# Precise measurement of renal filtration and vascular parameters using a two-compartment model for dynamic contrast-enhanced MRI of the kidney gives realistic normal values.

**ABSTRACT** To model the uptake phase of T(1)-weighted DCE-MRI data in normal kidneys and to demonstrate that the fitted physiological parameters correlate with published normal values.

The model incorporates delay and broadening of the arterial vascular peak as it appears in the capillary bed, two distinct compartments for renal intravascular and extravascular Gd tracer, and uses a small-vessel haematocrit value of 24%. Four physiological parameters can be estimated: regional filtration K ( trans ) (ml min(-1) [ml tissue](-1)), perfusion F (ml min(-1) [100 ml tissue](-1)), blood volume v ( b ) (%) and mean residence time MRT (s). From these are found the filtration fraction (FF; %) and total GFR (ml min(-1)). Fifteen healthy volunteers were imaged twice using oblique coronal slices every 2.5 s to determine the reproducibility.

Using parenchymal ROIs, group mean values for renal biomarkers all agreed with published values: K ( trans ): 0.25; F: 219; v ( b ): 34; MRT: 5.5; FF: 15; GFR: 115. Nominally cortical ROIs consistently underestimated total filtration (by ~50%). Reproducibility was 7-18%. Sensitivity analysis showed that these fitted parameters are most vulnerable to errors in the fixed parameters kidney T(1), flip angle, haematocrit and relaxivity.

These renal biomarkers can potentially measure renal physiology in diagnosis and treatment.

• Dynamic contrast-enhanced magnetic resonance imaging can measure renal function. • Filtration and perfusion values in healthy volunteers agree with published normal values. • Precision measured in healthy volunteers is between 7 and 15%.

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**ABSTRACT:**Chronic kidney disease (CKD) is a common, deadly, and expensive threat to public health. Patients susceptible to the development of CKD are difficult to identify because there are few noninvasive clinical techniques and markers to assess early kidney dysfunction. Noninvasive imaging techniques are being developed to quantitatively measure kidney morphology and function in preclinical research and in clinical trials. Magnetic resonance imaging (MRI) techniques in particular have the potential to provide structural and functional information in the kidney. Novel molecular imaging techniques, using targeted magnetic nanoparticles that exploit the characteristics of the endogenous protein, ferritin, have been developed in conjunction with MRI to count every perfused glomerulus in the kidney and measure their individual volumes. This technique could open the door to the possibility of prospectively assessing and eventually reducing a patient's risk for progression to CKD. This review highlights the potential clinical benefits of early detection in patients predisposed to CKD and discusses technologic and regulatory hurdles to the translation of these molecular MRI techniques to provide early diagnosis of CKD.Advances in chronic kidney disease 11/2013; 20(6):479-87. · 2.42 Impact Factor - Jeff L Zhang, Glen Morrell, Henry Rusinek, Eric E Sigmund, Hersh Chandarana, Lilach O Lerman, Pottumarthi V Prasad, David Niles, Nathan Artz, Sean Fain, Pierre-Hugues Vivier, Alfred K Cheung, Vivian S Lee[show abstract] [hide abstract]

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**ABSTRACT:**Nuclear medicine and MRI provide information about renal perfusion, function (glomerular filtration rate), and drainage. Some tracers that are used in nuclear medicine (technetium-diethylene triamine pentaacetic acid ([(99m)Tc-DTPA] and (51)chromium-EDTA) and some contrast media (CM) that are used for MRI (gadolinium-DTPA for instance) share the same pharmacokinetic properties, though, detection techniques are different (low-spatial resolution 2-dimensional projection with a good concentration-to-signal linearity for nuclear medicine and high-resolution 3-dimensional localization with nonlinear behavior for MRI). Thus, though based on the same principles, the methods are not the same and they provide somewhat different information. Many MRI perfusion studies have been conducted; some of them were compared with nuclear medicine with no good agreement. Phase contrast can reliably assess global renal blood flow but not perfusion at a tissular level. Arterial spin labeling has not proven to be a reliable tool to measure renal perfusion. Techniques using CM theoretically can assess perfusion at the tissular level, but they have not proven to be precise. To assess renal function, many models have been proposed. Some MRI techniques using CM, both semiquantitative (Patlak) and quantitative, have shown ability to roughly assess relative function. Some quantitative methods (Annet's and Lee's methods) have even showed that they could roughly estimate absolute renal function, with better results than estimated glomerular filtration rate. Quantification of drainage has not been much studied using MRI.Seminars in nuclear medicine 03/2014; 44(2):82-92. · 3.96 Impact Factor

Page 1

MAGNETIC RESONANCE

Precise measurement of renal filtration

and vascular parameters using a two-compartment model

for dynamic contrast-enhanced MRI of the kidney

gives realistic normal values

Paul S. Tofts & Marica Cutajar &

Iosif A. Mendichovszky & A. Michael Peters &

Isky Gordon

Received: 29 August 2011 /Revised: 25 November 2011 /Accepted: 4 December 2011 /Published online: 14 March 2012

#European Society of Radiology 2012

Abstract

Objective To model the uptake phase of T1-weighted DCE-

MRI data in normal kidneys and to demonstrate that the

fitted physiological parameters correlate with published nor-

mal values.

