Convolution Q: Ca(t) & R(t) for CT Attenuation Model

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In summary, the conversation is about a user apologizing for hijacking a thread and starting their own to ask a question about convolution. They provide the functions Ca(t) and R(t) and ask for help understanding how to perform the convolution. They also mention the context of the convolution in determining CT tissue attenuation in tracer kinetics modelling. Another user asks if they want to do the convolution manually or with a tool and the first user corrects a small error in their previous statement.
  • #1
SORRY, I figured it was wrong of me to hijack someone elses thread with my query so I will start my own thread, my apologizes in advance for I will also post a thread within the Math forum...

I am new here and I bring a similar question for my first post...

I want to know how to do a convolution where the two functions are:

Ca(t) - arbitrary Input function
(it actually represents the time activity curve of a CT contrast bolus injection in the blood)

R(t) - a piecewise function defined as follows:

R(t) = 1, 0<t<Tm
and E*(exp)^(kt), t>Tm

(this R(t) is called the Impulse Residue Function for the Johnson WIlson model for capillary tracer exchange)

so therefore:

Ca(t)*R(t) = (from 0 to Tm){Ca(t) convolved with 1} + (from Tm to t)E*{Ca(t) convolved with (*exp)^(-kt)}

Can anyone shed some light on this please?!

If anyone is curious the context of this convolution is for determining the representation of CT tissue attenuation in tracer kinetics modelling, considering a distributive parameter model. A background link for those interested is below.
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  • #2
do u want to do it manually or do u want to use a tool??
  • #3
manual, hand solution..

btw I realized I typed a small error...

the second part of R(t) should read E*(exp)^(-k(t-Tm)), t>Tm

where t is the variable of integration and Tm is a constant

Related to Convolution Q: Ca(t) & R(t) for CT Attenuation Model

1. What is the purpose of the Convolution Q: Ca(t) & R(t) for CT Attenuation Model?

The purpose of this model is to predict the attenuation of a material using the convolution of two functions, Ca(t) and R(t), which represent the composition and density of the material, respectively. This model is commonly used in computed tomography (CT) imaging to accurately measure the attenuation of X-rays as they pass through different materials.

2. How is the Convolution Q: Ca(t) & R(t) for CT Attenuation Model used in CT imaging?

In CT imaging, the model is used to determine the attenuation coefficients of different materials within the body. These coefficients are then used to generate cross-sectional images of the body, allowing for the detection and diagnosis of various medical conditions.

3. What are the benefits of using the Convolution Q: Ca(t) & R(t) for CT Attenuation Model?

One of the main benefits of this model is its ability to accurately predict the attenuation of X-rays in different materials, allowing for more precise and detailed CT images. Additionally, this model is relatively simple and can be applied to a wide range of materials, making it a valuable tool in medical imaging.

4. Are there any limitations to the Convolution Q: Ca(t) & R(t) for CT Attenuation Model?

While this model is widely used and has proven to be effective in CT imaging, it does have some limitations. It assumes that the material being imaged is homogeneous and has a constant density, which may not always be the case in real-world scenarios. Additionally, this model does not take into account other factors that may affect attenuation, such as beam hardening or scatter.

5. How is the Convolution Q: Ca(t) & R(t) for CT Attenuation Model related to other imaging techniques?

This model is specifically used in CT imaging, which uses X-rays to generate images of the body. However, the concept of convolution is also used in other imaging techniques, such as magnetic resonance imaging (MRI) and positron emission tomography (PET), to improve image quality and accuracy.

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