Introduction<\/strong><\/p>\nExtensive work on hemodynamic and blood movement and its nature has been conducted and theories have been put forward. The already existing theory states skeletal muscle contraction as a factor for venous return but our ambition is to mathematically find out how acceleration due to gravity and force of gravity affects the flow of blood. We would also try to investigate the shear which makes blood as a continuum to move. If we take up a clinical case where the subject has paralysis and skeletal muscles are not contracting, we fail to explain how blood is returning. If considerable amount of blood fails to return it means that it is antagonistic to Fick\u2019s principle. Considering the fluid mechanics, it is obvious that since blood is a continuum, it has to have a shear for movement. Shear here are the unbalanced force components which make fluid movement possible. In 1895 Leonard Hill from University College, London, Department of Physiology made a detailed study on effect of gravitation on blood and cerebrospinal fluid in his paper \u201cTHE INFLUENCE OF THE FORCE OF GRAVITY ON THE CIRCULATION OF THE BLOOD\u201d. But he failed to provide mathematical explanations to the phenomenon he investigated.<\/p>\n
This simple principle can be well grounded and applied in health economics for production ethics in vaccination and medicinal recovery process. According to Austrian school of economics, liberal guidance in ethics as well as CSR drives the foundation for development of health related recovery which can be fastened if the theoretical foundations of classical economics can be suppressed. Hayek further argues on the need for retrospective knowledge based on logic and deductive theory rather than induction in various journals. The need for CSR and ethics in the production campaign of haemodynamical solution to recovery analysis can be far outreached and sustained.<\/p>\n
Jason H Hamann, Zoran Valic, John B Buckwalter and Philip S Clifford of Medical College of Wisconsin and Veterans Affairs Medical Center, Milwaukee, Wisconsin, in their paper titled \u2018Muscle pump does not enhance blood flow in exercising skeletal muscle\u201d; 2002 made several experiments on hind-limb blood flow in Mongrel dog. They too concluded that venous blood from cutanoeus branches maybe affected by skeletal muscle contraction but venous return as a whole cannot be affected by it. Although they suspected that unsynchronized muscle contraction may help venous return to sudden extent during exercises. This is because during exercise or any kind of physical activity like walking, jogging, etc. the suction-impulse pumps (SIP) and muscle-venous-articular pumps (MVAP) are stimulated (Comparison of reduction of edema after rest and after muscle exercises in treatment of chronic venous insufficiency; Belczak Cleusa Ema Quilici, Cavalheri Gildo, Jr, Jose Maria Pereira de Godoy, corresponding author- Belczak Sergio Quilici, and Caffaro Roberto Augusto) <\/p>\n
The authors effectively kept their points because synchronized skeletal muscle contraction means two force vectors acting towards each other; resulting into zero movement of fluid in between by nullifying each other. But unsynchronized skeletal muscle contraction will give rise to unbalanced force vectors which might help the fluid in movement. <\/p>\n
Therefore, it is obvious that skeletal muscle contraction is a minor factor to influence the movement. In this regard, effect of gravitation has to be studied and understood as well gravitation.<\/p>\n
