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The human heart: Application of the golden ratio and angle
International Journal of Cardiology 150 (2011) 239–242
Contents lists available at ScienceDirect
International Journal of Cardiology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i j c a r d
Editorial
The human heart: Application of the golden ratio and angle Michael Y. Henein a,⁎ and the Golden Ratio Collaborators: Ying Zhao a,b,1, Rachel Nicoll a,1, Lin Sun b,1, Ashraf W. Khir c,1, Karl Franklin a,1, Per Lindqvist a,1 a b c
Department of Public Health and Clinical Medicine, and Heart Centre, Umea University, Sweden Ultrasound department, Beijing Anzhen Hospital, Capital Medical University, Beijing, China Brunel Institute for Bioengineering, Brunel University, Middlesex, UK
a r t i c l e
i n f o
Article history: Received 27 April 2011 Accepted 14 May 2011 Available online 23 June 2011 Keywords: human heart golden ratio
a b s t r a c t The golden ratio, or golden mean, of 1.618 is a proportion known since antiquity to be the most aesthetically pleasing and has been used repeatedly in art and architecture. Both the golden ratio and the allied golden angle of 137.5° have been found within the proportions and angles of the human body and plants. In the human heart we found many applications of the golden ratio and angle, in addition to those previously described. In healthy hearts, vertical and transverse dimensions accord with the golden ratio, irrespective of different absolute dimensions due to ethnicity. In mild heart failure, the ratio of 1.618 was maintained but in end-stage heart failure the ratio significantly reduced. Similarly, in healthy ventricles mitral annulus dimensions accorded with the golden ratio, while in dilated cardiomyopathy and mitral regurgitation patients the ratio had significantly reduced. In healthy patients, both the angles between the mid-luminal axes of the pulmonary trunk and the ascending aorta continuation and between the outflow tract axis and continuation of the inflow tract axis of the right ventricle approximate to the golden angle, although in severe pulmonary hypertension, the angle is significantly increased. Hence the overall cardiac and ventricular dimensions in a normal heart are consistent with the golden ratio and angle, representing optimum pump structure and function efficiency, whereas there is significant deviation in the disease state. These findings could have anatomical, functional and prognostic value as markers of early deviation from normality. © 2011 Elsevier Ireland Ltd. All rights reserved.
The golden ratio (also known as the golden mean or divine proportion) is so called when on a line consisting of a longer length (A) plus a shorter length (B), the proportion (A + B)/A equals A/B. This was described by Pythagoras as ‘the small is to the large as the large is to the whole’. It is expressed as an irrational mathematical constant phi (φ) of approximately 1.618. This proportion is said to be the most aesthetically pleasing and many artists and architects have incorporated it into their work [1,2]. The golden ratio was also known to Euclid, known as the father of geometry, and the mathematician Fibonacci; in the Fibonacci sequence, where each number is the sum of the two previous numbers, if the calculation above is applied to any two consecutive numbers, the mean result approximates to1.618 [3,4]. The golden ratio has also been found within the proportions of the human body, including limbs, facial features, teeth and the DNA molecule, and quantum field theory [5–8]. Allied to this is the golden angle, calculated by sectioning the circumference of a circle (c) into two arcs, such that the ratio of the length of the larger arc (a) to the length of the smaller arc (b) is the same as the ratio of the full cir-
⁎ Corresponding author at: Heart Centre and Department of Public Health and Clinical Medicine, Umeå University, Sweden. Tel.: + 46 90 785 0000; fax: + 46 90 137 633. E-mail address: [email protected] (M.Y. Henein). 1 The Golden Ratio Collaborators. 0167-5273/$ – see front matter © 2011 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.ijcard.2011.05.094
cumference to the length of the larger arc (i.e. c/a = a/b). The golden angle is the angle subtended by the smaller arc (b) and measures 137.5°. The golden ratio decimal (0.618) equates to an angle of 222.5°, the reverse of the golden angle, with the sum adding up to 360°. It has been found among branches and leaves on plant stems [7]. We hypothesised that since the golden ratio and angle are so deeply rooted in nature, they must have some correspondence in the healthy heart structure and function. Accordingly, we studied the heart dimensions and function using 2D or 3D cardiac ultrasound imaging technology at end-diastole, the resting phase of the heart cycle or CT scanning. Individual study data were compared using the unpaired t test. Study 1A: To assess the effect of ethnicity on the left ventricle, the most important heart pump, we took vertical and transverse cardiac measurements from 30 normal Swedish adults (Fig. 1) and compared these with measurements from 30 healthy matched Chinese. We found that Chinese left ventricle dimensions were smaller by 8±1 mm and 5± 0.5 mm, compared to Swedes but the golden ratio was maintained at approximately 1.618. Our findings suggest that it is the dimensions ratio that is important for healthy functioning and not the absolute dimensions. Study 1B: To assess whether the ratio of these dimensions differs in heart failure, we tested them in 40 Swedish patients; 20 with mild heart failure and 20 with end-stage heart failure. In mild heart failure, the left ventricular ratio was reduced (p b 0.001) but the right
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Fig. 1. An echocardiographic example of normal 4 chambers of the heart showing measurements of the vertical and transverse cardiac axes. LV, left ventricle and RV, right ventricle.
