A Program for FittingCompartmental Models toRegion-of-InterestDynamic Emission Tomography Data
RFIT is a package of Fortran-77 code to simulate compartmental models and estimate their parameters based on kinetic region-of-interest (ROI) data from a positron emission tomograph (PET). A compartmental model proposes that the injected isotope exists in the body in a xed number of physical or chemical states. Each compartment represents the isotope in one state.
RFIT is a package of Fortran-77 code to simulate compartmental models and estimate their parameters based on kinetic region-of-interest (ROI) data from a positron emission tomograph (PET). A compartmental model proposes that the injected isotope exists in the body in a xed number of physical or chemical states. Each compartment represents the isotope in one state.
By convention in the nuclear medicine literature, the rst compartment is the blood pool and its measured values serve as inputs to the dierential equation for the model. In the example below, the isotope is distributed between the blood pool, the interstitial uid and the intracellular space. Inside the cells it is either free or bound (to a receptor or protein,
Each compartmental model has an associated dierential equation, as described below. By a simulation we mean a calculation of the perfect (noise-free) data that would result from solving the equations for a particular model with a given blood input function and a given set of parameters. Parameters of the model are the transfer rates between compartments and, in some cases, additional parameters reecting measurement constraints (e.g. time delays between blood and tissue data).
Each compartmental model has an associated dierential equation, as described below. By a simulation we mean a calculation of the perfect (noise-free) data that would result from solving the equations for a particular model with a given blood input function and a given set of parameters. Parameters of the model are the transfer rates between compartments and, in some cases, additional parameters reecting measurement constraints (e.g. time delays between blood and tissue data).
At the present time the software described here runs on Sun/UNIX, Vax/VMS, and Cray/UNICOS systems. It is available, via anonymous ftp, along with illustrative examples as well as these notes from CFI.LBL.Gov. Please direct any correspondance to RHHuesman@LBL.Gov, GJKlein@LBL.Gov, or BWReutter@LBL.Gov. The Command Line RFIT is invoked with the command rfit bloodfile -sR -i uptakefile -u# outfile ?errfile where R is the (optional) scaling factor which adjusts for the dierent units and counting eciencies of the blood and uptake data, and # is the number of the region of interest in uptakefile.
The default scaling factor is 1. bloodfile and uptakefile are required on the command line. If the blood measurement was calculated as a PET region of interest, the blood and uptake data can be read from the same le:
rfit uptakefile -i# -u# outfile ?errfile
In this case, no scaling factor is necessary. The last three les: infile, outfile, and errfile, are optional. infile, provides a list of commands which direct RFIT to carry out desired simulations, parameter ts, etc. Principal commands are summarized in the sections below. If no input le is included in the command line, RFIT will wait for commands to be entered interactively. outfile and errfile are, respectively, depositories for the results and the error messages generated by RFIT.
In this case, no scaling factor is necessary. The last three les: infile, outfile, and errfile, are optional. infile, provides a list of commands which direct RFIT to carry out desired simulations, parameter ts, etc. Principal commands are summarized in the sections below. If no input le is included in the command line, RFIT will wait for commands to be entered interactively. outfile and errfile are, respectively, depositories for the results and the error messages generated by RFIT.
If they are not included in the command line, then the information will be written to the display terminal. Speci cation of the Model At present, RFIT can t parameters of linear or bilinear compartmental models with as many as nine distinct compartments. The time evolution of the distribution of an injected radioactive isotope (often attached as a label to a complex molecule) is described by a dierential equation:
d~Q(t) dt = A~Q(t) + nQ2(t) hB~Q(t)i2; :::;Qn(t) hB~Q(t)inoT+ ~F1(t) (1)
where: f(t) is the amount of isotope in compartment 1, the blood input function,
~Q(t) is the vector [Q2(t);Q3(t); :::;Qn(t)]
T containing the amount of isotope in compartments 2 to n
A is the matrix of linear transfer rates between tissue compartments, B is the matrix of coecients of the bilinear transfer terms, ~F is a vector of transfer rates from the input to each compartment. The structure of the model is speci ed in RFIT with the user command upmod. For example,
upmod 12 21 23 32 r43 corresponds to the compartmental model shown in the introduction. The 32 indicates that there is a linear transfer rate, with transfer coecient denoted k32, from compartment 2 to compartment 3.
upmod 12 21 23 32 r43 corresponds to the compartmental model shown in the introduction. The 32 indicates that there is a linear transfer rate, with transfer coecient denoted k32, from compartment 2 to compartment 3.
In the nuclear medicine convention, the rst compartment represents the blood pool which is the input f(t). The 21 indicates that compartment 2 is fed with a linear rate k21f(t) from the blood pool and therefore the rst entry of ~F is nonzero. The r in front of 43 means that compartment 4 represents a saturable receptor and the rate from compartment 3 to compartment 4 is bilinear of the form
[b4 Q4(t)]k43Q3(t).
