Pharmacokinetic Model

When we want to attempt to quantify the observed contrast agent kinetics in terms of physiologically meaningful parameters we first need to define the elements of the tumor or tissue structure and the functional processes that affect the distribution of the tracer (the contrast agent). It is customary to represent tissue as comprising three or four compartments, each of which is a bulk tissue characteristic (we are unable to observe these compartments at their natural microscopic scale, but we can observe their aggregate effects at the image voxel scale or in a region of interest).

These compartments are the vascular plasma space, the extracellular extravascular space (EES), and the intracellular space (Fig. 1.4).

Fig 1.4 Three compartments in tracer kinetic model. [2]

All clinically utilized MRI contrast agents, and most experimental agents, are not pass into the intracellular space of the tissue due to their size, inertness, and non- lipophilicity, making the intracellular space un-probable using DCE-MRI; for this reason, the intracellular and other volumes are usually lumped together as a loosely defined intracellular space. According to fig 1.4 we can get the relationship between these compartments:

sses that accompany the faster growth rate of man

fraction occupied by the intracellular space, Vb is the fraction occupied by whole blood, and Hct is the haematocrit (typically about 0.4).

The functional parameters, or delivery mec

ribution in the intravascular space and the EES are usually assumed to be restricted to blood flow F and the endothelial permeability surface area product PS, which describe how leaky a capillary wall is.

There are two physiological proce

y tumors: an increased number of vessels and along with an increased permeability.

Therefore, one could expect an increased overall signal enhancement in the vicinity of tumors due to increasing vascular volume, vessel permeability, and increased flow.

ould des

ed semi quantitatively using parameters der

models were developed from Nuclear Medicine quantitative studi

.4.1 General Kinetic Model

el (GKM) is one approach to understanding the complex In the simplest model of tissue signal enhancement characteristics one c cribe 3 parameters: maximum signal enhancement, the rate at which this initial enhancement occurs“wash-in”, and the rate at which this increased signal decays

“wash-out”. However, it is important to consider this contrast dynamic with respect to the concentration of contrast agent in the vascular system as it perfusion the tissues. In simple graphical wash-in wash-out, it is assumed that the contrast agent immediately reaches equilibrium in the vascular system [6].

The data obtained with DCE-MRI is report

ived from pharmacokinetic models. Quantitative techniques that are often combined with rapid temporal sampling have been used together with simple pharmacokinetic models of tissues, obtaining parameters such as the transfer constant (K^trans), the rate constant (kep), and ve.

Most MRI kinetic

es but the limitations of MRI dictated specific modifications. A number of these MRI models are in current use, the primarily differentiated based on the way that they model the “arterial input function” (AIF).

1

The General Kinetic Mod

kinetics of contrast enhancement. The physiological processes of GKM are described in Figure 1.5, where the GKM simplifies the anatomy of the tumor into two functional components, the vascular space and the EES and one non-functional component, the intracellular space.

Fig 1.5 Illustration of General Kinetic Model [6]

A contrast agent, spec weight agent which

rem

ifically a highly diffusible low molecular

ains extracellular, when introduced into the vascular space will leak into the EES at a characteristic rate and then will leak back into the vessel at another rate. Thus the net

change in concentration in the tumor can be described as:

trans t

p ep t

dC K C k C

dt

= − (1.3) where K^trans is a factor related to “wash in” and kep is a factor related to “wash-out” and the relationship between these parameters was the volume of extravascular-extracellular space ve. Furthermore, we can numerically evaluate these parameters for a variable concentration input function. This expression is mathematically described as a convolution integral.

(1.4)

relationship of the rat at any g

.4.2 Patlak Model

h to determining vessel permeability from time concentration cur

C t_t( )=K^trans[C t_p( )⊗e^{k tep} ]

This equation gives the e of change in tumor concentration iven time after contrast administration to the plasma and tumor concentration at that time. A numerical solution for the GKM (K^trans and kep) can be obtained by a nonlinear fitting algorithm for this expression. Subsequent models are based on this general model, but use various assumptions to work around the convolution integral.

1

Another approac

ves was proposed by Patlak [7]. It uses a graphical method to estimate permeability surfaces and fractional vascular space based on the slope and intercept of a derived line.

Figure 1.6 describes the physiological processes of the Patlak model in a block format.

Vascular Space Vascular

Fig 1.6 Illustration of Patlak Model [7]

In this method, flow from the tissue space to the vascular space is assumed negligible and flow is assumed to be unidirectional. In this model, the contrast agent in

( ) ( )

trans t

p p p

K

∫

C τ τd v C t (1.5)

wher lasma volume. The term is sim e.

the tumor can be expressed as:

C t_t( )= +

0

e vp is the fractional p ilar in concept to the term v Dividing both sides of the equation by Cp(t) yields:

0^t _p( )

The Patlak approach utilizes a simpler approach than the standard pharm mod

.4.3 Brix Model

is also a two compartment model in which the arterial input curve is assume

acokinetic el. A major advantage of the Patlak model is based in its incorporation of AIF.

