Question $3(40pts).$ A simple three$-$element model of active cardiac muscle contraction consists

of a passive non$-$linear spring in parallel with a contractile and passive series element, as shown

in the diagram. The model structure and equations have been modified from Fung.

I

The total tension $T$ in the muscle is given by the sum of tensions in the parallel and series

elements:

$T=T_p+T_s$

A simple three$-$element model of active cardiac muscle contraction consists of a passive non$-$

linear spring in parallel with a contractile and passive series element, as shown in the diagram.

The model structure and equations have been modified from Fung.

The total tension $T$ in the muscle is given by the sum of tensions in the parallel and series

elements:

$T=T_p+T_s$

where $T_p$ and $T_s,$ the tensions in the parallel and series elements respectively, are

given by:

$T_p=(e^{(L-L_0)}-1),T_s=(e^{L_s}-1)$

where $,$ are parameters and $L0$ denotes the resting length of the muscle. For the

contractile element, the velocity of its shortening is described by:

$\frac{d{L}_c}{(d)t}=\frac{a[T_s-S_{0}f(t)]}{T_s+S_0}$

where $a,,s0$ are muscle parameters and $f(t)$ is the muscle activation function

given by

with to and ti$,p$ constants defining activation phase offset and the time to peak isometric

contraction respectively.

Model parameters for a cardiac papillary muscle specimen are given in the table below:

Note that in the relaxed state, the length of the series element $LS$ is $0.$

$($a$)$ Using Python, solve for and plot the total tension $T$ against time during an isometric

contraction in which the muscle is clamped at its resting length $L0.$

$($b$)$ Solve for and plot muscle length $L$ against time during an isotonic contraction in which the

muscle is allowed to freely contract with no imposed load $($i$.$e$.T=0).$