Methods The model incorporates delay and broadening of

the arterial vascular peak as it appears in the capillary bed,

two distinct compartments for renal intravascular and extra-

vascular Gd tracer, and uses a small-vessel haematocrit

value of 24%. Four physiological parameters can be esti-

mated: regional filtration Ktrans(ml min−1[ml tissue]−1),

perfusion F (ml min−1[100 ml tissue]−1), blood volume vb

(%) and mean residence time MRT (s). From these are found

the filtration fraction (FF; %) and total GFR (ml min−1).

Fifteen healthy volunteers were imaged twice using oblique

coronal slices every 2.5 s to determine the reproducibility.

Results Using parenchymal ROIs, group mean values for

renal biomarkers all agreed with published values: Ktrans:

0.25; F: 219; vb: 34; MRT: 5.5; FF: 15; GFR: 115. Nomi-

nally cortical ROIs consistently underestimated total filtra-

tion (by ~50%). Reproducibility was 7–18%. Sensitivity

analysis showed that these fitted parameters are most vul-

nerable to errors in the fixed parameters kidney T1, flip

angle, haematocrit and relaxivity.

Conclusions These renal biomarkers can potentially mea-

sure renal physiology in diagnosis and treatment.

Key Points

• Dynamic contrast-enhanced magnetic resonance imaging

can measure renal function.

• Filtration and perfusion values in healthy volunteers

agree with published normal values.

• Precision measured in healthy volunteers is between 7 and

15%.

Keywords DCE-MRI.Kidney.GFR.Quantification.

Modeling

Introduction

Dynamic imaging of renal uptake of a contrast agent is an

established way of assessing renal physiology using nuclear

medicine, dynamic computed tomography and magnetic res-

onance imaging (MRI), and estimation of quantitative param-

eters is possible [1–8]. Several reviews are available [9–11].

Electronic supplementary material The online version of this article

(doi:10.1007/s00330-012-2382-9) contains supplementary material,

which is available to authorized users.

P. S. Tofts:M. Cutajar:A. M. Peters

Brighton and Sussex Medical School,

Falmer, Sussex BN1 9PX, UK

P. S. Tofts

UCL Institute of Neurology,

London WC1N 3BG, UK

M. Cutajar:I. Gordon

UCL Institute of Child Health,

London WC1N 3JH, UK

I. A. Mendichovszky

Imaging Science and Biomedical Engineering,

University of Manchester,

Manchester M20 3LJ, UK

P. S. Tofts (*)

48 Rugby Road,

Brighton BN1 6EB, UK

e-mail: bsms@paul-tofts.org.uk

Eur Radiol (2012) 22:1320–1330

DOI 10.1007/s00330-012-2382-9

Page 2

Dynamic contrast-enhanced (DCE) MRI of the kidneys is

now clinically feasible with the advent of fast sequences.

This paper builds on the compartmental modelling ap-

proach that has been so successful in characterising capillary

leakage in tumours [12, 13], with the addition of a delay and

broadening of the arterial vascular peak as is observed in the

kidney. This is most likely caused by a small delay along the

renal artery and a non-zero residence time in the renal

capillary bed. The renal model, applicable to cortical and

parenchymal ROIs, captures the essential features of the

dynamic data, yet remains simple.

The published modelling work in normal subjects gener-

ally shows little systematic effort to reconcile the measure-

ments with published values from other (non-MRI)

methods, and there are few systematic measurements of

precision (repeatability). Very often DCE images are just

reported visually (i.e. qualitatively). In this quantitative

study, testing the new model using data from healthy vol-

unteers shows that key renal physiological parameters relat-

ed to filtration and perfusion are estimated with good

reproducibility and give measurements close to published

normal values. Preliminary versions of this work have been

presented orally [14–16]. This paper should be read with the

Electronic Supplementary Material (ESM).

Materials and methods

Pharmacokinetic and MRI model

The model consists of two parts (full mathematical details

are given in the Appendix). The pharmacokinetic part (see

Fig. 1 and Eqs. 1–5 below) is applicable to the cortex and

parenchyma. It describes the intra-renal concentrations of

intravascular (IV; largely glomerular) and extravascular

(EV; mostly tubular) Gd contrast agent (also called tracer)

in renal tissue (Fig. 1). The arterial vascular peak is delayed

and broadened (dispersed) before arrival at the IV compart-

ment (the delay is apparent in the data, see Fig. 3a below).

This process is described by a vascular impulse response

function (VIRF); thus, the IV plasma concentration in the

kidney is the convolution of the arterial plasma concentra-

tion with the VIRF (Eq. 1 below). The VIRF is the response

in the renal vasculature to a very short pulse (in fact a

mathematical delta function) of Gd in the aorta (see Fig. 6

below). From the VIRF renal perfusion F (ml blood min−1

(100 ml tissue)−1) was estimated. Uptake or flux from the IV

to the EV compartment (by the process of filtration, where-

by Gd is removed from capillaries) is F1¼ KtransCkid

unit volume of tissue, where Ktransis the regional filtration

(GFR per unit volume of kidney; formally the volume

transfer constant [12] from plasma), Ckid

varying tracer plasma concentration in the IV compartment,

and F1has units of mmol min−1(ml tissue)−1. This trans-

portation of Gd into the ROI by vascular means is appro-

priate in a cortical or parenchymal ROI, but not in a

medullary ROI.