ventricular ratio was increased due to the enlarged left ventricle encroaching on the right ventricular dimensions (p b 0.001); overall the cardiac ratio remained at around 1.64. However, in end-stage heart failure, the left ventricular ratio was significantly reduced (p b 0.01) and this distorted the cardiac ratio to around 1.4 (p b 0.005), consistent with pathological remodeling resulting in a more spherical shape to the heart. These patients had 50% mortality over the course of 36 months' follow up, compared with 100% survival in the mild group (Fig. 2a and b). Consequently, it appears that as long as the cardiac ratio is maintained at approximately 1.618, survival is likely but a significant deviation from the golden ratio is associated with a poor
outcome. This downward spiral is known to result in mitral and tricuspid valve regurgitation and enlargement of the atria [9]. Study 2: We have previously shown that the normal mitral valve annulus dimensions ratio equates to 1.6 [10]. We then evaluated the left ventricular inlet length and width (mitral annulus dimensions) in 15 normal individuals and 12 age matched patients with dilated cardiomyopathy and functional mitral regurgitation. In normals, the relative mitral annulus circumference length and width proved to be 6.8 ± 0.8 cm and 4.2 ± 0.8 cm respectively, giving a ratio of 1.62, compared with the dilated cardiomyopathy patients, whose dimensions were 8.3 ± 1.8 cm and 5.8 ± 1.1 cm respectively, giving a ratio of 1.42 (p b 0.001). This reduced ratio in patients with dilated left ventricle is consistent with disturbance in mitral valve function and the development of mitral regurgitation, known for its clinical complications and poor prognosis [11–13]. Furthermore, others have shown that the structure and function of the tricuspid and mitral annuli are not independent of each other but are based around the golden ratio of 1.618 [14,15]. The same ratio applies to the extent of the amplitude of the longitudinal motion of the tricuspid and mitral annuli, a clinical measure taken to reflect right and left ventricular function, respectively [16]. We have also previously demonstrated that the golden ratio is seen in normal foetal myocardial function development, where the heart muscle diastolic function matures at a rate of 1.6 mm/s per week between 20 and 40 weeks of age [17], while a healthy adult heart muscle loses velocity of 1.6 cm/s each decade between age 20 and 80 years [18], indicating a close relationship between ageing and ventricular myocardial function. Finally, others have shown that the end product of normal cardiac function, the blood pressure ratio of 120/80 mmHg, also approximates to the golden ratio of 1.618 and carries significant prognostic value [19]. Additionally, myocardial muscle fibre orientation takes a helical shape, with the coils of each
Fig. 2. Echocardiograms (a) from 2 patients with heart failure, mild (left) and severe (right). Graph showing statistical measurements differences between the two heart failure groups and normals (error bars represent standard error).
