SYSTEM AND METHOD FOR CLEANING NOISY GENETIC DATA AND DETERMINING CHROMOSOME COPY NUMBER

Abstract

Disclosed herein is a system and method for increasing the fidelity of measured genetic data, for making allele calls, and for determining the state of aneuploidy, in one or a small set of cells, or from fragmentary DNA, where a limited quantity of genetic data is available. Poorly or incorrectly measured base pairs, missing alleles and missing regions are reconstructed using expected similarities between the target genome and the genome of genetically related individuals. In accordance with one embodiment, incomplete genetic data from an embryonic cell are reconstructed at a plurality of loci using the more complete genetic data from a larger sample of diploid cells from one or both parents, with or without haploid genetic data from one or both parents. In another embodiment, the chromosome copy number can be determined from the measured genetic data, with or without genetic information from one or both parents.

Claims

1. A method for determining genetic data for DNA from cancer cells, comprising: obtaining cell-free DNA from a blood sample, wherein the cell-free DNA comprises DNA from cancer cells comprising one or more chromosome segments; ligating at least one adapter to the chromosome segments, wherein the at least one adapter comprises a universal amplification sequence; performing targeted amplification using a universal primer that binds to the universal amplification sequence, and target-specific primers, to generate amplified nucleic acid molecules; and determining genetic data for the cell-free DNA from cancer cells by performing next-generation sequencing.

2. The method of claim 1, wherein the method further comprises detecting a point mutation, insertion or deletion.

3. The method of claim 1, wherein the next-generation sequencing is performed using sequencing-by-synthesis.

4. The method of claim 1, wherein the targeted amplification is targeted PCR.

5. The method of claim 1, wherein the targeted amplification amplifies SNP loci.

6. A method for preparing a deoxyribonucleic acid (DNA) fraction from a biological sample useful for analyzing genotypes for DNA from cancer cells, the method comprising: (a) extracting cell-free DNA from the biological sample; (b) producing a fraction of the DNA extracted in (a) by amplifying a plurality of target loci from the DNA to obtain amplification products; (c) analyzing the fraction of the DNA produced in (b) by (i) sequencing the amplification products by sequencing-by-synthesis to obtain genetic data for the plurality of target loci; and (ii) determining the most likely genotypes for DNA from cancer cells based on allele frequencies at the plurality of target loci in the genetic data.

7. The method of claim 6, wherein the biological sample is a blood sample.

8. The method of claim 6, wherein the amplification comprises targeted amplification to amplify the target loci.

9. The method of claim 6, wherein the amplification further comprises universal amplification.

10. The method of claim 6, wherein the sequencing-by-synthesis comprises clonal amplification and measurement of sequences of the clonally amplified DNA.

11. The method of claim 6, wherein the target loci comprise SNP loci.

12. The method of claim 11, wherein the confidence that each SNP is correctly called is at least 95%.

13. The method of claim 11, wherein the confidence that each SNP is correctly called is at least 99%.

14. The method of claim 6, wherein the genetic data obtained is noisy and comprises allele drop out errors.

15. The method of claim 6, wherein the genetic data obtained is noisy and comprises measurement bias.

16. The method of claim 6, wherein the genetic data obtained is noisy and comprises incorrect measurements.

17. The method of claim 6, further comprising normalizing the genetic data for differences in amplification and/or measurement efficiency between the loci.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0041] FIG. 1: determining probability of false negatives and false positives for different hypotheses.

[0042] FIG. 2: the results from a mixed female sample, all loci hetero.

[0043] FIG. 3: the results from a mixed male sample, all loci hetero.

[0044] FIG. 4: Ct measurements for male sample differenced from Ct measurements for female sample.

[0045] FIG. 5: the results from a mixed female sample; Taqman single dye.

[0046] FIG. 6: the results from a mixed male; Taqman single dye.

[0047] FIG. 7: the distribution of repeated measurements for mixed male sample.

[0048] FIG. 8: the results from a mixed female sample; qPCR measures.

[0049] FIG. 9: the results from a mixed male sample; qPCR measures.

[0050] FIG. 10: Ct measurements for male sample differenced from Ct measurements for female sample.

[0051] FIG. 11: detecting aneuploidy with a third dissimilar chromosome.

[0052] FIGS. 12A and 12B: an illustration of two amplification distributions with constant allele dropout rate.

[0053] FIG. 13: a graph of the Gaussian probability density function of alpha.

[0054] FIG. 14: matched filter and performance for 19 loci measured with Taqman on 60 pg of DNA.

[0055] FIG. 15: matched filter and performance for 13 loci measured with Taqman on 6 pg of DNA.

[0056] FIG. 16: matched filter and performance for 20 loci measured with qPCR on 60 pg of DNA.

[0057] FIG. 17: matched filter and performance for 20 loci measured with qPCR on 6 pg of DNA.

[0058] FIG. 18A: matched filter and performance for 20 loci using MDA and Taqman on 16 single cells.

[0059] FIG. 18B: Matched filter and performance for 11 loci on Chromosome 7 and 13 loci on Chromosome X using MDA and Taqman on 15 single cells.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

Conceptual Overview of the System

[0060] The goal of the disclosed system is to provide highly accurate genomic data for the purpose of genetic diagnoses. In cases where the genetic data of an individual contains a significant amount of noise, or errors, the disclosed system makes use of the expected similarities between the genetic data of the target individual and the genetic data of related individuals, to clean the noise in the target genome. This is done by determining which segments of chromosomes of related individuals were involved in gamete formation and, when necessary where crossovers may have occurred during meiosis, and therefore which segments of the genomes of related individuals are expected to be nearly identical to sections of the target genome. In certain situations this method can be used to clean noisy base pair measurements on the target individual, but it also can be used to infer the identity of individual base pairs or whole regions of DNA that were not measured. It can also be used to determine the number of copies of a given chromosome segment in the target individual. In addition, a confidence may be computed for each call made. A highly simplified explanation is presented first, making unrealistic assumptions in order to illustrate the concept of the invention. A detailed statistical approach that can be applied to the technology of today is presented afterward.

[0061] In one aspect of the invention, the target individual is an embryo, and the purpose of applying the disclosed method to the genetic data of the embryo is to allow a doctor or other agent to make an informed choice of which embryo(s) should be implanted during IVF. In another aspect of the invention, the target individual is a fetus, and the purpose of applying the disclosed method to genetic data of the fetus is to allow a doctor or other agent to make an informed choice about possible clinical decisions or other actions to be taken with respect to the fetus.

Definitions

[0062] SNP (Single Nucleotide Polymorphism): a single nucleotide that may differ between the genomes of two members of the same species. In our usage of the term, we do not set any limit on the frequency with which each variant occurs.

[0063] To call a SNP: to make a decision about the true state of a particular base pair, taking into account the direct and indirect evidence.

[0064] Locus: a particular region of interest on the DNA of an individual, which may refer to a SNP, the site of a possible insertion or deletion, or the site of some other relevant genetic variation. Disease-linked SNPs may also refer to disease-linked loci.

[0065] To call an allele: to determine the state of a particular locus of DNA. This may involve calling a SNP, or determining whether or not an insertion or deletion is present at that locus, or determining the number of insertions that may be present at that locus, or determining whether some other genetic variant is present at that locus.

[0066] Correct allele call: An allele call that correctly reflects the true state of the actual genetic material of an individual.

[0067] To clean genetic data: to take imperfect genetic data and correct some or all of the errors or fill in missing data at one or more loci. In the context of this disclosure, this involves using genetic data of related individuals and the method described herein.

[0068] To increase the fidelity of allele calls: to clean genetic data.

[0069] Imperfect genetic data: genetic data with any of the following: allele dropouts, uncertain base pair measurements, incorrect base pair measurements, missing base pair measurements, uncertain measurements of insertions or deletions, uncertain measurements of chromosome segment copy numbers, spurious signals, missing measurements, other errors, or combinations thereof.

[0070] Noisy genetic data: imperfect genetic data, also called incomplete genetic data.

[0071] Uncleaned genetic data: genetic data as measured, that is, where no method has been used to correct for the presence of noise or errors in the raw genetic data; also called crude genetic data.

[0072] Confidence: the statistical likelihood that the called SNP, allele, set of alleles, or determined number of chromosome segment copies correctly represents the real genetic state of the individual.

[0073] Parental Support (PS): a name sometimes used for the any of the methods disclosed herein, where the genetic information of related individuals is used to determine the genetic state of target individuals. In some cases, it refers specifically to the allele calling method, sometimes to the method used for cleaning genetic data, sometimes to the method to determine the number of copies of a segment of a chromosome, and sometimes to some or all of these methods used in combination.

[0074] Copy Number Calling (CNC): the name given to the method described in this disclosure used to determine the number of chromosome segments in a cell.

[0075] Qualitative CNC (also qCNC): the name given to the method in this disclosure used to determine chromosome copy number in a cell that makes use of qualitative measured genetic data of the target individual and of related individuals.

[0076] Multigenic: affected by multiple genes, or alleles.

[0077] Direct relation: mother, father, son, or daughter.

[0078] Chromosomal Region: a segment of a chromosome, or a full chromosome.

[0079] Segment of a Chromosome: a section of a chromosome that can range in size from one base pair to the entire chromosome.

[0080] Section: a section of a chromosome. Section and segment can be used interchangeably.

[0081] Chromosome: may refer to either a full chromosome, or also a segment or section of a chromosome.

[0082] Copies: the number of copies of a chromosome segment may refer to identical copies, or it may refer to non-identical copies of a chromosome segment wherein the different copies of the chromosome segment contain a substantially similar set of loci, and where one or more of the alleles are different. Note that in some cases of aneuploidy, such as the M2 copy error, it is possible to have some copies of the given chromosome segment that are identical as well as some copies of the same chromosome segment that are not identical.

[0083] Haplotypic Data: also called ‘phased data’ or ‘ordered genetic data;’ data from a single chromosome in a diploid or polyploid genome, i.e., either the segregated maternal or paternal copy of a chromosome in a diploid genome.

[0084] Unordered Genetic Data: pooled data derived from measurements on two or more chromosomes in a diploid or polyploid genome, i.e., both the maternal and paternal copies of a chromosome in a diploid genome.

[0085] Genetic data ‘in’, ‘of’, ‘at’ or ‘on’ an individual: These phrases all refer to the data describing aspects of the genome of an individual. It may refer to one or a set of loci, partial or entire sequences, partial or entire chromosomes, or the entire genome.

[0086] Hypothesis: a set of possible copy numbers of a given set of chromosomes, or a set of possible genotypes at a given set of loci. The set of possibilities may contain one or more elements.

[0087] Target Individual: the individual whose genetic data is being determined. Typically, only a limited amount of DNA is available from the target individual. In one context, the target individual is an embryo or a fetus.

[0088] Related Individual: any individual who is genetically related, and thus shares haplotype blocks, with the target individual.

[0089] Platform response: a mathematical characterization of the input/output characteristics of a genetic measurement platform, such as TAQMAN or INFINIUM. The input to the channel is the true underlying genotypes of the genetic loci being measured. The channel output could be allele calls (qualitative) or raw numerical measurements (quantitative), depending on the context. For example, in the case in which the platform's raw numeric output is reduced to qualitative genotype calls, the platform response consists of an error transition matrix that describes the conditional probability of seeing a particular output genotype call given a particular true genotype input. In the case in which the platform's output is left as raw numeric measurements, the platform response is a conditional probability density function that describes the probability of the numeric outputs given a particular true genotype input.

[0090] Copy number hypothesis: a hypothesis about how many copies of a particular chromosome segment are in the embryo. In a preferred embodiment, this hypothesis consists of a set of sub-hypotheses about how many copies of this chromosome segment were contributed by each related individual to the target individual.

Technical Description of the System

A Allele Calling: Preferred Method

[0091] Assume here the goal is to estimate the genetic data of an embryo as accurately as possible, and where the estimate is derived from measurements taken from the embryo, father, and mother across the same set of n SNPs. Note that where this description refers to SNPs, it may also refer to a locus where any genetic variation, such as a point mutation, insertion or deletion may be present. This allele calling method is part of the Parental Support (PS) system. One way to increase the fidelity of allele calls in the genetic data of a target individual for the purposes of making clinically actionable predictions is described here. It should be obvious to one skilled in the art how to modify the method for use in contexts where the target individual is not an embryo, where genetic data from only one parent is available, where neither, one or both of the parental haplotypes are known, or where genetic data from other related individuals is known and can be incorporated.

[0092] For the purposes of this discussion, only consider SNPs that admit two allele values; without loss of generality it is possible to assume that the allele values on all SNPs belong to the alphabet A={A,C}. It is also assumed that the errors on the measurements of each of the SNPs are independent. This assumption is reasonable when the SNPs being measured are from sufficiently distant genic regions. Note that one could incorporate information about haplotype blocks or other techniques to model correlation between measurement errors on SNPs without changing the fundamental concepts of this invention.

