Hyperpolarized 129-Xe is an important resource in many fields of medical physics and MRI research. The physics of the efficient production of hyperpolarized 129-Xe is therefore equally worth investigation. The main process of hyperpolarizing 129-Xe is Spin Exchange Optical Pumping (SEOP) and is dependent on several physical factors that can be difficult to constantly change in a lab setting. Physical modelling of 129-Xe hyperpolarization allows for the more efficient testing of hyperpolarization physics in a wide array of experimental setups to better determine the optimal values for hyperpolarization. This research project attempted to create a working model for 129-Xe hyperpolarization using MATLAB and Simulink with the physical modelling Simscape library, finding varying levels of success when compared to experimentally determined values

Abiad, Jacob F

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Scholarship@Western

SimulatingXe-129HyperpolarizationUsing MATLAB & Simulink withSimscape to model the SpinExchange Optical Pumping (SEOP) ofXe-129 for use in MRI imaging. Thermal ModelMATLAB's Simulink programming environmentutilizes graphical programming in the form ofblock diagrams to simplify the process ofcreating simulations and physical models. Tomodel any potential laser heating of the gaswithin the cell, the Simscape Thermal libraryand Gas library for Simulink was used. Sincemost polarizer cells are on the scale of ~1m orless in length, and the temperature gradientthroughout the cell is expected to be relativelysmall, conduction heating was employedrather than convection heating even thoughthe material being heated is in a gaseous state.The model cell is divided into 25 smallersegments that are defined by the Gas propertyblocks in Simscape, and are all treated as solidblocks of uniform temperature.Polarization ModelThe initial polarization model created wassolely used to model the optical pumping rateand spin exchange rate, leaving out the spindestruction rates of Rb and Xe. The goal of thissimplified model was to ensure the simulationwas accurate enough to achieve the expectedresult of 100% Xe polarization, which is to beexpected after a long enough pumping periodin a stopped flow polarizer with nodepolarization. Here, two separate modelswere also created, one that was connected tothe previous thermal model and one thatignored any possible heating above the celloven temperature. These models were createdby adding in basic Simulink function blocks foreach of the 25 thermal segments. Each thermalsegment is then divided into 10 more discretedistance values, with the optical pumping rateand Rb polarization calculated for each.Mathematical TheoryWhat is SEOP and What's its Purpose?by Jacob AbiadModelling ProcessSpin Exchange Optical Pumping (SEOP) is amethod for hyperpolarizing the nuclei of anoble gas. "Hyperpolarizing" a gas simplymeans that a significant portion of the gasesnuclei have been forced into a certain physicalstate (a spin state) artificially, far past whatwould naturally occur. SEOP involves the useof a specifically tuned laser and alkali metalvapor, the latter generally being Rubidium (Rb)for its low melting point, to force a sample ofgas such as 129-Xe to become hyperpolarized.The Rb vapor and Xe gas are mixed into aheated glass canister, or cell, that the laser canshoot through the length of. The laser first isabsorbed by the Rb, hyperpolarizing it. This isthe "Optical Pumping" part of SEOP. The Rbatoms then collide with the Xe atoms andswap their spin states with the Xe spin states,i.e. they undergo "Spin Exchange" whichtransfers the hyperpolarization to the Xe. Hyperpolarization is a very delicate state for agas to be in, and as such a gas can be easilydepolarized. Because of this, the gas mixture isoften not just Rb vapor and 129-Xe, but alsoNitrogen gas (N2) and sometimes Helium (He).These gases both allow more Rb to staypolarized for longer than it would otherwise,thus allowing for more Xe polarizationbuildup. There are two different kinds of SEOPmethods that can be utilized as well:continuous flow (CF) and stopped flow (SF)polarization. CF polarizers pump Xe gasthrough the cell carrying the rest of the gasmixture, which is done to minimize theamount of time available for polarized Xeatoms to depolarize themselves. SFpolarization, which is the method presented inthe simulation here, sacrifices the limiting ofdepolarization mechanisms in an attempt topolarize larger quantities of Xe at once, andsimplify the methods in which Xe is separatedand collected from the rest of the gas mixture.Hyperpolarized 129-Xe gas has importantmedical applications, specifically within therealm of magnetic resonance imaging (MRI).MRI of many bodily tissues such as the lungsand brain can be enhanced by the use of ahyperpolarized substance artificially added tothe bloodstream. While 3-He was initially themost commonly studied substance for use inthis method of MRI, the rising shortage of 3-Heis beginning to make its use uneconomical.129-Xe appears to be the best substitute as itis much more naturally abundant, whilesharing similar levels of hyperpolarizedstability unique to 129-Xe alone, which make itmore practical for medical use over otherpotential candidates such as 131-Xe and 83-Kr.Simulating SEOP processes first requires thecalculation of the optical pumping rate, whichis given by:The integral is calculated with respect tofrequency across a Gaussian distribution offrequencies that represents the laser, for manydiscrete values of z which represent severalpoints in the cell. The integral can beapproximated accurately using the trapezoidalrule computed over a very large range ofintervals. Due to the exponential containingthe lambda and z term, we can see that opticalpumping is expected to drop as the distanceinto the cell and the amount of Rb alreadypolarized increases.The amount of polarized Rb is given by:Where the capital gamma represents the spindestruction rate of the Rb:Once the Rb polarization at every discrete zvalue is calculated, the spatially averaged Rbpolarization value is found and used tocalculate the resulting xenon polarization:This is also dependent on the spin exchangerate which is given below, and the Xe spindestruction rate which is generally just aconstant that is