A Bayesian Decision-Theoretic Optimization Model for Personalized Timing of Non-Invasive Prenatal Testing Based on Maternal BMI

Abstract

Non invasive prenatal testing, NIPT, is widely used for fetal aneuploidy screening, but its clinical utility depends on gestational timing and maternal characteristics. Low fetal fraction can lead to unreportable tests and increased false negative risk, while GC-content-related sequencing bias may contribute to both false positive and false negative findings. We propose a Bayesian decision-theoretic optimization framework to recommend personalized NIPT timing across maternal body mass index (BMI) strata, explicitly incorporating test credibility and detection errors. We performed a retrospective analysis of de-identified NIPT records from a hospital in Guangdong Province, China, covering 1 January 2023 to 18 February 2024, including 1082 male fetus tests. Y chromosome concentration was used as a proxy for test reportability, with a 4 percent reporting threshold. Detection state proportions were empirically summarized from clinical reference information, with false positives at 10.35 percent and false negatives at 2.77 percent. A logistic regression model quantified the probability of obtaining a reportable result as a function of gestational week, maternal age, height, and weight, and the estimated probabilities were used to parameterize the Bayesian risk model. The optimized BMI-stratified schedule produced six BMI groups with recommended testing weeks ranging from 11 to 16, and the overall expected risk converged to 0.531. These results indicate a nonlinear BMI–timing relationship and suggest that a single universal testing week is suboptimal. The proposed framework provides quantitative decision support for BMI-stratified NIPT scheduling in clinical practice.

1. Introduction

Chromosomal defects are major causes of infant mortality and disability, including trisomy 21, trisomy 18, trisomy 13, triploidy, and sex chromosome aneuploidies. Non-invasive prenatal testing (NIPT) is widely used for prenatal screening, yet its performance may vary across individuals and gestational timing. It provides a highly accurate and safe method for screening fetal chromosomal abnormalities by analyzing placental-derived cell-free DNA in maternal peripheral blood [1,2]. Its core advantage lies in avoiding the miscarriage risk associated with traditional invasive diagnostics (such as amniocentesis), while enabling testing as early as 10 weeks gestation [2]. However, NIPT is fundamentally a screening technology, and positive results still require invasive diagnosis for definitive confirmation [2,3].

Currently, NIPT’s clinical applications have rapidly expanded from initial screening for common trisomies (Trisomy 21, 18, 13) to include sex chromosome aneuploidies (SCAs), rare autosomal trisomies (RATs), copy number variations (CNVs), and even single-gene disorders [2,4,5]. This “expanded NIPT” or “whole-genome NIPT” aims to provide more comprehensive fetal genetic information [5,6,7]. A modeling study indicates that as a first-line screening test, whole-genome NIPT detects 20% more chromosomal abnormalities than traditional first trimester combined screening while reducing the need for invasive procedures by 45% [6]. However, expanded testing also presents performance challenges. For instance, the positive predictive value (PPV) for sex chromosome abnormalities is significantly lower than for common trisomies, with PPV for Turner syndrome potentially as low as 20% [8]. Studies in high-risk twin pregnancies indicate that expanded NIPT demonstrates high detection rates for common trisomies but limited detection for sex chromosome abnormalities and microdeletions/duplications [7]. A meta-analysis of over 750,000 NIPT results further reveals substantial variation in false positive rates across different testing indications, with a false positive to false negative ratio as high as 27:1 [9].

The accuracy of NIPT results is influenced by a complex interplay of biological and technical factors. Placental mosaicism is a significant cause of both false negatives and false positives [6,10]. Case reports indicate that placental trisomy 21 mosaicism rates vary significantly across different regions (13% to 88%), directly contributing to false negative results [11]. Maternal genomic background variation, particularly maternal-derived CNVs, represents another major factor causing false positives, especially prominent during screening for syndromes like 22q11.2 deletion syndrome [12]. Emerging bioinformatics methods successfully increased the PPV for detecting 22q11.2 deletions from 87% to 94% by distinguishing whether CNVs originated from the fetus, the mother, or were shared between both [12]. Furthermore, sequencing depth and choice of analysis software critically impact detection performance. Higher sequencing depths (e.g., 0.4X vs. 0.15X) significantly improve CNV detection rates and PPV [13,14]. Alternative technologies like digital PCR (dPCR) serve as vital complements to next-generation sequencing (NGS) due to their high precision, low cost, and adaptability to resource-limited settings [15,16]. Meanwhile, cell-based NIPT (cbNIPT) offers more comprehensive fetal genomic information, demonstrating potential as a supplement or alternative to cfDNA-NIPT [17].

