CardioMath Equations Info

Dr. Chi-Ming Chow MDCM MSc FRCPC

St. Michael’s Hospital, University of Toronto
Last updated: 1st August, 2005

(I) Echo Equations

1. Aortic Valve Dimensionless Index

Also known as the Doppler Velocity Index (DVI). It is a useful index for
assessing and following aortic prosthetic valve function. It is a ratio of the
subvalvular velocity obtained by pulsed-wave Doppler and the maximum
velocity obtained by continuous-wave Doppler across the prosthetic valve.
This formula allows for autocorrection for subvalvular velocity especially in
high flow state. It is useful as a screening test for valve obstruction
particularly when the valve size is not known. A ratio <0.23 indicates
significant prosthetic stenosis.

V1
DI =
V2
V1 Subvalvular velocity [m/s]
V2 Maximum velocity across the valve [m/s]

DI Dimensionless Index [no unit]

! Chafizadeh ER, Zoghbi WA: Doppler echocardiographic assessment of the
St. Jude Medical prosthetic valve in the aortic position using the continuity
equation. Circulation 1991; 83:213-223.

2. Aortic Valve Area (Continuity Equation using VTI)

Aortic valve area can be calculated by using the principle of conservation of

mass. “What comes in must go out”.

VTI1 Subvalvular velocity time integral [cm]

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 VTI2 Maximum velocity time integral across the valve [cm]
LVOT Left ventricular outflow tract diameter [cm]

AVA Aortic Valve Area [cm2]

" * (LVOT /2) 2 *VTI1

AVA =
VTI2
Definition of Aortic Stenosis Severity

Normal 3.0-4.0 cm2

Mild: >1.3 cm2
Moderate: 0.75-1.3 cm2
! Severe: ≤0.75 cm2

Zoghbi WA, et al. Accurate noninvasive quantification of stenotic aortic valve

area by Doppler echocardiography. Circulation 1986; 73:452-459.

Grayburn PA, et al. Pivotal role of aortic valve area calculation by the
continuity equation for Doppler assessment of aortic stenosis in patients with
combined aortic stenosis and regurgitation. Am J. Cardiol 1988; 61:376-381.

3. Aortic Valve Area (Continuity Equation using VMax)

Aortic valve area can be calculated by using the principle of conservation of

mass. “What comes in must go out”. Although VTI is a more accurate
method, using maximum velocities is simpler and can generate a very close
result.

" * (LVOT /2) 2 *V1

AVA =
V2
V1 Subvalvular velocity [m/s]
V2 Maximum velocity across the valve [m/s]
LVOT Left ventricular outflow tract diameter [mm]

AVA Aortic Valve Area [cm2]

!
Chi-Ming Chow MD www.CardioMath.com Page 2 of 35
Definition of Aortic Stenosis Severity

Normal 3.0-4.0 cm2

Mild: >1.3 cm2
Moderate: 0.75-1.3 cm2
Severe: ≤0.75 cm2

Zoghbi WA, et al. Accurate noninvasive quantification of stenotic aortic valve

area by Doppler echocardiography. Circulation 1986; 73:452-459.

Grayburn PA, et al. Pivotal role of aortic valve area calculation by the
continuity equation for Doppler assessment of aortic stenosis in patients with
combined aortic stenosis and regurgitation. Am J. Cardiol 1988; 61:376-381.

4. dP/dt Isovolumic Phase Index of LV Contractility

The LV contractility dP/dt can be estimated by using time interval between 1

and 3m/sec on MR velocity CW spectrum during isovolumetric contraction
i.e. before aortic valve opens when there is no significant change in LA
pressure.

dP/dt = 32/T

T Time (1 to 3 m/sec MR) [s]

dP/dt dP/dt [mmHg/s]

Normal ≥ 1200 mmHg/s

Barigiggia GS et al. A. new method for estimating left ventricular dP/dt by

continuous wave Doppler echocardiography: Validation studies at cardiac
catheterization. Circulation 1989; 80: 1287-1292.

Chung NS et al. Measurement of left ventricular dP/dt by simultaneous

Doppler echocardiography and cardiac catheterization. J A, Soc
Echocardiogr 1992; 5:147-152.

5. dP/dt Isovolumic Phase Index of RV Contractility

The RV contractility dP/dt can be estimated by using time interval between 1

and 2m/sec on TR velocity CW spectrum during isovolumetric contraction

Chi-Ming Chow MD www.CardioMath.com Page 3 of 35

i.e. before pulmonic valve opens when there is no significant change in RA
pressure.

dP/dt (mmHg/sec) = 12/T

T Time (1 to 2 m/sec TR) [s]

dP/dt dP/dt [mmHg/s]

Oh JK et al. The Echo Manual. 2nd ed. p. 70, 1999.

6. LAP Estimation by Mitral Regurgitation Jet

Left atrial pressure (LAP) can be estimated by measuring the systolic blood
pressure and the maximum mitral regurgitation velocity by spectral Doppler
provided that there is no significant gradient across the aortic valve.

LAP = SBP - 4 (MR VMax)2

MR Vmax Maximum mitral regurgitation velocity [m/sec]

SBP Systolic blood pressure [mmHg]

LAP Left Atrial Pressure [mmHg]

Gorcscan J et al. Noninvasive estimation of left atrial pressure in patients

with congestive heart failure and mitral regurgitation by Doppler
echocardiography. Am. Heart J., 121:858, 1991.

Nishimura RA et al. Determination of left-sided pressure gradients by

utilizing Doppler aortic and mitral regurgitant signals: validation by
simultaneous dual catheter and Doppler studies. JACC 11:317, 1988.