It is known [4, 17] that no filtered tracer leaves the kidney

for a certain time after arrival in the blood capillaries (if a

parenchymal region is used, this time is usually about 90 s);

the model is used to analyse data over this relatively short

time period, and therefore it can be assumed that there is no

efflux from the EV compartment.

The MRI part of the model (Eqs. 6–9) defines how Gd

concentration enhances the MRI signal. It relies on knowl-

edge of several fixed parameter values (see Table 1 below):

T1of blood and kidney, Gd relaxivity, haematocrit, and the

imaging parameters TR and flip angle FA.

In the fitting procedure, the model is adjusted by varying

the free parameters in the model until the difference between

pðtÞ per

pðtÞ is the time-

Fig. 1 Two-compartment model for renal filtration. The intravascular

(IV) compartment is primarily glomerular; the extravascular (EV)

compartment is primarily tubular. The dotted box represents an ROI

or pixel in the kidney. This model is used soon after Gd bolus arrival,

before there is time for efflux from the EV compartment

Table 1 Filtration values and fit residuals from the uptake phase of

repeated imaging of 15 healthy volunteers

ROIVIRFKtrans(min−1) RMS residual

MeanSDa

ISDb

(Mean; %)

Cortical Delayed

exponential

Gaussian

Delayed

exponential

Gaussian

0.290.0580.046 3.9

0.30

0.25

0.060

0.058

0.046

0.043

4.8

4.0Parenchymal

0.250.0590.0434.5

VIRF 0 vascular impulse response function, ISD 0 instrumental stan-

dard deviation, RMS 0 root-mean-square

aGroup SD

bInstrumental SD (from repeated imaging)

Eur Radiol (2012) 22:1320–13301321

Page 3

the model and the measured signal data is minimised. The

free parameters are the filtration Ktrans, the blood volume vb

and the VIRF parameters (one for broadening, and if neces-

sary a second parameter to describe delay). Thus the instant

exponential VIRF was defined by only one free parameter,

whilst the delayed exponential and Gaussian VIRFs re-

quired two free parameters (see Appendix). For each fit

the RMS (root-mean-square) residual (i.e. difference between

a data point and the model value) was found. The local

filtration fraction is the ratio of regional filtration to regional

renal plasma flow, and was calculated using Eq. 10.

MRI acquisition and analysis

Fifteen healthy volunteers were imaged twice (about 13 days

apart) at 1.5 T with a temporal resolution of 2.5 s. A half-

dose of Gd-DPTA was used. No T1values were measured.

Image datasets were registered to remove in-plane move-

ment [18]. Regions of interest (ROIs) were placed over the

aorta and kidney. A parenchymal ROI was defined semi-

automatically on the perfusion image as proposed by Peters

et al. [19]. An ROI of the whole cortex was drawn manually

(Fig. 2). Full details of MRI acquisition and ROI generation

are given in the ESM.

In the model analysis, two phases (i.e. time periods) of

data were fitted: (1) the perfusion phase (up to the kidney

signal minimum after the first bolus passage) when Gd is

largely IV, and (2) the uptake (filtration) phase (up to 90 s

after bolus arrival, before any Gd has left the parenchymal

ROI). Renal filtration and vascular parameters (including

local filtration, filtration fraction, perfusion, blood volume

and mean residence time) were estimated. We hypothesised

that vascular parameters might be better estimated from

perfusion phase data than from uptake phase data; fits were

compared on the basis of reproducibility and residuals.

Three plausible VIRFs (instant exponential, delayed ex-

ponential and Gaussian) were compared in cortical and

parenchymal ROIs on the basis of the quality of the model

fit and the reproducibility of perfusion estimates. The values

of the seven fixed tissue parameters (see Table 5) were

initially set at: Hctlarge0Hctsmall041% [20]; rblood

riv

[22]; Tkidney

10

¼ 1:2s (average of cortical and medullary

values [23]).

1

¼ rd

¼ 1:4s

1¼

1¼ 4:5s?1mM?1(the in vitro value [21]); Tblood

10

Instrumental SD (ISD) For each kidney, the differences in

repeated measures of model parameters were calculated, to

give the standard deviation in a single measurement, using

the method of Bland and Altman [24]. In this approach, the

differences between repeated measurements in the same

subject are found; the standard deviation of these differences

are calculated and then divided by 1.4 to find the standard

deviation in a single measurement [25].

Total filtration The filtration parameter found by the model

(GFR per unit volume of tissue, or Ktrans) is an intensive

quantity (thus density and temperature are intensive quantities

as they do not automatically increase with size, as mass does).