M.Y. Henein et al. / International Journal of Cardiology 150 (2011) 239–242
segment of the apical loop being of different lengths and having a harmonic proportion which conforms to the golden ratio [20]. When these myocardial fibres become horizontal or transverse, as is the case in heart failure, they weaken overall ventricular function [21]. Study 3A: We assessed the anatomical angle between the right ventricular inlet axis and outflow tract axis in 16 healthy subjects and 19 patients with severe pulmonary hypertension and right heart failure. We found that the normal angle between the axes is approximately 43 ± 3°. The angle between the outflow tract axis and continuation of the inflow tract axis is therefore 138 ± 4°, which is approximately 137.5°. This angle increased significantly to 160 ± 4° (p N 0.001) in right heart failure patients, as the right ventricle became cylindrical and lost its native shape, reducing the inflow-outflow tract angle to approximately 20° from 43° (unpublished). Hence it can be seen that the complex anatomy of the right ventricle, as well as its myocardial fibre architecture, support the unique role of this angle in preserving the peristaltic circulation inside the right ventricle. With severe disease, right ventricular cavity enlargement and remodelling combine to produce loss of the normal golden angle, contributing to the intractable right ventricular pump decompensation. Study 3B: We then measured the angle between the mid-luminal axes of the pulmonary trunk and the proximal ascending aorta in the same 16 healthy subjects. This angle approximated to a mean of 39.5 ± 3.6°, making the angle between the pulmonary trunk and the continuation of the ascending aorta approximately 139 ± 3°, which corresponds with the golden angle of 137.5° (unpublished) (Fig. 3). Normal systolic pressure in the aorta is almost six times that in the pulmonary artery, hence the need for a pressure recovery space between the two arteries to avoid pressure transmission across their thin walls. In hypertension, the pressure difference between the aorta and the pulmonary artery could increase to 10 times the norm. Had the aorta and pulmonary artery been closely parallel, this could potentially have created a significant afterload to the right ventricle, which is avoided by the space provided by the angle between the axes. In addition to these two great arteries, the same angle seems to be adopted by the central and peripheral arterial tree; early anatomical studies demonstrated the branching pattern of the coronary and peripheral arteries to be compatible with the golden angle [22,23]. It appears that the golden angle is not only important for coronary vasculature packing but also for optimising the branching pattern so that perfusion of the myocardial bed is maximised [12]. Electrical function of the heart also seems to follow a parallel pattern [24]: the range of electrical potential of a normal heart lay between −30° and + 110°, representing a total angle of 140°, or approximately 137.5°. An overall optimum healthy heart function has also been observed when there is a graph of convergence of the
Fig. 3. Diagrams showing the golden angle between the ascending aorta and pulmonary trunk (left), and between inlet and outflow tract axes of the right ventricle (right). AO: aorta; PA: pulmonary artery; RA: right atrium; RV: right ventricle.
241
Fibonacci numbers and a phi relationship between the T waves on the surface electrocardiogram [9,10]. Our findings show that the overall cardiac and ventricular dimensions in a normal heart are consistent with an approximation of the golden ratio of 1.618 and the golden angle of 137.5°, representing optimum pump structure and function efficiency. These findings are not only of philosophical interest but may have a significant anatomical, functional and prognostic value. Application of these ratios and angles in our daily clinical practice may lead to development of simple and reliable markers of early deviation from normality, which could be treated before irreversible alteration in cardiac structure and function develops. Acknowledgements Dr. Ying Zhao is partially supported by Umeå University and Swedish Heart and Lung Foundation. The authors of this manuscript have certified that they comply with the Principles of Ethical Publishing in the International Journal of Cardiology [25]. References [1] Maor E. Trigonometric delights. Princeton, N.J: Oxford: Princeton University Press; 2002. [2] Padovan R. Proportion: science, philosophy, architecture. London: E & FN Spon; 1999. [3] Euclid. Similar figures, space, and solids : being a new geometry of the subjectmatter of Euclid, Books V., VI., and XI. London: W.B. Clive; 1926. [4] Tattersall JJ. Elementary number theory in nine chapters. 