[0093] Let e=(e.sub.1,e.sub.2) be the true, unknown, ordered SNP information on the embryo, e.sub.1,e.sub.2 ϵA.sup.n. Define e.sub.1 to be the genetic haploid information inherited from the father and e.sub.2 to be the genetic haploid information inherited from the mother. Also use e.sub.i=(e.sub.1i,e.sub.2i) to denote the ordered pair of alleles at the i-th position of e. In similar fashion, let f=(f.sub.1,f.sub.2) and m=(m.sub.1,m.sub.2) be the true, unknown, ordered SNP information on the father and mother respectively. In addition, let g.sub.1 be the true, unknown, haploid information on a single sperm from the father. (One can think of the letter g as standing for gamete. There is no g.sub.2. The subscript is used to remind the reader that the information is haploid, in the same way that f.sub.1 and f.sub.2 are haploid.) It is also convenient to define r=(f,m), so that there is a symbol to represent the complete set of diploid parent information from which e inherits, and also write r.sub.i=(f.sub.im.sub.i)=((f.sub.1i,f.sub.2i),(m.sub.1i,m.sub.2i)) to denote the complete set of ordered information on father and mother at the i-th SNP. Finally, let ê=(ê.sub.1, ê.sub.2) be the estimate of e that is sought, ê.sub.1, ê.sub.2ϵA.sup.n.

[0094] By a crossover map, it is meant an n-tuple θϵ{1,2}.sup.n that specifies how a haploid pair such as (f.sub.1,f.sub.2) recombines to form a gamete such as e.sub.1. Treating θ as a function whose output is a haploid sequence, define θ(f).sub.i=θ(f.sub.1,f.sub.2).sub.i=f.sub.θi,i. To make this idea more concrete, let f.sub.1=ACAAACCC, let f.sub.2=CAACCACA, and let θ=11111222. Then θ(f.sub.1,f.sub.2)=ACAAAACA. In this example, the crossover map θ implicitly indicates that a crossover occurred between SNPs i=5 and i=6.

[0095] Formally, let θ be the true, unknown crossover map that determines e.sub.1 from f, let ϕ be the true, unknown crossover map that determines e.sub.2 from m, and let ψ be the true, unknown crossover map that determines g.sub.1 from f. That is, e.sub.1=θ(f), e.sub.2ϕ(m), g.sub.1=ψ(f). It is also convenient to define X=(θ,ϕ,ψ) so that there is a symbol to represent the complete set of crossover information associated with the problem. For simplicity sake, write e=X(r) as shorthand for e(θ(f),ϕ(m)); also write e.sub.i=X(r.sub.i) as shorthand for e.sub.i=X(r).sub.i

[0096] In reality, when chromosomes combine, at most a few crossovers occur, making most of the 2.sup.n theoretically possible crossover maps distinctly improbable. In practice, these very low probability crossover maps will be treated as though they had probability zero, considering only crossover maps belonging to a comparatively small set Ω. For example, if Ω is defined to be the set of crossover maps that derive from at most one crossover, then |Ω|=2n.

[0097] It is convenient to have an alphabet that can be used to describe unordered diploid measurements. To that end, let B={A,B,C,X}. Here A and C represent their respective homozygous locus states and B represents a heterozygous but unordered locus state. Note: this section is the only section of the document that uses the symbol B to stand for a heterozygous but unordered locus state. Most other sections of the document use the symbols A and B to stand for the two different allele values that can occur at a locus. X represents an unmeasured locus, i.e., a locus drop-out. To make this idea more concrete, let f.sub.1=ACAAACCC, and let f.sub.2=CAACCACA. Then a noiseless unordered diploid measurement off would yield {tilde over (f)}=BBABBBCB .

[0098] In the problem at hand, it is only possible to take unordered diploid measurements of e, f, and m, although there may be ordered haploid measurements on g.sub.1. This results in noisy measured sequences that are denoted {tilde over (e)} ϵB.sup.n, {tilde over (f)} ϵB.sup.n, {tilde over (m)} ϵB.sup.n, and {tilde over (g)}.sub.1 ϵA.sup.n respectively. It will be convenient to define {tilde over (r)}=({tilde over (f)}, {tilde over (m)}) so that there is a symbol that represents the noisy measurements on the parent data. It will also be convenient to define {tilde over (D)}=({tilde over (r)}, {tilde over (e)}, {tilde over (g)}.sub.1) so that there is a symbol to represent the complete set of noisy measurements associated with the problem, and to write {tilde over (D)}.sub.i=({tilde over (r)}.sub.i, {tilde over (e)}.sub.i, {tilde over (g)}.sub.1i)=({tilde over (f)}.sub.i, {tilde over (m)}.sub.i,{tilde over (e)}.sub.i, {tilde over (g)}.sub.1i) to denote the complete set of measurements on the i-th SNP. (Please note that, while f.sub.i is an ordered pair such as (A,C), {tilde over (f)}.sub.i is a single letter such as B.)

[0099] Because the diploid measurements are unordered, nothing in the data can distinguish the state (f.sub.1, f.sub.2) from (f.sub.2, f.sub.1) or the state (m.sub.1, m.sub.2) from (m.sub.2, m.sub.1). These indistinguishable symmetric states give rise to multiple optimal solutions of the estimation problem. To eliminate the symmetries, and without loss of generality, assign θ.sub.1=ϕ.sub.1=1.

[0100] In summary, then, the problem is defined by a true but unknown underlying set of information {r, e, g.sub.1, X}, with e=X(r). Only noisy measurements {tilde over (D)}=({tilde over (r)}, {tilde over (e)}, {tilde over (g)}.sub.1) are available. The goal is to come up with an estimate ê of e, based on {tilde over (D)}.

[0101] Note that this method implicitly assumes euploidy on the embryo. It should be obvious to one skilled in the art how this method could be used in conjunction with the aneuploidy calling methods described elsewhere in this patent. For example, the aneuploidy calling method could be first employed to ensure that the embryo is indeed euploid and only then would the allele calling method be employed, or the aneuploidy calling method could be used to determine how many chromosome copies were derived from each parent and only then would the allele calling method be employed. It should also be obvious to one skilled in the art how this method could be modified in the case of a sex chromosome where there is only one copy of a chromosome present.

Solution Via Maximum a Posteriori Estimation

[0102] In one embodiment of the invention, it is possible, for each of the n SNP positions, to use a maximum a posteriori (MAP) estimation to determine the most probable ordered allele pair at that position. The derivation that follows uses a common shorthand notation for probability expressions. For example, P(e′.sub.i,{tilde over (D)}|X′) is written to denote the probability that random variable e.sub.i takes on value e′.sub.i and the random variable {tilde over (D)} takes on its observed value, conditional on the event that the random variable X takes on the value X′. Using MAP estimation, then, the i-th component of ê, denoted ê.sub.i=(ê.sub.1i, ê.sub.2i) is given by

[00001]$\text{?}$$\text{?}\text{indicates text missing or illegible when filed}$

[0103] In the preceding set of equations, (a) holds because the assumption of SNP independence means that all of the random variables associated with SNP i are conditionally independent of all of the random variables associated with SNP j, given X; (b) holds because r is independent of X; (c) holds because e.sub.i and {tilde over (D)}.sub.i are conditionally independent given r.sub.i and X (in particular, e.sub.i=X(r.sub.i)); and (*) holds, again, because e.sub.i=X(r.sub.i), which means that P(e′.sub.i|X′,r′.sub.i) evaluates to either one or zero and hence effectively filters r′.sub.i to just those values that are consistent with e′.sub.i and X′.

[0104] The final expression (*) above contains three probability expressions: P(X′), P(r′.sub.j), and P({tilde over (D)}.sub.j|X′,r′.sub.j). The computation of each of these quantities is discussed in the following three sections.

Crossover Map Probabilities

[0105] Recent research has enabled the modeling of the probability of recombination between any two SNP loci. Observations from sperm studies and patterns of genetic variation show that recombination rates vary extensively on kilobase scales and that much recombination occurs in recombination hotspots. The NCBI data about recombination rates on the Human Genome is publicly available through the UCSC Genome Annotation Database.

[0106] One may use the data set from the Hapmap Project or the Perlegen Human Haplotype Project. The latter is higher density; the former is higher quality. These rates can be estimated using various techniques known to those skilled in the art, such as the reversible-jump Markov Chain Monte Carlo (MCMC) method that is available in the package LDHat.

[0107] In one embodiment of the invention, it is possible to calculate the probability of any crossover map given the probability of crossover between any two SNPs. For example, P(θ=11111222) is one half the probability that a crossover occurred between SNPs five and six. The reason it is only half the probability is that a particular crossover pattern has two crossover maps associated with it: one for each gamete. In this case, the other crossover map is θ=22222111.

[0108] Recall that X=(θ,ϕ,ψ), where e.sub.1=θ(f), e.sub.2ϕ(m), g.sub.1=ψ(f). Obviously θ, ϕ, and ψ result from independent physical events, so P(X)=P(θ)P(ϕP(ψ). Further assume that P.sub.θ(⋅)=P.sub.ϕ(⋅)=P.sub.ψ(⋅), where the actual distribution P.sub.θ(⋅) is determined in the obvious way from the Hapmap data.

Allele Probabilities

[0109] It is possible to determine P(r.sub.i)=P(f.sub.i)P(m.sub.i)=P(f.sub.i1)P(f.sub.i2)P(m.sub.i1)P(m.sub.i 2) using population frequency information from databases such as dbSNP. Also, as mentioned previously, choose SNPs for which the assumption of intra-haploid independence is a reasonable one. That is, assume that

[00002]$P\left(r\right)=\underset{i}{.Math.}P\left(\mathrm{ri}\right)$

Measurement Errors

[0110] Conditional on whether a locus is heterozygous or homozygous, measurement errors may be modeled as independent and identically distributed across all similarly typed loci. Thus:

[00003]$\begin{array}{c}P\left(\tilde{D}.Math.X,r\right)=\underset{i}{.Math.}P\left({\tilde{D}}_i.Math.X,r_i\right)\\ =\underset{i}{.Math.}P\left({\tilde{f}}_i,{\tilde{m}}_i,{\tilde{e}}_i,{\tilde{g}}_{1i}.Math.X,f_i,m_i\right)\\ =\underset{i}{.Math.}P\left({\tilde{f}}_i.Math.f_i\right)P\left({\tilde{m}}_i.Math.m_i\right)P\left({\tilde{e}}_i.Math.\theta \left(f_i\right),\varphi \left(m_i\right)\right)P\left({\tilde{g}}_{1i}.Math.\Psi \left(f_i\right)\right)\end{array}$

where each of the four conditional probability distributions in the final expression is determined empirically, and where the additional assumption is made that the first two distributions are identical. For example, for unordered diploid measurements on a blastomere, empirical values p.sub.d=0.5 and p.sub.a=0.02 are obtained, which lead to the conditional probability distribution for P({tilde over (e)}.sub.i|e.sub.i) shown in Table 1.

[0111] Note that the conditional probability distributions mentioned above, P({tilde over (f)}.sub.i|f.sub.i), P({tilde over (m)}.sub.i|m.sub.i), P({tilde over (e)}.sub.i|e.sub.i), can vary widely from experiment to experiment, depending on various factors in the lab such as variations in the quality of genetic samples, or variations in the efficiency of whole genome amplification, or small variations in protocols used. Therefore, in a preferred embodiment, these conditional probability distributions are estimated on a per-experiment basis. We focus in later sections of this disclosure on estimating P({tilde over (e)}.sub.i|e.sub.i), but it will be clear to one skilled in the art after reading this disclosure how similar techniques can be applied to estimating P({tilde over (e)}.sub.i|f.sub.i) and P({tilde over (m)}.sub.i|m.sub.i). The distributions can each be modeled as belonging to a parametric family of distributions whose particular parameter values vary from experiment to experiment. As one example among many, it is possible to implicitly model the conditional probability distribution P({tilde over (e)}.sub.i|e.sub.i) as being parameterized by an allele dropout parameter p.sub.d and an allele dropin parameter p.sub.a. The values of these parameters might vary widely from experiment to experiment, and it is possible to use standard techniques such as maximum likelihood estimation, MAP estimation, or Bayesian inference, whose application is illustrated at various places in this document, to estimate the values that these parameters take on in any individual experiment. Regardless of the precise method one uses, the key is to find the set of parameter values that maximizes the joint probability of the parameters and the data, by considering all possible tuples of parameter values within a region of interest in the parameter space. As described elsewhere in the document, this approach can be implemented when one knows the chromosome copy number of the target genome, or when one doesn't know the copy number call but is exploring different hypotheses. In the latter case, one searches for the combination of parameters and hypotheses that best match the data are found, as is described elsewhere in this disclosure.

[0112] Note that one can also determine the conditional probability distributions as a function of particular parameters derived from the measurements, such as the magnitude of quantitative genotyping measurements, in order to increase accuracy of the method. This would not change the fundamental concept of the invention.

[0113] It is also possible to use non-parametric methods to estimate the above conditional probability distributions on a per-experiment basis. Nearest neighbor methods, smoothing kernels, and similar non-parametric methods familiar to those skilled in the art are some possibilities. Although this disclosure focuses parametric estimation methods, use of non-parametric methods to estimate these conditional probability distributions would not change the fundamental concept of the invention. The usual caveats apply: parametric methods may suffer from model bias, but have lower variance. Non-parametric methods tend to be unbiased, but will have higher variance.