A relationship can be established between Ktransand GFR,

which is thequantityusedclinically. FromthemeanKtransina

ROI, the total filtration inthatROI can befound (GFR_ROI 0

Ktrans×ROIvolume).AsincreasinglylargeROIsareused,we

expect to see a plateau in GFR_ROI, as all of the functioning

kidney in that slice is included. ROIs of increasing size were

created manually, from a small piece of cortex up to ROIs that

were larger than (‘over included’) all of the parenchyma in a

slice. (Here ‘over-inclusion’ means that all the partial-volume

pixels that could possibly be part of the parenchyma were

included.) This object strength approach overcomes partial

volume effects and the difficulty of drawing precise ROIs,

and has been used to characterise objects with indistinct

borders [26]. Eight kidney datasets were analysed (2 subjects,

each with repeated imaging).

Model sensitivity analysis The vulnerability of the various

tissue parameter estimates (Ktrans

vb

etc., see Table 5 below) to

Fig. 2 Cortical and parenchymal ROIs

1322 Eur Radiol (2012) 22:1320–1330

Page 4

error in the fixed parameters (r1, T10etc.) was found by

calculating the error propagation ratio (EPR) [27]. In this

sensitivity analysis, the model is used to determine by what

percentage a tissue parameter estimate will change as a

result of a 1% error in a fixed parameter. For example a

+1% error in Tkid

Ktrans; thus the EPR from Tkid

Normal values for kidney filtration and vascular parame-

ters were calculated from our measurements, and com-

pared with other (non-MRI) normal values from the

literature (see Table 4). Total kidney volume was calcu-

lated from the pre-Gd T1-weighted images for the 15

normal subjects. Values standardised to a body surface area

(BSA) of 1.73 m2were also found; BSAwas estimated from

BSA(m²)00.0235 height(cm)0.422weight(kg)0.514[28].

10will cause an approximately -1% error in

10to Ktransis −1.0.

Results

Typical fits for the two phases of data are shown in Fig. 3.

Sixty kidney image datasets were each fitted in two phases

and with three VIRFs, giving a total of 360 fitting opera-

tions. The cubic interpolation of the aortic blood data gives a

convincing description between the data points (Fig. 4) and

enables timing parameters (Δ, Tg, Tfwhm, MRT) to be

calculated with precision well below that of the imaging

temporal resolution. Delayed exponential and Gaussian

VIRFs fitted well. The onset of efflux from both the cortical

and parenchymal regions could be detected (see for example

Fig. 3a).

Comparing the three VIRFs in terms of the residuals from

fits, for cortical and parenchymal regions in the perfusion

phase, it was clear that the instant exponential VIRF per-

formed significantly less well (RMS residual 4.3–6.5%)

than the delayed exponential and Gaussian VIRFs (residuals

2.6–3.6%). Fractional blood volume vbmean values and

reproducibility (from the repeated examinations) were

approximately the same for all VIRFs, for cortical and

parenchymal ROIs and in both perfusion and uptake phases.

The instant exponential VIRF was therefore excluded from

subsequent analysis.

Comparing perfusion and uptake phases (for the estima-

tion of vb) showed that reproducibility was always slightly

better using uptake phase data, for both cortical and paren-

chymal ROIs (although the residuals from the fits were

slightly higher in the uptake phase). The hypothesis that

the perfusion phase would estimate perfusion parameters

better than the uptake phase, and justify the extra complex-

ity of carrying out two fits, was therefore disproved, and

perfusion phase data were not further analysed.

Measured normal values, residuals and measurement pre-

cision (ISD) are summarised in Tables 1 and 2.

Total filtration GFR values for increasingly large ROIs

reached a plateau (Fig. 5). The datasets analysed showed

similar behaviour, although some had a less obvious

Fig. 4 Estimated arterial plasma concentration from blood signal,

using cubic interpolation; note detail in peak not present in raw data

Fig. 3 Example fits of model to data from a single healthy kidney,

showing two phases of data, with delayed Gaussian vascular impulse

response function (VIRF). a Parenchymal ROI, uptake phase; b corti-

cal ROI, perfusion phase. Residuals are shown for fitting period only

(vertically offset in the plot, for clarity). Note evidence of efflux after

the end of the fitting period (data dip below model). Separate IV and

EV contributions to the model are shown

Eur Radiol (2012) 22:1320–1330 1323

Page 5

plateau, with GFR_ROI increasing beyond the nominal

parenchymal outline. The plateau value was 11.2±

2.8 ml min−1(mean ± SD; 2 subjects, 8 kidneys). Typical

values for a GFR_ROI calculation in a nominally cortical

ROI were: Ktrans00.23 min−1; ROI size0361 pixels; voxel

volume 73 mm3; GFR_ROI06.1 ml min−1. The effective

number of slices per kidney was 5.8 (taking into account

that peripheral slices contain smaller volumes of kidney);

thus the parenchymal data give an estimate for total GFR of

about 130 ml min−1by this method, in agreement with other

methods (see Table 4).

Sensitivity analysis is shown in Table 3. This confirms

the large influence that the fixed parameters haematocrit,

relaxivity, T10and FA have on the estimated values of tissue

parameters (principally filtration, blood volume and perfu-

sion). Mean residence time is unaffected by fixed parameter

errors, whilst the filtration fraction is more robust than

filtration or perfusion. Small deviations in fixed parameters

were used (~1–3%) to avoid non-linear effects, and the

resulting EPR values had small random errors (~1%); thus

a measured EPR of 0.99 could in fact be 1.00.