The rapid advancement and diverse application models of NIPT technologies have sparked profound ethical, societal, and practical discussions and challenges. A core concern is that the “routinization” of testing may undermine reproductive autonomy and generate social pressure [18,19,20]. However, empirical research indicates that when provided with adequate information and social support, most women can make informed choices based on their personal values and do not perceive undergoing NIPT as a moral obligation to be a “good mother [19,21]”. Some argue that “routinization” itself is not inherently harmful; its ethical acceptability depends on whether specific implementation safeguards free and informed decision-making [22]. Academic debate remains intense regarding whether NIPT coverage should be restricted based on a “severity” threshold for conditions. Opponents contend that defining “severity” is subjective and arbitrary, and that forced restrictions infringe on reproductive autonomy while potentially exacerbating discrimination against certain disabilities [23]. A more appropriate approach involves enhancing clinical counseling and informed consent processes to support expectant parents in making personalized decisions aligned with their values [23,24]. Interviews with healthcare professionals reveal that balancing standardized information delivery with respecting patient-led decision-making remains a critical challenge in current practice [24]. “Incidental findings” from whole-genome NIPT—such as rare trisomies—often surprise and deeply distress expectant mothers, particularly when their clinical significance remains unclear. This underscores the urgent need to optimize prenatal genetic counseling [25,26].

In clinical validation, large-scale, long-term follow-up data must be accumulated for expanded testing items [15,17,27]—particularly RATs, CNVs, and in special pregnancies like vanishing twin syndrome—to clarify their clinical utility, positive predictive value, and association with pregnancy outcomes [10,28,29]. At the health policy and economics level, formal cost–benefit analyses encompassing long-term health consequences, healthcare resource utilization, and psychosocial impacts are needed to provide evidence for optimizing screening strategies across varying resource settings [21,28,30]. At the ethics and practice level, there is an urgent need to establish patient-centered, standardized frameworks for decision support and genetic counseling. This includes enhancing training for healthcare professionals in non-directive counseling, shared decision-making, and the use of inclusive language (particularly regarding fetal sex prediction) [24,31]. The ultimate goal is to ensure that NIPT, a powerful technology, maximizes its medical value while effectively safeguarding and promoting individual reproductive autonomy, and achieving more equitable and responsible clinical application globally [2,18,23,30].

The aim of this study is as follows. In this study, the dependent variable in the statistical analysis is fetal Y chromosome concentration in maternal plasma, used as a proxy for fetal fraction and test reportability. The independent variables are maternal characteristics available in routine clinical records, including gestational week at testing, maternal age, height, and weight. Building on these associations, we further develop a Bayesian decision-theoretic optimization model to recommend personalized NIPT timing across BMI ranges, aiming to minimize population-level expected risk while maintaining clinical efficiency.

We built upon the critical prior finding that NIPT detection efficacy is influenced by individual maternal variations (particularly BMI), achieving a paradigm shift from “phenomenon description” to “protocol design” [3,4,10,15,32]. Specifically, we moved beyond the qualitative conclusion that “high BMI should delay testing” by constructing a mathematical optimization model grounded in Bayesian decision theory. This transformed the clinical problem into a quantifiable, solvable risk minimization problem [8,14,25,30,33,34,35]. We innovatively integrated multidisciplinary approaches: using principal component analysis and logistic regression to establish quantitative relationships between physiological indicators and fetal DNA concentration; employing Monte Carlo simulation to address data distribution imbalances; and applying dynamic programming and simulated annealing algorithms to determine optimal grouping and testing timepoints. Ultimately, we not only derived specific optimal gestational weeks for testing across different BMI ranges but also revealed the underlying nonlinear “risk–benefit tradeoff” mechanism. Crucially, we explained the decision threshold where extremely high BMI groups should undergo earlier testing. This work advances previous observational findings into a clinically actionable personalized decision support system.

2. Optimization Model Based on Bayesian Decision Theory

First, we assume that Y chromosome concentration positively correlates with fetal fraction (FF) in maternal blood. We neglect coupling effects between indicators influenced by Y chromosome concentration and assume the additional cost of delayed testing weeks is constant.

2.1. Notation

The following symbols in

Table 1. Notation used in the Bayesian decision-theoretic model.