[0114] Note that it should be obvious to one skilled in the art, after reading this disclosure, how one could use quantitative information instead of explicit allele calls, in order to apply the PS method to making reliable allele calls, and this would not change the essential concepts of the disclosure.

B Factoring the Allele Calling Equation

[0115] In a preferred embodiment of the invention, the algorithm for allele calling can be structured so that it can be executed in a more computationally efficient fashion. In this section the equations are re-derived for allele-calling via the MAP method, this time reformulating the equations so that they reflect such a computationally efficient method of calculating the result.

Notation

[0116] X*,Y*,Z*ϵ{A,C}.sup.n×2 are the true ordered values on the mother, father, and embryo respectively.

[0117] H*ϵ{A,C}.sup.n×h are true values on h sperm samples.

[0118] B*ϵ{A,C}.sup.n×b×2 are true ordered values on b blastomeres.

[0119] D={x,y,z,B,H} is the set of unordered measurement data on father, mother, embryo, b blastomeres and h sperm samples. D.sub.i={x.sub.i,y.sub.i,z.sub.i,H.sub.i,B.sub.i} is the data set restricted to the i-th SNP.

[0120] rϵ{A,C}.sup.4 represents a candidate 4-tuple of ordered values on both the mother and father at a particular locus.

[0121] {circumflex over (Z)}.sub.iϵ{A,C}.sup.2 is the estimated ordered embryo value at SNP i.

[0122] Q=(2+2b+h) is the effective number of haploid chromosomes being measured, excluding the parents. Any hypothesis about the parental origin of all measured data (excluding the parents themselves) requires that Q crossover maps be specified.

[0123] χϵ{1,2}.sup.n×Q is a crossover map matrix, representing a hypothesis about the parental origin of all measured data, excluding the parents. Note that there are 2.sup.nQ different crossover matrices. χ.sub.i χ.sub.i, is the matrix restricted to the i-th row. Note that there are 2.sup.Q vector values that the i-th row can take on, from the set χϵ{1,2}.sup.Q.

[0124] f(x; y, z) is a function of (x, y, z) that is being treated as a function of just x. The values behind the semi-colon are constants in the context in which the function is being evaluated.

PS Equation Factorization

[0125] [00004]$\text{?}$$\text{?}\text{indicates text missing or illegible when filed}$

[0126] The number of different crossover matrices χ is 2.sup.nQ. Thus, a brute-force application of the first line above is U(n2.sup.nQ). By exploiting structure via the factorization of P(χ) and P(z.sub.i,D|χ), and invoking the previous result, final line gives an expression that can be computed in O(n2.sup.2Q).

C Quantitative Detection of Aneuploidy

[0127] In one embodiment of the invention, aneuploidy can be detected using the quantitative data output from the PS method discussed in this patent. Disclosed herein are multiple methods that make use of the same concept; these methods are termed Copy Number Calling (CNC). The statement of the problem is to determine the copy number of each of 23 chromosome-types in a single cell. The cell is first pre-amplified using a technique such as whole genome amplification using the MDA method. Then the resulting genetic material is selectively amplified with a technique such as PCR at a set of n chosen SNPs at each of m=23 chromosome types.

[0128] This yields a data set [t.sub.ij], i=1 . . . n, j=1 . . . m of regularized ct (ct, or CT, is the point during the cycle time of the amplification at which dye measurement exceeds a given threshold) values obtained at SNP i, chromosome j. A regularized ct value implies that, for a given (i,j), the pair of raw ct values on channels FAM and VIC (these are arbitrary channel names denoting different dyes) obtained at that locus are combined to yield a ct value that accurately reflects the ct value that would have been obtained had the locus been homozygous. Thus, rather than having two ct values per locus, there is just one regularized ct value per locus.

[0129] The goal is to determine the set {n.sub.j} of copy numbers on each chromosome. If the cell is euploid, then n.sub.j=2 for all j; one exception is the case of the male X chromosome. If n≠2 for at least one j, then the cell is aneuploid; excepting the case of male X.

Biochemical Model

[0130] The relationship between ct values and chromosomal copy number is modeled as follows: α.sub.ijn.sub.jQ2.sup.βijtij−Q.sub.T In this expression, n.sub.1 is the copy number of chromosome j. Q is an abstract quantity representing a baseline amount of pre-amplified genetic material from which the actual amount of pre-amplified genetic material at SNP i, chromosome j can be calculated as α.sub.ijn.sub.jQ. α.sub.ij is a preferential amplification factor that specifies how much more SNP i on chromosome j will be pre-amplified via MDA than SNP 1 on chromosome 1. By definition, the preferential amplification factors are relative to α.sub.I11.

[0131] β.sub.ij is the doubling rate for SNP i chromosome j under PCR. t.sub.ij is the ct value. Q.sub.T is the amount of genetic material at which the ct value is determined. T is a symbol, not an index, and merely stands for threshold.

[0132] It is important to realize that α.sub.ij, β.sub.ij, and Q.sub.T are constants of the model that do not change from experiment to experiment. By contrast, n.sub.j and Q are variables that change from experiment to experiment. Q is the amount of material there would be at SNP 1 of chromosome 1, if chromosome 1 were monosomic.

[0133] The original equation above does not contain a noise term. This can be included by rewriting it as follows:

[00005]$\text{(*}){\beta }_{\mathrm{ij}}{t}_{\mathrm{ij}}=\log \frac{Q_T}{{\alpha }_{\mathrm{ij}}}-{\log n}_j,\log Q+Z_{\mathrm{ij}}$

[0134] The above equation indicates that the ct value is corrupted by additive Gaussian noise Z.sub.ij. Let the variance of this noise term be σ.sub.ij.sup.2.

Maximum Likelihood (ML) Estimation of Copy Number

[0135] In one embodiment of the method, the maximum likelihood estimation is used, with respect to the model described above, to determine n.sub.j. The parameter Q makes this difficult unless another constraint is added:

[00006]$\frac{1}{m}\underset{j}{.Math.}\log n_j=1$

[0136] This indicates that the average copy number is 2, or, equivalently, that the average log copy number is 1. With this additional constraint one can now solve the following ML problem:

[00007]$\hat{Q},{\hat{n}}_j=\underset{Q,n_j}{\mathrm{argmax}}\underset{\mathrm{ij}}{.Math.}\text{?}$$\text{?}\text{indicates text missing or illegible when filed}$

The last line above is linear in the variables log n.sub.j and log Q, and is a simple weighted least squares problem with an equality constraint. The solution can be obtained in closed form by forming the Lagrangian

[00008]$\underset{\mathrm{ij}}{\text{?}=.Math.}\frac{1}{{\sigma }_{\mathrm{ij}}^2}{\left(\begin{array}{c}\log n_j+\log Q-\\ \left(\log \frac{\mathrm{Qr}}{{\alpha }_{\mathrm{ij}}}-{\beta }_{\mathrm{ij}}{1}_{\mathrm{ij}}\right)\end{array}\right)}^2+\lambda \underset{j}{.Math.}\log n_j$$\text{?}\text{indicates text missing or illegible when filed}$

and taking partial derivatives.
Solution when Noise Variance is Constant

[0137] To avoid unnecessarily complicating the exposition, set σ.sub.ij.sup.2=1. This assumption will remain unless explicitly stated otherwise. (In the general case in which each σ.sub.ij.sup.2 is different, the solutions will be weighted averages instead of simple averages, or weighted least squares solutions instead of simple least squares solutions.) In that case, the above linear system has the solution:

[00009]$\log Q_j\stackrel{\Delta }{=}\frac{1}{n}\underset{i}{.Math.}\left(\log \frac{Q_T}{{\alpha }_{\mathrm{ij}}}-{\beta }_{\mathrm{ij}}{\mathrm{.Math.}}_{\mathrm{ij}}\right)\log Q=\frac{1}{m}\underset{j}{.Math.}\log Q_j-1\log n_j=\log Q_j-\log Q=\log \frac{Q_j}{Q}$

The first equation can be interpreted as a log estimate of the quantity of chromosome j. The second equation can be interpreted as saying that the average of the Q.sub.j is the average of a diploid quantity; subtracting one from its log gives the desired monosome quantity. The third equation can be interpreted as saying that the copy number is just the ratio

[00010]$\frac{Q_j}{Q}.$

Note that n.sub.j is a ‘double difference’, since it is a difference of Q-values, each of which is itself a difference of values.

Simple Solution

[0138] The above equations also reveal the solution under simpler modeling assumptions: for example, when making the assumption α.sub.ij=1 for all i and j and/or when making the assumption that β.sub.ij=β for all i and j. In the simplest case, when both α.sub.ij=1 and β.sub.ij=β, the solution reduces to

[00011]$\text{(*}\text{*)}\log n_j=1+\beta \left(\frac{1}{\mathrm{mn}}\underset{\mathrm{ij}}{.Math.}{\mathrm{.Math.}}_{\mathrm{ij}}-\frac{1}{n}\underset{i}{.Math.}{t}_{\mathrm{ij}}\right)$

The Double Differencing Method

[0139] In one embodiment of the invention, it is possible to detect monosomy using double differencing. It should be obvious to one skilled in the art how to modify this method for detecting other aneuploidy states. Let {t.sub.ij} be the regularized ct values obtained from MDA pre-amplification followed by PCR on the genetic sample. As always, t.sub.ij is the ct value on the i-th SNP of the j-th chromosome. Denote by t.sub.j the vector of ct values associated with the j-th chromosome. Make the following definitions:

[00012]$\stackrel{\_}{t}\stackrel{\Delta }{=}\frac{1}{\mathrm{mn}}\underset{\mathrm{ij}}{.Math.}{t}_{\mathrm{ij}}{\tilde{t}}_j\stackrel{\Delta }{=}{t}_j-\tilde{t}1$

Classify chromosome j as monosomic if and only if f.sup.T{tilde over (t)}.sub.i is higher than a certain threshold value, where f is a vector that represents a monosomy signature. f is the matched filter, whose construction is described next.

[0140] The matched filter f is constructed as a double difference of values obtained from two controlled experiments. Begin with known quantities of euploid male genetic data and euploid female genetic material. Assume there are large quantities of this material, and pre-amplification can be omitted. On both the male and female material, use PCR to sequence n SNPs on both the X chromosome (chromosome 23), and chromosome 7. Let {t.sub.ij.sup.X}, i=1 . . . n, jϵ{7, 23} denote the measurements on the female, and let {t.sub.ij.sup.Y} similarly denote the measurements on the male. Given this, it is possible to construct the matched filter f from the resulting data as follows:

[00013]${\stackrel{\_}{t}}_7^X\stackrel{\Delta }{=}\frac{1}{n}\underset{i}{.Math.}{l}_{i,7}^X\text{?}\stackrel{\Delta }{=}\frac{1}{n}\underset{i}{.Math.}\text{?}{\Delta }^X\stackrel{\Delta }{=}\text{?}-\text{?}{\Delta }^Y\stackrel{\Delta }{=}\text{?}-\text{?}f\stackrel{\Delta }{=}{\Delta }^Y-{\Delta }^X\text{?}\text{indicates text missing or illegible when filed}$

In the above, equations, t.sub.7.sup.X and t.sub.7.sup.Y are scalars, while Δ.sup.X and Δ.sup.Y are vectors. Note that the superscripts X and Y are just symbolic labels, not indices, denoting female and male respectively. Do not to confuse the superscript X with measurements on the X chromosome. The X chromosome measurements are the ones with subscript 23.

[0141] The next step is to take noise into account and to see what remnants of noise survive in the construction of the matched filter f as well as in the construction of {tilde over (t)}.sub.j. In this section, consider the simplest possible modeling assumption: that β.sub.ij=β for all i and j, and that α.sub.ij=1 for all i and j. Under these assumptions, from (*) above: βt.sub.ij=log Q.sub.T−log Q+Z.sub.ij Which can be rewritten as:

[00014]$t_{\mathrm{ij}}=\frac{1}{\beta }\log Q_T-\frac{1}{\beta }\log n_j-\frac{1}{\beta }\log Q+Z_{\mathrm{ij}}$

In that case, the i-th component of the matched filter f is given by:

[00015]$f_i\stackrel{\Delta }{=}{\Delta }_i^Y-{\Delta }_i^X=\left\{t_{i,23}^Y-{\stackrel{\_}{t}}_7^Y\right\}-\left\{\left(t_{i,23}^X-{\stackrel{\_}{t}}_7^X\right)\right\}=\left\{\left(\frac{1}{\beta }\log Q_T-\frac{1}{\beta }\log n_{23}^Y-\frac{1}{\beta }\log Q^Y+Z_{i,23}^Y\right)-\frac{1}{n}\underset{i}{.Math.}\left(\frac{1}{\beta }\log Q_T-\frac{1}{\beta }\log n_7^Y-\frac{1}{\beta }\log Q^Y+Z_{i,7}^Y\right)\right\}-\left\{\left(\frac{1}{\beta }\log Q_T-\frac{1}{\beta }\log n_{23}^X-\frac{1}{\beta }\log Q^X+Z_{i,23}^X\right)-\frac{1}{n}\underset{i}{.Math.}\left(\frac{1}{\beta }\log Q_T-\frac{1}{\beta }\log n_7^X-\frac{1}{\beta }\log Q^X+Z_{i,7}^X\right)\right\}-\left\{\left(\frac{1}{\beta }+Z_{i,23}^Y\right)-\frac{1}{n}\underset{i}{.Math.}{Z}_{i,7}^Y\right\}-\left\{Z_{i,23}^X-\frac{1}{n}\underset{i}{.Math.}{Z}_{i,7}^Y\right\}$

Note that the above equations take advantage of the fact that all the copy number variables are known, for example, n.sub.23.sup.Y=1 and that n.sub.23.sup.X=2.

