Page 147 - Advances in Biomechanics and Tissue Regeneration

8.2 CARDIAC MECHANICS AND MODEL 143

FIG. 8.1 Three-dimensional fiber distribution across the biventricle heart.
Following several studies made on rat cardiac tissue, Guccione et al. isolated the sarcomere length and the calcium ion
concentration as the principal features that dictate the contraction mechanism and introduced them via a modified Hill
model. The computation of the active contraction force, T active , along the fibers of the heart is carried out using the
following equation:
Ca 2 0
active
T ¼ T max 2 2 C t , (8.7)
Ca + ECa
0 50
where Ca 0 is the peak intracellular calcium ion concentration, ECa 50 the extracellular calcium ion concentration when
50% of peak active contraction force is achieved, and T max is the isometric tension force under maximal activation. C t is
the variable that defines its time transient as

1
C t ¼ ð1 cosωÞ: (8.8)
2
ω is a function of time and is defined over three ranges:
t
8
π
> when 0 t < t 0 ,
>
>
< t 0
π when t 0 t t 0 + t r ,
ω ¼ t t 0 + t r (8.9)
>
t r
>
>
0 when t 0 + t r t,
:
where the range 0 t < t 0 represents the time span to linearly reach maximal activation from rest; the range t 0 t t 0 + t r ,
the relaxation time to reach zero contraction from its peak value; and finally the range t 0 + t r t when no contraction is
present. t is the current time, t 0 is the time at which peak tension is reached, and t r is the time taken for the tension force to
dissipate. For the effect of the sarcomere stretch to be accounted for, the relaxation time, t r , is calculated using the actual
sarcomere length, l sarcomere :
t r ¼ ml sarcomere + b, (8.10)
where constant m is the slope of linear relaxation linking relaxation duration to sarcomere length and b its time inter-
cept. The calcium concentration at 50% tension, ECa 50 , in Eq. (8.7) is defined as
max
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: (8.11)
ðCa 0 Þ
ECa 50 ¼ p sarcomere
e Bðl sarcomere l 0 Þ 1
sarcomere
(Ca 0 ) max represents the highest intracellular calcium ion concentration, l 0 is the sarcomere length below which
no active tension occurs, and B is a constant governing the shape of the peak isometric tension-length relation.

I. BIOMECHANICS