Symbols	Explanations
$w$	Gestational week.
$d$	Fetal disease status, a 0–1 variable.
$\Psi (w,d)$	Treatment cost function when fetal condition is d at w weeks of pregnancy.
$k$	The detection results are categorized into four types: TN/TP/FN/FP.
$j$	A 0–1 variable indicating whether the NIPT test result is credible.
$L(w,k)$	Loss under detection state k at week w.
$\Phi (w,j,k)$	Treatment risk function when fetal condition is k at w weeks gestation and test result reliability is j.
$R\left(w\right)$	Expected risk for the pregnant woman at week w of pregnancy.
${\Omega }_1$	Sample space for chromosomal abnormalities
${\Omega }_2$	Sample space for Y chromosome concentration meeting standards
$P({\Omega }_1=k)$	Probability function for fetal chromosomal abnormalities
$P({\Omega }_2=k)$	Probability function for fetal Y chromosome concentration meeting standards at w weeks of pregnancy
$m$	Total number of BMI groups for pregnant women
$G_i$	Group to which the pregnant woman with BMI value i belongs

are used throughout the text and form the core of the formula.

2.2. Determining the Expected Risk Function for Treatment Using a Bayesian Decision Model

Consulting relevant literature, we selected a Bayesian decision model to determine the optimal NIPT timing for pregnant women across different BMI categories. Its core objective is to minimize classification error rates by integrating prior probabilities with observational data, calculating posterior probabilities via Bayes’ theorem, and making optimal decisions based on this criterion [36].

We know that in actual NIPT, early detection of unhealthy fetuses can reduce risks associated with a shortened treatment window, and early detection carries lower risks; risks increase with detection at later stages. In summary, we can conclude that the greater the gestational age at which a pregnant woman undergoes NIPT, the higher the risk of fetal disease. Therefore, we employed a standard sigmoid function for translation and scaling to define a treatment cost function suitable for Bayesian decision-making, with the specific expression as follows [36].

$$\Psi (w,d)=d{\displaystyle \frac{k_1}{1+e^{-0.3(w-20)}}}$$

The translation and scaling coefficients in the exponential form represent the gestational week data processing. Here, $\mathrm{d}$ is a 0–1 variable indicating fetal condition. A treatment window risk exists only when $\mathrm{d} = 1$; when $\mathrm{d} =$0, the cost is 0. This is because $\mathrm{d} =$0 indicates a healthy fetus and yields zero treatment cost at any gestational week. In contrast, $\mathrm{d} = 1$ indicates a fetus with chromosomal abnormality, and the treatment cost increases with gestational week to reflect the reduced clinical decision window at later weeks. The parameter ${\mathrm{k}}_1>0$ is a scaling coefficient controlling the magnitude of the treatment cost. The above equation represents the treatment cost function $\mathrm{w}$ for different fetal health conditions at different gestational weeks.

However, the derivation of the expected risk function above does not account for the effects of false negatives and false positives. Mechanistic analysis indicates that in non-invasive prenatal testing, both excessively high and low GC content reduce sequencing efficiency, altering read counts and thereby introducing bias in Z-score calculations. Therefore, GC content correction can mitigate false-positive and false-negative results. Using de-identified real-world NIPT records provided by a hospital in Guangdong Province, China, we retrospectively summarized the four detection states for tests performed from 1 January 2023 to 18 February 2024. The data are not publicly available due to privacy and ethical restrictions; access may be granted upon reasonable request as stated in the Data Availability Statement. The detection states were defined based on the available reference information in the clinical records (e.g., confirmatory diagnosis or pregnancy outcome follow-up). Statistical analysis of the four detection states yields the proportions shown in

Table 2. Detection Error State Proportion.

Detection State	True Positive	False Positive	True Negative	False Negative
Percentage	0.32%	10.35%	86.55%	2.77%

.

That is, the results of chromosomal abnormality detection are not entirely accurate, with false positives and false negatives occurring. The proportion of false positives and false negatives reaches 13.12%, which introduces some detection errors. Therefore, the risk function in the Bayesian decision is modified to update the expected risk function as follows:

$$L(w,k)=\left\{\begin{array}{c}0, k = 0, True negative\\ \Psi (w,1), k=1, True positive\\ 1, k = 2, False negative\\ 0, k = 3, False positive\end{array}\right.$$

In NIPT test results, the findings are only considered reliable when the Y chromosome concentration is greater than or equal to 4%. Therefore, a binary variable (0–1) is defined to determine the reliability of NIPT test results:

$$j=\left\{\begin{array}{c}1, Concentration meets standards; results are reliable\\ 0, Concentration did not meets standards; results are unreliable\end{array}\right.$$

