Theoretical development of the changes produced in the porosity of gels irradiated with X-rays

J E Dávila-Pérez1,2,3, R Martín Landrove1.2
1 School of Physics, Faculty of Science, Central University of Venezuela, Caracas, Venezuela
2Física Médica C.A., Caracas, Venezuela
3 Gurve Group of Radiotherapy, Caracas, Venezuela

Abstract. There is no abstract…..
1. Introduction

Modern radiotherapy treatments have included complex computational systems which have allowed to achieve more complex dose distributions conformed to the tumor volume allowing dose escalation. All this implies the need for greater quality control in the verification of treatment plans prior to delivery.
Three-dimensional dosimeters constructed with gels have emerged as an ideal tool for the verification of these treatment plans and their development is increased every day using different materials for their construction and methods for their evaluation.
The irradiation of these gels using X-rays with potentials of 6 MV and doses between 30 and 300 Gy generates morphological changes in their tertiary structure and porosity [1] which entails changes that can be analyzed with different spectroscopic and imaging techniques that allow detecting and quantifying these changes produced relating them to the dose delivered.
This paper makes a theoretical development of the changes produced in the porosity of gels built with gelatin and fructose and irradiated with X-rays using potentials of 6 MV from magnetic resonance imaging with diffusion technique

2. Materials and Methods
2.1. Gel formulation and sample preparation
The gel formulation to be considered for half a litter volume is a mixture of type B and 250 Bloom gelatin (20g), fructose (80g) and de-ionized distilled water (500cc). This particular choice for gelatin and fructose concentrations provides the highest response for MR images.
The first material utilized in combination with gelatin was sodium chloride (NaCl) in low concentrations and the irradiated samples were scanning with MR images (T1 and T2-FLAIR maps) and not appreciable changes were found in the images with respect to the initial image without irradiation. Subsequently, glucose was used varying the concentrations, but as with the sodium chloride samples there was no response.
From this moment we decided to change to fructose using as a reference the works of Bunn and Higgins (1981) and McPherson and collaborators (McPherson and et al, 1988) which propose that fructose is much more reactive than glucose with respect to its participation in glycosylation reactions.
For a sample preparation we consider two major stages:
1. After mechanical agitation gelatin was completely dissolved in water (250cc) at room temperature. The same was done with fructose, which was also completely dissolved in water (250cc) at 40oC. The next step was to add under mechanical agitation the fructose solution to the gelatin solution until a homogeneous mixture is attained.
2. A slow cooling process took place where room temperature was reached after 4 hours and then the sample was placed in a refrigerator at 4oC for 24 hours prior to irradiation.

2.2. Sample irradiation protocol

The irradiation of the samples was carried out according to the procedure used in irradiation in clinical cases. For this, arcs were chosen in an isocentric way by rotating the gantry of the linear accelerator an arc of 240.0 degrees on its axis of rotation. For irradiation, a cone of 2.0 cm in diameter was used coupled to the collimator and the doses delivery varied between 30 Gy and 300 Gy.

3. Theorical Development

3.1. Properties and structure of diffusion coefficient profile for irradiated samples as a function of absorbed dose
It will then be shown that to be a homogeneous function [2] with respect to its dependence on the dose profile . To do this we start from a profile of the diffusion coefficient and we see that
Ec. 1
Ec. 1 becomes the linear transformation required to have the results of fig. 4 if the quotient is a constant that does not depend on the dose profile involved. In our case the dose profile is proportional and then we can write
Ec. 2
and consequently is indeed a homogeneous function [2] with respect to its dose dependence. The quotient becomes a dose escalation parameter and the function is not arbitrary since it must have the form [2, 3, 4]
Ec. 3
where the parameter is called the degree of homogeneity. Not every homogeneous function corresponds to a power law, but power laws all are clearly homogeneous function. Next, we will show that is does indeed take the form of a power law, i.e.
Ec. 4
To do this we proceed to make the adjustment of the average profile of the diffusion coefficient using
Ec. 5
Where the dependency is assumed to be of power law with exponent and is the profile given by
Ec 6
In the same way is the dose profile provided by the treatment planning system, which is normalized so that the maximum dose is dimensionless and correspond to 100%. In principle corresponds to the dose on the glass wall at the left end of the profile, which for the purpose of this adjustment will be absorbed as a constant in . The temporary dependency does not count for what we are going to do and is also included in the constant and . The results in shown in fig. 4 and we have that the exponent is given by the adjustment as with a correlation coefficient of , which indicates that the adjustment is of excellent quality and then we can conclude that indeed the dependence is that of a power law.
Knowing that the effect of radiation on the gel in this case correspond to the denaturation of the protein by breaking bonds followed by an incomplete repair process where we will also have pore sealing when cross-linking occurs due to the action of the radiation products, the transport of water molecules in the gel will also be altered and in particular the path of water molecules which is linked to the diffusion process in the medium that form the gel.
The problem can be addressed with percolation theory where we can visualize the gel as a very large network that forms a medium where in the porous matrix circulate a liquid that transport ions where de most abundant are associated with the happens with water molecules.