Normal values for kidney filtration and vascular parame-

ters, measured using other (non-MRI) methods, were taken

from the literature (see Table 4), and our measurements were

comparedwiththese.KidneyvolumeVkidwasestimatedfrom

a published value of mass mkidas follows. The mass was

measured post-mortem and therefore excludes most of the

blood (which would drain out after removal). If a fraction of

blood α remains in the parenchyma, then Vkid¼

where ρ is the kidney density. Using mkid0150 g [29], ρ0

1.03 g ml−1, α010%, vb035% gives Vkid0213 ml, close to a

published value of 218 ml [30].

The small vessel haematocrit value Hctsmallis much

lower than Hctlarge(red blood cells have difficulty entering

mkid

ð

ρ 1? 1?a

Þvb

ðÞ,

Fig. 5 In a central kidney slice, glomerular filtration rate (GFR) in

progressively larger ROIs increases to a plateau value of about

11 ml min−1; for five slices per kidney this gives a total GFR of

110 ml min−1(single kidney GFR055). Nominal cortical (C0350

pixels) and parenchymal (P0600 pixels) ROIs are shown. Mean Ktrans

values for small cortical ROIs vary, then decrease progressively for

ROIs larger than the cortex

Table 2 Perfusion parameters

from the uptake phase of repeat-

ed imaging of 15 healthy volun-

teers. Residuals are shown in

Table 1 and symbols in Table 5

MRT 0 mean residence time

aFrom peak and MRTof delayed

exponential or Gaussian VIRF;

units: ml blood min−1(100 ml

tissue)−1

bGroup SD

cInstrumental SD (from repeated

imaging)

vb(%)MRT (s)Fpeak a

FMRT a

Mean SDb

ISDc

MeanSDISDMean SDISDMeanSD ISD

Cortical ROI, VIRF 0 delayed exponential

41 107

Cortical ROI, VIRF 0 Gaussian

40108

Parenchymal ROI, VIRF 0 delayed exponential

45 128

Parenchymal ROI, VIRF 0 Gaussian

44118

5.200.660.35 542150104482 12777

4.680.680.37 310 80 44523 13871

5.890.670.4847716468 46514166

5.490.730.40284 8971 495156 42

Table 3 Sensitivity analysis showing how fitted (free) parameters

(Ktransetc.) are affected by chosen value of fixed parameter (e.g.

Hctlarge). Values are error propagation ratio (EPR; i.e. percentage

change in fitted parameter for 1% change in fixed parameter) [27].

Symbols are defined in Table 5

Ktrans b

vb

a

Fb

MRTFF

Fixed tissue parameters

Hctlarge

Hctsmall

r1blood

r1d

r1iv

r1iv0r1blood

T10blood

T10kidney

Fixed instrumental parameters

θ

TR

−0.72

0

+0.98

−0.97

0

+0.99c

+1.04

−1.06

−0.69

+0.69

+1.03

0

−0.98

0

+1.24

−1.08

−0.70

+0.69

+1.03

0

−0.98

0

+1.29

−1.14

0

0

0

0

0

0

0.02

0

0

0

0

−0.99

+0.99

+0.99

−0.22

+0.22

−0.07

0

+0.25

−0.13

+0.29

−0.15

0.03

0

−0.38

+0.16

aAverage from delayed exponential and Gaussian VIRFs

bAverage of 4 F values from delayed exponential and Gaussian VIRFs,

using both peak and MRT (see Eqs. 15, 16, 18, 19 in ESM)

cr1ivfixed 0 r1blood; both altered together

1324Eur Radiol (2012) 22:1320–1330

Page 6

small channels); values of 24% (dog heart) [31], 31% (hu-

man brain) [32], 25% [33] and 8–20% [34] have been

reported. This is related to the Fahraeus effect; in small

vessels red blood cells travel faster than plasma [34, 35].

The renal vasa recta (10–20 μm in diameter) have a reduced

haematocrit of 40–50% compared with a large vessel [36];

the network Fahraeus effect can further reduce this by as

much as 20% [34].

The MRI values were recalculated using a value of 24%

for Hctsmall(Table 4, right hand column); blood volume

values then agreed with literature values. Left-sided cortical

values of vband F were significantly higher than right-sided

values (P<10−8).

The effect on our normal values of altering the fixed

tissue parameters revealed the following. Reducing Tkid

from 1.2 s to 1.1 s (a plausible value for a parenchymal

ROI dominated by cortical uptake; see Discussion in ESM)

increased values of Ktrans, vband F by about 9%, whilst

leaving FF unaltered. Reducing rd

indicated by rat studies [37] (e.g. 2.0 s−1mM−1; see Dis-

cussion in ESM) gave unrealistically high values of Ktrans

(0.56 min−1), whilst leaving vb, F and FF unaltered.

10

1towards the low values

Discussion

A much fuller discussion is given in the Electronic Supple-

mentary Material (ESM).