2nd ed. Cambridge: Cambridge University Press; 2005. [5] Hemenway P. Divine proportion: phi in art, nature, and science. New York: Published by Sterling Pub. Co.; 2005. [6] Livio M. The golden ratio : the story of phi, the world's most astonishing number. New York, NY: Broadway Books; 2002. [7] Stakhov AP, Sluchenkova A. International Club of the Golden Section. Museum of Harmony and the Golden Section : mathematical connections in nature, science and art. Vinnitsa: ITI; 2003. [8] Affleck I. Solid-state physics: golden ratio seen in a magnet. Nature 2010;464: 362–3. [9] Calcutteea ALP, Maras D, Lei W, Hodson M, Henein M. Myocardial determinants of right ventricular ejection in pulmonary hypertension: 2D strain rate and spectral doppler study. Eur J Echocardiogr 2008;9:S53. [10] Chung RDA, Calcuttea A, Pura B, Li W, Pepper JR, Henein MY. Mitral annulus shape change in dilated cardiomyopathy by real-time 3D transthoracic echocardiography. Eur J Echocardiogr 2008;9:S131. [11] Fattouch K, Sampognaro R, Speziale G, et al. Impact of moderate ischemic mitral regurgitation after isolated coronary artery bypass grafting. Ann Thorac Surg 2010;90:1187–94. [12] Ennezat PV, Marechaux S, Le Tourneau T, et al. Myocardial asynchronism is a determinant of changes in functional mitral regurgitation severity during dynamic exercise in patients with chronic heart failure due to severe left ventricular systolic dysfunction. Eur Heart J 2006;27:679–83. [13] Breithardt OA, Sinha AM, Schwammenthal E, et al. Acute effects of cardiac resynchronization therapy on functional mitral regurgitation in advanced systolic heart failure. J Am Coll Cardiol 2003;41:765–70. [14] Tei C, Pilgrim JP, Shah PM, Ormiston JA, Wong M. The tricuspid valve annulus: study of size and motion in normal subjects and in patients with tricuspid regurgitation. Circulation 1982;66:665–71. [15] Ormiston JA, Shah PM, Tei C, Wong M. Size and motion of the mitral valve annulus in man. I. A two-dimensional echocardiographic method and findings in normal subjects. Circulation 1981;64:113–20. [16] Henein MY, Xiao HB, Brecker SJ, Gibson DG. Berheim “a” wave: obstructed right ventricular inflow or atrial cross talk? Br Heart J 1993;69:409–13. [17] Gardiner HM, Pasquini L, Wolfenden J, et al. Myocardial tissue Doppler and long axis function in the fetal heart. Int J Cardiol 2006;113:39–47. [18] Henein M, Lindqvist P, Francis D, Morner S, Waldenstrom A, Kazzam E. Tissue Doppler analysis of age-dependency in diastolic ventricular behaviour and filling: a cross-sectional study of healthy hearts (the Umea General Population Heart Study). Eur Heart J 2002;23:162–71. [19] Ulmer H, Diem G, Bischof HP, Ruttmann E, Concin H. Recent trends and sociodemographic distribution of cardiovascular risk factors: results from two population surveys in the Austrian WHO CINDI demonstration area. Wien Klin Wochenschr 2001;113:573–9. [20] Buckberg GD. Basic science review: the helix and the heart. J Thorac Cardiovasc Surg 2002;124:863–83. [21] Buckberg GD, Coghlan HC, Torrent-Guasp F. The structure and function of the helical heart and its buttress wrapping. V. Anatomic and physiologic
242
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considerations in the healthy and failing heart. Semin Thorac Cardiovasc Surg 2001;13:358–85. [22] Gibson CM, Gibson WJ, Murphy SA, et al. Association of the Fibonacci Cascade with the distribution of coronary artery lesions responsible for ST-segment elevation myocardial infarction. Am J Cardiol 2003;92:595–7. [23] Harvey W, Sylvius Z. The anatomical exercises of Dr. William Harvey, professor of physick, and physician to King Charles the first concerning the motion of the heart and blood : with the preface of Zachariah Wood, physician of Roterdam : to which is
added, Dr. James de Back, his Discourse of the heart, physician in ordinary to the town of Roterdam. London: Printed for Richard Lowndes …, and Math. Gilliflower … 1673. [24] Einthoven W. The different forms of the human electrocardiogram and their signification. Lancet 1912;179:853–61. [25] Shewan LG, Coats AJ. Ethics in the authorship and publishing of scientific articles. Int J Cardiol 2010;144:1–2.