The aim of the study was to investigate an explanation for such phenomenon of return of blood to heart without contraction of skeletal muscle and effect of force of gravity in this regard.<\/u><\/p>\n
Methods<\/strong><\/u><\/p>\nFor centuries, there persists unsolved mystery of human physiology and rheumatology. It is still unclear how venous return of blood inside human veins happen and how being in gravitational field, our blood flow is affected by gravitational force. Theories have been put up but with discrepancies and fallacies. A deep mathematical study and dynamical derivations have been carried on to show how gravitation influences blood flow and what exactly is the shear equation of the blood. Equation has also been derived for calculating blood pressure of any blood vessel at a particular moment. Based on the mathematical findings have come up with futuristic economical plan and technology development scheme to enhance and apply the theory in practical clinical application around the globe. Our work provides the far future outlook about extra-terrestrial human settlement as well. Futuristic economic scheme have been proposed based on the equations.<\/p>\n
We took<\/p>\n
\n- The principles of hemodynamic as a fundamental instrument of investigation.<\/li>\n
- Calculus was used as a mathematical tool.<\/li>\n
- Study of blood\u2019s surface tension and thus its nature of flow was investigated.<\/li>\n
- General mechanics to clarify blood movement through bended blood vessels.<\/li>\n<\/ul>\n
As per law given by George Simon Ohm for electrical circuits (tr., The Galvanic Circuit Investigated Mathematically<\/em>) (1827) <\/p>\ndV = I x R; where we calculate V = potential difference<\/p>\n
I = current passing through the conductor and<\/p>\n
R = resistance<\/p>\n
In hemodynamics same analogical relationship is used to derive the relationship between pressure, rate of flow and resistance of vessel.<\/p>\n
dP = Q x R; where dP = pressure difference<\/p>\n
Q = flow of blood through vessel<\/p>\n
R = resistance of vessel<\/p>\n
But force exerted by blood on blood vessel in not being calculated here. By using relationships of general mechanics, we may come to few conclusions. Although above relationship does successfully describes the resistance factor.<\/p>\n
We know,<\/p>\n
P = F\/A (where, P = pressure, F = force & A = area)<\/p>\n
We, will consider area of a cylinder since blood vessels are almost cylindrical in shape and not area of a circle.<\/p>\n
Thus; P = F \/ (2\u03c0r Sf<\/sub> + 2\u03c0r2<\/sup>)<\/p>\n= F \/ 2\u03c0r (Sf<\/sub> +r) \u2026\u2026\u2026\u2026\u2026 equation (1)<\/p>\nHence, from equation (1) we get that;<\/p>\n
At constant Force; P \u221e 1 \/ r (Sf<\/sub> + r)<\/p>\nPressure in small vessels > Pressure in big vessels<\/p>\n
Here it is to be noted that the pressure maintained has to be considered as per the radius change. (Central blood pressure: current evidence and clinical importance; Carmel M. McEniery, John R. Cockcroft, Mary J. Roman, Stanley S. Franklin, and Ian B.Wilkinson)<\/p>\n
\n- Maiti (Department of applied mathematics IIT BHU, India) and J. C Mishra (Institute of Technical Education & Research, Siksha \u2019O\u2019 Anusandhan University, Bhubaneswar, India) describes that radius of blood vessel is not constant for a single vessel but doesn\u2019t describes the relation between pressure and changing radius. This maybe concluded in the following manner;<\/li>\n<\/ol>\n
For an infinitesimally small change in area of vessel to pressure, we will be considering infinitesimally small area, and will consider area of a circle as area of vessel.<\/p>\n
Hence<\/p>\n
P = F \/ A \u2026\u2026 equation (2) (where A = area is a vector)<\/p>\n