found experimentally:1 2Fitting the Data andDebuggingOnce the basic structure of the simulation iscompletely built, the next step is to conformthe values used in the simulation to someexperimental setup, and compare the resultsof the simulation to the experimental results.It is vital to keep this step in mind from thestart of the project. If the universal valuesused in calculations for every segment of codearen't based in a global repository (such as aMATLAB workspace populated by a MATLABscript dedicated to defining universal values),changing each individual value becomesimpractical. Physical simulations are generally susceptible two major bugs: unit errors andimplementation errors. Unlike papercalculations, keeping all the values beingcomputed in the proper units for everyequation used can become very difficult.Values reported in the incorrect units can havelarge scale effects on the simulation that canperpetuate themselves into other unrelatedcalculations due to the amount ofinterconnection. Implementation errorsbroadly cover actual programming errorscaused by incorrectly translating themathematical equations and physicalreality/interactions of a real physicalsystem into code form. This is something thatmakes Simulink extremely valuable - thegraphical element makes it much easier tointuitively see what parts of your simulationare interacting with each other. DepolarizationMechanismsEach of the 25 thermal segments then haveadditional Simulink blocks added to calculatethe Rb and Xe depolarization mechanisms,which generally expose further flaws with anymodelling for the previous two calculatedrates. While this stage is fairly brief in terms ofimplementation, it leads to the longest andmost intensive part of building the simulation.Results and Future Developments3 4A grouping of 5 connected thermal segments in SimulinkFully finished polarizer model shown in Simulink block formThe two main experimental properties tested at various values were the temperature of the cell andthe power of the laser. The laser heating was calculated by calculating the amount of photonsabsorbed at each time step and finding out how much total energy per each second was beingabsorbed. This approach calculated little to no effects from laser heating, so the laser heatingmodel had no apparent difference in results compared to the model that ignored laser heating. For the purposes of this report, 5 different experimental setups were calculated. All setups onlyvaried with temperature and laser power, with all other experimental values set to:Cell Pressure - 1.32 atmCell Geometry - CylindricalCell Length - 25 cmCell Radius - 2.54 cmLaser Linewidth - 0.3 nmGas Mixture Composition - 50% Xe, 45% N2, 5% HeThe following are the steady state Xe polarization levels for the five different setups:130 W, 343.15 K - 9.79%100 W, 343.15 K - 7.77%160 W, 343.15 K - 11.7%130 W, 313.15 K - 6.72%130 W, 373.15 K - 1.65%These results confirm the expected experimental trends one would expect with these two varyingproperties. 343.15 K is roughly the peak efficiency cell temperature that will yield the highestpolarization values, and we can see that polarization deteriorates at higher and lower temperatureregimes. Lower temperatures also increase the time needed to reach steady state polarization whilehigher temperatures decrease the time required, which was also expected based on previouslyacquired data. Also as expected, an increase in power seemed to proportionally increase thepolarization yields for the range of realistic laser power regimes explored.Polarization (%) vs. Time (s) graphs for the varied laser power setups. On the left is at 100 W and on the right is 160 W Polarization (%) vs. Time (s) graph for the base setup with 130 W laser power and 343.15 K cell temperature Polarization (%) vs. Time (s) graphs for the varied cell temperature setups. On the left is at 313.15 K and on the right is 373.15 KAlthough the general trends exhibited by the simulation over variations of key values match theexpected behavior, there are still two main problems currently presenting themselves with thesimulation. Firstly, the level of Rb polarization in each segment is unexpectedly low, leading to anequally low level of Xe polarization. This potentially indicates an issue with the implementation ofthe optical pumping function or Rb spin destruction calculations. Secondly, the times required toreach steady state Xe polarization are unexpectedly long for all setups. This could be an issue witheither the optical pumping function or the spin exchange function, and is potentially a tangentialissue to the first, with both stemming from the same flaw. These two discrepancies will guide thenext steps in increasing the simulations accuracy.ReferencesSkinner JG, Ranta K, Whiting N, Coffey AM, Nikolaou P, Rosen MS, Chekmenev EY, Morris PG, BarlowMJ, Goodson BM."High Xe density, high photon flux, stopped-flow spin-exchange optical pumping: Simulations versusexperiments." J Magn Reson. 2020 Mar;312:106686. doi: 10.1016/j.jmr.2020.106686. Epub 2020 Jan 16.Graham Norquay, Steven R. Parnell, Xiaojun Xu, Juan Parra-Robles, and Jim M. Wild,"Optimized production of hyperpolarized129Xe at 2 bars for in vivo lung magnetic resonanceimaging", Journal of Applied Physics 113, 044908(2013)https://doi.org/10.1063/1.4776763Michele Kelley, Alex Burant, and Rosa Tamara Branca,"Resolving the discrepancy between theoretical and experimental polarization ofhyperpolarized 129Xe using numerical simulations and in situ optical spectroscopy", Journal ofApplied Physics 128, 144901(2020)https://doi.org/10.1063/5.0019074B.M. Goodson, K. Ranta, J.G. Skinner, A.M. Coffey, P. Nikolaou, M. Gemeinhardt, D. Anthony, S.Stephenson, S. Hardy, J. Owers-Bradley, M.J. Barlow, E.Y. Chekmenev, Chapter 2 - The Physics of Hyperpolarized Gas MRI, Editor(s): Mitchell S. Albert, Francis T. Hane, Hyperpolarized and Inert Gas MRI, Academic Press, 2017, Pages 23-46, ISBN 9780128036754, https://doi.org/10.1016/B978-0-12-803675-4.00002-6. (https://www.sciencedirect.com/science/article/pii/B9780128036754000026)Contact:Jacob Abiadjabiad@uwo.ca

Simulating 129-Xe Hyperpolarization

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Simulating 129-Xe Hyperpolarization

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