Contents lists available at ScienceDirect
International Journal of Cardiology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i j c a r d
Editorial
The human heart: Application of the golden ratio and angle Michael Y. Henein a,⁎ and the Golden Ratio Collaborators: Ying Zhao a,b,1, Rachel Nicoll a,1, Lin Sun b,1, Ashraf W. Khir c,1, Karl Franklin a,1, Per Lindqvist a,1 a b c
Department of Public Health and Clinical Medicine, and Heart Centre, Umea University, Sweden Ultrasound department, Beijing Anzhen Hospital, Capital Medical University, Beijing, China Brunel Institute for Bioengineering, Brunel University, Middlesex, UK
a r t i c l e
i n f o
Article history: Received 27 April 2011 Accepted 14 May 2011 Available online 23 June 2011 Keywords: human heart golden ratio
a b s t r a c t The golden ratio, or golden mean, of 1.618 is a proportion known since antiquity to be the most aesthetically pleasing and has been used repeatedly in art and architecture. Both the golden ratio and the allied golden angle of 137.5° have been found within the proportions and angles of the human body and plants. In the human heart we found many applications of the golden ratio and angle, in addition to those previously described. In healthy hearts, vertical and transverse dimensions accord with the golden ratio, irrespective of different absolute dimensions due to ethnicity. In mild heart failure, the ratio of 1.618 was maintained but in end-stage heart failure the ratio significantly reduced. Similarly, in healthy ventricles mitral annulus dimensions accorded with the golden ratio, while in dilated cardiomyopathy and mitral regurgitation patients the ratio had significantly reduced. In healthy patients, both the angles between the mid-luminal axes of the pulmonary trunk and the ascending aorta continuation and between the outflow tract axis and continuation of the inflow tract axis of the right ventricle approximate to the golden angle, although in severe pulmonary hypertension, the angle is significantly increased. Hence the overall cardiac and ventricular dimensions in a normal heart are consistent with the golden ratio and angle, representing optimum pump structure and function efficiency, whereas there is significant deviation in the disease state. These findings could have anatomical, functional and prognostic value as markers of early deviation from normality. © 2011 Elsevier Ireland Ltd. All rights reserved.
The golden ratio (also known as the golden mean or divine proportion) is so called when on a line consisting of a longer length (A) plus a shorter length (B), the proportion (A + B)/A equals A/B. This was described by Pythagoras as ‘the small is to the large as the large is to the whole’. It is expressed as an irrational mathematical constant phi (φ) of approximately 1.618. This proportion is said to be the most aesthetically pleasing and many artists and architects have incorporated it into their work [1,2]. The golden ratio was also known to Euclid, known as the father of geometry, and the mathematician Fibonacci; in the Fibonacci sequence, where each number is the sum of the two previous numbers, if the calculation above is applied to any two consecutive numbers, the mean result approximates to1.618 [3,4]. The golden ratio has also been found within the proportions of the human body, including limbs, facial features, teeth and the DNA molecule, and quantum field theory [5–8]. Allied to this is the golden angle, calculated by sectioning the circumference of a circle (c) into two arcs, such that the ratio of the length of the larger arc (a) to the length of the smaller arc (b) is the same as the ratio of the full cir-
⁎ Corresponding author at: Heart Centre and Department of Public Health and Clinical Medicine, Umeå University, Sweden. Tel.: + 46 90 785 0000; fax: + 46 90 137 633. E-mail address: [email protected] (M.Y. Henein). 1 The Golden Ratio Collaborators. 0167-5273/$ – see front matter © 2011 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.ijcard.2011.05.094
