Page 75 - Computational Modeling in Biomedical Engineering and Medical Physics

62 Computational Modeling in Biomedical Engineering and Medical Physics

proportional to the group K B t 1=2 = t 1 1 t 2 Þ. The metabolic rate of the animal
ð
emerges as a constraint, as in Eq. (2.21)
1=2
t
2 BK constantÞ:
t 1 1 t 2 ð ð2:25Þ
The power consumption of the heart is then proportional to the inverse of
1=2 _
t 2 1 2 Kt . It follows that W-N for t 2 - 0; K 2 2f g, and at the intermediate
22
value t 2;opt B 4=9 K it exhibits a relatively sharp minimum. The constraint K on

the mass transfer provides the contraction time t 1;opt B 2=9 K 22 . Interesting enough
(Bejan, 2000a,b)
21=2
AD ΔC
22
t 1 ; t 2 Þ BK 5 ; t 1;opt =t 2;opt B1=2;
_ m
ð opt ð2:26Þ
whichare valid for animalsinawide rangeofsizes (Peters, 1983; Schmidt-Nielsen, 1984).
It may be inferred the existence of an eigen frequency of the heartbeats that mini-
mizes the mechanical power consumed by the heart, subject to the constraints of the
interface and the mass transfer, or metabolic rate of O 2 . This maximization with
respect to the active diffusion time interval is the foundation of all optimal pulsations
in the engineered and natural systems.

Coupled rhythms in the cardio-pulmonary system
Within the framework of the thermodynamics with finite speed (TFS), the cardio-
pulmonary system (CPS), may be considered as an ensemble of two biological machines,
naturally designed and optimized: the heart, a “naturally designed” blood (liquid) pump,
and the lungs, “naturally designed” air compressors (Petrescu et al., 2018). Studies based
on a large number of measurements, for stationary states related to different positions:
walking, sitting, laying, repetitive exercise, etc., have shown that the two frequencies—
for heart, f H , and lung, f L —are correlated, for a healthy person, through

f H 5 f L 2 1 N=4 : ð2:27Þ

Here N is an integer, called quantum number of the interaction between the heart and
lung in a stationary state. It is thought of as the interaction parameter that links the
two biological machines (heart and lungs), namely the difference in phase between
them (Petrescu et al., 2018). The analogy with classical thermodynamics indicates that
Eq. (2.27) relates three state parameters (f H , f L , N), of which obviously only two may
be independent. Three particular processes are then identified: (1) iso-pulse
(f H 5 const.); (2) iso-rhythm (f L 5 const.); and (3) iso-quantum (N 5 const.). Moreover, it
has been observed that CPS works properly (healthy) only if certain coordination in