[0142] Given that all the noise terms are zero mean, the ideal matched filter is 1/β1. Further, since scaling the filter vector doesn't really change things, the vector 1 can be used as the matched filter. This is equivalent to simply taking the average of the components of {tilde over (t)}.sub.j. In other words, the matched filter paradigm is not necessary if the underlying biochemistry follows the simple model. In addition, one may omit the noise terms above, which can only serve to lower the accuracy of the method. Accordingly, this gives:

[00016]${\tilde{t}}_{\mathrm{ij}}\stackrel{\Delta }{=}{t}_j-\stackrel{\_}{t}=\left\{\frac{1}{\beta }\log Q_T-\frac{1}{\beta }\log n_i-\frac{1}{\beta }\log Q+Z_{\mathrm{ij}}\right\}-\frac{1}{\mathrm{mn}}\underset{i,j}{.Math.}\left\{\frac{1}{\beta }\log Q_T-\frac{1}{\beta }\log n_i-\frac{1}{\beta }\log Q+Z_{\mathrm{ij}}\right\}=\frac{1}{\beta }\left(1-\log n_j\right)+Z_{\mathrm{ij}}-\frac{1}{\mathrm{mn}}\underset{i,j}{.Math.}{Z}_{\mathrm{ij}}$

In the above, it is assumed that

[00017]$\frac{1}{\mathrm{mn}}\underset{i,j}{.Math.}\log n_j=1.$

that is, that the average copy number is 2.
Each element of the vector is an independent measurement of the log copy number (scaled by 1/β), and then corrupted by noise. The noise term Z.sub.ij cannot be gotten rid of: it is inherent in the measurement. The second noise term probably cannot be gotten rid of either, since subtracting out t is necessary to remove the nuisance term

[00018]$\frac{1}{\beta }\log Q.$

Again, note that, given the observation that each element of {tilde over (t)}.sub.j, is an independent measurement of

[00019]$\frac{i}{\beta }\left(1-\log n_j\right),$

it is clear that a UMVU (uniform minimum variance unbiased) estimate of

[00020]$\frac{1}{\beta }\left(1-\log n_j\right)$

is just the average of the elements of {tilde over (j)}.sub.j. (In the case in which each σ.sub.ij.sup.2 is different, it will be a weighted average.) Thus, performing a little bit of algebra, the UMVU estimator for log n.sub.j is given by:

[00021]$\frac{1}{n}\underset{i}{.Math.}{\tilde{t}}_{\mathrm{ij}}\approx \frac{1}{\beta }\left(1-\log n_j\right).Math.\log n_j\approx 1-\beta .Math.\frac{1}{n}\underset{i,j}{.Math.}{\tilde{t}}_{\mathrm{ij}}=1-\beta \left(\frac{1}{n}\underset{i}{.Math.}{t}_{\mathrm{ij}}-\frac{1}{\mathrm{mn}}\underset{i,j}{.Math.}{t}_{\mathrm{ij}}\right)$

Analysis Under the Complicated Model

[0143] Now repeat the preceding analysis with respect to a biochemical model in which each β.sub.ij and α.sub.ij is different. Again, take noise into account and to see what remnants of noise survive in the construction of the matched filter f as well as in the construction of {tilde over (t)}.sub.j. Under the complicated model, from (*) above:

[00022]${\beta }_{\mathrm{ij}}{t}_{\mathrm{ij}}=\log \frac{Q_T}{{\alpha }_{\mathrm{ij}}}-\log n_j-\log Q+Z_{\mathrm{ij}}$

Which can be rewritten as:

[00023]$\text{(*}\text{**)}{t}_{\mathrm{ij}}=\frac{1}{{\beta }_{\mathrm{ij}}}\log \frac{Q_T}{{\alpha }_{\mathrm{ij}}}-\frac{1}{{\beta }_{\mathrm{ij}}}\log n_j-\frac{1}{{\beta }_{\mathrm{ij}}}\log Q+Z_{\mathrm{ij}}$

The i-th component of the matched filter f is given by:

[00024]$f_i\stackrel{\Delta }{=}{\Delta }_i^Y-{\Delta }_i^X-\left\{{\mathrm{.Math.}}_{i,23}^Y-{\stackrel{.Math.}{\mathrm{.Math.}}}_7^Y\right\}-\{\left({\mathrm{.Math.}}_{i,23}^X-{\stackrel{.Math.}{\mathrm{.Math.}}}_7^X\right)=\left\{\left(\frac{1}{{\beta }_{i,23}}\log \frac{Q_T}{{\alpha }_{i,23}}-\frac{1}{{\beta }_{i,23}}\log n_{23}^Y-\frac{1}{{\beta }_{i,23}}\log Q^Y+Z_{i,23}^Y\right)-\frac{1}{n}\underset{i}{.Math.}\left(\frac{1}{{\beta }_{i,7}}\log \frac{Q_T}{{\alpha }_{i,7}}-\frac{1}{{\beta }_{i,7}}\log n_7^n-\frac{1}{{\beta }_{i,7}}\log Q^Y+Z_{i,7}^Y\right)\right\}-\left\{\left(\frac{1}{{\beta }_{i,23}}\log \frac{Q_T}{{\alpha }_{i,23}}-\frac{1}{{\beta }_{i,23}}\log n_{23}^X-\frac{1}{{\beta }_{i,23}}\log Q^Y+Z_{i,23}^X\right)-\frac{1}{n}\underset{i}{.Math.}\left(\frac{1}{{\beta }_{i,7}}\log \frac{Q_T}{{\alpha }_{i,7}}\log n_7^X-\frac{1}{{\beta }_{i,7}}\log Q^Y+Z_{i,7}^X\right)\right\}=\frac{1}{{\beta }_{i,23}}+\left(\frac{1}{{\beta }_{i,23}}-\left(\frac{1}{n}\underset{i}{.Math.}\frac{1}{{\beta }_{i,7}}\right)\right)\log \frac{Q^Y}{Q^X}+\left\{Z_{i,23}^Y-Z_{i,23}^X+\frac{1}{n}\underset{i}{.Math.}{Z}_{i,7}^X-\frac{1}{n}\underset{i}{.Math.}{Z}_{i,7}^Y\right\}$

Under the complicated model, this gives:

[00025]${\tilde{I}}_{\mathrm{ij}}\stackrel{\Delta }{=}{I}_j-\stackrel{.Math.}{t}=\left\{\frac{1}{B_{\mathrm{ij}}}\log \frac{Q_T}{{\alpha }_{\mathrm{ij}}}-\frac{1}{{\beta }_{\mathrm{ij}}}\log n_j-\frac{1}{{\beta }_{\mathrm{ij}}}\log Q+Z_{\mathrm{ij}}\right\}-\frac{1}{\mathrm{mn}}\underset{i,j}{.Math.}\left\{\frac{1}{{\beta }_{\mathrm{ij}}}\log \frac{Q_T}{{\alpha }_{\mathrm{ij}}}-\frac{1}{{\beta }_{\mathrm{ij}}}\log n_j-\frac{1}{{\beta }_{\mathrm{ij}}}\log Q+Z_{\mathrm{ij}}\right\}$

An Alternate Way to Regularize CT Values

[0144] In another embodiment of the method, one can average the CT values rather than transforming to exponential scale and then taking logs, as this distorts the noise so that it is no longer zero mean. First, start with known Q and solve for betas. Then do multiple experiments with known n_j to solve for alphas. Since aneuploidy is a whole set of hypotheses, it is convenient to use ML to determine the most likely n_j and Q values, and then use this as a basis for calculating the most likely aneuploid state, e.g., by taking the n_j value that is most off from 1 and pushing it to its nearest aneuploid neighbor.

Estimation of the Error Rates in the Embryonic Measurements.

[0145] In one embodiment of the invention, it is possible to determine the conditional probabilities of particular embryonic measurements given specific underlying true states in embryonic DNA. In certain contexts, the given data consists of (i) the data about the parental SNP states, measured with a high degree of accuracy, and (ii) measurements on all of the SNPs in a specific blastomere, measured poorly.

[0146] Use the following notation: U—is any specific homozygote, Ū is the other homozygote at that SNP, H is the heterozygote. The goal is to determine the probabilities (p.sub.ij) shown in Table 2. For instance p.sub.11 is the probability of the embryonic DNA being U and the readout being U as well. There are three conditions that these probabilities have to satisfy:

p.sub.11+p.sub.12+p.sub.13+p.sub.14=1   (1)

p.sub.21+p.sub.22+p.sub.23+p.sub.24=1   (2)

p.sub.21=p.sub.23   (3)

The first two are obvious, and the third is the statement of symmetry of heterozygote dropouts (H should give the same dropout rate on average to either U or Ū).

[0147] There are 4 possible types of matings: U×U, U×U, U×H, H×H. Split all of the SNPs into these 4 categories depending on the specific mating type. Table 3 shows the matings, expected embryonic states, and then probabilities of specific readings (p.sub.ij). Note that the first two rows of this table are the same as the two rows of the Table 2 and the notation (p.sub.ij) remains the same as in Table 2.

[0148] Probabilities p.sub.3i and p.sub.4i can be written out in terms of p.sub.1i and p.sub.2i.

p.sub.31=½[p.sub.11+p.sub.21]  (4)

p.sub.32=½[p.sub.12+p.sub.22]  (5)

p.sub.33=½[p.sub.13+p.sub.23]  (6)

p.sub.34=½[p.sub.14+p.sub.24]  (7)

p.sub.41=¼[p.sub.11+2p.sub.21+p.sub.13]  (8)

p.sub.42=½[p.sub.12+p.sub.22]  (9)

p.sub.43=¼[p.sub.11+2p.sub.23+p.sub.13]  (10)

p.sub.44=½[p.sub.14+p.sub.24]  (11)

These can be thought of as a set of 8 linear constraints to add to the constraints (1), (2), and (3) listed above. If a vector P=[p.sub.11, p.sub.12, p.sub.13, p.sub.14, p.sub.21 . . . , p.sub.44].sup.T (16×1 dimension) is defined, then the matrix A (11×16) and a vector C can be defined such that the constraints can be represented as:

AP=C   (12)

C=[1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0].sup.T. Specifically, A is shown in Table 4, where empty cells have zeroes.

[0149] The problem can now be framed as that of finding P that would maximize the likelihood of the observations and that is subject to a set of linear constraints (AP=C). The observations come in the same 16 types as p.sub.ij. These are shown in Table 5. The likelihood of making a set of these 16 n.sub.ij observations is defined by a multinomial distribution with the probabilities p.sub.ij and is proportional to:

[00026]$\begin{array}{cc}L\left(P,n_{\mathrm{ij}}\right)\propto \underset{\mathrm{ij}}{.Math.}{p}_{\mathrm{ij}}^{n_{\mathrm{ij}}}& \left(13\right)\end{array}$

Note that the full likelihood function contains multinomial coefficients that are not written out given that these coefficients do not depend on P and thus do not change the values within P at which L is maximized. The problem is then to find:

[00027]$\begin{array}{cc}\underset{P}{\max }\left[L\left(P,n_{\mathrm{ij}}\right)\right]=\underset{P}{\max }\left[\ln \left(L\left(P,n_{\mathrm{ij}}\right)\right)\right]=\underset{P}{\max }\left(\underset{\mathrm{ij}}{.Math.}{n}_{\mathrm{ij}}\ln \left(p_{\mathrm{ij}}\right)\right)& \left(14\right)\end{array}$

subject to the constraints AP=C.
Note that in (14) taking the In of L makes the problem more tractable (to deal with a sum instead of products). This is standard given that value of x such that f(x) is maximized is the same for which ln(f(x)) is maximized. P(n.sub.j,Q,D)=P(n.sub.j)P(Q)P(D.sub.j|Q,n.sub.j)P(D.sub.k≠j|Q).
D MAP Detection of Aneuploidy without Parents

[0150] In one embodiment of the invention, the PS method can be applied to determine the number of copies of a given chromosome segment in a target without using parental genetic information. In this section, a maximum a-posteriori (MAP) method is described that enables the classification of genetic allele information as aneuploid or euploid. The method does not require parental data, though when parental data are available the classification power is enhanced. The method does not require regularization of channel values. One way to determine the number of copies of a chromosome segment in the genome of a target individual by incorporating the genetic data of the target individual and related individual(s) into a hypothesis, and calculating the most likely hypothesis is described here. In this description, the method will be applied to ct values from TAQMAN measurements; it should be obvious to one skilled in the art how to apply this method to any kind of measurement from any platform. The description will focus on the case in which there are measurements on just chromosomes X and 7; again, it should be obvious to one skilled in the art how to apply the method to any number of chromosomes and sections of chromosomes.

Setup of the Problem

[0151] The given measurements are from triploid blastomeres, on chromosomes X and 7, and the goal is to successfully make aneuploidy calls on these. The only “truth” known about these blastomeres is that there must be three copies of chromosome 7. The number of copies of chromosome X is not known.