cumference to the length of the larger arc (i.e. c/a = a/b). The golden angle is the angle subtended by the smaller arc (b) and measures 137.5°. The golden ratio decimal (0.618) equates to an angle of 222.5°, the reverse of the golden angle, with the sum adding up to 360°. It has been found among branches and leaves on plant stems [7]. We hypothesised that since the golden ratio and angle are so deeply rooted in nature, they must have some correspondence in the healthy heart structure and function. Accordingly, we studied the heart dimensions and function using 2D or 3D cardiac ultrasound imaging technology at end-diastole, the resting phase of the heart cycle or CT scanning. Individual study data were compared using the unpaired t test. Study 1A: To assess the effect of ethnicity on the left ventricle, the most important heart pump, we took vertical and transverse cardiac measurements from 30 normal Swedish adults (Fig. 1) and compared these with measurements from 30 healthy matched Chinese. We found that Chinese left ventricle dimensions were smaller by 8±1 mm and 5± 0.5 mm, compared to Swedes but the golden ratio was maintained at approximately 1.618. Our findings suggest that it is the dimensions ratio that is important for healthy functioning and not the absolute dimensions. Study 1B: To assess whether the ratio of these dimensions differs in heart failure, we tested them in 40 Swedish patients; 20 with mild heart failure and 20 with end-stage heart failure. In mild heart failure, the left ventricular ratio was reduced (p b 0.001) but the right
240
M.Y. Henein et al. / International Journal of Cardiology 150 (2011) 239–242
Fig. 1. An echocardiographic example of normal 4 chambers of the heart showing measurements of the vertical and transverse cardiac axes. LV, left ventricle and RV, right ventricle.
ventricular ratio was increased due to the enlarged left ventricle encroaching on the right ventricular dimensions (p b 0.001); overall the cardiac ratio remained at around 1.64. However, in end-stage heart failure, the left ventricular ratio was significantly reduced (p b 0.01) and this distorted the cardiac ratio to around 1.4 (p b 0.005), consistent with pathological remodeling resulting in a more spherical shape to the heart. These patients had 50% mortality over the course of 36 months' follow up, compared with 100% survival in the mild group (Fig. 2a and b). Consequently, it appears that as long as the cardiac ratio is maintained at approximately 1.618, survival is likely but a significant deviation from the golden ratio is associated with a poor
outcome. This downward spiral is known to result in mitral and tricuspid valve regurgitation and enlargement of the atria [9]. Study 2: We have previously shown that the normal mitral valve annulus dimensions ratio equates to 1.6 [10]. We then evaluated the left ventricular inlet length and width (mitral annulus dimensions) in 15 normal individuals and 12 age matched patients with dilated cardiomyopathy and functional mitral regurgitation. In normals, the relative mitral annulus circumference length and width proved to be 6.8 ± 0.8 cm and 4.2 ± 0.8 cm respectively, giving a ratio of 1.62, compared with the dilated cardiomyopathy patients, whose dimensions were 8.3 ± 1.8 cm and 5.8 ± 1.1 cm respectively, giving a ratio of 1.42 (p b 0.001). This reduced ratio in patients with dilated left ventricle is consistent with disturbance in mitral valve function and the development of mitral regurgitation, known for its clinical complications and poor prognosis [11–13]. Furthermore, others have shown that the structure and function of the tricuspid and mitral annuli are not independent of each other but are based around the golden ratio of 1.618 [14,15]. The same ratio applies to the extent of the amplitude of the longitudinal motion of the tricuspid and mitral annuli, a clinical measure taken to reflect right and left ventricular function, respectively [16]. We have also previously demonstrated that the golden ratio is seen in normal foetal myocardial function development, where the heart muscle diastolic function matures at a rate of 1.6 mm/s per week between 20 and 40 weeks of age [17], while a healthy adult heart muscle loses velocity of 1.6 cm/s each decade between age 20 and 80 years [18], indicating a close relationship between ageing and ventricular myocardial function. Finally, others have shown that the end product of normal cardiac function, the blood pressure ratio of 120/80 mmHg, also approximates to the golden ratio of 1.618 and carries significant prognostic value [19]. Additionally, myocardial muscle fibre orientation takes a helical shape, with the coils of each
Fig. 2. Echocardiograms (a) from 2 patients with heart failure, mild (left) and severe (right). Graph showing statistical measurements differences between the two heart failure groups and normals (error bars represent standard error).