7. LV Ejection Fraction

Modified Quinones Method

The Left Ventricular Ejection Fraction (LV EF) can be determined without
planimetry by measuring 8 averaged LV internal dimensions at different
levels of the LV in the parasternal long-axis, apical 4 chamber, and long-
axis views at end-diastole (LVEDD) and end-systole (LVESD) adjusting

Chi-Ming Chow MD www.CardioMath.com Page 4 of 35

 for contraction in long-axis. Limited by important wall motion
abnormalities and distorted ventricles, e.g. focal aneurysms.

LVEDD2 # LVESD2
2
%"D =
LVEDD2
LVEF = (%"D 2 ) + [(1# %"D 2 )(%"L)]

LVEDD Averaged LV end-diastolic dimension [mm]

LVESD Averaged LV end-systolic dimension [mm]
%ΔL Correction for apical contraction
! 15% Normal apex
5% Hypokinetic apex
0% Akinetic apex
-5% Slightly dyskinetic apex
-10% Frankly dyskinectic apex

LVEF Ejection Fraction [%]

Normal > 60%

Quinones MA et al. A new, simplified and accurate method for

determining ejection fraction with two-dimensional echocardiography.
Circulation 64:744, 1981.

Simplified Method

An even simpler adaptation uses the commonly measured end-diastolic

and end-systolic diameters of the LV in the parasternal long axis view,
with a correction of the apex as follows …

LVEDD 2 " LVESD 2

LVEF = *100% + K
LVEDD 2
LVEDD LV end-diastolic dimension [mm]
LVESD LV end-systolic dimension [mm]
!
Chi-Ming Chow MD www.CardioMath.com Page 5 of 35
 K Correction for apical contraction
10% Normal apex
5% Hypokinetic apex
0% Akinetic apex
-5% Dyskinetic apex
-10% Apical aneurysm

EF Ejection Fraction [%]

Normal > 60%

Weyman AE.Principles and practice of echocardiography. 2ed.1994 p.605-6

8. LV Fractional Shortening

Percent change in LV cavity dimensions at the base with systolic

contraction. Limited by significant apical wall motion abnormalities.

LVEDD - LVESD
FS = *100%
LVEDD
LVEDD LV end-diastolic dimension [mm]
LVESD LV end-systolic dimension [mm]

FS
!
Fractional Shortening [%]

Normal >25%

Lewis RP et al. Relationship between changes in the left ventricular

dimension and ejection fraction in man. Circulation 44:548; 1971.

9. LV Mass and LV Mass Index

Left ventricular mass and left ventricular mass indexed to body surface
area estimated by LV cavity dimension and wall thickness at end-diastole.

LV Mass(g) = 1.04[([LVEDD + IVSd + PWd]3 " LVEDD3 )] "13.6

!
Chi-Ming Chow MD www.CardioMath.com Page 6 of 35
 LVEDD LV end-diastolic dimension [mm]
PWd Posterior wall thickness at end-diastole [mm]
IVSd Interventricular septal thickness at end-diastole [mm]
1.04 Specific gravity of the myocardium [g/cm3]

H Height
W Weight

LV MI LV Mass Indexed to Body Surface Area [g/m2]

Upper limit Men 134 g/m2

Women 110 g/m2

Devereux R et al. Echocardiographic determination of left ventricular

mass in man. Circulation 55:613,1977.

Hammond IW et al. The prevalence and correlates of echocardiographic

left ventricular hypertrophy among employed patients with uncomplicated
hypertension. J Am Coll Cardiol 7:639,1986.

10. Mitral Valve Area (Pressure Half-Time)

Mitral valve area is inversely proportional to the pressure half time (PHT).
PHT is the time interval for the peak pressure gradient to reach its half level.
Underestimation occurs when PHT is shortened by concomitant
significant aortic insufficiency, decreased ventricular compliance,
immediately after mitral balloon valvuloplasty, and atrial septal defect.

220
MVA =
PHT
PHT Pressure Half Time [ms]

MVA Mitral Valve Area [cm2]

!
Definition of Mitral Stenosis Severity

Normal 4.0-6.0 cm2

Mild: 1.6-2.0 cm2
Moderate: 1.1-1.5 cm2

Chi-Ming Chow MD www.CardioMath.com Page 7 of 35

Severe: ≤1.0 cm2

Hatle L, Angelsen B: Doppler Ultrasound in Cardiology: Physical

Principles and Clinical Applications. Philadelphia, Lea & Febiger, 1985.

11. Mitral Valvuloplasty Score (MGH)

In patients undergoing mitral valvuloplasty for mitral stenosis, an
echocardiographic scoring system based on 1) leaftlet mobility, 2) valve
thickening, 3) calcification, 4) subvalvular thickening can be used to
predict the procedural outcome. Each item is graded from 1 (normal) to 4
which yield a score from 4 to 16. A score of 8 or less predicts a more
favorable outcome than those with a higher score. However, a score higher
than 8 does not exclude a patient from having a mitral valvuloplasty.
Commissural calcification or fusion is another important predictor for
poor outcome after mitral valvuloplasty.