Or, P = F \/ \u03c0r2<\/sup> (since, A is area of circle)<\/p>\nIntegrating both sides as a function of radius \u201cr\u201d we get \u2013<\/p>\n
P\uab4d dr = F \/ \u03c0\uab4d (1\/ r2<\/sup>) dr<\/p>\n Or, P (r) = – F \/ (\u03c0r) + C (where, C = integration constant) \u2026\u2026.. Equation (3)<\/p>\n
At unit force applied on unit radius, when F = 1, r = 1, then P = (1\/\u03c0) from equation (2).<\/p>\n
Taking mod function in equation (3) i.e Pressure in dA area, we get C = 0<\/p>\n
Therefore, P = – (F \/ \u03c0r2<\/sup>) \u2026\u2026\u2026. Equation (4) <\/p>\nWe would consider -F = F4<\/sub> for convenience of further calculations.<\/p>\nNegative sign in equation (4) suggests that every fluid has negative force acting against increasing or decreasing pressure gradient to prevent the change of state of motion of fluid molecules, i.e. analogous to a force providing inertia.<\/p>\n
Here it is important to note that F4<\/sub> acts in a direction opposite to the direction of increasing Pressure and Pressure will increase with shortening radius.<\/p>\nThe equation (4) can be proven in another analogical fashion. It is as follows ;-<\/p>\n
If we take two points with pressure and radius (P1<\/sub>, r1<\/sub>) and (P2<\/sub>, r2<\/sub>) where (P1 <\/sub>> P2<\/sub>) and (r1<\/sub> > r2<\/sub>) then taking;-<\/p>\nP = F \/ (\u03c0r2<\/sup>)<\/p>\nFinding difference between two points;<\/p>\n
(P2<\/sub> \u2013 P1<\/sub>) = (F\/\u03c0) (1\/r2<\/sub>2<\/sup> \u2013 1\/r1<\/sub>2<\/sup>)<\/p>\nOr, dP = (F\/\u03c0) [ (r2<\/sub> \u2013 r1<\/sub>) (r2<\/sub>+r1<\/sub>) \/ (r2<\/sub>2<\/sup>)(r1<\/sub>2<\/sup>)]<\/p>\nOr, dP = (F\/\u03c0) (r2<\/sub>+r1<\/sub>)\/(r2<\/sub>2<\/sup>)(r1<\/sub>2<\/sup>) dr<\/p>\nIntegrating both sides of the above equation we get;-<\/p>\n
\uab4d dP = (F\/\u03c0) [\uab4d (r1<\/sub>)\/(r2<\/sub>2<\/sup>r1<\/sub>2<\/sup>) dr + \uab4d (r2<\/sub>)\/(r2<\/sub>2<\/sup>r1<\/sub>2<\/sup>) dr] \u2026\u2026. Equation (i)<\/p>\nLet us consider; (r1<\/sub>)\/(r2<\/sub>2<\/sup>r1<\/sub>2<\/sup>) = i1<\/sub> and (r2<\/sub>)\/(r2<\/sub>2<\/sup>r1<\/sub>2<\/sup>) = i2<\/sub> ; using integral function by parts with the principle; \uab4duv dx = u \uab4dv dx – \uab4d[du\/dx \uab4d v dx] dx; we get ;-<\/p>\ni1<\/sub> = \uab4d r2<\/sub>-2<\/sup>r1<\/sub>-1<\/sup> dr = (r2<\/sub>-2<\/sup>) \uab4d r1<\/sub>-1<\/sup> dr – \uab4d [d(r2<\/sub>-2<\/sup>)\/dr \uab4d (r1<\/sub>-1<\/sup>) dr] dr = r2<\/sub>-2<\/sup> ln r1<\/sub> – \uab4d[(-2r2<\/sub>-3<\/sup>) ln r1<\/sub>] dr = (ln r1<\/sub>)\/r2<\/sub>2<\/sup> + 2 \uab4d(r2<\/sub>-3<\/sup> ) (ln r1<\/sub>) dr \u2026 equation (ii)<\/p>\nNow let us consider \uab4d(r2<\/sub>-3<\/sup> ) (ln r1<\/sub>) dr = i3<\/sub>\u2019<\/p>\nI3<\/sub>\u2019 = ln r1<\/sub> \uab4d r2<\/sub>-3<\/sup> dr – \uab4d [d(ln r1<\/sub>)\/dr \uab4d r2<\/sub>-3<\/sup> dr] dr = ln r1<\/sub> (r2<\/sub>-2<\/sup>)\/(-2) – \uab4d (r2<\/sub>-2<\/sup>\/-2r1<\/sub>)dr = – ln r1<\/sub>\/(2r2<\/sub>2<\/sup>) + \u00bd \uab4d (1\/r1<\/sub>r2<\/sub>2<\/sup>)dr = – ln r1<\/sub>\/(2r2<\/sub>2<\/sup>) +1\/2 \uab4d(r1<\/sub>-1<\/sup>)(r2<\/sub>-2<\/sup>)dr = – ln r1<\/sub> (2r2<\/sub>2<\/sup>) + \u00bd (i1<\/sub>) \u2026\u2026. Equation (iii)<\/p>\nSo, if we consider ln r1<\/sub>\/r2<\/sub>2<\/sup> = \u03b1 and ln r2<\/sub>\/r1<\/sub>2<\/sup> = \u03b2<\/p>\nHence<\/p>\n
i3<\/sub>\u2019 = – \u03b1\/2 + i1<\/sub>\/2 \u2026\u2026. Equation (iv)<\/p>\nSimilarly it can be stated; i2<\/sub> = ln r2<\/sub>\/(r1<\/sub>2<\/sup>) + 2 \uab4d( r1<\/sub>-3<\/sup>) (ln r2<\/sub>) dr<\/p>\nConsidering; \uab4d( r1<\/sub>-3<\/sup>) (ln r2<\/sub>) dr = i4<\/sub>\u2019<\/p>\nIf we consider; [\uab4d (r1<\/sub>)\/(r2<\/sub>2<\/sup>r1<\/sub>2<\/sup>) dr + \uab4d (r2<\/sub>)\/(r2<\/sub>2<\/sup>r1<\/sub>2<\/sup>) dr] = i<\/p>\nThen<\/p>\n
