Commonly Used Formulas and Calculations

Introduction

This appendix presents a list of calculations commonly used by emergency physicians. It is not all-inclusive and purposely does not include decision tools or scales that do not require algebraic calculations. Many of these equations can be found in online calculators on websites such as https://www.mdcalc.com or http://reference.medscape.com/guide/medical-calculators .

English-to-Metric Conversions

Patients frequently express common figures such as weight, temperature, and volume in standard measurements. Please see Table A.1 for conversion factors and simple examples.

TABLE A.1

Standard-to-Metric Calculations

Fahrenheit to Celsius °C = (°F − 32)/1.8	A patient's temperature is 100.4°F (100.4°F − 32)/1.8 = 38°C
Pounds to kilograms kg = lbs/2.2	A patient reports his weight to be 154 lb 154 lb/2.2 = 70 kg
Inches to centimeters cm = inches × 2.54	The patient's height is 72 inches 72 inches × 2.54 = 182.9 cm
Fluid ounces to milliliters mL = oz × 30	The child drinks 4 oz of formula at a sitting 4 oz × 30 = 120 mL

Mean Arterial Pressure

Calculation of mean arterial pressure (MAP) provides a weighted average of systolic blood pressure (SBP) and diastolic blood pressure (DBP). It is a determination of tissue perfusion pressure and is normally 70 to 100 mm Hg in adults. To determine MAP:

MAP=[SBP+(2×DBP)]/3 $MAP = [SBP + (2 \times DBP)] / 3$

Example: A hypertensive emergency is diagnosed in an elderly, hypertensive patient. Current recommendations are to reduce MAP by 10% to 20% in the first hour. His blood pressure is 240/120. To calculate the current MAP,

MAP=[240+(2×120)]/3=160mm Hg $MAP = [240 + (2 \times 120)] / 3 = 160 mm Hg$

QT and QTC Intervals

The QT interval on the electrocardiogram (ECG) represents the period of ventricular electrical activity from activation to repolarization. The most important determinant of the QT interval is the heart rate. As the heart rate increases, the QT interval shortens. Many ECG machines calculate this; however, the presence of U waves or other ECG abnormalities can result in inappropriate readings. To calculate the rate-corrected QT interval (QTc) using Bazett's formula, divide the QT interval by the square root of the R-R interval (the interval between the R wave on two consecutive QRS complexes). The R-R interval may be measured from the ECG or calculated as 60 divided by the heart rate (in beats/min). The interval is represented in seconds or milliseconds:

$QTc = QT / \sqrt{(R-R)}$

The QTc is normally less than 0.46 second in men and 0.44 second in women. This may be calculated online at: http://www.medical-calculator.nl/calculator/QTc/ .

The list of drugs that can cause QTc prolongation is quite lengthy. The clinical significance of QTc prolongation is often unclear. The following Internet websites offer up-to-date information on this ever-changing topic:

https://crediblemeds.org/
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1767957/ .

Table A.2 shows the normal range of the QT interval in adults.

TABLE A.2

Normal Range of the QT Interval in Adults

HEART RATE (beats/min)	NORMAL QT RANGE (sec)
40	0.42–0.53
50	0.37–0.48
60	0.34–0.44
70	0.31–0.41
80	0.29–0.38
90	0.28–0.36
100	0.27–0.34
110	0.25–0.32
120	0.24–0.31
130	0.23–0.30
140	0.22–0.29
150	0.21–0.28

Example: A 21-year-old man ingested a large quantity of amitriptyline (tricyclic antidepressant) tablets. His ECG revealed a QT interval of 0.37 second and a heart rate of 120 beats/min. The QTc is calculated as:

$R-R interval = 60 / 120 = 0.50$

$QTc = 0.37 / \sqrt{0.5} = 0.523 = 523 msec$

The patient's QTc is significantly prolonged for his heart rate and indicates significant cardiac effects from overdose of a tricyclic antidepressant.

Predicted Peak Expiratory Flow Rate

The peak expiratory flow rate (PEFR), measured in liters per minute, is a useful means of assessing airway obstruction. It is measured by having a patient exhale maximally through a peak-flow meter. Normal values range from 350 to 600 L/min. Comparison between the initial and posttreatment PEFR in patients with exacerbations of asthma helps determine the degree of severity. Patients with an initial PEFR of less than 20% of predicted or with a subsequent value of less than 60% of predicted after initial therapy may require further evaluation, treatment, or both. Many patients monitor PEFR on themselves and are able to state a personal best, which is the preferred standard for that individual. Other patients may be monitored with an estimated PEFR. Estimations are based primarily on a patient's gender, age, and height. Although graphs and tables are available to provide values across a range of ages and heights, the PEFR can also be approximated by using the following formulas:

$Adults : PEFR = 13 \times (Height (in) - 40) + 110$

$\begin{matrix} Female children / adolescents : \\ PEFR = Height (m) \times 5.5 - Age \times 0.03 - 1.11 \end{matrix}$

$\begin{matrix} Male children / adolescents : \\ PEFR = Height (m) \times 6.14 - Age \times 10.043 + 0.15 \end{matrix}$

These may also be found online in many places including: http://www.mdcalc.com/estimatedexpected-peak-expiratory-flow-peak-flow/ .

Endotracheal Tube Size

Adults

Select the largest-diameter endotracheal tube (ETT) that can be tolerated for adults. A 7.5-mm cuffed ETT is well tolerated by most adult female patients. An 8.0-mm cuffed ETT is well tolerated by most adult male patients.

Pediatrics

An uncuffed ETT should be used for children younger than 8 years. A number of techniques are available for estimating the appropriate size of ETT in children. Commonly used formulas for estimating tube size and depth of insertion use age in years and are as follows:

$ETT size = (Age + 16) / 4 or (Age / 4) + 4$

To estimate depth of insertion for a child older than 2 years:

$Depth of insertion = 3 \times Internal diameter of the ETT$

Ventilator Settings

The recommended initial ventilator settings follow. Adjustments in these ventilator settings may be made according to the patient's clinical situation:

V t is often based on ideal body weight, and there is greater recognition that a lower V t may be beneficial, such as 5 to 10 mL/kg. (Adjust V _T to limit inflation or plateau pressures to 30 cm H ₂ O or lower. Lower tidal volumes are recommended to limit ventilator-induced lung injury.)

$\begin{matrix} Rate = 10 to 12 breaths / min for adults, 16 to 20 breaths / min \\ for children, and 20 to 30 breaths / min for infants . \end{matrix}$

$\begin{matrix} Inspiratory-to-expiratory (I / E) ratio = 1 : 2 . To allow \\ complete exhalation, the I / E ratio should be at least 1 : 2 . \end{matrix}$

V̇ _E will vary with the patient's mass and disease process; the normal adult range is approximately 7 to 10 L/min.

Example: A 6-year-old girl with asthma has respiratory distress and altered mental status and requires endotracheal intubation. Her weight is 20 kg. To prepare for intubation and mechanical ventilation, use the following equipment and settings:

$ETT size (mm) : (6 + 16) / 4 = 5.5 -mm ETT (uncuffed) .$

$Depth of insertion : 3 \times 5.5 = 16.5 cm .$

$Respiratory rate = 16 breaths / min .$

$I / E ratio = 1 : 2 .$

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