This allows us to define the loss function under the state space ${\Omega }_1=\{0,1,2,3\}$ and solution space $\mathrm{w}=\{11,12,\dots ,30\}$. Accordingly, we define the treatment risk loss function conditional on gestational week $w$, test credibility $j$, and detection state $k\in {\Omega }_1$ as follows:

$$\Phi (w,j,k)=\left\{\begin{array}{c}L(w,k), j=1,k\in {\Omega }_1\\ min\left\{R\right(w+i\left)\right\}+\epsilon , j=0, k\in {\Omega }_1\end{array}\right.$$

When $\mathrm{j}=0$, the current test is not reportable (unreliable), and a repeat test is required; therefore, the loss does not depend on the specific detection state $\mathrm{k}$ and is represented by the same value for all $\mathrm{k}\in {\mathsf{\Omega }}_1$. Therefore, the following formula applies:

$$\Phi (w,j)=\bigcup _{k=0}^3\Phi (w,j,k)P({\Omega }_1=k)$$

When the test result is credible ($j=1$), the incurred loss depends on the detection state $k\in {\Omega }_1=\{0,1,2,3\}$ through the loss function $L(w,k)$. Therefore, the aggregated treatment risk at gestational week $w$ under credible results can be obtained by marginalizing over the detection states:

$$\Phi (w,1)=\bigcup _{k=0}^3\Phi (w,j,k)P({\Omega }_1=k)=\sum _{k=0}^{3}L(w,k)P({\Omega }_1=k)$$

When the test result is unreliable ($j=0$), the above risk function becomes inapplicable. Considering fetal health needs, the pregnant woman should opt for a subsequent NIPT test at a later time point. To minimize risk, an expected risk function is introduced, selecting the time point with the lowest risk value for the repeat NIPT test. Thus, the treatment risk function for the current scenario is constructed as

$$\Phi (w,0)=min\left\{R\right(w+i\left)\right\}+\epsilon$$

Here, $R(w+i)$ represents the expected risk at the subsequent week $i$, while $\epsilon$ represents the time, economic, and other losses incurred by the failure of this NIPT test. Considering the greater importance of fetal health, this value is set to be relatively small.

To quantify total losses across different NIPT timing points, we introduce an expected risk function describing the total expected risk under varying chromosomal conditions and detection reliability levels [36].

First, reviewing relevant literature, we note that pregnant women beyond 30 weeks gestation cannot undergo NIPT, implying fetal abnormalities are fixed [37]. Since NIPT is unavailable after 30 weeks, we consider only true positive and true negative scenarios. Combining the treatment cost function $L(w,k)$ and chromosomal abnormality probabilities $P({\Omega }_1=1)$, the expected risk function for NIPT at $w$ weeks under these conditions is

$$R(w)=\sum _{d=0}^1\Psi (w,d)P(d) (w\ge 30,w\in Z)$$

For pregnant women before 30 weeks gestation $(w=11,\dots ,29)$, the overall expected risk at week www is obtained by marginalizing over the test credibility indicator $\mathrm{j}\in \{0,1\}$ using the law of total expectation. Specifically, the credibility-specific expected loss is $\Phi (w,j)$, and it is weighted by the week-dependent credibility probability $P({\Omega }_2=j\left|w\right.)$. Thus, the specific expression is as follows:

$$R(w)=\sum _{j=0}^1\Phi (w,j)P({\Omega }_2=j\left|w\right.) (w=11,\dots ,29)$$

2.3. Establishing a Dual-Objective Optimization Model for Determining the Optimal Detection Timing

We first establish the objective function for the optimization model. Based on the definition of the optimal timing, minimizing the detection risk for pregnant women and minimizing the time cost of hospital testing are set as the optimization directions for the model. To minimize the detection risk for pregnant women, we first cluster pregnant women with similar expected risk function distributions based on their BMI values into groups, forming a total of $m$ groups according to the distribution of the NIPT expected risk function for pregnant women with different BMIs [37]. The group $n (n=1,\dots )$ encompasses $n_l$ pregnant women carrying male fetuses with different BMI values. To minimize the average risk for pregnant women undergoing NIPT, our objective function can be defined as follows: after determining the NIPT time for each BMI group, minimize the risk value of the shortened therapeutic window across all BMI categories. This is specifically expressed as

$$min\sum _{j=1}^n{R}_i\left(w_j\right)$$

where $R_i\left(w_j\right)$ represents the expected risk for pregnant women in the $j$, $n$ group at $w$ weeks.