Imaging biomarkers and renal function

The kidney has many functions in maintaining homeostasis,

one of which, glomerular filtration, is critical in both clinical

nephrology and kidney research. The kidney’s filtration

fraction (FF) is defined as the ratio of GFR to renal plasma

flow (RPF); however, RPF is difficult to measure (both

experimentally and clinically). The mathematical model

presented in this paper provides parameters that can give

quantitative single kidney renal perfusion and GFR values

as well as measurements of blood volume and FF (Table 4).

Existing MRI techniques

Various models [1–11] have been proposed for analysis of

renal filtration using DCE-MRI. A critical review [38]

showed that none of the published methods was sufficiently

Table 4 Comparison of normal parenchymal renal parameters esti-

mated using DCE-MRI with published values. Thirty normal kidneys

were each measured twice, using the uptake phase and Gaussian VIRF.

Using a lower value for small-vessel haematocrit dramatically reduces

values for blood volume (right-hand column)

MRI

Hctsmall041%

Mean (SD)

Instrumental SD (CV) Literature value MRI

Hctsmall024%

Mean (SD)

Filtration (min−1)

Mean residence time (s)

Blood volume (%)

Perfusiondml blood min−1(100 ml tissue)−1

Filtration fraction (%)g

Absolute single kidney volume (ml)

Standardised single kidney volume (ml)j

Total GFR (ml min−1)

Ktrans

MRT

vb

F

FF

Vkid

Vkid*

GFR

0.25 (0.05)

5.5 (0.7)

44 (11)

284 (89)

15.5 (2.9)

230 (28)

214 (20)

115 (27)k

0.04 (18%)

0.4 (7%)

8 (18%)

72 (14%)e

1.5 (9%)

Not measured

Not measured

Not measured

0.28a

6.5b

35c

264f

15–20h

213i

213i

125h

0.25 (0.05)

5.5 (0.7)

34 (8)

219 (67)

15.5 (2.9)

230 (28)

214 (20)

115 (27)k

CV Coefficient of Variation

aGFR/(2Vkid*)

bMeasured using MRI by Sourbron [6]: ‘plasma transit time TP’; SD01.3 s

cFrom CT [43]

dFrom peak of Gaussian VIRF Fgausspeak; average perfusion over parenchymal ROI; plasma perfusion is independent of Hctsmall(see text)

eISD of cortical perfusion is better (14%; see Table 2)

fMean parenchymal perfusion 0 RBF/2*Vkid; RBF01.1 l min [1, 46]

gUsing Eq. 10

hTypical for young adult males [46]

iEstimated using mass0150 g (see text)

jI.e. corrected for body surface area (see text)

k2 KtransVkid

Eur Radiol (2012) 22:1320–13301325

Page 7

accurate to be clinically accepted. The Patlak model and

graphical analysis approach have been proposed for renal

MRI [4, 17, 39–41], and comparisons have been made [1, 5]

with more modern models.

Computed tomography

In DCE-CT imaging [41–43] the relationship between tracer

concentration and signal intensity is linear. However partial

volume effects and the estimation of small vessel haematoc-

rit are problems for both CT and MRI.

Benefits of this model

Analysis of the early (uptake) phase of DCE-MRI using this

model measures a few critical biomarkers of kidney func-

tion, namely vascular parameters and filtration. By restrict-

ing the time period over which the data are analysed (i.e.

excluding efflux of tracer from the ROI), a simpler model

can be used, designed specifically to measure vascular

parameters and filtration. By analysing parenchymal (not

cortical) ROIs, this analysis period has been extended to

90 s. This gives a time window where the domination of

signal behaviour by filtration can be exploited using an

optimal model, which is simple and therefore precise. Only

a single free parameter is required to characterise renal

function. The model represents the complexity of the actual

kidney DCE-MRI signal data (Fig. 3) whilst avoiding undue

further complexity; this probably contributes to being able

to estimate the model parameters reliably [44]. This is an

example of Occam’s razor, which states that if a variety of

explanations of a phenomenon are available, then in the

absence of any other information, the simplest one is to be

preferred.

Two additional kidney parameters are produced in this

analysis. The filtration fraction (Eq. 10) is a valuable pa-

rameter in diabetes and other clinical diseases. The mean

residence time (MRT) is very stable (Table 4), and its

physiological significance needs to be evaluated. Although

the MRT is measured from the aortic ROI, transit along the

renal artery is rapid [45].

Shortcomings of DCE-MRI measurements

Uncertainty in the haematocrit value affects some of the

tissue parameters and a realistic value for small vessel

haematocrit (Hctsmall) is required. The relationship between

renal blood and plasma flow is not straightforwardly given

by the large vessel haematocrit (as implied in text books

[46]), and CT measurements would be equally affected. The

higher values in the left-sided kidneys are probably an

artefact, for which a possible explanation is poor slice

profile [47].

Tissue parameter estimates are vulnerable to errors in

haematocrit, tissue relaxivity, T10and flip angle (Table 3).

Estimates of renal plasma flow Fp(and also plasma volume

vp) are independent of Hctsmall, and Fpmight be a more

useful tissue parameter than F. Our normal parenchymal

values (from Table 4) are Fp0167 ml plasma min−1

(100 ml tissue)−1, vp026%.

Relaxivity can alter in vivo from the in vitro values [37]

and is probably the largest source of systematic error in

DCE-MRI studies, as well as being unavoidable [48–50].

