by Andrew McGovern

Introduction

• What the model is

• What it does

• How it was implemented

• The model in action

• Results

Purpose of the model

• To describe the function of the kidney mathematically.

• To enable renal function to be measure from imaging.

• A number of attempts to do this with limited success.1-3

• Tofts’ model measures: renal filtration, renal blood volume and blood flow.4,5

Cpart Cp

glom Cd

A B C

Composition of the model

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T

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fwhm 0

))(( 2

)(2ln2

B and C: )exp()()( deffluxglom

pd

trans

d vtKtCv

KtC

Implementing the model

Maths: Images:

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cos1

sin)1()(

)(

)(

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1

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TRtR

e

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sin)(

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d vtKtCv

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Implementing the modelboolean step5() {

double t = Global.flipangle * Math.PI/180;double r1;float ct[] = new float[Global.size];Global.modsigEV = new float[Global.size];Global.modsigIV = new float[Global.size];Global.modkidneysig = new float[Global.size];float s0 = (float)(Global.precontrenalsig*(1.000-Math.cos(t)*Math.exp(-0.001*Global.trms/Global.t10kidney))/((1.000-Math.exp(-0.001*Global.trms/Global.t10kidney))*Math.sin(t)));int i;for (i = 0; i < Global.size; i++) {

ct[i] = (float)(Global.vb*(1.00 - 0.01*Global.hctsmall)*Global.cpren[i]);r1 = 1/Global.t10kidney + Global.r1kidney*Global.vdcd[i];Global.modsigEV[i] = (float)(s0*(1.00-Math.exp(-0.001*Global.trms*r1))*Math.sin(t)/(1-Math.exp(-0.001*Global.trms*r1)*Math.cos(t)));r1 = 1/Global.t10kidney + Global.r1kidney*ct[i];Global.modsigIV[i] = (float)(s0*(1.00-Math.exp(-0.001*Global.trms*r1))*Math.sin(t)/(1-Math.exp(-0.001*Global.trms*r1)*Math.cos(t)));ct[i] += Global.vdcd[i];r1 = 1/Global.t10kidney + Global.r1kidney*ct[i];Global.modkidneysig[i] = (float)(s0*(1.00-Math.exp(-0.001*Global.trms*r1))*Math.sin(t)/(1-Math.exp(-0.001*Global.trms*r1)*Math.cos(t)));

}return true;

}/** Model input selection function */boolean getToftsInputs() {

boolean error;String GIRFoptions[] = {"Gaussian", "Delayed exponential"};String effluxoptions[] = {"Off", "On"};String input, effout = effluxoptions[0], girfout = " ";if(Global.effluxoptions) effout = effluxoptions[1];switch(Global.girfoption) {

case(1):girfout = GIRFoptions[0];break;

case(2):girfout = GIRFoptions[1];break;

}/** Creates an input dialog box */do {

GenericDialog d = new GenericDialog("Tofts' Model: Input box", IJ.getInstance());d.addNumericField("End fit time (seconds after bolus arrives): ", Global.endfittime, 0, 6, "s");d.addChoice("Inpulse response function", GIRFoptions, girfout);d.addChoice("Efflux: ", effluxoptions, effout);

Results

Results

Filtration rate Blood volume

The future

• A few mathematical discrepancies to fix

• Publish the program on the internet

• Test the model with diseased kidneys

• Use as a research tool

• Use in clinical practice

References

1. David LB, Ala'a ES, Ching MC, Andrew PJ, Hari M, Philip AK. Measurement of single kidney function using dynamic contrast-enhanced MRI: Comparison of two models in human subjects. J Magn Reson Imaging. 2006;24(5):1117-23.

2. Miles KA, Leggett DA, Bennett GA. CT derived Patlak images of the human kidney. Br J Radiol. 1999 Feb;72(854):153-8.

3. Daghini E, Juillard L, Haas JA, Krier JD, Romero JC, Lerman LO. Comparison of mathematic models for assessment of glomerular filtration rate with electron-beam CT in pigs. Radiology. 2007 Feb;242(2):417-24.

4. Tofts PS, Cutajar M, Mendichovszky IA, Peters AM, Miles KA, Buckley DL, et al. Estimation of renal filtration and vascular parameters using a simple three-compartment model for dynamic contrast-enhanced MRI of the kidney. Unpublished Work: Brighton and Sussex Medical School 2010.

5. Tofts PS, Cutajar M, Mendichovszky IA, Gordon I. Accurate and precise measurement of renal filtration and vascular parameters using DCE-MRI and a 3-compartment model. International Society for Magnetic Resonance in Medicine Conference. Stockholm 2010.