[0152] The strategy here is to use MAP estimation to classify the copy number N.sub.7 of chromosome 7 from among the choices {1,2,3} given the measurements D. Formally that looks like this:

[00028]${\hat{n}}_7=\underset{n_7\in \left\{1,2,3\right\}}{\mathrm{argmax}}P\left(n_7,D\right)$

Unfortunately, it is not possible to calculate this probability, because the probability depends on the unknown quantity Q. If the distribution f on Q were known, then it would be possible to solve the following:

[00029]${\hat{n}}_7=\underset{n_7\in \left\{1,2,3\right\}}{\mathrm{argmax}}\int f\left(Q\right)P\left(n_7,D|Q\right)dQ$

In practice, a continuous distribution on Q is not known. However, identifying Q to within a power of two is sufficient, and in practice a probability mass function (pmf) on Q that is uniform on say {2.sup.1,2.sup.2 . . . , 2.sup.40} can be used. In the development that follows, the integral sign will be used as though a probability distribution function (pdf) on Q were known, even though in practice a uniform pmf on a handful of exponential values of Q will be substituted.

[0153] This discussion will use the following notation and definitions: N.sub.7 is the copy number of chromosome seven. It is a random variable. n.sub.7 denotes a potential value for N.sub.7. N.sub.X is the copy number of chromosome X. n.sub.X denotes a potential value for N.sub.X. N.sub.j is the copy number of chromosome-j, where for the purposes here jϵ{7,X}. n.sub.j denotes a potential value for N.sub.j. D is the set of all measurements. In one case, these are TAQMAN measurements on chromosomes X and 7, so this gives D={D.sub.7,D.sub.X}, where D.sub.j={t.sub.ij.sup.A,t.sub.ij.sup.C} is the set of TAQMAN measurements on this chromosome. t.sub.ij.sup.A is the ct value on channel-A of locus i of chromosome-j. Similarly, t.sub.ij.sup.C is the ct value on channel-C of locus i of chromosome-j. (A is just a logical name and denotes the major allele value at the locus, while C denotes the minor allele value at the locus.) Q represents a unit-amount of genetic material such that, if the copy number of chromosome-j is n.sub.j, then the total amount of genetic material at any locus of chromosome-j is n.sub.jQ. For example, under trisomy, if a locus were AAC, then the amount of A-material at this locus would be 2Q, the amount of C-material at this locus is Q, and the total combined amount of genetic material at this locus is 3Q. (n.sup.A,n.sup.C) denotes an unordered allele patterns at a locus when the copy number for the associate chromosome is n. n.sup.A is the number of times allele A appears on the locus and n.sup.C is the number of times allele C appears on the locus. Each can take on values in 0, . . . , n, and it must be the case that n.sup.A+n.sup.C=n. For example, under trisomy, the set of allele patterns is {(0,3), (1,2), (2,1), (3,0)}. The allele pattern (2,1) for example corresponds to a locus value of A.sup.2C, i.e., that two chromosomes have allele value A and the third has an allele value of C at the locus. Under disomy, the set of allele patterns is {(0,2), (1,1), (2,0)}. Under monosomy, the set of allele patterns is {(0,1), (1,0)}. [0154] Q.sub.T is the (known) threshold value from the fundamental TAQMAN equation Q.sub.02.sup.βr=Q.sub.T. [0155] β is the (known) doubling-rate from the fundamental TAQMAN equation Q.sub.02.sup.βt=Q.sub.T. [0156] ⊥ (pronounced “bottom”) is the ct value that is interpreted as meaning “no signal”. [0157] f.sub.Z(χ) is the standard normal Gaussian pdf evaluated at χ. [0158] σ a is the (known) standard deviation of the noise on TAQMAN ct values.

MAP Solution

[0159] In the solution below, the following assumptions have been made:

[0160] N.sub.7 and N.sub.χ are independent.

[0161] Allele values on neighboring loci are independent.

[0162] The goal is to classify the copy number of a designated chromosome. In this case, the description will focus on chromosome 7. The MAP solution is given by:

[00030]$\begin{array}{c}{\hat{n}}_7=\underset{n_7\in \left\{1,2,3\right\}}{\mathrm{argmax}}\int f\left(Q\right)P\left(n_7,D|Q\right)dQ\\ =\underset{n_7\in \left\{1,2,3\right\}}{\mathrm{argmax}}\int f\left(Q\right)\underset{n_X\in \left[1,2,,3\right]}{.Math.}P\left(n_7,n_X,D|Q\right)\mathrm{dQ}\\ =\underset{n_7\in \left\{1,2,3\right\}}{\mathrm{argmax}}\int f\left(Q\right)\\ \underset{n_X\in \left[1,2,,3\right]}{.Math.}P\left(n_7\right)P\left(n_X\right)P\left(D_7|Q,n_7\right)P\left(D_X|Q,n_X\right)\mathrm{dQ}\\ =\underset{n_7\in \left\{1,2,3\right\}}{\mathrm{argmax}}\int f\left(Q\right)\left(P\left(n_7\right)P\left(D_7|Q,n_7\right)\right)\\ \left(\underset{n_X\in \left[1,2,3\right]}{.Math.}P\left(n_X\right)P\left(D_X|Q,n_X\right)\right)\mathrm{dQ}\\ =\underset{n_7\in \left\{1,2,3\right\}}{\mathrm{argmax}}\int f\left(Q\right)\left(P\left(n_7\right)\underset{i}{.Math.}P\left(t_{i,7}^A,t_{i,7}^C|Q,n_7\right)\right)\\ \left(\underset{n_Z\in \left[1.2\text{.3}\right]}{.Math.}P\left(n_X\right)\underset{i}{.Math.}P\left(t_{i,X}^A,t_{i,X}^C|Q,n_X\right)\right)\mathrm{dQ}\text{(*})\\ =\underset{n_7\in \left[1,2,3\right]}{\mathrm{argmax}}\int f\left(Q\right)\\ \left(P\left(n_7\right)\underset{i}{.Math.}\underset{n^A+n^C=n_7}{.Math.}P\left(n^A,n^C|n_7,i\right)P\left(t_{i,7}^A|Q,n^A\right)P\left(t_{i,7}^C|Q,n^C\right)\right)\times \\ \left(\begin{array}{c}\underset{n_X\in \left[1,2,3\right]}{.Math.}P\left(n_X\right)\underset{i}{.Math.}\underset{n^A+n^C=n_X}{.Math.}\\ P\left(n^A,n^C|n_X,i\right)P\left(t_{i,X}^A|Q,n^A\right)P\left(t_{i,X}^C|Q,n^C\right)\end{array}\right)\mathrm{dQ}\end{array}$

Allele Distribution Model

[0163] Equation (*) depends on being able to calculate values for P(n.sup.A,n.sup.Cln.sub.7,i) and P(n.sup.A,n.sup.Cln.sub.X,i). These values may be calculated by assuming that the allele pattern (n.sup.A,n.sup.C) is drawn i.i.d (independent and identically distributed) according to the allele frequencies for its letters at locus i. An example should suffice to illustrate this. Calculate P((2,1)|n.sub.7=3) under the assumption that the allele frequency for A is 60%, and the minor allele frequency for C is 40%. (As an aside, note that P((2,1)|n.sub.7−2)−0, since in this case the pair must sum to 2.) This probability is given by

[00031]$P\left(\left(2,1\right)|n_7=3\right)=\left(\begin{array}{c}3\\ 2\end{array}\right){\left(\text{.60}\right)}^Z\left(\text{.40}\right)$

The general equation is

[00032]$P\left(n^A.n^C|n_j,i\right)=\left(\begin{array}{c}n\\ n^A\end{array}\right){\left(1-p_{\mathrm{ij}}\right)}^{n^A}{\left(p_{\mathrm{ij}}\right)}^{n^C}$

Where p.sub.i,j is the minor allele frequency at locus i of chromosome j.

Error Model

[0164] Equation (*) depends on being able to calculate values for P(t.sup.A|Q,n.sup.A) and P(t.sup.C|Q,n.sup.C). For this an error model is needed. One may use the following error model:

[00033]$P\left(t^A|Q,n^A\right)=\left\{\begin{array}{cc}{p}_d& t^A=\perp \mathrm{and}{n}^A>0\\ \left(1-p_a\right)f_Z\left(\frac{1}{\sigma }\left(t^A-\frac{1}{\beta }\log \frac{Q_T}{n^{A}Q}\right)\right)& t^A\ne \perp \mathrm{and}{n}^A>0\\ 1-p_a& t^A=\perp \mathrm{and}{n}^A=0\\ 2p_a{f}_Z\left(\frac{1}{\sigma }\left(t^A-\perp \right)\right)& t^A\ne \perp \mathrm{and}{n}^A=0\end{array}\right\}$

[0165] Each of the four cases mentioned above is described here. In the first case, no signal is received, even though there was A-material on the locus. That is a dropout, and its probability is therefore p.sub.d. In the second case, a signal is received, as expected since there was A-material on the locus. The probability of this is the probability that a dropout does not occur, multiplied by the pdf for the distribution on the ct value when there is no dropout. (Note that, to be rigorous, one should divide through by that portion of the probability mass on the Gaussian curve that lies below ⊥, but this is practically one, and will be ignored here.) In the third case, no signal was received and there was no signal to receive. This is the probability that no drop-in occurred, 1−p.sub.a. In the final case, a signal is received even through there was no A-material on the locus. This is the probability of a drop-in multiplied by the pdf for the distribution on the ct value when there is a drop-in. Note that the ‘2’ at the beginning of the equation occurs because the Gaussian distribution in the case of a drop-in is modeled as being centered at ⊥. Thus, only half of the probability mass lies below ⊥ in the case of a drop-in, and when the equation is normalized by dividing through by one-half, it is equivalent to multiplying by 2. The error model for P(t.sup.C|Q,n.sup.C) by symmetry is the same as for P(t.sup.A|Q,n.sup.A) above. It should be obvious to one skilled in the art how different error models can be applied to a range of different genotyping platforms, for example the ILLUMINA INFINIUM genotyping platform.

Computational Considerations

[0166] In one embodiment of the invention, the MAP estimation mathematics can be carried out by brute-force as specified in the final MAP equation, except for the integration over Q. Since doubling Q only results in a difference in ct value of 1/β, the equations are sensitive to Q only on the log scale. Therefore to do the integration it should be sufficient to try a handful of Q-values at different powers of two and to assume a uniform distribution on these values. For example, one could start at Q=Q.sub.T2.sup.−20β, which is the quantity of material that would result in a ct value of 20, and then halve it in succession twenty times, yielding a final Q value that would result in a ct value of 40.

[0167] What follows is a re-derivation of a derivation described elsewhere in this disclosure, with slightly difference emphasis, for elucidating the programming of the math. Note that the variable D below is not really a variable. It is always a constant set to the value of the data set actually in question, so it does not introduce another array dimension when representing in MATLAB. However, the variables D.sub.j do introduce an array dimension, due to the presence of the index j.

[00034]$n_7=\underset{n_7\in \left\{1,2,3\right\}}{\mathrm{argmax}}P\left(n_7,D\right)$$P\left(n_7,D\right)=\underset{Q}{.Math.}P\left(n_7,Q,D\right)$$P\left(n_7,Q,D\right)=P\left(n_7\right)P\left(Q\right)P\left(D_7|Q,n_7\right)P\left(D_X|Q\right)$$P\left(D_j|Q\right)-\underset{n_j\in \left[1,2,3\right]}{.Math.}P\left(D_j,n_j|Q\right)$$P\left(D_j,n_j|Q\right)=P\left(n_j\right)P\left(D_j|Q,n_j\right)$$P\left(D_j|Q,n_j\right)=\underset{i}{.Math.}P\left(D_{\mathrm{ij}}|Q,n_j\right)$$P\left(D_{\mathrm{ij}}|Q,n_j\right)=\underset{n^A+n^C=n_j}{.Math.}P\left(D_{\mathrm{ij}},n^A,n^C|Q,n_j\right)$$P\left(\begin{array}{c}{D}_{\mathrm{ij}},n^A,\\ n^C|Q,n_j\end{array}\right)=P\left(n^A,n^C|n_j,i\right)P\left(t_{\mathrm{ij}}^A|Q,n^A\right)P\left(t_{\mathrm{ij}}^C|Q,n^C\right)$$P\left(n^A,n^C|n_j,i\right)=\left(\begin{array}{c}n\\ n^A\end{array}\right){\left(1-p_{\mathrm{ij}}\right)}^{n^A}{\left(p_{\mathrm{ij}}\right)}^{n^C}$$P\left(\begin{array}{c}{t}_{\mathrm{ij}}^A|Q,\\ n^A\end{array}\right)=\left\{\begin{array}{cc}{p}_d& t^A=\perp \mathrm{and}{n}^A>0\\ \left(1-p_a\right)f_Z\left(\frac{1}{\sigma }\left(t^A-\frac{1}{\beta }\log \frac{Q_T}{n^{A}Q}\right)\right)& t^A\ne \perp \mathrm{and}{n}^A>0\\ 1-p_a& t^A=\perp \mathrm{and}{n}^A=0\\ 2p_a{f}_Z\left(\frac{1}{\sigma }\left(t^A-\perp \right)\right)& t^A\ne \perp \mathrm{and}{n}^A=0\end{array}\right\}$

E MAP Detection of Aneuploidy with Parental Info

[0168] In one embodiment of the invention, the disclosed method enables one to make aneuploidy calls on each chromosome of each blastomere, given multiple blastomeres with measurements at some loci on all chromosomes, where it is not known how many copies of each chromosome there are. In this embodiment, the a MAP estimation is used to classify the copy number N.sub.j of chromosome where jϵ{1,2 . . . 22,X,Y}, from among the choices {0, 1, 2, 3} given the measurements D, which includes both genotyping information of the blastomeres and the parents. To be general, let jϵ{1,2 . . . m} where m is the number of chromosomes of interest; m=24 implies that all chromosomes are of interest. Formally, this looks like:

[00035]${\hat{n}}_j=\underset{n_j\in \left[1,2,3\right]}{\mathrm{argmax}}P\left(n_j,D\right)$

[0169] Unfortunately, it is not possible to calculate this probability, because the probability depends on an unknown random variable Q that describes the amplification factor of MDA. If the distribution f on Q were known, then it would be possible to solve the following:

[00036]${\hat{n}}_j=\underset{n_j\in \left[1,2,3\right]}{\mathrm{argmax}}\int f\left(Q\right)P\left(n_j,D|Q\right)\mathrm{dQ}$

[0170] In practice, a continuous distribution on Q is not known. However, identifying Q to within a power of two is sufficient, and in practice a probability mass function (pmf) on Q that is uniform on say {2.sup.1, 2.sup.2 . . . , 2.sup.40} can be used. In the development that follows, the integral sign will be used as though a probability distribution function (pdf) on Q were known, even though in practice a uniform pmf on a handful of exponential values of Q will be substituted.