M.Y. Henein et al. / International Journal of Cardiology 150 (2011) 239–242
segment of the apical loop being of different lengths and having a harmonic proportion which conforms to the golden ratio [20]. When these myocardial fibres become horizontal or transverse, as is the case in heart failure, they weaken overall ventricular function [21]. Study 3A: We assessed the anatomical angle between the right ventricular inlet axis and outflow tract axis in 16 healthy subjects and 19 patients with severe pulmonary hypertension and right heart failure. We found that the normal angle between the axes is approximately 43 ± 3°. The angle between the outflow tract axis and continuation of the inflow tract axis is therefore 138 ± 4°, which is approximately 137.5°. This angle increased significantly to 160 ± 4° (p N 0.001) in right heart failure patients, as the right ventricle became cylindrical and lost its native shape, reducing the inflow-outflow tract angle to approximately 20° from 43° (unpublished). Hence it can be seen that the complex anatomy of the right ventricle, as well as its myocardial fibre architecture, support the unique role of this angle in preserving the peristaltic circulation inside the right ventricle. With severe disease, right ventricular cavity enlargement and remodelling combine to produce loss of the normal golden angle, contributing to the intractable right ventricular pump decompensation. Study 3B: We then measured the angle between the mid-luminal axes of the pulmonary trunk and the proximal ascending aorta in the same 16 healthy subjects. This angle approximated to a mean of 39.5 ± 3.6°, making the angle between the pulmonary trunk and the continuation of the ascending aorta approximately 139 ± 3°, which corresponds with the golden angle of 137.5° (unpublished) (Fig. 3). Normal systolic pressure in the aorta is almost six times that in the pulmonary artery, hence the need for a pressure recovery space between the two arteries to avoid pressure transmission across their thin walls. In hypertension, the pressure difference between the aorta and the pulmonary artery could increase to 10 times the norm. Had the aorta and pulmonary artery been closely parallel, this could potentially have created a significant afterload to the right ventricle, which is avoided by the space provided by the angle between the axes. In addition to these two great arteries, the same angle seems to be adopted by the central and peripheral arterial tree; early anatomical studies demonstrated the branching pattern of the coronary and peripheral arteries to be compatible with the golden angle [22,23]. It appears that the golden angle is not only important for coronary vasculature packing but also for optimising the branching pattern so that perfusion of the myocardial bed is maximised [12]. Electrical function of the heart also seems to follow a parallel pattern [24]: the range of electrical potential of a normal heart lay between −30° and + 110°, representing a total angle of 140°, or approximately 137.5°. An overall optimum healthy heart function has also been observed when there is a graph of convergence of the
Fig. 3. Diagrams showing the golden angle between the ascending aorta and pulmonary trunk (left), and between inlet and outflow tract axes of the right ventricle (right). AO: aorta; PA: pulmonary artery; RA: right atrium; RV: right ventricle.
241
Fibonacci numbers and a phi relationship between the T waves on the surface electrocardiogram [9,10]. Our findings show that the overall cardiac and ventricular dimensions in a normal heart are consistent with an approximation of the golden ratio of 1.618 and the golden angle of 137.5°, representing optimum pump structure and function efficiency. These findings are not only of philosophical interest but may have a significant anatomical, functional and prognostic value. Application of these ratios and angles in our daily clinical practice may lead to development of simple and reliable markers of early deviation from normality, which could be treated before irreversible alteration in cardiac structure and function develops. Acknowledgements Dr. Ying Zhao is partially supported by Umeå University and Swedish Heart and Lung Foundation. The authors of this manuscript have certified that they comply with the Principles of Ethical Publishing in the International Journal of Cardiology [25]. References [1] Maor E. Trigonometric delights. Princeton, N.J: Oxford: Princeton University Press; 2002. [2] Padovan R. Proportion: science, philosophy, architecture. London: E & FN Spon; 1999. [3] Euclid. Similar figures, space, and solids : being a new geometry of the subjectmatter of Euclid, Books V., VI., and XI. London: W.B. Clive; 1926. [4] Tattersall JJ. Elementary number theory in nine chapters. 