2.7. Volume Fraction EstimationThe volume fraction is used to express the proportion of a phase selleck chemical Vandetanib or component within the whole structure. The volume fraction of an X phase within a Y reference volume is simply expressed as follows:VV(X,Y)=Volume??of??X??phase??in??Y??reference??spaceVolume??of??Y??reference??space.(2)Volume fraction ranges from 0 to 1 and is often expressed as a percentage [15�C17]. The volume fraction formula with the combination of a point-counting grid can be written asVV(X,Y)=��PX��PY.(3)In this formula, ����PX�� indicates the number of points hitting the X phase and ����PY�� the number of points hitting the reference space Y.We estimated the volume fraction of the herniated tonsils within the whole cerebellum by means of the following formula:VV(Herniation,Cerebellum)=��PHerniation��PCerebellum.

(4)In this formula, ��Pherniation is the total number of points hitting the components of the herniated tonsil, and ��Pcerebellum is the total number of points hitting sectioned surface of the cerebellum, including all its parts. The application of the described approaches for the estimation of volume and volume fraction is presented (Table 1):Table 1An example of the application of volume and the volume fraction method described in the present study. P(Y) = number of points hitting the cross section of the cerebellum. P(X) = number of points hitting the cross section of the herniated tonsil.V(Cerebellum)=T��[SU��dSL]2����P=0.5��[2��0.81.95]��375=124.55?cm3,V(Herniation)=T��[SU��dSL]2����P=0.15��[0.2��0.250.25]��173=1.

04?cm3,(5)CE=[kk?1��u2��u��u+��u2��v��v?2��uv��u��u]1/2,CE=[2121?18345(375)2+2207(173)2?23964(375��173)]1/2=0.10=10%.(6)The volume fraction of the phase Y was estimated =��PHerniation��PCerebellum=1.04?cm3124.55?cm3=0.01=1%.(7)2.8.??as:VV(Herniation,Cerebellum) Error Prediction for Point Counting Technique with Volume FractionAccordingly, the efficiency of sampling and the density of grid points were performed as documented in the literature [17, 18]. The coefficient of error (CE) was calculated following the formula:CE=[kk?1��u2��u��u+��u2��v��v?2��uv��u��u]1/2.(8)In this formula, there are k images, and each summation is over 1 to k. An example of the application of this type of calculation is given in Table 1. The coefficient of error (CE) of this estimate was approximated using (8).2.9. Statistical AnalysesThe statistical analyses were performed using statistical package for the Social Sciences for Windows (SPSS, Inc., Chicago, IL) 7.5 version software. Mean values are presented with their Batimastat standard deviations.