HCO3- = 14 mEq/L

Calculate the expected PaCO2 range using Winters’ formula.

**Calculation:**

Expected PaCO2 = (1.5 x HCO3-) + 8 ± 2

Expected PaCO2 = (1.5 x 14) + 8 ± 2

Expected PaCO2 = 29 ± 2

**Answer:**

Expected PaCO2 Range = 27–31 mmHg

## Final Thoughts

The integral role of **mathematics in respiratory therapy** cannot be overstated. From diagnostic evaluations to therapeutic interventions, the ability to perform calculations using specific formulas is crucial for every respiratory therapist (and student).

While mastering these equations may seem daunting at first, it’s clear that with practice, these calculations are not only manageable but essential for learning and understanding the concepts and principles of respiratory care.

Written by:

**John Landry, BS, RRT**

John Landry is a registered respiratory therapist from Memphis, TN, and has a bachelor's degree in kinesiology. He enjoys using evidence-based research to help others breathe easier and live a healthier life.

### References

- Maury, Bertrand. The Respiratory System in Equations (MS&A). 2013th ed., Springer, 2013.
- Chang, David. Respiratory Care Calculations. 4th ed., rtexam, 2018.
- Chang, David. Clinical Application of Mechanical Ventilation. 4th ed., Cengage Learning, 2013.
- Faarc, Kacmarek Robert PhD Rrt, et al. Egan’s Fundamentals of Respiratory Care. 11th ed., Mosby, 2016.
- Gertler R. Respiratory Mechanics. Anesthesiol Clin. 2021.