Consider minimizing the time cost of hospital testing. While finer BMI grouping reduces individual pregnant women’s perceived risk, different measurement timepoints impose significant time costs on hospitals. To enhance NIPT efficiency, hospital BMI grouping should minimize the number of categories, leading to the following objective:

$$min m$$

Next, constraints were imposed on the model based on practical requirements and model properties, resulting in the following three conditions: uniqueness, monotonic continuity of BMI groupings, and consistent testing timepoints within the same group [38]. Given the uniqueness of BMI value assignment—where each BMI value belongs to only one grouping category—the following restrictions apply:

$$G_i\in \{1, 2,\dots ,k\}$$

where $G_i$ represents the group containing the $i$-th BMI value. The maximum BMI value in Group 1 is less than the minimum BMI value in Group 2, and so on through Groups $k$.

Moreover, BMI grouping should satisfy monotonicity and be continuous:

$$G_i\le G_{i+1}$$

Combining this with actual operational processes, to enhance hospital testing efficiency, it is concluded that the optimal NIPT timing is consistent for pregnant women within the same BMI group, namely

$$w_i=w_{G_i}$$

Based on the above analysis, the optimized BMI grouping model that minimizes potential risks for pregnant women is established as follows:

$$min\sum _{j=1}^n{R}_i\left(w_j\right)$$

$$s.t.\left\{\begin{array}{c}{G}_i\in \{1, 2,\dots ,k\}\\ G_i\le G_{i+1}\\ w_i=w_{G_i}\end{array}\right.$$

3. Correlation Analysis

3.1. Identifying Key Indicators via Principal Component Analysis

To examine factors associated with fetal Y chromosome concentration in maternal plasma, we treated Y chromosome concentration as the outcome variable and considered 29 routinely recorded indicators as candidate predictors, excluding Y chromosome concentration itself. These indicators can be grouped into four domains: maternal demographics and anthropometrics, pregnancy and testing information, sequencing and alignment quality metrics, and GC-related and chromosome level metrics with clinical outcomes. Because these predictors are high-dimensional and partially correlated, directly fitting a multivariable regression using all variables may yield unstable estimates and hinder interpretation. We therefore used principal component analysis to reduce dimensionality and to guide the selection of a smaller set of informative predictors for subsequent regression modeling.

3.2. Further Analysis of the Relationship Between Indicators and Y Chromosome Concentration Using Logistic Regression Models

To further evaluate the impact of maternal and fetal indicators on Y chromosome concentration in male fetuses, we employed a logistic regression model to quantify how routine clinical covariates are associated with test reportability. Following common clinical practice, we treated a Y chromosome concentration of at least 4% as the minimum reporting threshold for sufficient fetal fraction. Accordingly, we modeled the probability that the Y chromosome concentration meets this threshold at gestational week w as $P({\Omega }_2=k)$, and examined its relationship with the selected indicators using logistic regression [38].

(1)

Establishing the Logistic Regression Model

Step 1: Let $k = 1, \dots , \mathrm{n}$ index the NIPT records. Define $j_k$ as a binary outcome indicating whether the fetal Y chromosome concentration meets the reporting threshold. We set $j_k=1$ if it meets the threshold and $j_k=0$ otherwise. Let $x_{k1},\dots ,x_{k4}$ denote the four predictors for record $k$, namely gestational week at testing, maternal age, maternal height, and maternal weight. The logistic regression linear predictor is defined as

$$f_k\left(x\right)={ \beta }_0+\sum _{i=1}^4{\beta }_i{x}_{\mathit{ki}}$$

where $x_{\mathit{ki}}$ represents the data value of the $i$-th physiological indicator for the male fetus of pregnant woman with ID $k$ and sample. $f_k\left(x\right)$ denotes the linear regression value of Y chromosome concentration for the male fetus of pregnant woman $k$ with ID. ${\beta }_i$ indicates the regression coefficient.

Step 2: Establish a continuous logistic regression function. Map the linear values calculated in the previous step to the (0, 1) interval using the logistic function, with the mapping defined as follows:

$$F_k=P(j_k=1/x_{k1}, \dots ,{ x}_{k4})={\displaystyle \frac{e^{f_k\left(x\right)}}{1+e^{f_k\left(x\right)}}}={\displaystyle \frac{1}{1+e^{-f_k\left(x\right)}}}$$

where $F_k$ denotes the logistic regression value for Y chromosome concentration. Its range is $(0, 1)$. The closer the value is to $1$, the higher the probability that logistic classification indicates Y chromosome concentration has reached the threshold $P({\Omega }_1=1)$. Conversely, the closer the value is to $0$, the higher the probability $P({\Omega }_1=0)$ that logistic classification indicates Y chromosome concentration has not reached the threshold.

