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Problem 3 (40 points) – Medical Guidewire A guidewire is a thin, flexible wire used in interventional procedures to navigate through blood vessels and guide devices like catheters to target sites inside the body. Unlike open surgery, which requires large incisions, interventional procedures are minimally invasive, using small entry points and imaging guidance for treatment. The guidewire must accurately transmit motion from the operator’s hand to its tip, making its torsional and bending dynamics critical for safe and precise navigation. Guidewire system can be modeled as a rotational mechanical system, where the torsional stiffness of the wire acts as a rotational spring, and damping arises from friction, deformation and blood viscosity within the vessel. The dynamic behaviour can be represented using standard elements such as rotational inertia, damping, and torsional springs, as shown in the figure. GUIDEWIRE NAVIGATION TO ARTERIAL BRANCHES GUIDEWIRE INSERTION THROUGH FEMORAL ARTERY Guidewire Tip Guidewire Blood vessel wall Viscous drag force Friction force Deformation energy Guiding catheter $\theta_o$ T, $\theta_i$ J k b a. Obtain the differential equations representing guidewire tip angle $\theta_o(t)$ when a certain torque is applied T(t). b. Obtain the transfer functions G(s) = $\theta_o(s)$/T(s) (assume zero initial conditions) c. Given the numerical values J = 0.1, b = 2 and k = 100. Assume that the physicians’ hand jerked for a moment with a torque T = 10Nm/s over a duration of 0.1 s; that the magnitude of the input is 10 x 0.1 = 1 N-m. Since the torque T applied to the rotor is of short duration, but relatively large amplitude, it can be approximated as an impulse input ($\delta$) – you do not have to calculate this specific input T, just use an ideal impulse ($\delta$) as input. Find the response $\theta_o(t)$ of the system and plot the response over the time interval 0 to 2 seconds. d. Assume the guidewire is lodged in the vasculature structure, such that now b = 20. Find the response $\theta_o(t)$ of the system of the system and plot the response over the time interval 0 to 2 seconds.

Problem 3 (40 points) – Medical Guidewire
A guidewire is a thin, flexible wire used in interventional procedures to navigate through blood vessels and guide devices like catheters to target sites inside the body. Unlike open surgery, which requires large incisions, interventional procedures are minimally invasive, using small entry points and imaging guidance for treatment. The guidewire must accurately transmit motion from the operator’s hand to its tip, making its torsional and bending dynamics critical for safe and precise navigation.
Guidewire system can be modeled as a rotational mechanical system, where the torsional stiffness of the wire acts as a rotational spring, and damping arises from friction, deformation and blood viscosity within the vessel. The dynamic behaviour can be represented using standard elements such as rotational inertia, damping, and torsional springs, as shown in the figure.
GUIDEWIRE NAVIGATION TO ARTERIAL BRANCHES
GUIDEWIRE INSERTION THROUGH FEMORAL ARTERY
Guidewire Tip
Guidewire
Blood vessel wall
Viscous drag force
Friction force
Deformation energy
Guiding catheter
$\theta_o$
T, $\theta_i$
J
k
b
a. Obtain the differential equations representing guidewire tip angle $\theta_o(t)$ when a certain torque is applied T(t).
b. Obtain the transfer functions G(s) = $\theta_o(s)$/T(s) (assume zero initial conditions)
c. Given the numerical values J = 0.1, b = 2 and k = 100. Assume that the physicians’ hand jerked for a moment with a torque T = 10Nm/s over a duration of 0.1 s; that the magnitude of the input is 10 x 0.1 = 1 N-m. Since the torque T applied to the rotor is of short duration, but relatively large amplitude, it can be approximated as an impulse input ($\delta$) – you do not have to calculate this specific input T, just use an ideal impulse ($\delta$) as input. Find the response $\theta_o(t)$ of the system and plot the response over the time interval 0 to 2 seconds.
d. Assume the guidewire is lodged in the vasculature structure, such that now b = 20. Find the response $\theta_o(t)$ of the system of the system and plot the response over the time interval 0 to 2 seconds.

Added by Timothy P.

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