Leaflet Mobility
1 Highly mobile
2 Reduced mobility
3 Basal leaflet motion only
4 Minimal motion

Valve thickening
1 Near normal (4-5mm)
2 Thickened tips
3 Entire leaflet thickened (5-8mm)
4 Marked leaflet thickening (>8-10 mm)

Calcification
1 Single area of brightness
2 Scattered areas at leaflet margins
3 Brightness extends to mid leaflets
4 Extensive leaflet brightness

Subvalvular thickening
1 Minimal chordal thickening
2 Chordal thickening up to 1/3
3 Distal third of chordae thickening
4 Extensive thickening to pap muscle

Chi-Ming Chow MD www.CardioMath.com Page 8 of 35

Mitral Valve Score = Leaflet Mobility + Valve Thickening +
Calcification + Subvalvular Thickening

Wilkins GT, Weyman AE, Abascal VM, Block PC, Palacios IF.
Percutaneous balloon dilatation of the mitral valve: analysis of
echocardiographic variables related to outcome and the mechanism of
dilatation. Br Heart J. 1988 Oct; 60(4):299-308.

Abascal VM, Wilkins GT, Choong CY, Thomas JD, Palacios IF, Block PC,
Weyman AE. Echocardiographic evaluation of mitral valve structure and
function in patients followed for at least 6 months after percutaneous
balloon mitral valvuloplasty. J Am Coll Cardiol 1988; 12:606-615

12. Modified Bernoulli Equation

Simplified formula for converting velocity difference obtained by spectral

Doppler to instantaneous pressure gradient. Proximal velocity, V1 should
be taken into account when > 1m/s, otherwise there is no important effect
on the result by using the distal velocity, V2 alone.

∆P = P2-P1 = 4(V22 – V12)

V1 Proximal velocity [m/s]

V2 Distal velocity [m/s]

∆P Instantaneous pressure gradient [mmHg]

Requarth JA. In vitro verification of Doppler prediction of transvalve

pressure gradient and orifice area in stenosis. Am J Cardiol
1984;53(9):1369-73.

13. LV Myocardial Performance Index (LV MPI)

Also known as the Tei index. It is an index that incorporates both systolic
and diastolic time intervals in expressing global systolic and diastolic
ventricular function. Systolic dysfunction prolongs prejection (isovolumic
contraction time, IVCT) and a shortening of the ejection time (ET). Both
systolic and diastolic dysfunction result in abnormality in myocardial
relaxation which prolongs the relaxation period (isovolumic relaxation

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 time, IVRT).

IVCT + IVRT MCOT " LVET

LV MPI = =
LVET LVET
MCOT Mitral Valve Closure to Opening Time [ms]
LVET LV Ejection time [ms]

LV MPI LV Myocardial Performance Index [no unit]

!
Normal 0.39 ± 0.05
Dilated cardiomyopathy 0.59 ± 0.10

Tei C, Ling LH, Hodge DO, et al. New index of combined systolic and
diastolic myocardial performance: a simple and reproducible measure of
cardiac function - a study in normals and dilated cardiomyopathy. J
Cardiol 1995;26:357-366.

Oh JK et al. The Echo Manual. 2nd ed. p. 55, 1999.

Chi-Ming Chow MD www.CardioMath.com Page 10 of 35

 14. RV Myocardial Performance Index (RV MPI)
Also known as the Tei index. It is an index that incorporates both systolic
and diastolic time intervals in expressing global systolic and diastolic
ventricular function. It has been shown that it is a good predictor for
discriminating patients with pulmonary hypertension from normal.

IVCT + IVRT TCOT " RVET

RV MPI = =
RVET RVET
TCOT Tricuspid Valve Closure to Opening Time [ms]
RVET RV Ejection time [ms]

RV MPI Myocardial Performance Index [no unit]

!
Normal 0.28 ± 0.04
Pulmonary Hypertension 0.93 ± 0.34

Tei C, Dujardin KS, Hodge DO, et al. Doppler echocardiographic index

for assessment of global right ventricular function. J Am Soc
Echocardiogt 1996;9:838-847.

Oh JK et al. The Echo Manual. 2nd ed. p. 55, 1999.

15. PCWP by E/Ea (Nagueh Formula)

Ea, the early diastolic velocity of mitral annulus as obtained by tissue
Doppler imaging (TDI) of the lateral mitral annulus behaves as a pre-load-
independent index of LV relaxation. Mitral inflow E velocity as obtained
by pulsed-wave (PW) Doppler when corrected for the influence of
relaxation by using E/Ea ratio correlates well to the mean pulmonary
capillary wedge pressure (PCWP) as obtained by simultaneous catheter
measurements.

E/Ea ratio < 8 is very specific for a PCWP < 15mmHg while a ratio > 15
is very specific for elevated pressures > 15mmHg. Between 8 and15, there
is a gray zone with overlapping of values for PCWP.

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 E
PCWP = 1.24 * + 1.9
Ea
E Mitral inflow E velocity [m/s]
Ea Early diastolic velocity of mitral annulus [m/s]

!
E/Ea E/Ea ratio [no unit]
PCWP Mean pulmonary capillary wedge pressure [mmHg]

Nagueh SF, Middleton KJ, Kopelen HA, Zoghbi WA, Quinones MA: Doppler tissue
imaging: a noninvasive technique for evaluation of left ventricular relaxation and
estimation of filling pressures. J Am Coll Cardiol 1997;30:1527-1533

16. PISA Method in Quantitating Mitral Regurgitation

Quantification of mitral regurgitation using the principle of conservation of

mass by analyzing the Proximal Isovelocity hemispheric Surface Area of the
flow convergence on the ventricular side.