(2)

Solving Logistic Regression Parameters via Maximum Likelihood Estimation

Step 1: Define decision output and probability density function. In the previous solution, we provided the 0–1 variable $j$ indicating whether Y chromosome concentration meets the standard. This serves as the decision result. Using maximum likelihood estimation to solve its probability density function, since $j$ follows a binomial distribution, its probability density function can be expressed as

$$P\left(j_k\right)={F_k}^{j_k}{(1-F_k)}^{1-j_k}$$

Step 2: Determine the likelihood function of the distribution.

Assuming the sample size is $n$, with each sample being independent and identically distributed, the likelihood function (joint probability) is

$$L=\prod _{K=1}^{n}P\left(j_k\right)={F_k}^{\sum _{k=1}^n{j}_k}{(1-F_k)}^{\sum _{k=1}^n{(1-j}_k)}$$

To simplify calculations, we take the logarithm of both sides of the likelihood function. The result is as follows:

$$ln(L)=\sum _{k=1}^n{j}_{k}ln\left(F_k\right)+\sum _{k=1}^n(1-j_k)ln(1-F_k)=\sum _{k=1}^n\left[j_{k}ln({\displaystyle \frac{F_k}{1-F_k}})+ln(1-F_k)\right]$$

Step 3: Solve for the estimated coefficients.

Take partial derivatives for coefficients ${\beta }_0$ and ${\beta }_{\mathrm{i}}$. When the partial derivative equals zero, the maximum point of the likelihood function is reached. Therefore, solving the simultaneous equations yields the estimated values for both coefficients.

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathsf{\partial }ln\left(L\right)}{\mathsf{\partial }{\beta }_0}}={\displaystyle \sum _{k=1}^n}\left[y_k-{\displaystyle \frac{e^{{\beta }_0+\sum _{i=1}^n{\beta }_i{x}_{\mathit{ki}}}}{1+e^{{\beta }_0+\sum _{i=1}^n{\beta }_i{x}_{\mathit{ki}}}}}\right]=0\\ {\displaystyle \frac{\mathsf{\partial }ln\left(L\right)}{\mathsf{\partial }{\beta }_i}}={\displaystyle \sum _{k=1}^n}{x}_{\mathit{ki}}\left[y_k-{\displaystyle \frac{e^{{\beta }_0+\sum _{i=1}^n{\beta }_i{x}_{\mathit{ki}}}}{1+e^{{\beta }_0+\sum _{i=1}^n{\beta }_i{x}_{\mathit{ki}}}}}\right]=0\end{array}\right.$$

4. Model Results

4.1. Correlation Analysis Results

Based on correlation analysis results, age, height, weight, gestational week at testing, and BMI were selected as initial indicators for PCA. The results of the PCA conducted using SPSS 27.0.1 software are shown in

Table 3. Principal Component Analysis Results.

	Age	Height	Weight	Gestational Age at Testing	BMI
Characteristic Value	2.11705	1.05837	0.97152	0.85157	0.00149
Eigenvalue Contribution Rate	42.341%	21.1674%	19.4304%	17.0314%	0.0298%
Cumulative Contribution Rate	42.341%	63.5084%	82.9388%	99.9702%	100.0000%

below:

The results indicate that age exhibits the highest characteristic value, meaning it exerts the greatest influence on Y chromosome concentration. BMI demonstrates the lowest characteristic value, with a contribution rate below 0.1%. Therefore, age, height, weight, and gestational week at testing are selected as the primary influencing factors in the analysis of their relationship with Y chromosome concentration.

4.2. Logistic Regression Results

Based on the above steps, the logistic regression parameters were solved using Python 3.11 programming, as shown in

Table 4. Logistic Regression Parameter Results.