[0171] This discussion will use the following notation and definitions:

[0172] N.sub.α is the copy number of autosomal chromosome α, where αϵ{1, 2 . . . 22}. It is a random variable. n.sub.α denotes a potential value for N.sub.α.

[0173] N.sub.X is the copy number of chromosome X. n.sub.X denotes a potential value for N.sub.X.

[0174] N.sub.j is the copy number of chromosome-j, where for the purposes here jϵ{1,2 . . . m}. n.sub.j denotes a potential value for N.sub.j.

[0175] m is the number of chromosomes of interest, m=24 when all chromosomes are of interest.

[0176] H is the set of aneuploidy states. hϵH. For the purposes of this derivation, let H={paternal monosomy, maternal monosomy, disomy, t1 paternal trisomy, t2 paternal trisomy, t1 maternal trisomy, t2 maternal trisomy}. Paternal monosomy means the only existing chromosome came from the father; paternal trisomy means there is one additional chromosome coming from father. Type 1 (t1) paternal trisomy is such that the two paternal chromosomes are sister chromosomes (exact copy of each other) except in case of crossover, when a section of the two chromosomes are the exact copies. Type 2 (t2) paternal trisomy is such that the two paternal chromosomes are complementary chromosomes (independent chromosomes coming from two grandparents). The same definitions apply to the maternal monosomy and maternal trisomies.

[0177] D is the set of all measurements including measurements on embryo D.sub.E and on parents D.sub.F,D.sub.M. In the case where these are TAQMAN measurements on all chromosomes, one can say: D={D.sub.1, D.sub.2 . . . D′.sub.m}, D.sub.E={D.sub.E,1, D.sub.E,2 . . . D.sub.E,m}, where D.sub.k=(D.sub.E,k, D.sub.F,k, D.sub.M, k), D.sub.Ej={t.sub.E,ij.sup.A,t.sub.E,ij.sup.C} is the set of TAQMAN measurements on chromosome j.

[0178] t.sub.E,ij.sup.A is the ct value on channel-A of locus i of chromosome-j. Similarly, t.sub.E,ij.sup.Cis the ct value on channel-C of locus i of chromosome-j. (A is just a logical name and denotes the major allele value at the locus, while C denotes the minor allele value at the locus.)

[0179] Q represents a unit-amount of genetic material after MDA of single cell's genomic DNA such that, if the copy number of chromosome-j is n.sub.1, then the total amount of genetic material at any locus of chromosome-j is n.sub.jQ. For example, under trisomy, if a locus were AAC, then the amount of A-material at this locus is 2Q, the amount of C-material at this locus is Q, and the total combined amount of genetic material at this locus is 3Q.

[0180] q is the number of numerical steps that will be considered for the value of Q.

[0181] N is the number of SNPs per chromosome that will be measured.

[0182] (n.sup.A,n.sup.C) denotes an unordered allele patterns at a locus when the copy number for the associated chromosome is n. n.sup.A is the number of times allele A appears on the locus and n.sup.C is the number of times allele C appears on the locus. Each can take on values in 0, . . . , n, and it must be the case that n.sup.A+n.sup.C=n. For example, under trisomy, the set of allele patterns is {(0,3),(1,2),(2,1),(3,0)}. The allele pattern (2,1) for example corresponds to a locus value of A.sup.2C, i.e., that two chromosomes have allele value A and the third has an allele value of C at the locus. Under disomy, the set of allele patterns is {(0,2),(1,1),(2,0)}. Under monosomy, the set of allele patterns is {(0,1),(1,0)}.

[0183] Q.sub.T is the (known) threshold value from the fundamental TAQMAN equation Q.sub.02.sup.βt=Q.sub.T.

[0184] β is the (known) doubling-rate from the fundamental TAQMAN equation Q.sub.02.sup.βt=Q.sub.T.

[0185] ⊥ (pronounced “bottom”) is the ct value that is interpreted as meaning “no signal”.

[0186] f.sub.Z(x) is the standard normal Gaussian pdf evaluated at x.

[0187] σ is the (known) standard deviation of the noise on TAQMAN ct values.

MAP Solution

[0188] In the solution below, the following assumptions are made:

[0189] N.sub.js are independent of one another.

[0190] Allele values on neighboring loci are independent.

[0191] The goal is to classify the copy number of a designated chromosome. For instance, the MAP solution for chromosome a is given by

[00037]$\begin{array}{c}{n}_j=\underset{n_j\in \left[1,2,3\right]}{\mathrm{argmax}}\int f\left(Q\right)P\left(n_j,D|Q\right)\mathrm{dQ}\\ =\underset{n_j\in \left[1,2,3\right]}{\mathrm{argmax}}\int f\left(Q\right)\underset{n_1\in \left[1,2,3\right]}{.Math.}.Math.\underset{n_{j-1}\in \left[1,2,3\right]}{.Math.}\underset{n_{j+1}\in \left[1,2,3\right]}{.Math.}.Math.\\ \underset{n_m\in \left[1,2,3\right]}{.Math.}P\left(n_1,.Math.n_m,D|Q\right)\mathrm{dQ}\\ =\underset{n_j\in \left[1,2,3\right]}{\mathrm{argmax}}\int f\left(Q\right)\underset{n_1\in \left[1,2,3\right]}{.Math.}.Math.\underset{n_{j-1}\in \left[1,2,3\right]}{.Math.}\underset{n_{j+1}\in \left[1,2,3\right]}{.Math.}.Math.\\ \underset{n_m\in \left[1,2,3\right]}{.Math.}\underset{k=1}{\overset{m}{.Math.}}P\left(n_k\right)P\left(D_k|Q,n_k\right)\mathrm{dQ}\\ =\underset{n_j\in \left[1,2,3\right]}{\mathrm{argmax}}\int f\left(Q\right)\left(P\left(n_j\right)P\left(D_j|Q,n_j\right)\right)\\ \left(\underset{k\ne j}{.Math.}\underset{n_k\in \left[1,2,3\right]}{.Math.}P\left(n_k\right)P\left(D_k|Q,n_k\right)\right)\mathrm{dQ}\\ =\underset{n_j\in \left[1,2,3\right]}{\mathrm{argmax}}\int f\left(Q\right)\left(P\left(n_j\right)\underset{h\in H}{.Math.}P\left(D_j|Q,n_j,h\right)P\left(h|n_j\right)\right)\\ \left(\underset{k+j}{.Math.}\underset{n_k\in \left[1,2,3\right]}{.Math.}P\left(n_k\right)\underset{h\in H}{.Math.}P\left(D_k|Q,n_k,h\right)P\left(h|n_k\right)\right)\mathrm{dQ}\\ =\underset{n_j\in \left[1,2,3\right]}{\mathrm{argmax}}\int f\left(Q\right)\\ \left(P\left(n_j\right)\underset{h\in H}{.Math.}P\left(h|n_j\right)\underset{i}{.Math.}P\left(t_{E,\mathrm{ij}}^A,t_{E,\mathrm{ij}}^C,D_{F,\mathrm{ij}}{D}_{M,\mathrm{ij}}|Q,n_j,h\right)\right)\times \\ (\underset{k\ne j}{.Math.}\underset{n_k\in \left[1,2,3\right]}{.Math.}P\left(n_k\right)\underset{h\in H}{.Math.}P\left(h|n_k\right)\\ \underset{i}{.Math.}P\left(t_{E,\mathrm{ik}}^A,t_{E,\mathrm{ik}}^C,D_{F,\mathrm{ik}}{D}_{M,\mathrm{ik}}|Q,n_k,h\right))\mathrm{dQ}\\ =\underset{n_j\in \left[1,2,3\right]}{\mathrm{argmax}}\int f\left(Q\right)\\ (\begin{array}{c}P\left(n_j\right)\underset{h\in H}{.Math.}P\left(h|n_j\right)\underset{i}{.Math.}\underset{n_M^A+n_M^C=2}{\underset{n_F^A+n_F^C=2}{.Math.}}P\left(n_F^A,n_F^C,n_M^A,n_M^C\right)\\ P\left(t_{E,\mathrm{ij}}^A,t_{E,\mathrm{ij}}^C,D_{F,\mathrm{ij}}{D}_{M,\mathrm{ij}}|Q,n_j,h,n_F^A,n_F^C,n_M^A,n_M^C\right)\end{array})\times \\ (\underset{k\ne j}{.Math.}\underset{n_k\in \left[1,2,3\right]}{.Math.}P\left(n_k\right)\underset{h\in H}{.Math.}P\left(h|n_k\right)\\ \underset{i}{.Math.}\underset{n_F^A+n_F^C=2}{.Math.}P\left(n_F^A,n_F^C,n_M^A,n_M^C\right)\\ n_M^A+n_M^C=2\\ P\left(t_{E,\mathrm{ik}}^A,t_{E,\mathrm{ik}}^C,D_{F,\mathrm{ik}}{D}_{M,\mathrm{ik}}|Q,n_k,h,n_F^A,n_F^C,n_M^A,n_M^C\right))\mathrm{dQ}\\ =\underset{n_1\in \left[1,2,3\right]}{\mathrm{argmax}}\int f(Q\}(P\left(n_j\right)\underset{h\in H}{.Math.}P\left(h|n_j\right)\\ \underset{i}{.Math.}\underset{n_M^A+n_M^C=2}{\underset{n_F^A+n_F^C=2}{.Math.}}P\left(n_F^A,n_F^C,n_M^A,n_M^C\right)\\ P\left(t_{F,\mathrm{ij}}^A|n_F^A{Q}^{\prime }\right)P\left(t_{F,\mathrm{ij}}^C|n_F^C{Q}^{\prime }\right)P\left(t_{M,\mathrm{ij}}^A|n_M^A{Q}^{\prime }\right)\\ P\left(t_{M,\mathrm{ij}}^C|n_N^C{Q}^{\prime }\right)\times \underset{n^A+n^C=n_j}{.Math.}P\left(n^A,n^C|n_j,h,n_F^A,n_F^C,n_M^A,n_M^C\right)\\ P\left(t_{E,\mathrm{ij}}^A|Q,n^A\right)P\left(t_{E,\mathrm{ij}}^C|Q,n^C\right))\times \\ (\underset{k\ne j}{.Math.}\underset{n_k\in \left[1,2,3\right]}{.Math.}P\left(n_k\right)\underset{h\in H}{.Math.}P\left(h|n_k\right)\\ \underset{i}{.Math.}\underset{n_M^A+n_M^C=2}{\underset{n_F^A+n_F^C=2}{.Math.}}P\left(n_F^A,n_F^C,n_M^A,n_M^C\right)P\left({\tau }_{F,\mathrm{ik}}^A|n_F^A{Q}^{\prime }\right)\\ P\left({\tau }_{F,\mathrm{ik}}^C|n_F^C{Q}^{\prime }\right)P\left({\tau }_{M,\mathrm{ik}}^A|n_M^A{Q}^{\prime }\right)P\left({\tau }_{m,\mathrm{ik}}^C|n_M^C{Q}^{\prime }\right)\times \\ \underset{n^A+n^C=n_1}{.Math.}P\left(n^A,n^C|n_k,h,n_F^A,n_F^C,n_M^A,n_M^C\right)P\left({\tau }_{E,\mathrm{ik}}^A|n^{A}Q\right)\\ P\left(t_{E,\mathrm{ik}}^C|n^{C}Q\right))\mathrm{dQ}\text{(*})\end{array}$

Here it is assumed that Q′, the Q are known exactly for the parental data.