VFR = 2πr2 * Vr
VFR
ERO =
VMax
RVol = ERO * VTI

2πr2 Proximal isovelocity hemispheric surface area at

! a radial distance r from the orifice
Vr Aliasing velocity at the radial distance r [cm/s]
VMax Peak velocity of the mitral regurgitant jet [m/s]
VTI VTI of the mitral regurgitant jet [cm]

VFR Volume Flow Rate [mL/s]

ERO Effective Regurgitant Orifice [mm2]
RVol Regurgitant Volume [mL]

Mitral Regurgitation Grade of Severity

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 ERO [mm2] RVol [mL]
Mild <20 <30
Mild-Moderate 20-29 30-44
Moderate-Severe 30-39 45-59
Severe ≥40 ≥60

Enriquez-Sarano M et al. Effective mitral regurgitation orifice area: clinical

use and pitfalls of the proximal isovelocity surface area method. JACC 1995;
23(3):703-9.

Utsonomiya T et al. Doppler color flow “proximal isovelocity surface area”

method for estimating volume flow rate: effect of orifice shape and machine
factors. JACC 1991;17:1103.

Zoghbi WA et al. Recommendations for evaluation of the severity of native

valvular regurgitation with two-dimensional Doppler Echocardiography. J
Am Soc Echocardiogr 2003; 16(7):777-799.

17. PISA Method in Quantitating Mitral Stenosis

Estimation of the mitral valve area using the principle of conservation of

mass by analyzing the Proximal Isovelocity hemispheric Surface Area of the
flow convergence on the atrial side. Due to funnel shape of the proximal
convergence, it is important to take into account the angle correction, α when
the aliasing velocity (Vr<40 cm/s). If the aliasing velocity is high (Vr>40
cm/s), it is not as important since the PISA approximates a hemisphere.

Vr #o
2
MVA = 2"r * *
Vmax 180 0
2πr2 Proximal isovelocity hemispheric surface area at
a radial distance r from the orifice
Vr Aliasing velocity at the radial distance r [cm/s]
Vmax Peak mitral stenosis velocity by CW [m/s]
! α Angle between two mitral leaflets on the atrial side [degree]

MVA Mitral valve area [cm2]

Chi-Ming Chow MD www.CardioMath.com Page 13 of 35

Definition of Mitral Stenosis Severity

Normal 4.0-6.0 cm2

Mild: 1.6-2.0 cm2
Moderate: 1.1-1.5 cm2
Severe: ≤1.0 cm2

Rodriguez L. et al. Validation of the proximal flow convergence method.

calculation of orifice area in patients with mitral stenosis. Circulation. 1993
Sep; 88(3):1157-65.

18. Pulmonary-Systemic Flow Ratio (Qp/Qs)

Qp/Qs can be estimated by using 2-D echo and spectral Doppler

measurements in patients who have intra- or extra-cardiac shunts e.g. atrial or
ventricular septal defects.

LVOT Left ventricular outflow tract diameter [mm]

LVOT VTI LVOT subvalvular velocity time integral [cm]
RVOT Right ventricular outflow tract diameter [mm]
RVOT VTI RVOT subvalvular velocity time integral [cm]

Qp/Qs Pulmonary-Systemic Shunt Ratio

Qp = RVOT VTI * π (RVOT/2) 2

Qs = LVOT VTI * π (LVOT/2) 2

Qp/Qs Ratio = Qp/Qs

Sanders SP, et al. Measurement of systemic and pulmonary blood flow and
Qp/Qs ratio using Doppler and two-dimensional echocardiography. Am J.
Cardiol, 51:952, 1983.

19. RVSP estimation by Tricuspid Regurgitation Jet

Right ventricular systolic pressure can be estimated by measuring the TR jet

maximum velocity by spectral Doppler. The RA pressure can be estimated by
the respiratory collapsibility of the inferior vena cava. If there is no
significant stenosis at the right ventricular outflow tract, or the pulmonic
valve, the RVSP is equivalent to the pulmonary artery systolic pressure.

Chi-Ming Chow MD www.CardioMath.com Page 14 of 35

RVSP = 4(TR Vmax)2 + RA pressure

TR Vmax TR Max Jet Velocity [m/sec]

RA pressure Choose between 10 or 15 mmHg, or other [mmHg]

RVSP Right ventricular systolic pressure [mmHg]

Yock PG et al. Noninvasive estimation of right ventricular systolic pressure

by Doppler ultrasound in patients with tricuspid regurgitation. Circulation.
1984; 70(4):657-62.

20. Stroke Volume, Cardiac Output

The Doppler VTI method in estimating stroke volume and cardiac output
correlates well with results of concurrent thermodilution cardiac output
determinations in patients without significant left-sided valvular
regurgitation.

SV = π (LVOT/2) 2 *VTI1
CO = SV * HR / 1000

LVOT Left ventricular outflow tract diameter [mm]

VTI1 LVOT subvalvular velocity time integral [cm]
HR Heart rate [bpm]

SV Stroke Volume [mL]

CO Cardiac Output [L/min]

Huntsman LL, et al. Noninvasive Doppler determination of cardiac output in

man-clinical validation. Circulation 1983; 106:1057-1065.

Chi-Ming Chow MD www.CardioMath.com Page 15 of 35

 (II) EKG

1. Heart Rate & Cycle Length Calculation

Converts between heart rate in beats per minute to cycle length in msec
and vice versa.
60000
Cycle Length =
Heart Rate
Heart Rate[bpm]
Cycle Length[msec]

! 2. QTc (Bazett Formula)

Also known as the Bazett formula which allows adjustment of QT interval

for heart rate.
QT
QTc =
RR
QT QT Interval [msec]
RR RR Interval [msec]

! QTc QT Corrected Interval[msec]

Upper limit: Man 370 msec

Woman 410 msec

Most clinical studies does not consider QT interval

abnormally long until it is more than 440 msec.

