"# <center>Using Statistical Models in Perfusion Kinetic Modeling</center>\n",
"\n",
"<center>by Ethan Tu</center>"
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"Currently, statistical models are not used for modeling perfusion kinetics or used to determine the parameters that dictate flow or the flux of contrast agent (CA) within the body. This is simply because perfusion kinetics in relation to dynamic contrast-enhanced imaging does not require a set of statistical assumptions concerning the generation of sample data. Normally, passage and uptake of the CA then cause time-dependent changes in signal intensity, which, in turn, allow for the determination of time-resolved CA concentrations. From these time-resolved contrast agent concentrations, a range of hemodynamic parameters can then be derived using the well-known principles of tracer-kinetic theory. Put another way, the uptake of contrast agent by surrounding tissue is unique to every person, and in a research setting we account for this on a person to person basis. Using a statisical model to represent the tissue response is possible, but has largely been ignored because it is not the purpose of such modeling.\n",
"\n",
"There is, perhaps one application of statisical modeling that could be used for kinetics. Statisical models have been widely used for turbulence. Famously, turbulence has been an unsolved physics problem; a smooth solution to the Navier-Stokes equation has yet to be found in turbulent settings. Statisical models are used for turbulence because the behavior of particles, their collisions, dispersion, and clustering, are all statisical in nature. Turbulence, and the Navier-Stokes equation in general, are used to describe the motion of fluid in space. Conveniently, perfusion kinetics deals with trying to describe the motion of blood/plasma in blood vessels. Therefore, if we want to overcomplicate our normal compartmental models, we can describe the influx and efflux of our tracer not in terms of ordinary differential equations, but partial differential equations governed by the Navier-Stokes equation.\n",
"\n",
"Previously described statistical models such as ones used for heavy particles or intertial particles in a homogeneous isotropic turbulence environment can be used to describe the flow of blood/plasma.$^1$$^2$ In normal differential equations as explained in the previous report, we have a constant value for flow that needs to be optimized. Instead, in this theoretical hybrid model, we would instead use the statistical model as our input function Cin to describe this flow. This is not a novel idea. Holway in 1966 described new methods to describe kinetic theory with statistical modeling.$^3$ Outlined in his paper, the method is only applicable to quite general systems in nonequilibrium statistical mechanics. The major drawbacks of this hybrid method is that it is unnecessarily complicated, computationally expensive, and may not even yield more accurate results. While we *can* use statistical models to define flow, we can already simply measure flow directly in the patient's blood vessels. Why simulate it when we can already measure it? Large statistical models, especially those with thousands of particles to represent fluids, are notoriously computationally expensive. While yes, it may be able to complete in a day, the purpose of perfusion kinetics in dynamic contrast ehanced imaging is to give near instantaneous point of care results. There's no reason to use this hybrid method, even if it is slightly more accurate, if it will take hours to simulate.\n",
"\n",
"In conclusion, statistical models are not currently used in the field of dynamic contrast enhanced imaging. While it is possible, the drawbacks are too significant to allow for mainstream use."
]
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"---\n",
"# References\n",
"\n",
"[1]\tK. Gustavsson and B. Mehlig, “Statistical models for spatial patterns of heavy particles in turbulence,” Advances in Physics, vol. 65, no. 1, pp. 1–57, Jan. 2016, doi: 10.1080/00018732.2016.1164490.\n",
"\n",
"[2]\tL. I. Zaichik, O. Simonin, and V. M. Alipchenkov, “Two statistical models for predicting collision rates of inertial particles in homogeneous isotropic turbulence,” Physics of Fluids, vol. 15, no. 10, pp. 2995–3005, Sep. 2003, doi: 10.1063/1.1608014.\n",
"\n",
"[3]\tL. H. Holway, “New Statistical Models for Kinetic Theory: Methods of Construction,” The Physics of Fluids, vol. 9, no. 9, pp. 1658–1673, Sep. 1966, doi: 10.1063/1.1761920."