i = i1<\/sub> + i\u00ad2<\/sub> = [\u03b1 + 2(-\u03b1 + i1<\/sub>\/2)] + [\u03b2 +2 (-\u03b2 + i2<\/sub>\/2)] = (\u03b1 – 2\u03b1 + i1<\/sub>) + (\u03b2 -2\u03b2 + i2<\/sub>) = – \u03b1 + i1<\/sub> \u2013 \u03b2 + i2<\/sub> = – [ (\u03b1+\u03b2) – (i1<\/sub> + i2<\/sub>)] = – \u03da<\/p>\nWhere \u03da signifies integral continuity of radius.<\/p>\n
Thus, by equation (i)<\/p>\n
P = (F\/\u03c0) [ \u2013 {(\u03b1+\u03b2) – (i1<\/sub> + i2<\/sub>)} ] = – (F\/\u03c0) \u03da<\/p>\nThe type of flow of blood and its rheological property is described by Reynold\u2019s equation. But a change in the equation maybe done and reframed as \u2013<\/p>\n
Re<\/sub> = \u03f1 d v \/ \u03b7 (where; Re<\/sub> = Reynold\u2019s number, \u03f1 = density, d = diameter of vessel, v = velocity of blood flow &\u03b7 = viscosity of blood)<\/p>\nFrom equation (1) and from relation [P = (\u03f1 Sf<\/sub> g)] we get;-<\/p>\nRe<\/sub> = (P 2r v) \/ Sf<\/sub> g \u03b7<\/p>\nOr, Re<\/sub> = F 2r v \/ 2\u03c0r (Sf<\/sub> + r) Sf<\/sub> g \u03b7<\/p>\nOr, Re<\/sub> = F v \/ \u03c0 Sf<\/sub> g \u03b7 (Sf<\/sub> + r) \u2026\u2026\u2026\u2026\u2026 Equation (5)<\/p>\nWhen, length of vessel and radius of vessel are constant i.e.<\/p>\n
[\u03c0 Sf<\/sub> g (Sf<\/sub> + r)] = \u0416; constant.<\/p>\nThus; Re <\/sub>= F.v \/\u0416 \u03b7 (F and v both are vector quantities)<\/p>\nHence<\/p>\n
Re <\/sub>= |F||v| cos\u03f4 \/ \u0416 \u03b7 \u2026\u2026\u2026. Equation (5)<\/p>\nIt is to be noted that in Non-Newtonian fluids like blood viscosity; \u03b7 is variable and in Newtonian fluid it is constant.<\/p>\n
Here it can be noted that Human beings cannot be settled on planets like Mars (gmars<\/sub> = gearth <\/sub>\/ 2.5 approximately) or satellites like Moon (gmoon<\/sub> = gearth<\/sub> \/ 6 approximately). It also indicates another reason for not development of any higher life-forms in any other celestial bodies apart from Earth. It signifies that in such places the flow of blood will become highly turbulent which will lead to production of Eddy currents in blood and rupture vessel wall; leading to death.<\/p>\nWe also know that;<\/p>\n
F = G (Mm\/R2<\/sup>)<\/p>\n[Where; F = force of gravity, G = gravitation constant, M = mass of planet, m = mass of subjected body and R = radius of the celestial body; none of which are changeable]<\/p>\n
For two parallel flat areas of fluid of the same size A which are separated by a distance dx (Fig. 1). If the plates are moving in the same direction at different velocities V1 and V2, then the force required to maintain the speed is proportional to the difference in speed through the liquid, or the velocity gradient <\/p>\n
F\/A = \u03b7 dy\/dx ; where \u03b7 is the viscosity of the fluid.<\/p>\n
The shear rate describes the shearing which the liquid experiences. In other words, the velocity gradient, dy\/dx, which is a measure of the change in speed at which the intermediate layers move with respect to each other is called the shear rate (\u03b3) and its unit of measure, is the reciprocal second (s-1<\/sup>).<\/p>\nThe term F\/A indicates the force per unit area and is the shearing action or shear stress (\u03c4). The unit of measurement of shear stress is dynes per square centimeter (dynes\/cm2<\/sup>) or Newton per square meter (N\/m2<\/sup>). According to the above statement, viscosity can be defined as shear stress over shear rate:<\/p>\n\u03b7 = \u03c4 \/\u03b3 (Where \u03c4 = V\/A and \u03b3 = F\/A)<\/p>\n
We clearly may conclude that F4<\/sub> acts on the opposite direction of increasing pressure gradient.<\/p>\n\n\n\n