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[EQUATION] Let us furthermore expand these in [MATH] since it cannot appear in final equation. We have [EQUATION] where [EQUATION] |
We use the symbol [MATH] for equalities that should hold modulo an acceptable equation. For a set of equations the equality should be modulo the same equation, which we call [MATH] |
Since [MATH] does not contain [MATH] , the Bäcklund condition [MATH] implies three equations [EQUATION] For a genuine eqBT these equations should not all be satisfied automatically but rather for at least one equation the LHS should factor with an acceptable equation as a factor. |
5.3 Deriving the Lax condition It is interesting to compare eqBT with conditions derived from a Lax pair. The starting point is still ( 24 ), but now we replace [MATH] . Thus for example |
[EQUATION] that is [EQUATION] Then the commutativity condition [EQUATION] implies [EQUATION] The entries of this matrix equation are |
[EQUATION] using the previous notation. By taking ratios we get [EQUATION] that is, polynomial equations [EQUATION] These equations should either vanish, or factorize with the desired equation as a factor. |
When comparing ( 30 ) with ( 27 ) it is easy to verify that ( 30 ) implies ( 27 ). However, in the other direction one finds that in addition to |
[EQUATION] from ( 27a ), ( 27c ), one finds [EQUATION] from ( 27b ) after eliminating [MATH] . This suggests a possible new and different type of solution with [MATH] This, however, does not work: for the equations under consideration the expression [MATH] never vanishes, and furthermore it cannot be used to generate a... |
5.4 BT for the results obtained Above we have described how the eqBT computations lead to equations 27 ) which should not hold automatically, but only after using [MATH] , in other words, [MATH] should factorize, with [MATH] as one of the factors. We can of course do these computations for all three choices of side equ... |
1. back-left-front-right equations producing bottom and top equations, 2. bottom-back-top-front equations producing left and right equations, |
3. bottom-left-top-right equations producing back and front equation. We have done these computations for all equations found to have CAC (one representative per orbit) as listed in Appendix . The results are given in Tables 1-3. In the table a “0” means the corresponding eqBT produces nothing, i.e., the equations 27 )... |
dependence on the [MATH] , while 2* stands for the cases where the two equations are essentially different and the extra equation is a version of [MATH] . Finally, if one of the resulting equations is not quadratic its degree is given as subscript. |
Here are some examples For equation {58,10,10} the bottom-back-top-front side equations produce for the left equation a rational expression whose numerator factorizes as |
[EQUATION] The factor in round brackets is the expected left equation. Note that for the alternate equation in square brackets the [MATH] dependence is superfluous since the back equation should only depend on [MATH] |
For equation {10,15,58} the the bottom-back-top-front side equations produce [EQUATION] The term in square brackets is the proper left equation, the extra factor [MATH] is simpler and appears often. For this case the same type of result is found for right, back and front equations. All “2*” cases are like this. |
For {63,59,31}-1 eqBT produces the bottom equation and as an alternative, [MATH] , while for the top equation the alternative is [MATH] |
Summary All the equations discussed here satisfy the consistency-around-the-cube condition. The equations on the consistency cube can also provide a Lax pair or produce a Bäcklund transformation in which the “side equations” may generate the bottom and top equations. It turns out that many of the equations produce a fa... |
Discussion We have searched for all homogeneous quadratic triplets of multilinear irreducible equations that satisfy the Consistency-Around-a-Cube condition with uniform embedding (translation invariance of quad equations). The results were given in Appendix and classified in Section . The three equations forming the t... |
The ansatz for the equations contained coefficients that depended on the two lattice parameters associated with the quadrilateral in question. The results can be divided into two classes: those that cannot have arbitrary dependence on the two lattice parameters and those that can. The arbitrary functions can sometimes ... |
possibility of free functions is essential. The results can be arranged into sub-cases, some relations are given in Figures . The equations that cannot have free functions are all reductions of [MATH] of the ABS list |
Recently it has been noted that some equations that pass the CAC test can have fake Lax pairs , and independently the possibility of “weak Lax pairs” has also been observed |