Fig. 4. Graf of logarithm versus of logarithm of the diffusion coefficient profile as a function of the normalized dimensionless dose so that the maximum corresponds to 100% and after having subtracted the baseline. The linear behavior indicates that as assumed it is a power law with exponent 1.75  0.01 and correlation coefficient squared of 0.987. The error in the adjustment was 1.42%. The result is completely independent of the spatial distribution of the delivered dose.
These ions correspond to , which is formed by solvation of , and . In this way we can speak of a conductivity and a diffusion coefficient for the i-th species of ion, which are related by the Nernst-Einstein equation as
Ec. 7
where is the density of ions with charge . In what follows, in order to make the argument more transparent, we will ignore the subscript since the relevant quantities scale in the same way for all ions. We can consider the correlation length for this system, given by [5] (see fig. 5, although the gel as a porous medium is in a three-dimensional space, we can resort to two-dimensional examples that allow us to visualize from the qualitative point of view what happens although the quantitative behavior is different according to the number of dimensions to be handle)
Ec. 8

Fig. 5. Diagram of the situation obtained by simulation in a finite two-dimensional network (25×25) where a portion of the connections is shown for a finite aggregate or “cluster” where its typical size coincides with the correlation length (taken from Redner, 2011).
where is the probability that there is a connection between pores, its complement is the probability for the absence of that connection and for the correlation length is very large or infinite, that is, when the probability that there is a connection reaches that value the correlation length becomes comparable to the size of the system. We also have the complementary space where there is no connection, with probability of occurrence also complementary and the correlation length is given by the Ec. 8 with the replacement , where . The region that represents the gel can be divided into subregions small enough to be able to assume that the irradiation is uniform in each one and at the same time large enough so that in them aggregates or “cluster” can be formed where their typical sizes coincide with the correlation length. To those subregions we are going to take them as the system to consider and if it is their typical size of the system, the quotient can be seen as measured in units of or simply as a way to compare the length of correlation with the size of the system and must decrease by the increase of with the dose, since it approaches and the opposite occurs with and .
Using Ec. 8 we see that if the exact value corresponding to [5] is used, it can be written as
Ec. 9
Consequently, the conductance is given by [5]
Ec. 10
where we take the exponent with the value 2 because the upper bound is known and nearby values such as [6] have been found in all numerical treatments. For conductivity and ion density we arrive at
Ec. 11
where the conductivity is obtained from the Ec. 10 for the conductance and where it was taken into account that the relationship between them is given through a multiplicative factor in the first power, since the conductance is the product of the conductivity by an area (which has Euclidean dimension and is proportional to ) between a length (also with Euclidean dimension and is proportional to ). Obtaining the density of Ec. 11 requires the introduction of the concept of “backbone” of an aggregate or “cluster”, which we will see below and where is the fractal dimension of the “backbone”.
Fig. 6 shows a portion of and infinite aggregate o “cluster” of segments connected in an equally infinite network in two dimensions [7], we can be used to represent what happens to the transport of liquid in the gel and has been used extensively in the literature to study conduction in non-homogenous media. There we can see that there are sets of segments that can be separate by cutting a single connection.