2nd ed. Cambridge: Cambridge University Press; 2005. [5] Hemenway P. Divine proportion: phi in art, nature, and science. New York: Published by Sterling Pub. Co.; 2005. [6] Livio M. The golden ratio : the story of phi, the world's most astonishing number. New York, NY: Broadway Books; 2002. [7] Stakhov AP, Sluchenkova A. International Club of the Golden Section. Museum of Harmony and the Golden Section : mathematical connections in nature, science and art. Vinnitsa: ITI; 2003. [8] Affleck I. Solid-state physics: golden ratio seen in a magnet. Nature 2010;464: 362–3. [9] Calcutteea ALP, Maras D, Lei W, Hodson M, Henein M. Myocardial determinants of right ventricular ejection in pulmonary hypertension: 2D strain rate and spectral doppler study. Eur J Echocardiogr 2008;9:S53. [10] Chung RDA, Calcuttea A, Pura B, Li W, Pepper JR, Henein MY. Mitral annulus shape change in dilated cardiomyopathy by real-time 3D transthoracic echocardiography. Eur J Echocardiogr 2008;9:S131. [11] Fattouch K, Sampognaro R, Speziale G, et al. Impact of moderate ischemic mitral regurgitation after isolated coronary artery bypass grafting. Ann Thorac Surg 2010;90:1187–94. [12] Ennezat PV, Marechaux S, Le Tourneau T, et al. Myocardial asynchronism is a determinant of changes in functional mitral regurgitation severity during dynamic exercise in patients with chronic heart failure due to severe left ventricular systolic dysfunction. Eur Heart J 2006;27:679–83. [13] Breithardt OA, Sinha AM, Schwammenthal E, et al. Acute effects of cardiac resynchronization therapy on functional mitral regurgitation in advanced systolic heart failure. J Am Coll Cardiol 2003;41:765–70. [14] Tei C, Pilgrim JP, Shah PM, Ormiston JA, Wong M. The tricuspid valve annulus: study of size and motion in normal subjects and in patients with tricuspid regurgitation. Circulation 1982;66:665–71. [15] Ormiston JA, Shah PM, Tei C, Wong M. Size and motion of the mitral valve annulus in man. I. A two-dimensional echocardiographic method and findings in normal subjects. Circulation 1981;64:113–20. [16] Henein MY, Xiao HB, Brecker SJ, Gibson DG. Berheim “a” wave: obstructed right ventricular inflow or atrial cross talk? Br Heart J 1993;69:409–13. [17] Gardiner HM, Pasquini L, Wolfenden J, et al. Myocardial tissue Doppler and long axis function in the fetal heart. Int J Cardiol 2006;113:39–47. [18] Henein M, Lindqvist P, Francis D, Morner S, Waldenstrom A, Kazzam E. Tissue Doppler analysis of age-dependency in diastolic ventricular behaviour and filling: a cross-sectional study of healthy hearts (the Umea General Population Heart Study). Eur Heart J 2002;23:162–71. [19] Ulmer H, Diem G, Bischof HP, Ruttmann E, Concin H. Recent trends and sociodemographic distribution of cardiovascular risk factors: results from two population surveys in the Austrian WHO CINDI demonstration area. Wien Klin Wochenschr 2001;113:573–9. [20] Buckberg GD. Basic science review: the helix and the heart. J Thorac Cardiovasc Surg 2002;124:863–83. [21] Buckberg GD, Coghlan HC, Torrent-Guasp F. The structure and function of the helical heart and its buttress wrapping. V. Anatomic and physiologic
242
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considerations in the healthy and failing heart. Semin Thorac Cardiovasc Surg 2001;13:358–85. [22] Gibson CM, Gibson WJ, Murphy SA, et al. Association of the Fibonacci Cascade with the distribution of coronary artery lesions responsible for ST-segment elevation myocardial infarction. Am J Cardiol 2003;92:595–7. [23] Harvey W, Sylvius Z. The anatomical exercises of Dr. William Harvey, professor of physick, and physician to King Charles the first concerning the motion of the heart and blood : with the preface of Zachariah Wood, physician of Roterdam : to which is
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