This leadsto nonzero entries in A (the linear coecient b4k43) and in B
(the bilinear coecient k43). Note that a receptor compartment introduces an additional parameter b which represents the maximum possible amount of isotope in the compartment. The associated system of dierential equations (see equation 1) is:
dQ2 dt (t) = (k12 + k32)Q2(t) + k23Q3(t) + k21f(t) (2)
[b4 Q4(t)]k43Q3(t).
This leadsto nonzero entries in A (the linear coecient b4k43) and in B
(the bilinear coecient k43). Note that a receptor compartment introduces an additional parameter b which represents the maximum possible amount of isotope in the compartment. The associated system of dierential equations (see equation 1) is:
dQ2 dt (t) = (k12 + k32)Q2(t) + k23Q3(t) + k21f(t) (2)
dQ3 dt (t) = k32Q2(t) fk23 + [b4 Q4(t)] k43gQ3(t) (3)
dQ4 dt (t) = [b4 Q4(t)] k43Q3(t) (4)
Each positive term in the equation for Qi corresponds to an arrow into compartment i and each negative term to an arrow out.
Simulation of PET Measurements One of the principal functions of RFIT is to simulate PET measurements for a given model. Each PET measurement represents an accumulation of counts of radioactive emissions over an interval of time [tl; tl+1].
Simulation of PET Measurements One of the principal functions of RFIT is to simulate PET measurements for a given model. Each PET measurement represents an accumulation of counts of radioactive emissions over an interval of time [tl; tl+1].
It is assumed that some fraction (fv) of the counts from a particular region of interest originate in the blood pool rather than the tissue. The integral nature of the data and the necessity of the vascular fraction have been previously discussed in [1]. Thus the measurement function for the system is given by:
yl = (1 fv)[1; 1; :::; 1] Z tl+1tl~Q(t)dt + fv Z tl+1tlf(t)dt (5)
The entries of A, Bi, and ~F can be set to nonzero values only when they correspond to the transfer rates speci ed with the upmod command (see above). For the model (12 21 23 32 r43) the values would be assigned by the following commands: In RFIT commands, ij is shorthand for kij . Here the values of k12, k32, k23, and b4k43 will form entries of A, k21 will be the rst entry of ~F, and the value of k43 will be placed in B.
State and Regional Impact The annual market for RFID products is estimated to exceed $5 billion by 2008. The global research firm IDTechEx predicts that item-level RFID tagging alone will expand from a $160 million market in 2006 to $13 billion in 2016 for systems and tags. In 2006, 200 million items were tagged worldwide.
The entries of A, Bi, and ~F can be set to nonzero values only when they correspond to the transfer rates speci ed with the upmod command (see above). For the model (12 21 23 32 r43) the values would be assigned by the following commands: In RFIT commands, ij is shorthand for kij . Here the values of k12, k32, k23, and b4k43 will form entries of A, k21 will be the rst entry of ~F, and the value of k43 will be placed in B.
State and Regional Impact The annual market for RFID products is estimated to exceed $5 billion by 2008. The global research firm IDTechEx predicts that item-level RFID tagging alone will expand from a $160 million market in 2006 to $13 billion in 2016 for systems and tags. In 2006, 200 million items were tagged worldwide.
By 2016, the company estimates that 550 billion items may carry RFID tags. The Texas RFIT Center will develop new and innovative technologies, set standards on national policy, and train the future work force to meet the R&D demand. Estimates show that the center can help attract more than 50% ($2.5 billion) of the annual RFID market to the Dallas-Fort Worth region. In fact, in the last 12 months, RFID companies from California, Washington, D.C., and Virginia have contacted UT Arlington to discuss moving their businesses to Dallas-Fort Worth because of the area’s rich RFID knowledge base.
Job growth is an important component of this project. In addition to undergraduate and graduate level engineering education with hands-on experience gained through industry projects, UT Arlington is teaming with North Lake College, a predominately Hispanic-serving institution, to carry out its state of Texas Certified RFID Technician Training program. This program is designed to meet the growing demand of Dallas-Fort Worth companies for RFID-trained technicians.
Estimates indicate that these companies will need to hire more than 10,000 engineers and technicians during the next five to eight years.
UT Arlington is contributing $1 million in cash and in-kind support towards this effort. An additional $500,000 in matching funds has been provided by industry within the Dallas-Fort Worth area and from RFID companies acrossthe nation. Impact to Area Companies With comprehensive expertise in RF chip design, antenna design, security protocols, and national and international
RFID standards, the Texas RFIT Center will provide R&D capabilities, address the concerns about RFID privacyand security, and provide a work force of engineers and technicians to the growing number of Dallas-Fort Worth RFID companies.
UT Arlington is contributing $1 million in cash and in-kind support towards this effort. An additional $500,000 in matching funds has been provided by industry within the Dallas-Fort Worth area and from RFID companies acrossthe nation. Impact to Area Companies With comprehensive expertise in RF chip design, antenna design, security protocols, and national and international
RFID standards, the Texas RFIT Center will provide R&D capabilities, address the concerns about RFID privacyand security, and provide a work force of engineers and technicians to the growing number of Dallas-Fort Worth RFID companies.