Bazett HC. An analysis of the time-relations of electrocardiograms. Heart

7:353, 1920.

Chi-Ming Chow MD www.CardioMath.com Page 16 of 35

(III) Epidemiology & Biostatistics Equations
1. Risks & Numbers Need to Treat
Relative Risk (RR)
The ratio of the risk for an adverse outcome among those receiving the treatment of
interest compared with that among those not receiving it.

Relative Risk Reduction (RRR)

The extent to which treatment reduces the risk in comparison with patients not
receiving the treatment of interest.

Absolute Risk Reduction (ARR)

The absolute difference in rates of adverse events in treated and comparison patients.

Number Needed to Treat (NTT)

Number of patients one needs to treat to prevent 1 adverse outcome

RR = Risk1 / Risk2
RRR = 1 - RR
ARR = Risk1 - Risk2
NTT = 1 / ARR

Risk 1 Risk of Treatment Outcome[%]

Risk 2 Risk of Control Outcome[%]

RR Relative Risk[%]
RRR Relative Risk Reduction[%]
ARR Absolute Risk Reduction[%]
NTT Numbers Needed to Treat[%]

2. Test Performance

Sensitivity
Proportion of subjects with the disease who have a positive test.

Specificity
Proportion of subjects without the disease who have a negative test.

Positive Predictive Value (PPV)

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Probability that a person with a positive result actually has the disease.

Negative Predictive Value (NPV)

Probability that a person with a negative result actually does not have the disease.

Sensitivity = TP / (TP + FN)

Specificity = TN / (TN + FP)

PPV = TP / (TP + FP)

NPV = TN / (TN + FN)

TP True Positive
FP False Positive
FN False Negative
TN True Negative

Gold Std Positive Gold Std Negative

Test Positive TP FP TP+FP
Test Negative FN TN FN+TN
TP+FN FP+TN TP+FP+TP+FN

Chi-Ming Chow MD www.CardioMath.com Page 18 of 35

 (IV) Exercise Stress Test Equations
1. Duke Treadmill Score
The Duke Treadmill Score (DTS) is a weighted index combining treadmill
exercise time using standard Bruce protocol, maximum net ST segment
deviation (depression or elevation), and exercise-induced angina. It was
developed to provide accurate diagnostic and prognostic information for
the evaluation of patients with suspected coronary heart disease. The
typical observed range of DTS is from -25 (highest risk) to +15 (lowest
risk). A low DTS is actually better at excluding ischemic heart disease in
women than men.

DTS = Exercise Time - (5 x Max ST) - (4 x Angina Index)

Ex Time Treadmill exercise time [min]

Max ST Maximum net ST deviation (except aVR) [mm]
Angina Index Treadmill Angina Index [no unit]
! 0. No angina during exercise
1. Non-limiting angina
2. Exercise limited angina

DTS Duke Treadmill Score [no unit]

Risk >=+5 Low risk
+4 to -10 Moderate risk
<= -11 High risk

Chi-Ming Chow MD www.CardioMath.com Page 19 of 35

DTS 1-Yr No 1 VD 2 VD 3VD
Risk Mortality Stenosis >=75% >=75% >=75%
Category >=75% or LM
>=75%
Men
Low 0.9% 52.6% 22.4% 13.6% 11.4%
Mod 2.9% 17.8% 15.6% 27.9% 38.7%
High 8.3% 1.8% 9.1% 17.5% 71.5%
Women
Low 0.5% 80.9% 9.4% 6.2% 3.5%
Mod 1.1% 65.1% 14.2% 8.3% 12.4%
High 1.8% 10.8% 18.9% 24.3% 46%
VD = vessel disease; LM = left main

Mark DB, Hlatky MA, Harrell FE, Lee KL, Califf RM, Pryor DB. Exercise treadmill
score for predicting prognosis in coronary artery disease. Ann Int Med 1987; 106:793-
800.

Mark DB, Shaw L, Harrell FE Jr. et al. Prognostic value of a treadmill exercise score
in outpatients with suspected coronary artery disease. N Engl J Med 1991; 325:849-
53.

Alexander KP, Shaw L, Delong ER, Mark DB, Peterson ED. Value of exercise
treadmill testing in women. J Am Coll Cardiol 1998; 32:1657-64.

2. Max. Predicted Heart Rate by Age

Maximal expected heart rate according to age as calculated by …

100% * (220-Age)
85% * (220-Age)

3. Max. Predicted Heart Rate by Date of Birth

Maximal expected heart rate according to age calculated by date of birth to
today’s date.

100% * (220-Age)
85% * (220-Age)

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 4. % Maximal Predicated Heart Rate Achieved

Max HR Achieved
%MPHR =
(220 " Age)
Age [Years]
Maximal HR Achieved [BPM]

% of MPHR
!

5. METs & VO2 according to Treadmill Speed & Grade

This calculator allows you to estimate the METs and VO2 given the
treadmill speed and the grade. It is useful for situations when you want to
use the non-standard treadmill protocol to find out exactly how much the
patient can do.

Speed = Treadmill Speed * 26.8

Horizontal Component (HC) = Speed * 0.1
Vertical Component (VC) = Speed * 1.8 * Grade

VO2 = HC + VC + Rest = HC + VC + 3.5 mL/kg/min

VO 2
METs =
3.5
Treadmill Speed [mph]
Treadmill Grade [%]

! Metabolic Equivalents [METs]

Effort Capacity Mets

Excellent >=13
Good 10 -12
Fair 7-9
Reduced 5-6
Poor <= 4

Chi-Ming Chow MD www.CardioMath.com Page 21 of 35

6. Rate Pressure Product

This allows you to calculate the internal workload or hemodynamic

response, as represented by rate pressure product, given the maximum
heart rate and maximum blood pressure achieved.