In disease, T10is often raised, and should be measured

explicitly if possible. In the absence of a measurement, then

published values must be used [23, 51, 52]. Flip angle errors

are likely at 3 T; a simple B1mapping technique [53] takes

only a few minutes.

Cortical and parenchymal ROIs For estimation of filtration

the entire parenchyma needs to be included in the ROI as the

filtrate progresses relentlessly down the tubules into the

medulla and back up to the cortex. Thus, the nominally

cortical ROIs seriously underestimate uptake (Fig. 5). The

Fig. 6 Vascular impulse response functions (VIRFs). All fitted the

data shown in Fig. 3a (parenchymal ROI, uptake phase) and have unit

area. The instant exponential VIRF modelled the delayed perfusion

peak badly. Differing peak values give rather different estimates for

perfusion (see Table 2), although mean residence times are similar for

both delayed VIRFs

1326Eur Radiol (2012) 22:1320–1330

Page 8

variable ROI definition is probably a major contributor to

within-subject variation [54].

VIRF shapes None of the VIRFs used in this work (see Fig.

6) is completely satisfactory; this is unsurprising given the

complexity of renal anatomy, and the precise form is unim-

portant for estimation of filtration (and hence GFR) or blood

volume (Table 1). An instant exponential model of the renal

VIRF does not fit the vascular peak as well as a delayed

exponential or Gaussian, although it has been used in the

two-compartment exchange model (2CXM) [55]. The

2CXM may be too simple a model for kidney vascula-

ture. Work is in progress to identify appropriate VIRFs.

The model has been extended to cover efflux, by adding a

single extra free parameter [14] (see ESM).

Clinical studies will establish whether the DCE-MRI

parameters will be sensitive to alterations in disease state such

as focal renal damage in reflux nephropathy, disease states

that affect the kidneys asymmetrically (obstruction or

stone disease) or following renal transplantation. If these

DCE-MRI biomarkers are shown to be reproducible then

the clinician will have a non-invasive tool that avoids

any radiation burden, and a quick single test will provide

both anatomy and physiological parameters of each kid-

ney separately. Recent publications show the advantage

of quantitative renal perfusion measurements over con-

ventional MRA [56, 57]. Our study has established the

feasibility of using our model to measure both blood

flow parameters and filtration with good reproducibility

and reasonable accuracy in normal volunteers (Table 4).

Table 5 Parameters used for modelling

Quantity SymbolUnitsType

Concentration in aortic blood

Concentration in extravascular space

Concentration in aortic plasma

Concentration in kidney plasma

Concentration in kidney tissue

VIRF delay

Perfusion

Filtration fraction

Flow into extravascular space per unit volume of tissue

Flow out of extravascular space per unit volume of tissuea

Flip angle

Haematocrit in large vessels

Haematocrit in capillaries

Filtration (0 GFR per unit volume of tissue)

Mean residence timec

T1relaxivity in blood

T1relaxivity in extravascular space

T1relaxivity in intravascular space

Pre-Gd blood signal

Pre-Gd tissue signal

T1of blood

T1of kidney

Gaussian VIRF width

Exponential VIRF width

TR

Fractional blood volume in kidney

Fractional plasma volume in kidney

Cb

Cd

Cpaorta

Cpkid

Ct

Δ

F

FF

F1

F2

θ

Hctlarge

Hctsmall

Ktrans

MRT

r1blood

r1d

r1iv

S(0)blood

S(0)kidney

T10blood

T10kidney

Tfwhm

Tg

TR

vb

vp

mM

mM

mM

mM

mM

s

ml blood (100 ml tissue)−1min−1

%

mmol s−1ml−1

mmol s−1ml−1

Degrees

%

%

min−1 b

s

s−1mM−1

s−1mM−1

s−1mM−1

A.U.

A.U.

s

s

s

s

s

0<vb<1

0<vp<1

Free

Fixed (17o)

Fixed (41%)

Fixed (41% or 24%)

Free

Fixed (4.5 s−1mM−1) [21]

Fixed (4.5 s−1mM−1)

Fixed (4.5 s−1mM−1)

Pre-calculatedd

Pre-calculatedd

Fixed (1.4 s)

Fixed (1.2 s)

Free

Free

Fixed (1.6 ms)

Free

aUsed for model with efflux (Eqs. 20, 21)

bOr can be expressed in ml min−1(100 ml tissue)−1to be compatible with F

cFrom VIRF

dFor each dataset, found from pre-Gd blood and tissue signals (arbitrary units)

Eur Radiol (2012) 22:1320–13301327

Page 9

Conclusion

Our model will calculate perfusion, filtration, filtration

fraction, mean residence time, blood volume and single

kidney GFR. It has been simplified by using uptake–

phase time-domain data from parenchymal ROIs, it deals

with signal non-linearity, it uses a reasonably realistic

vascular impulse response function, and it recognises a

reduced value of small vessel haematocrit. This model now

needs to be evaluated on datasets from other centres, and in

patients.

Acknowledgements

generously provided insightful and detailed comments and discussion.

Dr Sheldon Cooper brought Occam’s razor to our attention. Prof.