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%% Cell type:markdown id: tags:
# <center>Using Statistical Models in Perfusion Kinetic Modeling</center>
<center>by Ethan Tu</center>
%% Cell type:markdown id: tags:
Currently, statistical models are not used for modeling perfusion kinetics or used to determine the parameters that dictate flow or the flux of contrast agent (CA) within the body. This is simply because perfusion kinetics in relation to dynamic contrast-enhanced imaging does not require a set of statistical assumptions concerning the generation of sample data. Normally, passage and uptake of the CA then cause time-dependent changes in signal intensity, which, in turn, allow for the determination of time-resolved CA concentrations. From these time-resolved contrast agent concentrations, a range of hemodynamic parameters can then be derived using the well-known principles of tracer-kinetic theory. Put another way, the uptake of contrast agent by surrounding tissue is unique to every person, and in a research setting we account for this on a person to person basis. Using a statisical model to represent the tissue response is possible, but has largely been ignored because it is not the purpose of such modeling.
There is, perhaps one application of statisical modeling that could be used for kinetics. Statisical models have been widely used for turbulence. Famously, turbulence has been an unsolved physics problem; a smooth solution to the Navier-Stokes equation has yet to be found in turbulent settings. Statisical models are used for turbulence because the behavior of particles, their collisions, dispersion, and clustering, are all statisical in nature. Turbulence, and the Navier-Stokes equation in general, are used to describe the motion of fluid in space. Conveniently, perfusion kinetics deals with trying to describe the motion of blood/plasma in blood vessels. Therefore, if we want to overcomplicate our normal compartmental models, we can describe the influx and efflux of our tracer not in terms of ordinary differential equations, but partial differential equations governed by the Navier-Stokes equation.
Previously described statistical models such as ones used for heavy particles or intertial particles in a homogeneous isotropic turbulence environment can be used to describe the flow of blood/plasma.$^1$$^2$ In normal differential equations as explained in the previous report, we have a constant value for flow that needs to be optimized. Instead, in this theoretical hybrid model, we would instead use the statistical model as our input function Cin to describe this flow. This is not a novel idea. Holway in 1966 described new methods to describe kinetic theory with statistical modeling.$^3$ Outlined in his paper, the method is only applicable to quite general systems in nonequilibrium statistical mechanics. The major drawbacks of this hybrid method is that it is unnecessarily complicated, computationally expensive, and may not even yield more accurate results. While we *can* use statistical models to define flow, we can already simply measure flow directly in the patient's blood vessels. Why simulate it when we can already measure it? Large statistical models, especially those with thousands of particles to represent fluids, are notoriously computationally expensive. While yes, it may be able to complete in a day, the purpose of perfusion kinetics in dynamic contrast ehanced imaging is to give near instantaneous point of care results. There's no reason to use this hybrid method, even if it is slightly more accurate, if it will take hours to simulate.
In conclusion, statistical models are not currently used in the field of dynamic contrast enhanced imaging. While it is possible, the drawbacks are too significant to allow for mainstream use.
%% Cell type:markdown id: tags:
---
# References
[1] K. Gustavsson and B. Mehlig, “Statistical models for spatial patterns of heavy particles in turbulence,” Advances in Physics, vol. 65, no. 1, pp. 1–57, Jan. 2016, doi: 10.1080/00018732.2016.1164490.
[2] L. I. Zaichik, O. Simonin, and V. M. Alipchenkov, “Two statistical models for predicting collision rates of inertial particles in homogeneous isotropic turbulence,” Physics of Fluids, vol. 15, no. 10, pp. 2995–3005, Sep. 2003, doi: 10.1063/1.1608014.
[3] L. H. Holway, “New Statistical Models for Kinetic Theory: Methods of Construction,” The Physics of Fluids, vol. 9, no. 9, pp. 1658–1673, Sep. 1966, doi: 10.1063/1.1761920.