RPP = Max HR * Max SBP

Max HR Maximum Heart Rate [bpm]

Max BP Maximum Systolic Blood Pressure [mmHg]

RPP Rate Pressure Product

Hemodynamic RPP
Response
High > 30,000
High Intermediate 25 - 30,000
Intermediate 20 - 25,000
Low Intermediate 15 - 20,000
Low 10 - 15,000

Chi-Ming Chow MD www.CardioMath.com Page 22 of 35

(V) General Medical Equations
1. Alveolar-arterial Oxygen Gradient

A-a oxygen gradient is determined by subtracting the arterial oxygen

tension from the calculated alveolar oxygen tension.

A-a Gradient = ( (PB-PH2O)*FIO2- PaCO2/RQ) - PaO2

PB Atmospheric pressure [mmHg] (760mmHg at sea level)

PH2O Partial Pressure of Water [mmHg] (47mmHg at 37 deg C)
FiO2 % of Oxygen inspired (21% with room air)
RQ Respiratory Quotient (approx 0.8, depends on food intake)
PaO2 Arterial O2 measured [mmHg]
PaCO2 Arterial CO2 measured [mmHg]

Normal = 7-14 mmHg on room air or

= 2.5 + 0.21 x age [yrs]

Mellemgaard, K. The alveolar-arterial oxygen difference. Size and

components in normal man. Acta Physiol Scand 1966; 67:10.

Kanber, GL et al. The alveolar-arterial oxygen gradient in young and

elderly men during air and oxygen breathing. Am Rev Respir Dis 1968;
97:376.

2. Anion Gap

Determination of plasma anion gap (AG) is an important step in the

differential diagnosis of metabolic acidosis. The expected normal values
for the AG must be adjusted downward in hypoalbuminemia, with the AG
reduced by 2.5 meq/L for every 1 g/dL drop in plasma albumin. Increased
AG metabolic acidosis includes lactic acidosis, ketoacidosis, ingestions,
and renal failure.

AG = Na+ - (Cl- + HCO3-)

Normal = 10-15

Chi-Ming Chow MD www.CardioMath.com Page 23 of 35

 Rose, BD, Post, TW. Clinical Phsiology of Acid-Base and Electrolyte
Disorders, 5th ed., McGraw-Hill, New York, 2001, p. 583-588.

Gabow, PA. Disorders associated with an altered anion gap. Kidney Int
1985; 27:472.

3. Body Mass Index

Body Mass Index = Weight(kg) / Height(m)2

A Body Mass Index between 25 and 29.9 is "overweight", greater than or equal to 30
is "obese" However, some very muscular people can have high Body Mass Indexes
and in adolescents, BMIs frequently result in overestimation of fatness. A better
classification of "overweight" is given in this table below.

Women Men
Underweight <19.1 <20.7
Ideal weight 19.1-25.8 20.7-26.4
Marginally overweight 25.8-27.3 26.4-27.8
Overweight 27.3-32.3 27.8-31.1
Very overweight or obese >32.3 >31.1

Clinical Guidelines on the Identification, Evaluation, and Treatment of Overweight

and Obesity in Adults. National Heart, Lung and Blood Institute. June 17, 1998

4. Body Surface Area (Mosteller Formula)

Ht [cm] * Wt [kg]
BSA [m2 ] =
3600

Mosteller RD: Simplified Calculation of Body Surface Area. N Engl J Med

1987 Oct 22;317(17):1098 (letter)
!

Chi-Ming Chow MD www.CardioMath.com Page 24 of 35

5. Body Water Deficit

The water deficit in the hypernatremic patient can be estimated by this formula.
This formula estimates the amount of positive water balance required to return
the plasma sodium concentration to 140 meq/L. Over rapid correction can lead to
cerebral edema which is potentially dangerous. In general plasma sodium
correction should be ≤ 0.5 meq/L per hour.

BWD = 0.6 * Weight[Kg] * ((NaMeasured / NaCorrected) -1)

Na (Measured) Plasma sodium measured

Na (Corrected) Plasma sodium corrected to (default 140)
Weight Body weight

BWD Body Water deficit

Rose, BD, Post, TW. Clinical Phsiology of Acid-Base and Electrolyte

Disorders, 5th ed., McGraw-Hill, New York, 2001, p. 775-784.

Adogue HJ et al. Hypernatremia. N Engl J Med 2000; 342:1493.

Blum D, et al. Safe oral rehydration of hypertonic dehydration. J Pediatr

Gastroenterol Nutr 1986; 5:232.

6. Creatinine Clearance (Cockroft & Gault Formula)

Dose adjustments most frequently required when creatinine clearance < 30

mL/min. This method is applicable for adults greater than 20 years old
and result will differ from the true value at extremes of body composition.
There is some controversy as to using the total body weight (TBW) versus
ideal body weight (IBW) in the calculation. Here we have included the
creatinine clearance using both weights. This formula may falsely estimate
creatinine clearance if patient is elderly, emaciated or with unstable renal
function.