. In order to characterize the equations further we studied how the Bäcklund transformation works on these equations. The result are given in Tables 1-3. One finds that all equations that can have free functions also have some failing BTs (or equivalently some fake Lax pairs). This is natural since it is not possible t... |
From our results we infer the following: 1. The CAC condition can only be a necessary condition of integrability. It cannot be sufficient as many equations with failing eqBTs pass this test. |
2. The CAC condition may be sufficient if accompanied with some other type of condition, but the tetrahedron condition is not enough for that purpose. |
3. We conjecture that if there is a unique BT for each direction of the cube then that should be sufficient for integrability. Non-uniqueness in some directions may be a signal of partial integrability while failure of the BT in one or more directions suggests non-integrability. |
4. Integrability may be lost during reductions. But note that the reduction here does not change dimension but just means simplifying the equation by some limit of the coefficient(s). |
Appendix A Results of the search We will now list the CAC equation triplets within the category of equations studied in this paper, namely those that are quadratic and acceptable by Definition . We list them following the order of the search process which was based on the triplet code. In Section |
they are grouped according to the “highest equation” and its reductions. During the search process sub-case inclusion was not considered except for obvious reductions. If the triplet found is a member of a symmetry orbit of more than one entry we also list the orbit. |
A.1 {10,10,10} The triplet [EQUATION] satisfies the CAC condition without any constraints on the parameters [MATH] . However, by the gauge transform ( 11 ) we can take the triplet into the simple form |
[EQUATION] with [EQUATION] which does not have the tetrahedron property. A.2 [MATH] This category contains the triplet {63,10,10} and its special cases. |
[EQUATION] That is, a completely arbitrary homogeneous quadratic equation is MDC with the side equations as given above. The triply shifted [MATH] is given by |
[EQUATION] and thus ( 33 ) has TET in the sub-case [MATH] , with code {53,10,10}. The sub-cases of {63,10,10} arrange themselves into orbits as follows: |
3 terms: {21,10,10},{37,10,10},{49,10,10},{52,10,10}, 2 terms: {5,10,10},{48,10,10}. In solving the CAC equations for these triplet codes we were usually led to conditions on the [MATH] terms of the back and left equation. For {63,10,10} it is not possible to have any free functions in the side equations, but as the {5... |
A.3 [MATH] In this case we arrange the bottom equation to be the simple one. The main result in this category is {10,58,15} [EQUATION] |
The triply shifted quantity is given by [EQUATION] This has sub-cases when one or two [MATH] in either or both equations vanish while keeping the equations irreducible. Missing [MATH] terms sometimes allow a more general bottom equation, but that freedom can be gauged away. The sub-case orbits are as follows: |
2,2,3 terms: {10,26,5},{10,42,5},{10,48,11},{10,48,14}, 2,2,2 terms: {10,48,5}. The number of terms in each equation is given in the first column, but that list is not ordered because some reflections exchange back and left equations. The sub-cases belonging to {X,10,10} are excluded. |
{10,15,58} In this category there is also an equation that is not a sub-case of 34 ): [EQUATION] with [EQUATION] This cannot be a sub-case of ( 14 ) because the equations do not contain the same shifted [MATH] variables. We can use gauge transformation to simplify this further. Choosing the gauge so that 37 ) simplifie... |
[EQUATION] the equations get the form (after redefining [MATH] [EQUATION] For an intuitive understanding of the difference between 34 ) and ( 39 ) we can look at the Figure |
. Note also that in ( 35 [MATH] is in the numerator while in ( 38 ) it is in the denominator. A.4 [MATH] Only {63,58,15} survives, all other cases lead to equations that factorize. |
[EQUATION] [EQUATION] A.5 [MATH] According to ( 8a ) we have by tilde-hat reflection and rotation [MATH] . Thus [MATH] and [MATH] , are not related by this reflection and may have different types of solutions. |
The main solution in this category is {53,15,58} , and after a gauge transformation the equation satisfying CAC can be written as |
[EQUATION] The triply shifted variable is [EQUATION] thus it has TET. All other equations in this category that satisfy CAC are sub-cases of the above, or lead to equations that factorize. The orbits for the sub-cases are |
4 terms: {53,15,58}, 3 terms: {21,15,58},{37,15,58},{49,15,58},{52,15,58}, 2 terms: {5,15,58},{48,15,58}. A.6 [MATH] or [MATH] By the tilde-hat reflection we have [MATH] and therefore there is one-to-one correspondence between the sets [MATH] and [MATH] and we only need to discuss one of them. |