Parameter	Constant	Gestational Age Estimation	Age	Height	Weight
Parameter Estimate	3.9609	0.0069	−0.0321	−0.0033	−0.0386
Std. Error	1.1863	0.0013	0.0141	0.0015	0.0098
z-value	3.3389	5.3077	−2.2766	−2.2001	−3.9388
p-value	0.0001	0.0000	0.0228	0.0278	0.0001
Odds Ratio	19.9576	1.0065	0.9637	0.9899	0.9573

below:

The logistic regression results in Table 4 indicate that gestational age at testing is positively associated with meeting the reporting threshold for fetal Y chromosome concentration. The estimated coefficient for gestational age is 0.0069 with a statistically significant z-value of 5.3077 and p-value below 0.001, corresponding to an odds ratio of 1.0065. This suggests that, as gestational age increases, the probability of achieving reportable Y chromosome concentration increases.

Maternal age, height, and weight show negative associations with the outcome. Maternal age is significant with an estimated coefficient of −0.0321, z-value of −2.2766, and p-value of 0.0228, with an odds ratio of 0.9637, indicating a modest decrease in odds with increasing maternal age. Maternal height is also significant with an estimated coefficient of −0.0033, z-value of −2.2001, and p-value of 0.0278, with an odds ratio of 0.9899. Maternal weight shows the strongest negative association among these variables, with an estimated coefficient of −0.0386, z-value of −3.9388, and p-value below 0.001, corresponding to an odds ratio of 0.9573, implying that higher maternal weight is linked to a lower likelihood of reaching the reporting threshold.

Overall, these results support the modeling assumption that later testing weeks tend to increase test reportability, whereas higher maternal body-size-related factors tend to reduce it, and they motivate the subsequent optimization model for personalized timing decisions across BMI ranges.

4.3. Monte Carlo Data Simulation

The attached data exhibits non-uniform distribution across different BMI and gestational weeks, significantly impacting the solution of the NIPT timing selection model based on historical pass rates in previous chapters. Therefore, we consider using Monte Carlo simulation with logistic regression to achieve uniform data distribution.

Based on the logistic regression parameters, we simulated 10,000 sets of gestational age, age, height, and weight data. We calculated their mean values across different BMIs and gestational weeks to replace the historical pass rate of Y chromosome concentration in previous chapters. The simulation data visualization results are shown in Figure 1:

Observation of the figure reveals that Y chromosome concentration generally decreases with increasing BMI, while the mean concentration increases with delayed gestational week at testing—consistent with the original hypothesis. These simulated results provide week-specific estimates of the probability that the fetal Y chromosome concentration meets the reporting threshold under different BMI levels. We then use these probability estimates as the key inputs for the downstream optimization, where BMI values are partitioned into a finite number of contiguous groups and one optimal testing week is assigned to each group. The resulting number of BMI groups and the corresponding recommended testing weeks are reported in the BMI grouping results table.

4.4. BMI Grouping Results

In this study, a BMI group refers to a contiguous interval of BMI values, and all individuals within the same interval share a single recommended NIPT week. Before presenting the optimization outcomes, we summarize the study cohort and the key variables used in the correlation modeling and optimization procedure in

Table 5. Model Optimization Results.

Variable	Unit	n	Mean	Range
Gestational age at testing	weeks	1082	15.2	11.0 to 29.0
Maternal age	years	1082	29.8	18.0 to 44.0
Maternal height	cm	1082	161.5	145.0 to 178.0
Maternal weight	kg	1082	66.2	42.0 to 110.0
Maternal BMI	kg per m2	1082	30.6	20.0 to 47.0
Fetal Y chromosome concentration	percent	1082	4.9	0.3 to 15.2
Test reportability indicator j	0 or 1	1082	0.87	0 to 1

to provide an overview of the data scale and variable ranges.

Based on dynamic programming, we first determine an optimal partition of the BMI domain into contiguous intervals. We then apply simulated annealing to identify the recommended NIPT week for each BMI interval. The iterative procedure converged, and the final sum of expected risks reached 0.531. The optimized BMI grouping and corresponding recommended testing weeks are reported in

Table 6. Model Optimization Results.

BMI Range	(20, 25]	(25, 29]	(29, 31]	(31, 39]	(39, 45]	(45, 47)
Optimal Timing	16 weeks	11 weeks	12 weeks	15 weeks	12 weeks	16 weeks

.

Due to insufficient data within the first BMI range, optimization results were unstable. Therefore, only the optimal NIPT timing results within the BMI range (25, 47) were considered. Overall, the optimal timing results show a trend of increasing, then decreasing, and then increasing again as BMI increases. Ideally, the optimal NIPT timing should be delayed as BMI increases. However, the results for the BMI range (39, 45] exhibit an anomalous pattern. When maternal BMI falls within (39, 45], the lysis of adipose tissue and trophoblast cells leads to an increase in fetal free DNA concentration in the blood. However, this increase occurs at a slow rate, and the risk associated with delaying testing is disproportionate to the benefit. Therefore, it is advisable to perform NIPT at the earliest stage that meets testing requirements to mitigate such risks.