Male:

When Serum Creatinine, SCr is given in mg/dL

Chi-Ming Chow MD www.CardioMath.com Page 25 of 35

 (140 " age) * (Wt)
CrCl =
72 *SCr

When Serum Creatinine, SCr is given in µmol/L

!
(140 " age) * (Wt)
CrCl = *1.23
SCr

N.B. Weight is in Kg

! Female:

CrCl [Female] = CrCl [Male] *0.85

Age [years]
Height [cm] or [inch]
Weight [kg] or [lb]
Serum Creatinine [µmol/L] or [mg/dL]

CrCl Creatinine Clearance [mL/min]

Normal:

Male: > 125 mL/min

Female: > 105 mL/min

Cockroft D and Gault H. Prediction of Creatinine Clearance from Serum

Creatinine. Nephron, 1976; 16: 33-41.

7. Ideal Body Weight

Originally published by BJ Devine in 1974. The “Devine formula” was
initially intended to be used to calculate dosage of certain medications
such as Digoxin, and Gentamicin. However, it has been adopted widely in
many clinical calculators for health & fitness. Caution has to been made

Chi-Ming Chow MD www.CardioMath.com Page 26 of 35

for application of this formula in women, as it may be too low for this
group particularly in shorter women.

Male:

IBW [kg] = 50 [kg] + (2.3 * (Height [inch] – 60))

Female:

IBW [kg] = 45.5 [kg] + (2.3 * (Height [inch] – 60))

Devine B.J. Gentamicin therapy. Drug Intelligence and Clinical Therapy;

8:65, 1974.

Pai MP et al. The origin of the “Ideal” body weight equations. Ann
Pharmacol 2000; 34:1066-69.

8. Lean Body Weight

Refers to the weight of all the body’s organs, bone and muscles, without
fat. It is used in calculation for certain medication dosages. It can also be
used for calculating % fat composition of the body.

Male:

Lean Body Weight [kg] = (1.10 x Weight[kg]) - 128 ( Weight[kg]2/(100 x Height(m))2)

Female:
Lean Body Weight [kg] = (1.07 x Weight[kg]) - 148 ( Weight[kg]2/(100 x Height(m))2)

9. LDL Estimation (Friedewald Equation)

The ultracentrifugal measurement of LDL is time consuming and

expensive and requires specialist equipment. For this reason, LDL-
cholesterol is most commonly estimated from quantitative measurements

Chi-Ming Chow MD www.CardioMath.com Page 27 of 35

of total and HDL-cholesterol and plasma triglycerides (TG) using the
empirical relationship of Friedewald et al.(1972).

The Friedewald equation should not be used under the following circumstances:
1. When chylomicrons are present
2. When plasma triglyceride concentration exceeds 400 mg/dL (4.52 mmol/L)
3. In patients with dysbetalipoproteinemia (type III hyperlipoproteinemia)

LDL is estimated if fasting triglycerides < 400 mg/dl

Units in mmol/L (SI)

LDL = Total Cholesterol - ([Triglycerides /2.2] + HDL)

Units in mg/dl (US)

LDL = Total Cholesterol - ([Triglycerides /5] + HDL)

10. Serum Osmolarity

SI Units
Serum Osmality[mmol/kg]= 2*(Na+ + K+) + BUN + Glu

US Units
Serum Osmality[mOsm/kg]= 2*(Na+ + K+) + BUN/2.8 + Glu/18

N = 280-300

Chi-Ming Chow MD www.CardioMath.com Page 28 of 35

 (VI) Hemodynamic Equations
1. Aortic Valve Area (Gorlin Formula)

CO/(SEP * HR)
AVA =
44.3 * MVG
CO Cardiac Output [mL/min]
SEP Systolic Ejection Period [sec/beat]
HR Heart Rate [beats/min]
MVG Mean Valvular Gradient [mmHg]
!
AVA Aortic Valve Area [cm2]

Normal 3.0- 4.0 cm2

Gorlin R, Gorlin SG: Hydraulic formula for calculation of stenotic mitral

valve, other cardiac valves, and central circulatory shunts. Am Heart J
41:1-29, 1951

2. Aortic Valve Area (Hakki Formula)

Simplified formula for estimating the aortic valve area using peak-to-peak
gradient across the valve. If heart rate is above 90 per minute for AS then
divide the Hakki's equation by 1.35.

Transvalvular Flow
Valve Area(cm2 ) =
Peak - to - Peak Gradient
CO
Aortic Valve Area(cm2 ) =
LV SP - Ao SP
!
CO Cardiac Output [L/min]
LV SP Left Ventricular Systolic Pressure [mmHg]
Ao SP Aortic Systolic Pressure [mmHg]
!
AVA Aortic Valve Area [cm2]

Chi-Ming Chow MD www.CardioMath.com Page 29 of 35

 Normal 2-4 cm2

Hakki AH, Iskandrian AS, Bemis CE, Kimbiris D, Mintz GS, Segal BL,
Brice C: A simplified valve formula for the calculation of stenotic cardiac
valve areas. Circulation 63:150-1055, 1981.

Angel J, Soler-Soler J, Anivarro I, Domingo E: Hemodynamic evaluation

of stenotic cardiac valves :II. Modification of the simplified formula for
mitral and aortic valve area calculation. Cathet Cardiovasc Diagn
11:127, 1985.

3. Mean Arterial Pressure (MAP)

MAP = 2/3 DAP + 1/3 SAP

PP = SAP – DAP

SBP Systolic Blood Pressure [mmHg]

DBP Diastolic Blood Pressure [mmHg]

MAP Mean Arterial Pressure [mmHg]

PP Pulse Pressure [mmHg]

4. Mitral Valve Area (Gorlin Formula)

The factor 0.85 is adjustment for the diastolic filling period.