There are types two solutions, the first one is {58,48,5} given by [EQUATION] with [EQUATION] One can change some of the signs by gauge but one cannot eliminate the freedom in [MATH] . This does not have TET except if [MATH] , in which case it becomes {48,48,5} ( 49 ). The reflected case is {15,48,5}. |
{26,15,59} The second solution is, after gauge and rotation [EQUATION] [EQUATION] The triplet ( 43 ) is symmetric under bar reversal and tilde reversal, but not under hat-reversal and therefore there are also the corresponding solution within three reversed categories. The orbit is {15,59,26},{15,62,42},{58,11,31},{58,... |
A.7 [MATH] There are five solutions in this category and they all have TET. {53,48,5} [EQUATION] where [MATH] . The signs [MATH] can be changed by gauge. |
[EQUATION] {53,5,48} [EQUATION] [EQUATION] The sub-case orbits are as for {53,15,58} ( 41 ). The difference between ( 44 ) and ( 45 ) is illustrated in Figure |
{53,53,53} [EQUATION] with triply shifted variable [EQUATION] We have added here some coefficients [MATH] and [MATH] and if they are nonzero they can be gauged to [MATH] which yields [MATH] of the ABS list. If a [MATH] |
vanishes we get a rotation of {53,49,21}, if a [MATH] vanishes we get a rotation of its reflection {53,52,37}, and if [MATH] for some [MATH] we get a rotation of a special case of {53,48,05}. |
{53,49,21} [EQUATION] [EQUATION] There is also the reflected case {53,52,37}. {53,11,26} [EQUATION] [EQUATION] Note that [MATH] is the same as for {53,5,48} ( 45 ). There is also the reflected case {53,14,42}. Sub-case orbits are as for {53,15,58} ( 15 ). |
The difference between ( 47 ) and ( 48 ) is illustrated in Figure A.8 [MATH] or [MATH] By the tilde-hat reflection we have [MATH] . The solutions in this category are as follows: |
{48,48,05} [EQUATION] [EQUATION] The signs can be controlled by gauge. In this orbit there is also {05,05,48}. {21,05,48} [EQUATION] |
[EQUATION] The full orbit of reflected cases is {21,5,48}, {37,5,48}, {49,5,48}, {52,5,48}. {63,48,5} [EQUATION] [EQUATION] The sign [MATH] cannot be eliminated by gauge. |
{5,11,26} [EQUATION] [EQUATION] The orbit is {5,11,26}, {48,11,26}, {5,14,42}, {48,14,42}. A.9 The rest: [MATH] {27,31,59} [EQUATION] |
[EQUATION] There are altogether 4 different reflections of this triplet: {27,31,59}, {43,31,62}, {30,47,59}, {46,47,62}. {21,11,26} |
[EQUATION] [EQUATION] There are 8 elements in the orbit {11,26,21}, {11,26,37}, {11,26,49}, {11,26,52}, {14,42,21}, {14,42,37}, {14,42,49}, {14,42,52}; |
The remaining cases were the difficult ones to solve. In the category {63,59,31} there are two solutions: {63,59,31}-1 [EQUATION] |
[EQUATION] Note that the back and left equations are not related by a cyclic variable change. {63,59,31}-2 [EQUATION] [EQUATION] |
There are also corresponding two reflected cases {63,62,47}. {63,63,63} In the category {63,63,63} there are also two solutions. The preliminary way of writing the result, after a gauge transformation, is the form |
[EQUATION] where we still have the conditions [EQUATION] If [MATH] and/or [MATH] are nonzero the result is awkward, but we can simplify the result by a transformation [MATH] . However, in order to connect with known results we we want the coefficient of the [MATH] be [MATH] after the transformation and a possible overa... |
{63,63,63}-1 If [MATH] then by a simple translation we get [MATH] , where [MATH] , but this sign can be eliminated by gauge. In order to connect with known results we redefine [MATH] after which we get |
[EQUATION] [EQUATION] This is in fact Q1( [MATH] ) in the ABS list. {63,63,63}-2 A different solution is obtained if [MATH] . Then by transforming |
[EQUATION] etc. one finds [EQUATION] and after clearing denominators (and renaming [MATH] ) we get [MATH] of the ABS list: [EQUATION] |
with [EQUATION] Note that the difference between Q1 and Q3 arises from the factorization properties of the [MATH] coefficient in 57 ). |
Acknowledgment I would like to thank Da-jun Zhang for useful comments on the manuscript. All computations were done using the REDUCE computer algebra system |
# Source: arxiv 1806.08552 # Title: Innovative method for reducing uninformative calls in non-invasive prenatal testing # Sections: all # Downloaded: 2026-03-02T08:41:41.541646+00:00 |
\epstopdfDeclareGraphicsRule .tifpng.pngconvert #1 \OutputFile \AppendGraphicsExtensions .tif Innovative method for reducing uninformative calls in non-invasive prenatal testing |
Abstract Motivation. Non-invasive prenatal testing or NIPT is currently among the top researched topic in obstetric care. While the performance of the current state-of-the-art NIPT solutions achieve high sensitivity and specificity, they still struggle with a considerable number of samples that cannot be concluded with... |
Results. We propose a supplementary method to traditional z-scores to reduce the number of such uninformative calls. The method is based on a novel analysis of the length profile of circulating cell free DNA which compares the change in such profiles when random-based and length-based elimination of some fragments is p... |