4.5. In-Depth Analysis and Discussion

Dynamic Risk–Benefit Tradeoff: The model results reveal that determining the optimal NIPT timing is fundamentally a dynamic balancing act between risks and benefits. For women with moderate-to-low BMI (e.g., 25–31), their physiological environment facilitates earlier release of fetal free DNA to reliable detection levels (>4%). Consequently, the model tends to recommend testing at 11–12 weeks gestation. This allows high-probability acquisition of reliable results when the accumulated risk of fetal abnormalities is still low, thereby maximizing the benefits of early intervention.

The “Waiting Paradox” and the critical point for High-BMI groups can be traced back to the analysis pipeline and the reported outputs. The overall trend that the fetal Y chromosome concentration decreases as BMI increases, while it increases with gestational week, is supported by the Monte Carlo simulation pattern shown in Figure 1, which is generated based on the fitted logistic regression in Table 4 and then used to obtain week-dependent test credibility probabilities. On this basis, the optimized timing in Table 6 is produced by minimizing the expected risk function defined in Section 2.2 across feasible weeks for each BMI interval. For women in the BMI interval (31, 39], the model recommends delaying testing to 15 weeks because earlier weeks have a higher probability of non-reportable results, which increases the expected loss through the repeat-testing component in the Bayesian risk formulation; therefore, waiting improves reportability sufficiently to reduce the overall expected risk. In contrast, for women in (39, 45], although additional waiting may still improve reportability, the gestational-week-dependent treatment window loss increases more rapidly at later weeks, so the marginal benefit from improved reportability is outweighed by the escalating delay-related loss, and the minimization of the expected risk therefore shifts the optimal strategy from further waiting to earlier testing. This mechanism explains the seemingly counterintuitive result in Table 6 where the optimal timing for (39, 45] is 12 weeks, earlier than the 15 weeks obtained for (31, 39], and clarifies that the “decision threshold” arises from the point at which the expected-risk reduction from improved credibility no longer compensates for the increased loss caused by delayed gestational age.

Model efficacy and clinical implications: This optimization model integrates maternal physiological indicators (proxied by BMI), inherent testing errors (false positive rate 10.35%, false negative rate 2.77%), and clinical risks across gestational weeks into a unified decision framework. The ultimately converged overall expected risk of 0.531 provides a quantitative benchmark for evaluating different screening strategies. The clinical implication is that a “one-size-fits-all” testing recommendation (e.g., uniformly at 12 weeks) is not optimal. Stratified management based on maternal BMI, coupled with a differentiated testing schedule, can systematically reduce the potential risk across the entire screening population. For example, women with a BMI around 30 are recommended to undergo testing at 12 weeks, those with a BMI around 35 at 15 weeks, and those with a BMI over 40 should be tested as soon as possible after a 12-week assessment.

5. Discussion

This paper constructs a mathematical optimization model that accounts for the impact of individual differences among pregnant women—particularly body mass index (BMI)—on the efficacy of non-invasive prenatal testing (NIPT). Based on Bayesian decision theory, this model integrates the prior probability of fetal chromosomal abnormalities, the expected risk associated with testing at different gestational ages, the credibility of test results (influenced by false positives/false negatives), and the time cost of hospital testing. It aims to determine the optimal NIPT time point for pregnant women in different BMI ranges to minimize the overall expected risk for the pregnant population. The model employs principal component analysis and logistic regression to quantify the relationship between maternal age, height, weight, gestational week at testing, and fetal DNA concentration (proxied by Y chromosome concentration). Monte Carlo simulation was used for data modeling to support parameter estimation. Model solutions reveal a nonlinear relationship between maternal BMI and optimal NIPT timing. For instance, the optimal timing for the (25, 29] BMI range occurs at 11 weeks gestation, while it shifts to 15 weeks for the (31, 39] range, illustrating a dynamic risk–benefit tradeoff. Numerical results demonstrate that this optimization model can systematically reduce the potential risk for the screened population, with the overall expected risk converging to 0.531. Based on these findings, this study provides theoretical justification and decision support for implementing personalized, precision-based NIPT timing recommendations in clinical practice, and can be further extended to more complex clinical decision-making scenarios.