CO/(DFP * HR)
MVA =
0.85 * 44.3 * MVG
CO Cardiac Output [mL/min]
DFP Diastolic Filling Period [s/beat]
HR Heart Rate [beats/min]
! MVG Mean Valvular Gradient [mmHg]

MVA Mitral Valve Area [cm2]

Normal 4.0- 6.0 cm2

Chi-Ming Chow MD www.CardioMath.com Page 30 of 35

 Gorlin R, Gorlin SG: Hydraulic formula for calculation of stenotic mitral
valve, other cardiac valves, and central circulatory shunts. Am Heart J
41:1-29, 1951

5. Mitral Valve Area (Hakki Formula)

Simplified formula for estimating the mitral valve area using peak-to-peak
gradient across the valve. If heart rate was below 75 per minute for MS
then divide the Hakki's equation by 1.35

Transvalvular Flow
Valve Area(cm2 ) =
Peak - to - Peak Gradient
CO
Mitral Valve Area(cm2 ) =
LA MP - LV DP
!
CO Cardiac Output [mL/min]
LV DP Left Ventricular Diastolic Pressure [mmHg]
LA MP Left atrial mean Pressure [mmHg]
!
MVA Mitral Valve Area [cm2]

Normal 4-6 cm2

Hakki AH, Iskandrian AS, Bemis CE, Kimbiris D, Mintz GS, Segal BL,
Brice C: A simplified valve formula for the calculation of stenotic cardiac
valve areas. Circulation 63:150-1055, 1981.

Angel J, Soler-Soler J, Anivarro I, Domingo E: Hemodynamic evaluation

of stenotic cardiac valves :II. Modification of the simplified formula for
mitral and aortic valve area calculation. Cathet Cardiovasc Diagn
11:127, 1985

6. Mixed Venous O2 Saturation

Chi-Ming Chow MD www.CardioMath.com Page 31 of 35

 Estimation of mixed venous oxygen saturation which can then be used for
calculation of pulmonary-to-systemic shunt ratio.

3*SVC O 2 + IVC O 2
MV O 2 =
4
SVC O2 SVC O2 Sat [%]
IVC O2 IVC O2 Sat [%]

! MV O2 Mixed venous Oxygen Sat [%]

7. Pulmonary Vascular Resistance (PVR)

In Wood Units:

MPAP - PCWP
PVR =
Qp
In dyne-sec/cm5:

MPAP - PCWP
PVR = * 80
! Qp
MPAP Mean Pulmonary Arterial Pressure [mmHg]
PCWP Pulmonary Capillary Wedge Pressure [mmHg]
Qp Pulmonary Flow [L/min]
!
PVR Pulmonary Vascular Resistance [Wood Units] or [dyne-
sec/cm5]

Normal 50-250 dyne-sec/cm5

8. Qp/Qs Shunt Calculation (Simplified)

Pulmonary-to-Systemic shunt ratio calculation using oxygen saturations obtained in

different locations.

Chi-Ming Chow MD www.CardioMath.com Page 32 of 35

 Qp SA O 2 " MV O 2
=
Qs PV O 2 " PA O 2
SA O2 Arterial O2 Sat [%]
MV O2 Mixed Venous O2 Sat [%]
PV O2 Pulmonary Venous O2 Sat [%]
PA O2 Pulmonary Arterial O2 Sat [%]
!
Qp/Qs Pulmonary-to-Systemic Ratio

9. Systemic Vascular Resistance (SVR)

In Wood Units:

MAP - CVP
SVR =
CO
In dyne-sec/cm5:

MAP - CVP
! SVR = * 80
CO
MAP Mean arterial pressure [mmHg]
CVP Central Venous Pressure [mmHg]
! CO Cardiac Output [L/min]

SVR Systemic Vascular Resistance [Wood Units] or [dyne-

sec/cm5]

Normal 800-1200 dyne-sec/cm5

Chi-Ming Chow MD www.CardioMath.com Page 33 of 35

(VII) Unit Conversion Equations
SI US Conversion Factor
Length cm in 0.3937
Weight kg lb 2.2
Temperature C F (9/5)X+32

Glucose mmol/L mg/dL 18.01

Cholesterol (Total/HDL/LDL) mmol/L mg/dL 38.67
Triglyceride mmol/L mg/dL 88.57
BUN mmol/L mg/dL 2.801
Creatinine umol/L mg/dL 0.0113

US SI Conversion Factor
Length in cm 2.54
Weight lb kg 0.4545
Temperature F C (5/9)(X-32)

Glucose mg/dL mmol/L 0.0555

Cholesterol (Total/HDL/LDL) mg/dL mmol/L 0.0259
Triglyceride mg/dL mmol/L 0.0113
BUN mg/dL mmol/L 0.357
Creatinine mg/dL umol/L 88.4

Laposata M. SI Unit Conversion Guide. The New England Journal Of

Medicine,1992
JAMA instructions for authors. JAMA. 283(1), 2000

Chi-Ming Chow MD www.CardioMath.com Page 34 of 35

Disclaimer (What Our Lawyer Want Us to Say)

The calculators provided are for information purposes only. They

are not meant to be a substitute for professional advise and are not to
be used for medical diagnosis. All calculations must be confirmed before
clinical use or diagnostic purposes by qualified medical professionals.
The authors make no claims of the accuracy of the information
contained herein. The authors make no claims whatsoever, expressed or
implied, about the authenticity, accuracy, reliability, completeness or
timeliness of the material, software, text, graphics and links given. In no
event shall the authors, its suppliers, affiliates or any third parties be
liable in any manner whatsoever for any damages arising upon use of
any information provided.

Chi-Ming Chow MD www.CardioMath.com Page 35 of 35