Availability and Implementation. A particular implementation of the proposed methods is not provided with the manuscript. Contact. Correspondence regarding the manuscript should be directed at Frantisek Duris (fduris@dcs.fmph.uniba.sk). |
Supplementary Information. No additional supplementary information is available. Keywords: Next-generation sequencing, Cell-free DNA, Uninformative result, Method, Trisomy, Prenatal testing |
Introduction Prenatal screening and diagnostics are important parts of obstetric care. Current methods of prenatal testing still involve most commonly invasive sampling of fetal material using procedures such as amniocentesis and chorionic villus sampling which are associated with a small but real risk of miscarriage [... |
(Mujezinovic and Alfirevic, 2007 . To prevent the risk of abortion associated with invasive sampling procedures, non-invasive prenatal testing (NIPT) based on fetal DNA analysis from maternal circulation has been developed. In 1997, the discovery of fetal cell-free DNA (cfDNA) in maternal plasma and serum revolutionize... |
While the performance of the current state-of-the-art NIPT solutions achieve high sensitivity and specificity (Bianchi et al., 2014 ; Koumbaris et al., 2016 , they still struggle with a considerable number of samples that cannot be concluded with certainty. The great source of such uninformative samples is in the natur... |
Other problems are represented by maternal DNA aberrations such as maternal mosaicism (Wang et al., 2014 , unidentified maternal tumors (Amant et al., 2015 , or copy number variations (Snyder et al., 2015 ; Zhou et al., 2017 . Hypothetically, a duplication of even a small part of maternal chromosome, which may not be d... |
There is a growing body of studies addressing this issue (Wang et al., 2014 2015b 2015a . A particularly interesting venue of research focuses on qualitative differences between fetal and maternal cfDNA fragments, namely their lengths. It was previously reported that fetal fragments are on average shorter than maternal... |
Extending the above mentioned work, in this paper we present a method to further boost the elimination of uninformative results which have a potential to cause needless stress to the parents, requiring repeated blood samplings and analyzes. The associated increase of expenses, in turn, lower viability of the NIPT produ... |
Material and methods 2.1 Sample acquisition We have collected altogether 2,621 samples with singleton pregnancy, of which 2,569 were negative for trisomy of chromosomes 13, 18 and 21, while 5 were confirmed as T13, 6 were confirmed as T18, and 39 were confirmed as T21. One negative sample (analyzed twice) was falsely r... |
2.2 Sample preparation and sequencing Blood from pregnant women was collected into EDTA tubes and kept at [MATH] temperature until plasma separation. Blood plasma was separated within 36 hours after collection and stored at [MATH] unit DNA isolation. DNA was isolated using Qiagen DNA Blood Mini kit. Standard fragment l... |
2.3 Mapping and read count correction Sequencing reads were aligned to the human reference genome (hg19) using Bowtie 2 algorithm (Langmead et al., 2009 . The first stage of data processing was carried out as previously described (Minarik et al., 2015 . NextSeq-produced fastq files (two per sample; R1 and R2) were dire... |
Furthermore, variability of human genome in population also contributes to the mapping bias, mainly in regions with common structural differences. At first, bin counts were transformed into a principal space, where the first component represented the highest variability across individuals in the control set. To normali... |
2.4 Reference z-score calculation The reference z-scores of samples were calculated as normalized chromosome values (NCV) according to Sehnert et al. ( 2011 . Given our training set, the optimal reference chromosomes were determined to be 1, 4, 8, 10, 19 and 20 for trisomy 21, 4, 7, 8, 9, 10 and 16 for trisomy 18, and ... |
2.5 Length score calculation We defined three novel statistics based on the fragment lengths, each building on the previous one. The basis for our first novel statistic, termed [MATH] -score, were read counts for the chromosome of interest (e.g., the usual [MATH] [MATH] or [MATH] chromosome). However, in contrast with ... |
More particularly, if [MATH] marks the chromosome of interest and [MATH] the number of reads mapped to the [MATH] chromosome of the tested sample, then let [MATH] be the number of reads mapped to the [MATH] chromosome that are of length at most [MATH] . Furthermore, let the total number of reads of the sample be [MATH]... |
For each sample and chromosome, we defined a series of [MATH] -scores as [EQUATION] where [MATH] ranges from 50 to 200 and [MATH] indicates a chromosome. Next, we defined a second novel statistic termed [MATH] -score as |