Copy Number Prior Probability

[0192] Equation (*) depends on being able to calculate values for P(n.sub.α) and P(n.sub.X), the distribution of prior probabilities of chromosome copy number, which is different depending on whether it is an autosomal chromosome or chromosome X. If these numbers are readily available for each chromosome, they may be used as is. If they are not available for all chromosomes, or are not reliable, some distributions may be assumed. Let the prior probability

[00038]$p\left(n_a=1\right)=P\left(n_a=2\right)=P\left(n_a=3\right)=\frac{1}{3}$

for autosomal chromosomes, let the probability of sex chromosomes being XY or XX be ½.

[00039]$P\left(n_x=0\right)=\frac{1}{3}\times \frac{1}{4}=\frac{1}{12}.P\left(n_x=1\right)=\frac{1}{3}\times \frac{3}{4}+\frac{1}{3}\times \frac{1}{2}+\frac{1}{3}\times \frac{1}{2}\times \frac{1}{4}=\frac{11}{24}=0.458,$

where ¾ is the probability of the monosomic chromosome being X (as oppose to Y), ½ is the probability of being XX for two chromosomes and ¼ is the probability of the third chromosome being Y.

[00040]$P\left(n_x=3\right)=\frac{1}{3}\times \frac{1}{2}\times \frac{3}{4}=\frac{1}{8}=0.125,$

where ½ is the probability of being XX for two chromosomes and ¾ is the probability of the third chromosome being X.

[00041]$P\left(n_x=2\right)=1-P\left(n_x=0\right)-P\left(n_x=1\right)-P\left(n_x=3\right)=\frac{4}{12}=0.333.$

Aneuploidy State Prior Probability

[0193] Equation (*) depends on being able to calculate values for P(h|n.sub.j), and these are shown in Table 6. The symbols used in the Table 6 are explained below

TABLE-US-00001 Symbol Meaning Ppm paternal monosomy probability Pmm maternal monosomy probability Ppt paternal trisomy probability given trisomy Pmt maternal trisomy probability given trisomy pt1 probability of type 1 trisomy for paternal trisomy, or P(type 1|paternal trisomy) pt2 probability of type 2 trisomy for paternal trisomy, or P(type 2|paternal trisomy) mt1 probability of type 1 trisomy for maternal trisomy, or P(type 1|maternal trisomy) mt2 probability of type 2 trisomy for maternal trisomy, or P(type 2|maternal trisomy)
Note that there are many other ways that one skilled in the art, after reading this disclosure, could assign or estimate appropriate prior probabilities without changing the essential concept of the patent.
Allele Distribution Model without Parents

[0194] Equation (*) depends on being able to calculate values for p(n.sup.A,n.sup.C|n.sub.α,i) and P(n.sup.A,n.sup.C|n.sub.X,i). These values may be calculated by assuming that the allele pattern (n.sup.A,n.sup.C) is drawn i.i.d according to the allele frequencies for its letters at locus i. An illustrative example is given here. Calculate P((2,1)|n.sub.7=3) under the assumption that the allele frequency for A is 60%, and the minor allele frequency for C is 40%. (As an aside, note that P((2,1)|n.sub.7=2)=0, since in this case the pair must sum to 2.) This probability is given by

[00042]$P(\left(2,1\right).Math.\text{?}$$\text{?}\text{indicates text missing or illegible when filed}$

The general equation is

[00043]$P\left(n^A,n^C.Math.n_j,i\right)=\left(\begin{array}{c}n\\ n^A\end{array}\right){\left(1-p_{\mathrm{ij}}\right)}^{n^A}{\left(p_{\mathrm{ij}}\right)}^{n^C}$

Where p.sub.ij is the minor allele frequency at locus i of chromosome j.

Allele Distribution Model Incorporating Parental Genotypes

[0195] Equation (*) depends on being able to calculate values for p(n.sup.A,n.sup.C|n.sub.j,h,T.sub.P,ijT.sub.M,ij) which are listed in Table 7. In a real situation, LDO will be known in either one of the parents, and the table would need to be augmented. If LDO are known in both parents, one can use the model described in the Allele Distribution Model without Parents section.

Population Frequency for Parental Truth

[0196] Equation (*) depends on being able to calculate p(T.sub.FAJT.sub.MAJ). The probabilities of the combinations of parental genotypes can be calculated based on the population frequencies. For example, P(AA,AA)=P(A).sup.4, and P(AC,AC)=P.sub.heteroz.sup.2 where P.sub.heteroz=2P(A)P(C) is the probability of a diploid sample to be heterozygous at one locus i.

Error Model

[0197] Equation (*) depends on being able to calculate values for P(t.sup.A|Q,n.sup.4) and P(t.sup.C|Q,n.sup.C). For this an error model is needed. One may use the following error model:

[00044]$\text{?}$$\text{?}\text{indicates text missing or illegible when filed}$

[0198] This error model is used elsewhere in this disclosure, and the four cases mentioned above are described there. The computational considerations of carrying out the MAP estimation mathematics can be carried out by brute-force are also described in the same section.

Computational Complexity Estimation

[0199] Rewrite the equation (*) as follows,

[00045]${\hat{n}}_j=\underset{n_j\in \left\{1,2,3\right\}}{\mathrm{argmax}}\int f\left(Q\right)\left(\begin{array}{c}P\left(n_j\right)\underset{i}{.Math.}\underset{n^A+n^C=n_j}{.Math.}P\left(n^A,n^C.Math.n_j,i\right)\\ P\left(t_{i,j}^A.Math.Q,n^A\right)P\left(t_{i,j}^C.Math.Q,n^C\right)\end{array}\right)\times \left(\begin{array}{c}\underset{k\ne j}{.Math.}\underset{n_k\in \left\{1,2,3\right\}}{.Math.}P\left(n_k\right)\underset{i}{.Math.}\underset{n^A+n^C=n_k}{.Math.}\\ P\left(n^A,n^C.Math.n_k,i\right)P\left(t_{i,k}^A.Math.Q,n^A\right)P\left(t_{i,k}^C.Math.Q,n^C\right)\end{array}\right)\mathrm{dQ}\text{(*})$

Let the computation time for P(n.sup.A,n.sup.C|n.sub.j,i) be t.sub.x, that for P(t.sub.i,j.sup.A|Q,n.sup.A) or P(t.sub.i,j.sup.C|Q,n.sup.C) be t.sub.y, Note that P(n.sup.A,n.sup.C|n.sub.j,i) may be pre-computed, since their values don't vary from experiment to experiment. For the discussion here, call a complete 23-chromosome aneuploidy screen an “experiment”. Computation of Π.sub.i Σ.sub.nA+nC=njP(n.sup.A,n.sup.C|n.sub.j,i)P(t.sub.i,j.sup.A|Q,n.sup.A)P(t.sub.i,j.sup.C|Q,n.sup.C) for 23 chromosomes takes [0200] if n.sub.j=1, (2+t.sub.x+2*t.sub.y)*2N*m [0201] if n.sub.j=2, (2+t.sub.x+2*t.sub.y)*3N*m [0202] if n.sub.j=3, (2+t.sub.x+2*t.sub.y)*4N*m
The unit of time here is the time for a multiplication or an addition. In total, it takes (2+t.sub.x+2*t.sub.y)*9N*m

[0203] Once these building blocks are computed, the overall integral may be calculated, which takes time on the order of (2+t.sub.x+2*t.sub.y)*9N*m*q. In the end, it takes 2*m comparisons to determine the best estimate for n.sub.j. Therefore, overall the computational complexity is O(N*m*q).

[0204] What follows is a re-derivation of the original derivation, with a slight difference in emphasis in order to elucidate the programming of the math. Note that the variable D below is not really a variable. It is always a constant set to the value of the data set actually in question, so it does not introduce another array dimension when representing in MATLAB. However, the variables D.sub.j do introduce an array dimension, due to the presence of the index j.

[00046]$\text{?}$$\text{?}\text{indicates text missing or illegible when filed}$

F Qualitative Chromosome Copy Number Calling

[0205] One way to determine the number of copies of a chromosome segment in the genome of a target individual by incorporating the genetic data of the target individual and related individual(s) into a hypothesis, and calculating the most likely hypothesis is described here. In one embodiment of the invention, the aneuploidy calling method may be modified to use purely qualitative data. There are many approaches to solving this problem, and several of them are presented here. It should be obvious to one skilled in the art how to use other methods to accomplish the same end, and these will not change the essence of the disclosure.

Notation for Qualitative CNC

[0206] 1. N is the total number of SNPs on the chromosome.

[0207] 2. n is the chromosome copy number.

[0208] 3. n.sup.M is the number of copies supplied to the embryo by the mother: 0, 1, or 2.

[0209] 4. n.sup.F is the number of copies supplied to the embryo by the father: 0, 1, or 2.

[0210] 5. p.sub.d is the dropout rate, and f(p.sub.d) is a prior on this rate.

[0211] 6. p.sub.a is dropin rate, and f(p.sub.a) is a prior on this rate.

[0212] 7. c is the cutoff threshold for no-calls.

[0213] 8. D=(x.sub.k,y.sub.k) is the platform response on channels X and Y for SNP k.

[0214] 9. D(c)={G(x.sub.k,y.sub.k);c}={ĝ.sub.k.sup.(c)} is the set of genotype calls on the chromosome. Note that the genotype calls depend on the no-call cutoff threshold c.

[0215] 10. ĝ.sub.k.sup.(c) is the genotype call on the k-th SNP (as opposed to the true value): one of AA, AB, BB, or NC (no-call).

[0216] 11. Given a genotype call ĝ at SNP k, the variables (ĝ.sub.x, ĝ.sub.y) are indicator variables (1 or 0), indicating whether the genotype g implies that channel X or Y has “lit up”. Formally, ĝ.sub.x=1 just in case ĝ contains the allele A, and ĝ.sub.Y=1 just in case contains the allele B.

[0217] 12. M={g.sub.k.sup.M} is the known true sequence of genotype calls on the mother. g.sup.M refers to the genotype value at some particular locus.

[0218] 13. F={g.sub.k.sup.F} is the known true sequence of genotype calls on the father. g.sup.F refers to the genotype value at some particular locus.

[0219] 14. n.sup.A,n.sup.B are the true number of copies of A and B on the embryo (implicitly at locus k), respectively. Values must be in {0,1,2,3,4}.

[0220] 15. c.sub.M.sup.A,c.sub.M.sup.B are the number of A alleles and B alleles respectively supplied by the mother to the embryo (implicitly at locus k). The values must be in {0, 1, 2}, and must not sum to more than 2. Similarly, c.sub.F.sup.A,c.sub.F.sup.B are the number of A alleles and B alleles respectively supplied by the father to the embryo (implicitly at locus k). Altogether, these four values exactly determine the true genotype of the embryo. For example, if the values were (1,0) and (1,1), then the embryo would have type AAB.

Solution 1: Integrate Over Dropout and Dropin Rates.

[0221] In the embodiment of the invention described here, the solution applies to just a single chromosome. In reality, there is loose coupling among all chromosomes to help decide on dropout rate p.sub.d, but the math is presented here for just a single chromosome. It should be obvious to one skilled in the art how one could perform this integral over fewer, more, or different parameters that vary from one experiment to another. It should also be obvious to one skilled in the art how to apply this method to handle multiple chromosomes at a time, while integrating over ADO and ADI. Further details are given in Solution 3B below.

[00047]$P\left(n.Math.D\left(c\right),M,F\right)=\text{?}$$\text{?}\text{indicates text missing or illegible when filed}$

The derivation other is the same, except applied to channel Y.

[00048]$P\left(\begin{array}{c}{n}^A,n^B.Math.n^M,\\ n^F,g^M,g^F,\end{array}\right)=\underset{\begin{array}{c}{c}_M^A+c_F^A=n^A\\ c_M^B+c_F^B=n^B\end{array}}{.Math.}P\left(\begin{array}{c}{c}_M^A,c_M^B.Math.\\ n^M,g^M\end{array}\right)P\left(\begin{array}{c}{c}_F^A,c_F^B.Math.\\ n^F,g^F\end{array}\right)$$P\left(c_M^A,C_M^B.Math.n^M,g^M\right)=\left(c_M^A+c_M^B=n^M\right)\{\begin{array}{c}(c_M^B=0,g^M=\mathrm{AA}\\ \left(c_M^A=0\right),g^M=\mathrm{BB}\\ \frac{1}{n^M+1},g^M=\mathrm{AB}\end{array}$

The other derivation is the same, except applied to the father.

Solution 2: Use ML to Estimate Optimal Cutoff Threshold c

Solution 2, Variation A

[0222] [00049]$\hat{c}=\underset{c\in (0,a]}{\mathrm{argmax}}P\left(D\left(c\right).Math.M,F\right)$$P\left(n\right)=\underset{\left(n^M,n^F\right),\in n}{.Math.}P\left(n^M,n^F.Math.D\left(\hat{c}\right),M,F\right)$

[0223] In this embodiment, one first uses the ML estimation to get the best estimate of the cutoff threshold based on the data, and then use this c to do the standard Bayesian inference as in solution 1. Note that, as written, the estimate of c would still involve integrating over all dropout and dropin rates. However, since it is known that the dropout and dropin parameters tend to peak sharply in probability when they are “tuned” to their proper values with respect to c, one may save computation time by doing the following instead:

Solution 2, Variation B

[0224] [00050]$\text{?}$$\text{?}\text{indicates text missing or illegible when filed}$

[0225] In this embodiment, it is not necessary to integrate a second time over the dropout and dropin parameters. The equation goes over all possible triples in the first line. In the second line, it just uses the optimal triple to perform the inference calculation.