Kenneth Miles contributed to the design of this project. Kidney Re-

search UK sponsored acquisition of the volunteer data.

Prof. David Buckley and Dr Steven Sourbron

Appendix: Two-compartment mathematical model with

broadening and delay of vascular peak

A1 Mathematical model (Fig. 1)

The perfusion peak (first pass of the bolus) in the kidney

tissue data is consistently delayed and broadened compared

with that in the arterial blood curve (see Fig. 2). Here the

intravascular plasma concentration in the kidney Ckid

modelled as a convolution of the arterial plasma concentra-

tion Cart

response function (VIRF) g(t):

Zt

pðtÞ is

pðtÞ with a simple normalised vascular impulse

Ckid

pðtÞ ¼ Cart

pðtÞ ? gðtÞ ¼

0

Cart

pðt ? tÞgðtÞdt

ð1Þ

Z1

Thus g(t) is the IV response to an arterial delta function.

Depletion of IV Gd by filtration is assumed to be small

(i.e. FF≪1). The VIRF gives a variable delay and

broadening. Several functions for g(t) were investigated

(see below).

The rate of uptake F1into the renal extravascular space

(filtration) is proportional to the IV concentration:

0

gðtÞdt ¼ 1

ð2Þ

vddCdðtÞ

dt

¼ F1¼ KtransCkid

pðtÞð3Þ

vdis the fractional volume of the renal EV space (0<vd<1),

Cd(t) is the time-dependent concentration in this space, and

F1is the flow rate of Gd into the EV compartment. Ktrans

here is the filtration (GFR) per unit volume of tissue; it is

formally the unidirectional local transfer constant [12] for

Gd from the IV space. The solution for Cd(t) is a simple

integral:

Zt

The concentration of IV tracer in tissue is vpCpkid,

where the fractional volume ofthe IVplasmaspacevp¼ vb

ð1 ? HctsmallÞ; vbis the fractional blood volume in the

kidney and Hctsmallis the haematocrit in small vessels

such as capillaries. The total Gd tissue concentration

Ct(t) is then the sum of the IV and EV contributions

(which we have in Eqs. 1 and 4):

vdCdðtÞ ¼ Ktrans

0

Ckid

pðtÞdt

ð4Þ

CtðtÞ ¼ vbð1 ? HctsmallÞCkid

The MRI signal enhancement from a given Gd concen-

tration C(t) in blood or tissue is straightforward (assuming

fast exchange for water, so that all the water in a voxel is

relaxed by all the Gd). The reduction in T1is given by:

p þ vdCdðtÞð5Þ

R1ðtÞ ¼ R10þ r1CtðtÞ

R1is the relaxation rate (R101/T1), R10(01/T10) is its native

(i.e. pre-Gd) value (before injection of contrast agent) and r1

is the relaxivity (change in relaxation rate per unit concen-

tration of Gd). The possibility of differing relaxivities in the

IV space, the EV space and the arterial blood can be incor-

porated into fuller versions of this equation for tissue and

blood:

ð6Þ

Rkidney

1

Rblood

1

ðtÞ¼Rkidney

ðtÞ ¼ Rblood

10

þ riv

þ rblood

1vbð1 ? HctsmallÞCkid

CbðtÞ

pðtÞ þ rd

1vdCdðtÞ

101

ð7Þ

where individual values riv

compartment (see Table 5).

The signal from a spoilt gradient echo sequence is:

1; rd

1; rblood

1

are used for each

SðtÞ ¼ S0ð1 ? e?R1ðtÞTRÞsinθ

1 ? e?R1ðtÞTRcosθ

where θ is the flip angle.

The Gd concentration in the artery Cart

Eq. 1) is obtained from the measured blood signal as fol-

lows. Given S(0)blood(measured before the arrival of Gd)

andTblood

10

we can findSblood

0

(from Eq. 8). From the observed

blood signal (time–intensity curve) S(t)bloodcan then be

found Rblood

1

ðtÞ (Eq. 8). We can find the Gd concentration

in arterial blood Cart

plasma is then related by:

ð8Þ

pðtÞ (required for

bðtÞ (using Eq. 7). The concentration in

CbartðtÞ ¼ ð1 ? HctlargeÞCpartðtÞ

where Hctlargeis the haematocrit in arteries.

ð9Þ

1328Eur Radiol (2012) 22:1320–1330

Page 10

The curve-fitting procedure, the discrete representation of

the continuous functions at a temporal resolution of 0.4 s,

spreadsheet implementation and interpolation of the AIF using

Everett’s formula for cubic interpolation [58] are all described

in the ESM.

A2: Vascular impulse response functions: estimation

of perfusion and filtration fraction

Three VIRFs were implemented as discrete functions (see

ESM), forced to zero for t<0 and normalised to have a unit

area over their finite duration (up to about 20 s), i.e.

P

time (MRT) was found from the first moment of g(t).

Perfusion F was estimated from the VIRF peak or MRT.

The filtration fraction is then simply the ratio of GFR to

renal plasma flow F(1-Hctsmall) :

i

giðtÞdt ¼ 1 (Fig. 6). For each, the mean residence

FF ¼

Ktrans

ð1 ? HctsmallÞF

ð10Þ

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