Fig. 6. Schematic of the situation in an infinite two-dimensional network where a portion of the connections is shown for an infinite aggregate or “cluster”. The aggregate or “cluster” has a fractal dimension that is different from the fractal dimension of the “backbone” which is represented by tick lines. Branches that do not belong to the “backbone” are those that can be separated by cutting off a single connection. The points mark the optimal path to go from the point at the upper end to the point at the lower end (taken from [7]).
When you finish cutting all the branches that have that property, there is still are interconnected set that is known as the backbone of the aggregate or cluster. Herrmann and Stanley (1984) [8] numerically determined the fractal dimension for the same type of three-dimensional system with the result . In that work they found that the “mass” (which can effectively represent a mass, but the same time can be used with other extensive quantities such as number of particles) of the “backbone” must scale as
Ec. 12
when the distance connecting two points in the network satisfies that , the fractal dimension take the aforementioned value and is the typical size of the system [9]. The “backbone” in general will be relevant in the description of transport properties and treatment of the fracturing phenomenon [9,10]. In our case it can be expected that and consequently we have that the number of ions that are circulating behaves as
Ec. 13
To find the density we consider the dimension associated with the volume as Euclidean space, which by dividing what is given in the Ec. 13 allows us to obtain what is presented in the Ec. 11. All ion densities in the liquid scale in the same way. Finally considering the Ec. 7 we arrive at that the diffusion coefficient behaves as
Ec. 14
Where and if we take we arrive at , which is in accordance with the experimental exponent found. If we take as the exact value , we get equally .
As indicated before, the correlation length must grow with the dose delivered and to see it we consider an aggregate or “cluster” of size within one of the aforementioned subregions, where irradiation can be taken as uniform and with a typical size . The formation of the aggregate by the action of radiation is partly due to the action of free radicals associated mainly with the radiolysis of water and also to the direct ionization by high-energy secondary electrons. For free radicals it will be difficult to penetrate the structure of the aggregate and with very high probability they will interact with its surface. In the same way they can also contribute to their initiation.
Secondary electrons will be generated anywhere in the subregions but their contribution to the formation of aggregate structures will take place outside then giving rise to an initiation or on the surface of them and in that sense their contribution will be similar to that due to free radicals. Inside the aggregate these electrons can in principle alter its structure but this leads to a situation that does not alter the transport in the sense that there are practically no important changes of the “backbone” through which water cannot circulate.
Based on this we can say that the fraction of change of the correlation length (where at the same time is the typical size of the aggregate) must be proportional to the surface/volume ratio of the aggregate. The volume of the aggregate, which has a Euclidean dimension, must be proportional to and the surface where the action of the radiation products will take place is also Euclidean in this case due to its high reactivity (the reaction take place preferentially at the outermost points not giving opportunity for them to penetrate something into the aggregate) with the result to being proportional to . The surface/volume ratio is therefore proportional to . At the same time as aggregates grow, there must be a saturation effect when their size becomes comparable to the typical size of the subregions we are considering. So, to take into account these two effects we can write the fraction of change of the correlation length for an absorbed dose between and as
Ec. 15
where is a constant representing the rate of increase in length per unit dose absorbed and is a quantity that characterizes the gel. The increase in size stops when it reaches . Ec. 15 can also be written as
Ec. 16
If we define , the Ec. 16 can be written in terms of dimensionless quantities and is reduced to
Ec. 17
and has as a solution
Ec. 18
The saturation property that is must have is clearly reflected in the Ec. 30 in the limit for high doses and that it must correspond to the percolation threshold.
This phenomenon is similar to that observed by Neamtu et al. (1999) [11] in a system made up of proteins, glycoproteins and lipids such as cell membranes. In this experiment, erythrocyte membranes are taken to which pores are previously produced by the action of an electric field. These pores where sealed by bombarding the membranes with and electron beam and the quality of the sealing process also depends of the absorbed dose. The critical value of the dose for the corresponded to 100 Gy, above which Neamtu et al. (1999) [11] report a significant drop in effect. It is expected that something similar will be induced by ionizing radiation in our case and the order of magnitude of the dose to reach saturation by geometric conformation must be in several hundred Gy. When this occurs in contiguous regions where aggregates of comparable sizes grow, this can give rise to what is referred to in the literature as explosive percolation where the correlation length in a collection of subregions is the size of the region that groups them. In the limit of low dose, the Ec. 20 leads us to
Ec. 19
Or equivalently
Ec. 20

For irradiations conditions such that it meets in the entire scanned dose range between 30 Gy and 300 Gy with
Ec. 21
or simple the dose is closed , as they can also be the same , you get
Ec. 22
and is the profile of the diffusion coefficient where the baseline was subtracted. With the idea of making and additional verification, the adjustment of the profile of the average diffusion coefficient was carried out again, but now with the expression
Ec. 23
where the free adjustment parameters are and , taking which is the value of the dose on the wall of the phantom when we irradiate at 30 Gy. The results of this procedure are shown in Fig. 7 and we have that the exponent in the new setting take the same value of and . As indeed, in addition to the fact that the behavior shown in Fig. 7 is independent of the spatial distribution of the dose delivered, we can conclude that the dependence corresponds to a power law.

Fig. 7. Profiles of the diffusion coefficient (experimental points, 4,17% error) and the dose delivered according to the planning system (for a maximum of 30Gy, although the profile presented corresponds to the normalized one with 100% maximum dose) depending on the position along the diameter of the spherical phantom.

4. Conclusion
As indeed, in addition to the fact that the behavior shown in Fig. 7 is independent of the spatial distribution of the dose delivered, we can conclude that the dependence corresponds to a power law.

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