[EQUATION] The bounds 125 and 145 were determined empirically. Finally, the [MATH] -scores were normalized into z-scores which approximately follow standard normal distribution for euploid samples. Then, for any sample, its normalized [MATH] -score value [MATH] was used as an alternative method for the prediction of an... |
2.6 Combining the scores In our analysis, we found no correlation between the reference ( [MATH] ) and length-based ( [MATH] ) scores measured in our training set of euploid samples [MATH] . Thus, we considered them as two independent random variables, each from a standard normal distribution. The sum of their squares ... |
On the other hand, there was a significant correlation between [MATH] (defined below in section 3.2 ) and [MATH] [MATH] . In this case, we combined them according to (Owen and Chmielewski, 1985 . Specifically, we first performed a principal component analysis of the pairs [MATH] . Then, we recalculated the z-scores alo... |
2.7 Statistical analysis The significance of our findings was evaluated using statistical tests implemented in Python scipy package (Jones et al., 2014 . The linear dependency of the two scores in negative samples was calculated with Pearson correlation. Since scores of aberrant samples were not normally distributed, W... |
Results 3.1 [MATH] score profiles First, we calculated series of [MATH] -scores for euploid and trisomic samples for chromosomes 13, 18 and 21, and length range from 50bp to 220bp. We observed that trisomic samples behaved differently than euploid samples (Figure ). This difference can be explained by fetal fragments b... |
At first, the [MATH] -scores of trisomic samples gradually increase with the elimination of longer fragments. This positive effect is however balanced by the negative effect of lower number of remaining fragments, and so, after while, the [MATH] -scores decline to the values expected by a random draw. We observed the h... |
Note that in contrast with the reference z-score [MATH] , the calculation of [MATH] -scores is not based on comparison of proportions of fragments with healthy population. An excessive number of fragments from a chromosome would therefore not result in the positive call, if these fragments do not have fetal length dist... |
3.2 In silico size selection Next, we examined the effect of the length-based fragment filtering on the reference z-scores. Particularly, we first calculated the reference z-scores [MATH] (section 2.4 ), and then we removed the fragments longer than 150bp. However, this considerably changed the read count for our sampl... |
On the other hand, when all aligned reads were considered (recall that up until now, all samples were restricted to the first 5 million raw alignments, see section 2.3 ), and the same procedure was applied again, the increase in z-scores of trisomic samples became statistically significant (average multiplicative incre... |
This finding indicates that the length-based fragment selection is beneficial only for samples with more than 5 million aligned fragments, at least for the reference z-scores. This is in accord with our previous findings that in silico size-based filtering of fragments (only reads up to 155bp were retained) did not lea... |
3.3 Combined scores as supplemental evaluation method The performance of the novel statistic [MATH] by itself was observed to be weaker than that of the traditional [MATH] (Figure ). Particularly, there was a substantial decrease in z-scores for trisomic samples. On the other hand, their combination [MATH] resulted in ... |
Furthermore, we observed a decrease of [MATH] scores of false positive and uninformative samples when compared with reference [MATH] , even though other samples that were previously classified as negative replaced them in the uninformative range. Additionally, [MATH] and [MATH] resulted both in false positive results i... |
3.4 Improved evaluation method In light of our findings, we propose a following improvement of the NIPT evaluation process. First, a reference [MATH] is calculated. If it is in the negative zone, then no further test is applied and the sample is closed as normal. If the [MATH] is in the uninformative zone, then [MATH] ... |
3.5 A false positive sample We observed a atypical sample from IVF pregnancy (analyzed twice) with high risk for trisomy 18 (Figure ). Although both analyzes were supported by solid [MATH] scores 6.82 and 6.31, the predicted aberration was not confirmed by invasive follow-up test. On the other hand, the fragment length... |
The most likely explanation is that this was a result of a maternal copy number variation on the [MATH] chromosome which is known to cause false positive results (Zhou et al., 2017 . Unfortunately, we were not able to perform a follow-up of maternal genotype. If this was the case, it would indicate that the proposed me... |
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