Solution 3: Combining Data Across Chromosomes

[0226] The data across different chromosomes is conditionally independent given the cutoff and dropout/dropin parameters, so one reason to process them together is to get better resolution on the cutoff and dropout/dropin parameters, assuming that these are actually constant across all chromosomes (and there is good scientific reason to believe that they are roughly constant). In one embodiment of the invention, given this observation, it is possible to use a simple modification of the methods in solution 3 above. Rather than independently estimating the cutoff and dropout/dropin parameters on each chromosome, it is possible to estimate them once using all the chromosomes.

Notation

[0227] Since data from all chromosomes is being combined, use the subscript j to denote the j-th chromosome. For example, D.sub.j(c) is the genotype data on chromosome j using c as the no-call threshold. Similarly, M.sub.j,F.sub.j are the genotype data on the parents on chromosome j.

Solution 3, Variation A: Use all Data to Estimate Cutoff Dropout/Dropin

[0228] [00051]$\text{?}$$\text{?}\text{indicates text missing or illegible when filed}$

Solution 3, Variation B:

[0229] Theoretically, this is the optimal estimate for the copy number on chromosome j.

[00052]${\hat{n}}_j=\underset{n}{\mathrm{argmax}}\underset{\left(n^M,n^F\right)\in n}{.Math.}\int \int f\left(p_d\right)f\left(p_a\right)P\left(D_j\left(\hat{c}\right)\right).Math.n^M,n^F,M_j,F_j,p_d,p_a)\text{?}$$\text{?}\text{indicates text missing or illegible when filed}$

Estimating Dropout/Dropin Rates from Known Samples

[0230] For the sake of thoroughness, a brief discussion of dropout and dropin rates is given here. Since dropout and dropin rates are so important for the algorithm, it may be beneficial to analyze data with a known truth model to find out what the true dropout/dropin rates are. Note that there is no single tree dropout rate: it is a function of the cutoff threshold. That said, if highly reliable genomic data exists that can be used as a truth model, then it is possible to plot the dropout/dropin rates of MDA experiments as a function of the cutoff-threshold. Here a maximum likelihood estimation is used.

[00053]$\hat{c},{\hat{P}}_d,{\hat{P}}_a=\text{?}$$\text{?}\text{indicates text missing or illegible when filed}$

In the above equation, ĝ.sub.jk.sup.(c), is the genotype call on SNP k of chromosome j, using c as the cutoff threshold, while g.sub.jk, is the true genotype as determined from a genomic sample. The above equation returns the most likely triple of cutoff, dropout, and dropin. It should be obvious to one skilled in the art how one can implement this technique without parent information using prior probabilities associated with the genotypes of each of the SNPs of the target cell that will not undermine the validity of the work, and this will not change the essence of the invention.

G Bayesian Plus Sperm Method

[0231] Another way to determine the number of copies of a chromosome segment in the genome of a target individual is described here. In one embodiment of the invention, the genetic data of a sperm from the father and crossover maps can be used to enhance the methods described herein. Throughout this description, it is assumed that there is a chromosome of interest, and all notation is with respect to that chromosome. It is also assumed that there is a fixed cutoff threshold for genotyping. Previous comments about the impact of cutoff threshold choice apply, but will not be made explicit here. In order to best phase the embryonic information, one should combine data from all blastomeres on multiple embryos simultaneously. Here, for ease of explication, it is assumed that there is just one embryo with no additional blastomeres. However, the techniques mentioned in various other sections regarding the use of multiple blastomeres for allele-calling translate in a straightforward manner here.

Notation

[0232] 1. n is the chromosome copy number.

[0233] 2. n.sup.M is the number of copies supplied to the embryo by the mother: 0, 1, or 2.

[0234] 3. n.sup.F is the number of copies supplied to the embryo by the father: 0, 1, or 2.

[0235] 4. p.sub.d is the dropout rate, and f(p.sub.d) is a prior on this rate.

[0236] 5. p.sub.a is the dropin rate, and f(p.sub.a) is a prior on this rate.

[0237] 6. D={ĝ.sub.k} is the set of genotype measurements on the chromosome of the embryo. ĝ.sub.k is the genotype call on the k-th SNP (as opposed to the true value): one of AA, AB, BB, or NC (no-call). Note that the embryo may be aneuploid, in which case the true genotype at a SNP may be, for example, AAB, or even AAAB, but the genotype measurements will always be one of the four listed. (Note: elsewhere in this disclosure ‘B’ has been used to indicate a heterozygous locus. That is not the sense in which it is being used here. Here ‘A’ and ‘B’ are used to denote the two possible allele values that could occur at a given SNP.)

[0238] 7. M={g.sub.k.sup.M} is the known true sequence of genotypes on the mother. g.sub.k.sup.m is the genotype value at the k-th SNP.

[0239] 8. F={g.sub.k.sup.F} is the known true sequence of genotypes on the father. g.sub.k.sup.F is the genotype value at the k-th SNP.

[0240] 9. S={ĝ.sub.k.sup.S} is the set of genotype measurements on a sperm from the father. ĝ.sub.k.sup.S is the genotype call at the k-th SNP.

[0241] 10. (m.sub.1,m.sub.2) is the true but unknown ordered pair of phased haplotype information on the mother. m.sub.1k is the allele value at SNP k of the first haploid sequence. m.sub.2k is the allele value at SNP k of the second haploid sequence. (m.sub.1,m.sub.2) ϵM is used to indicate the set of phased pairs (m.sub.1,m.sub.2) that are consistent with the known genotype M. Similarly, (m.sub.1,m.sub.2) ϵg.sub.k.sup.M is used to indicate the set of phased pairs that are consistent with the known genotype of the mother at SNP k.

[0242] 11. (f.sub.1,f.sub.2) is the true but unknown ordered pair of phased haplotype information on the father. f.sub.1k is the allele value at SNP k of the first haploid sequence. f.sub.2k is the allele value at SNP k of the second haploid sequence. (f.sub.1,f.sub.2) ϵF is used to indicate the set of phased pairs (f.sub.1,f.sub.2) that are consistent with the known genotype F. Similarly, (f.sub.1,f.sub.2) ϵg.sub.k.sup.F is used to indicate the set of phased pairs that are consistent with the known genotype of the father at SNP k.

[0243] 12. s.sub.1 is the true but unknown phased haplotype information on the measured sperm from the father. s.sub.1k is the allele value at SNP k of this haploid sequence. It can be guaranteed that this sperm is euploid by measuring several sperm and selecting one that is euploid.

[0244] 13. χ.sup.M={ϕ.sub.1, . . . , ϕ.sub.nM} is the multiset of crossover maps that resulted in maternal contribution to the embryo on this chromosome. Similarly, χ.sup.F={θ.sub.1, . . . , θ.sub.nF} is the multiset of crossover maps that results in paternal contribution to the embryo on this chromosome. Here the possibility that the chromosome may be aneuploid is explicitly modeled. Each parent can contribute zero, one, or two copies of the chromosome to the embryo. If the chromosome is an autosome, then euploidy is the case in which each parent contributes exactly one copy, i.e., χ.sup.M={ϕ.sub.1} and χ.sup.F={θ.sub.1}. But euploidy is only one of the 3×3=9 possible cases. The remaining eight are all different kinds of aneuploidy. For example, in the case of maternal trisomy resulting from an M2 copy error, one would have χ.sup.M={ϕ.sub.1ϕ.sub.1} and χ.sup.F={θ.sub.1}. In the case of maternal trisomy resulting from an M1 copy error, one would have χ.sup.M−{ϕ.sub.1,ϕ.sub.2} and χ.sup.F={θ.sub.1}. (χ.sup.M, χ.sup.F)ϵn will be used to indicate the set of sub-hypothesis pairs (χ.sup.N, χ.sup.F) that are consistent with the copy number n. χ.sub.k.sup.M will be used to denote {ϕ.sub.1,k, . . . , ϕ.sub.nM.sub.k}, the multiset of crossover map values restricted to the k-th SNP, and similarly for χ.sup.F.Math.χ.sub.k.sup.M(m.sub.1,m.sub.2) is used to mean the multiset of allele values {ϕ.sub.1,k(m.sub.1,m.sub.2), . . . , ϕ.sub.nM.sub.k(m.sub.1,m.sub.2)}={m.sub.ϕi,k, . . . , m.sub.ϕnM,k}. Keep in mind that ϕ.sub.l,kϵ{1,2}.

[0245] 14. ψ is the crossover map that resulted in the measured sperm from the father. Thus s.sub.1=ψ(f.sub.1,f.sub.2). Note that it is not necessary to consider a crossover multiset because it is assumed that the measured sperm is euploid. wk will be used to denote the value of this crossover map at the k-th SNP.

[0246] 15. Keeping in mind the previous two definitions, let {e.sub.1.sup.M, . . . , e.sub.n.sup.MM} be the multiset of true but unknown haploid sequences contributed to the embryo by the mother at this chromosome. Specifically, e.sub.1.sup.M=ϕ.sub.1(m.sub.1,m.sub.2), where ϕ.sub.l is the l-th element of the multiset χ.sup.m, and e.sub.1k.sup.M is the allele value at the k-th snp. Similarly, let {e.sub.l.sup.F, . . . , e.sub.nF.sup.F} be the multiset of true but unknown haploid sequences contributed to the embryo by the father at this chromosome. Then e.sub.l.sup.F−θ.sub.l(f.sub.1,f.sub.2), where θ.sub.i is the l-th element of the multiset χ.sup.F, and f.sub.1k.sup.M is the allele value at the k-th SNP. Also, {e.sub.l.sup.M, . . . , e.sub.nM.sup.M}=χ.sup.M(m.sub.1m.sub.2), and {e.sub.l.sup.F, . . . , e.sub.nF.sup.F}=χ.sup.F(f.sub.1,f.sub.2) may be written.

[0247] 16. P(ĝ.sub.k|χ.sub.k.sup.M(m.sub.1,m.sub.2), X.sub.k.sup.F(f.sub.1,f.sub.2),p.sub.d,p.sub.c) denotes the probability of the genotype measurement on the embryo at SNP k given a hypothesized true underlying genotype on the embryo and given hypothesized underlying dropout and dropin rates. Note that χ.sub.k.sup.M(m.sub.1,m.sub.2) and χ.sub.k.sup.F(f.sub.1,f.sub.2) are both multisets, so are capable of expressing aneuploid genotypes. For example, χ.sub.k.sup.M(m.sub.1,m.sub.2)={A,A} and χ.sub.k.sup.F(f.sub.1,f.sub.2)={B} expresses the maternal trisomic genotype AAB.

[0248] Note that in this method, the measurements on the mother and father are treated as known truth, while in other places in this disclosure they are treated simply as measurements. Since the measurements on the parents are very precise, treating them as though they are known truth is a reasonable approximation to reality. They are treated as known truth here in order to demonstrate how such an assumption is handled, although it should be clear to one skilled in the art how the more precise method, used elsewhere in the patent, could equally well be used.

Solution

[0249] [00054]$\hat{n}=\underset{n}{\mathrm{argmax}}P\left(n,D,M,F,S\right)$$\begin{array}{c}P\left(n,D,M,F,S\right)=\underset{\left(x^U,x^F\right)\in n}{.Math.}\underset{\psi }{.Math.}P\left({\chi }^M,{\chi }^F,\psi ,D,M,F,S\right)\\ =\underset{\left({\chi }^M,{\chi }^F\right)\in n}{.Math.}P\left({\chi }^M\right)P\left({\chi }^F\right)\underset{\psi }{.Math.}P\left(\psi \right)\int f\left(p_d\right)\int \\ f\left(p_d\right)\underset{k}{.Math.}P\left({\hat{g}}_k,g_k^M,g_k^F,{\hat{g}}_k^S|{\chi }_k^M,{\chi }_k^F,{\psi }_k,p_d,p_a\right)\\ =\underset{\left({\chi }^M,{\chi }^F\right)\in n}{\stackrel{{\mathrm{dp}}_d{\mathrm{dp}}_c}{.Math.}}P\left({\chi }^M\right)P\left({\chi }^F\right)\underset{\psi }{.Math.}P\left(\psi \right)\int f\left(p_d\right)\int f\left(p_c\right)\times \\ \underset{k}{.Math.}\underset{\left(f_1,f_2\right)=0_k^F}{.Math.}P\left(f_1\right)P\left(f_2\right)P\left(\text{?}|{\psi }_k\left(f_1,f_2\right),p_d,p_a\right)\\ \underset{\left(m_1,m_2\right)=0_k^M}{.Math.}P\left(m_1\right)P\left(m_2\right)P({\hat{g}}_k|{\chi }_k^M\left(m_1,m_2\right),{\chi }_k^F\left(f_1,f_2\right),\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$