However one limiting assumption of this model is that the contrast agent flows only into the tissue of interest. If the slope of the “Patlak” graph is not linear, then the assumption of no back flow is violated and the parameters generated would no longer be valid.

1

Brix model

d to be the result of a prolonged constant infusion that takes the shape of square wave (i.e. the contrast agent instantly reaches a plateau, remains constant for awhile and then instantly is over) which mixes in the vascular space and is slowly eliminated by renal excretion [8]. The input function is of magnitude Kin, the elimination constant is kel, and the rate constants describing the transfer of contrast agent from plasma to the

Central Compartment

Peripheral Compartment

K

_in

K

_el

K

₂₁

K

₁₂

Central Compartment

Peripheral Compartment

K

_in

K

_el

K

₂₁

K

₁₂

Fig 1.7 Illustration of Brix Model. [8]

The mathematical expression of the temporal response of SCM (t) / S0 is obtained:

^SCM_S^{( )t = +1 A v e}

{

^⎡⎣ ^{(k tel')−1⎤}⎦^{e(k tel)−u e⎡}⎣ ^{(k t21')−1⎤}⎦^{e(k t21)}

}

(1.7

0

)

W is the time–independent Gd-DTPA enhanced M s a

fitti

here SCM (t) RI signals, A i

ng parameter depending on the properties of the tissue of the sequence used, and of the infusion rate (Kin). Brix put forth a mathematical description that incorporated a term that allowed the adjustment of an AIF parameter.

.5 Tofts Model

es a different approach to the arterial input function (AIF), but reta

by diffusion transfer of contrast material between the vas

del to establish the time cou

to model the concentration of tracer with time. It con

1

Tofts model tak

ins the fundamental assumptions of the GKM (General Kinetic Model) [9],[10]. In the Tofts model, the input function is assumed to be the result of a pulse bolus injected into a two compartment system.

The arterial input is modified

cular space and body extravascular space; this system of compartments modifies the pulse bolus into a biexponential arterial input function [11].

This model consists of two parts: a compartmental mo

rse of the contrast agent (Gd-DTPA) tracer concentration in the tissue; and relate to observed MRI signal enhancement.

A compartmental model is used

sists of a plasma volume, connected to a large extracellular space which is distributed throughout most of the body (e.g., muscle). The kidneys drain tracer from the plasma, and hence from the extracellular space. We have modified this model by adding a fourth compartment, the lesion, which is connected to the plasma through a leaky membrane (Fig 1.8).

Fig 1.8 Illustration of Tofts Model.

Most methods of analyzin T1-weighted data have used a co

mpartmental analysis to obtain some combination of the three principle parameters:

the transfer constant (Ktrans), the extravascular extracellular space (EES) fractional volume (ve), and the rate constant (kep).

Symbol Preferred short name Full name

K^trans Volume transfer constant between blood plasma

and EES Transfer Constant

kep Rate Constant Rate constant between EES and blood plasma ve EES Volume of extravascular extracellular space per

unit volume of tissue tandard Kinetic Param

Table 1.1 Three S eters.

Most methods of analyzing dynamic contrast-enhanced T1-weighted data have used a co

mpartmental analysis to obtain some combination of the three principle parameters:

the transfer constant (K^trans), the extravascular extracellular space (EES) fractional volume(ve), and the rate constant (kep). The transferconstant and the EES relate to the fundamental physiology, whereas the rate constant is the ratio of the transfer constant to

the EES [10]:

h permeability, transfer constant is equal to the blood plasma flow per unit vol

The rate constant can be derived f ape of the tracer concentration vs time , whereas the transfer constant and EES require access to absolute values of tracer concentration. The transfer constant Ktrans has several physiologic interpretations, depending on the balance between capillary permeability and blood flow in the tissue of interest. Where F are Perfusion (or flow) of whole blood per unit mass o

the

racer flux is permeability (PS >> F)

f tissue,

ρ

means density of tissue, Hct represent for Hematocrit, P means total permeability of capillary wall, S means surface area per unit mass of tissue.

In the other limiting case of low permeability, where t

limited, the transfer constant is equal to the permeability surface area product between blood plasma and the EES, per unit volume of tissue [12]:

trans

K =PSρ (PS<<F) (1.10) Tracer flows passively from the blood pl eable capillary into the EES, thro

ate constant kep is formally the flux rate constant between the EES and blood plas

low-Limited Model (High Permeability)

con

d by setting the venous asma in a perm

ugh microscopic pores or defects in the capillary walls. It also called the interstitial space.

