becomes more dense and chaotic as tumours mature, thus restricting blood flow and conse-quently preventing agents from reaching their target. While smaller particles are then still able to penetrate tumours, nanoparticles with diameters of 100 nm or even larger do not penetrate tumours any more as they sequester near tumour blood vessels. Under this consideration of tu-mour biology and nanoparticle size, they were able to experimentally show a much higher GNP uptake by prostate tumours up to 40 % of the injected amount. This concept of “personalised nanomedicine” by tailoring nanoparticle designs according to the patient’s tumour characteris-tics, has the potential to improve the targeting process for the detection of tumours by over 50 %.
Apart from imaging applications, GNPs are also used in therapy, as e.g. shown in [43] where gold-silica nanoparticles were designed such that they absorb near-infrared light at wavelengths of high tissue transparency to provide a highly localised light-based strategy for the treatment of prostate cancer. This example shows that in case of a highly tuneable X-ray source, it is possi-ble to first use GNPs for the localisation of a tumour and secondly make use of their enhancing properties in tumour therapy without the necessity of moving the patient to different treatment stations.
The nanoparticles for the experiments reported in this thesis were manufactured at the Center for Hybrid Nanostructures (CHyN) which is part of the Institute of Nanostructure- and Solid State Physics (INF) of the University of Hamburg in the group of Prof. Dr. Parak. Research done in this group showed that the physiochemical properties of GNPs are mainly accounted for by their hydrodynamic diameter and their zeta potential (the measure of surface charge) which in turn determines the in vitro uptake and in vivo biodistribution of those particles [44]. For any scenario of an in vitro or in vivo application of GNPs, the nanoparticles would be exposed to proteins which eventually will adsorb to the surface of the particles and, furthermore, their kinetics depends on the interaction between nanoparticles and proteins [45, 46].
2.8 Data analysis
2.8.1 Fitting algorithm for photon spectra
Even if filters, collimators, and an optimal detector geometry are used for the measurements, the total counts registered by the detector are the sum of background coming from Compton scattered X-rays and fluorescence X-rays from the target element and other elements in the setup. Therefore, it is necessary to isolate the signal from the background by constructing an
2 Basics of X-ray physics, semiconductors, XFI and statistics
appropriate fit. The region selected for this analysis covers the strongest lines present, resulting from fluorescence processes in the target element, lead from the shielding and the electron beam dump and the detector material CdTe. The background non-peak region is fitted with a third-order polynomial function and interpolated to give an estimate of the background function in the region of the signal peaks. Gaussian functions are used to fit the fluorescence signals, limited by the following constraints:
• For the Gaussians fitted to the characteristic gold K-lines, the width is fixed to the values of the norm spectrum of a gold-foil.
• The width of the Gaussians fitted to the characteristic Gd K-lines is fixed to the values of the spectrum of 78 mg/ml Gd-solution.
• All peak positions are set to the corresponding peak energies taken from [47].
Finally, the number of fluorescence photons is given by the area underneath the Gaussian functions, divided by the bin width of the histogram.
If one wants to quantify the level of agreement between the data and a hypothesis, a test of significance [48] has to be carried out. The discrepancy between the data and the expectation under the assumption of a hypothesis H0 is quantified by the p-value. It is defined as the probability to find a defined statistic t in the region of equal or lesser compatibility with H0 than the level of compatibility observed with the actual data,
p=
∞
Z
tobs
f(t|H0)dt, (2.11)
wheretobsis the value of the statistic obtained in the experiment.
Often the p-value is converted into an equivalent significanceZ which is usually expressed in units of the standard deviationσ,
Z = Φ−1(1−p), (2.12)
withΦbeing the cumulative distribution of the standard Gaussian andΦ−1 its inverse function.
For a very large number of photons, the calculation of the significance can be simplified to Z ≈ nobs−nexp
√nexp . (2.13)
22
2.8 Data analysis
In order to make sure that the net signal was within the 99.7 % confidence interval, only signal values in the interval ±3σ around the Kα1 and Kα2 fluorescence lines are considered.
The evaluation of this combined significance is commonly used for the quantification of the signal quality in XFI measurements and will be used in this thesis as well.
The last statistical parameter of interest is the so-called goodness-of-fit, where one obtains the minimum value of the quantity χ2. As it is a Poisson distribution with variancesσ2i =µ2i, this quantity becomes Pearson’sχ2 statistic,
χ2 =
N
X
i=1
(ni−µi)2
µi . (2.14)
If the hypothesisµ= (µ1, ...., µN)is correct, and if the measurementsni can be treated as a Gaussian distribution, then theχ2 statistic will follow the probability density function with the number of degrees of freedom equal to the number of measurements N minus the number of fitted parameters. Aχ2value close to one implies that the fit can describe the data well, whereas higher values represent a poor fit.
2.8.2 Fitting algorithm for electron spectra
The electron spectra are fitted with a Gaussian model in MATLAB [27] which is given by y=
n
X
i=1
aie[−(
x−bi ci )2]
, (2.15)
where a is the amplitude, b is the centroid (location), c is related to the peak width, n is the number of peaks to fit, and 1 ≤ n ≤ 8. The corresponding two-term Gaussian model is therefore given by
y=a1e[−(
x−b1 c1 )2]
+a2e[−(
x−b2 c2 )2]
. (2.16)
It has to be noted that the definition used in MATLAB is different from the standard Gaussian distribution (also known as standard normal distribution). The simplest case with a meanµ= 0 and standard deviationσ= 1, is described by the probability density function
φ(x) = 1
√2πe−12x2. (2.17)
Every normal distribution is a version of the standard normal distribution whose domain has
2 Basics of X-ray physics, semiconductors, XFI and statistics
been stretched by a factorσand then translated byµas f(x|µ, σ2) = 1
σφ(x−µ
σ ). (2.18)
The probability density must be scaled by1/σso that the distribution remains normalised. If Z is a standard normal deviate, thenX =σZ+µwill have a normal distribution with expected value µand standard deviationσ. Conversely, ifX is a normal deviate with parametersµand σ2, thenZ = (X−µ)/σwill have a standard normal distribution.
If one wants to equate the standard deviation s (as defined in the MATLAB model) for a single Gaussian model, it can be computed by
s =c1/√
2, (2.19)
while the mean valueµis simply given by the parameterb1.