The r

ma. It’s always greater than the transfer constant K^trans. For a range of typical EES fractional volumes seen in tumors and multiple sclerosis (ve = 20% ~ 50%), kep is two to five times higher than K^trans [10].

F

Its first assumption is that arterial and venous blood have well-defined centrations, supplying and draining the tissue under study. Second, because permeability is high, venous blood leaves the tissue with a tracer concentration that is at all times in equilibrium with the tissue. Thus, soon after injection of the tracer, the arterial concentration is high, the venous concentration is low, and most of the tracer is being removed from the blood as it passes through the tissue.

For an extracellular tracer, the model can be extende

concentration equal to that of the EES. The effect of intravascular tracer on the MR signal can be ignored (ie, the vascular signal is small compared with the tissue signal).

In this case the following differential equation relating tissue concentration Ct to arterial

plasma concentration Cp can be obtained:

^dCt _{F (1}

considered as a single pool, with equal arte

bility surface area product of the cap

P

If flow is high, the blood plasma can be

rial and venous concentrations. The transport of tracer out of the vasculature is slow enough not to deplete the intravascular concentration.

The rate of uptake is then determined by the permea

illary wall and the difference between the blood plasma concentration and the EES concentration. If the contribution of tracer in the intravascular space is ignored, the

transport equation is

.6 Application in DCE-MRI

netic resonance imaging (DCE-MRI) is being used in

be used as a biomarker, the method for quantifying the assay has to

n illustration of parametric analysis of DCE-MRI images using an emp

oncology as a noninvasive method for measuring properties of the tumor microvasculature.

For DCE-MRI to

be defined. There are several goals to be weighed in optimizing the biomarker definition. The biomarker needs to (1) maximize the sensitivity to biologic changes caused by treatment; (2) capture tumor heterogeneity, which is an important as a biomarker [13].

Fig 1.9 is a

irical parameter, the SER, for a patient with locally-advanced breast cancer treated with doxorubicin-cyclophosphamide (AC) chemotherapy. MRI was performed before chemotherapy, 2 weeks after the first cycle of chemotherapy, and at the end of AC treatment, before surgery, using a three–time point DCE-MRI method.

Pharmacokinetic properties of the tumor were quantified by computi

el, defined as SER＝(S1-S0)/(S2-S0), where S0, S1 and S2 are the pre-contrast (baseline), early post-contrast and late post-contrast signal intensities.

DCE-MRI is a promising biomarker candidate for assessin

tment. Correlative studies performed in combination with therapeutic trials have

demonstrated proof of concept for DCEMRI as a biomarker; however they have not been powered to adequately evaluate biomarker performance [13].

Fig 1.9 Contrast-enhanced magnetic resonance images (top row) and signal enhancement ratio (SER) parametric maps (bottom row), acquired before treatment (A), 2 weeks after the first cycle o

onstrates a single slice imaging technique. The image acquisition is p

f doxorubicin-cyclophosphamide (B), and at the end of chemotherapy, before surgery (C), for a patient with locally advanced breast cancer. Blue, green, and red color coding corresponds to low, moderate, and high values, respectively. [13]

Another paper dem

erformed in less than 500 ms making it relatively insensitive to respiratory motion.

Data from phantom studies and a reproducibility study in solid human tumor. The reproducibility study showed a coefficient of variation (CoV) of 19.1% for Ktrans and

or two commonly used parameters, K^trans and IAUC (60

improved to 16 and 13.9% if tumor of diameter less than 3 cm were excluded. The individual repeatability was 30.6% for K^trans and 26.5% for IAUC for tumor which are greater than 3 cm diameter [14].

The individual patient data f

), calculated from R1 values, are given in Table 1.2. Although no correlation was seen between T2 signal intensity and enhancement parameters, the second case in Table 1.2 had very high T2 compared with the other cases, consistent with a cystic nature of the metastasis. Guidelines from a recent US national cancer institute workshop on DCE–MRI state that tumors in a fixed superficial location should be at least 2 cm in diameter and other tumors should be at 3 cm in diameter. This study shows a tendency for greater variability with reducing size, and excluding lesions less that 3 cm in diameter reduced CoV.

Table 1.2 Individual patient data showing tumour size, mean difference, coefficient of variation (CoV) and repeatability for K^trans and IAUC(60) for two scans.[14]

and repeatability values The colorectal liver metastases group also had lower CoV

ns 14.2 % and 26.5% and IAUC(60) 11 % and 21.3%, respectively), although this

may be related to the fact that this group had relatively larger tumor.

Another approach explore the randomized trials confirm these

ivalent survival for adjuvant and neo-adjuvant chemotherapy in patients with primary operable breast cancer [16-17]. A further benefit of neo-adjuvant chemotherapy is the opportunity to assess the chemo-responsiveness of the tumor. The overall response rates reported vary between 60% and 100%, with complete clinical responses ranging from 10% to almost 50%, avoiding mastectomy in most cases. Clinical responders have a better prognosis than do non-responders [18].

The prognostic importance of histo-pathologic response amon

-adjuvant chemotherapy for breast cancer is also recognized [19]. Patients who have complete pathologic response or pathologic minimal residual disease have a longer disease-free and overall survival compared with patients who have gross residual disease. The ability to identify non-responders early after the start of chemotherapy would be of major benefit because it would enable treatment to be adjusted or enable alternative and possibly more efficacious treatments, such as other types of chemotherapy or early surgery, to be offered as soon as possible [20].

Fig 1.10 shows the change in transfer constant in perimenopaus

gra

(middle row), an increase in the transfer constant med

de 3 infiltrating ductal carcinoma of the left breast not responding to mitoxantrone and methotrexate chemotherapy.

After one cycle of treatment

ian and range is seen (57% and 34%, respectively), compared with a 10% decrease in tumor size. After two treatments (bottom row), a further increase in the transfer constant median and range is seen (186% and 181%, respectively) on the transfer constant histogram, compared with a 11% increase in tumor size [21].

Fig 1.10 Columns show anatomic subtraction images, corresponding Transfer constant maps, and histograms from pixel data. Row shows data before treatment and after one and two cycles of mitoxantrone and methotrexate chemotherapy, respectively. [21]

Chapter 2 Theory in Segmentation

.1 Segmentation in normalized cuts method

a digital image into multiple reg

rpose algorithms and techniques have been developed for image seg

ed by Shi and Malik in 1997[23] . In this

2

Segmentation refers to the process of partitioning

ions. The goal of segmentation is to simplify or change the representation of an image into something that is more meaningful and easier to analyze [22]. Image segmentation is typically used to locate objects and boundaries (lines, curves, etc.) in images. The result of image segmentation is a set of regions that collectively cover the entire image, or a set of contours extracted from the image. Each of the pixel in a region are similar with respect to some characteristic or computed property, such as color, intensity, or texture.

Several general-pu

mentation. Since there is no general solution to the image segmentation problem, these techniques often have to be combined with domain knowledge in order to effectively solve an image segmentation problem.

The “normalized cuts” method was first propos

method, the image being segmented is modeled as a weighted undirected graph.

Each pixel is a node in the graph, and an edge is formed between every pair of pixels.

The weight of an edge is a measure of the similarity between the pixels. The image is

l dissimilarity between the

partitioned into disjoint sets by removing the edges connecting the segments. The optimal partitioning of the graph is the one that minimizes the weights of the edges that were removed (the “cut”). Shi’s algorithm seeks to minimize the “normalized cut”, which is the ratio of the “cut” to all of the edges in the set.

The normalized cut criterion measures both the tota

erent groups as well as the total similarity within the groups. The grouping algorithm consists of the following steps:1. Given an image or image sequence, set up a weighted graph G = (V,E) and set the weight on the edge connecting two nodes to be a measure of the similarity between the two nodes. 2. Solve …(D-W)x = λDx for eigenvectors with the smallest eigenvalues. 3. Use the eigenvector with the second smallest eigenvalue to bipartition the graph. 4. Decide if the current partition should be subdivided and recursively repartition the segmented parts if necessary.

A graph G = (V,E) can be partitioned into two dis

oving edges connecting the two parts. The degree of dissimilarity between these two pieces can be computed as total weight of the edges that have been removed. In graph

theoretic language, it is called the cut :

cut A B)( ,

∑

( , ) (2.1)

e one that minimizes this cut value. Although there are an

The optimal of a graph is th

exponential number of such partitions, finding the minimum cut of a graph is a

d by recursively finding the minimum cuts that bise

cut criteria favo

well-studied problem and there exist efficient algorithms for solving it. Wu and Leahy [24] proposed a clustering method based on this minimum cut criterion. In particular, they seek to partition a graph into k-subgraphs such that the maximum

cut across the subgroups is minimized.

This problem can be efficiently solve

ct the existing segments. As shown in Wu and Leahy's work, this globally optimal criterion can be used to produce good segmentation on some of the images.

However, as Wu and Leahy also noticed in their work, the minimum

rs cutting small sets of isolated nodes in the graph. This is not surprising since the cut defined in (1) increases with the number of edges going across the two partitioned

rs cutting small sets of isolated nodes in the graph. This is not surprising since the cut defined in (1) increases with the number of edges going across the two partitioned

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