# Profile of a Multivariate Observation under Destructive Sampling—A Monte Carlo Approach to a Case of Spina Bifida

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Details

_{5}. We use a model-oriented endeavor to build the profiles. The method is outlined and implemented in Section 2.2.

#### 2.2. Statistical Methods

_{1}= roughness at zero weeks. After the patch is dipped in AF (or PBS), let X

_{2}= roughness at four weeks, X

_{3}= roughness at eight weeks, X

_{4}= roughness at twelve weeks, and X

_{5}= roughness at sixteen weeks.

_{1}, X

_{2}, X

_{3}, X

_{4}, X

_{5}) is not observable in its entirety for any patch. This means, for example, if X

_{1}is observed for a patch, X

_{2}, X

_{3}, X

_{4}, and X

_{5}are not observable. In the experiment, three measurements were obtained on each X

_{i}independently from a total of 15 patches. Let (μ

_{1}, μ

_{2}, μ

_{3}, μ

_{4}, μ

_{5}) be the population mean vector of (X

_{1}, X

_{2}, X

_{3}, X

_{4}, X

_{5}). The homogeneity of the means was tested by the ANOVA (analysis of variance) method. The null hypothesis of homogeneity of means was rejected for patches soaked in AF (p < 0.001). The homogeneity of population variances was tested by the Bartlett test (p = 0.247). The hypothesis of homogeneity of variances was not rejected. An estimate of the common variance was given as 517. The normality and homoskedasticity were checked out to be valid (Wilk–Shapiro test: p = 0.705). Similar results hold for patches soaked in PBS (homogeneity of means: p = 0.005; normality and homoscedasticity: Wilk–Shapiro Test: p = 0.364; homogeneity of variances: Bartlett test: p = 0.253). Estimate of the common variance = 490.

_{i}can be taken to be normally distributed. It is reasonable to assume that (X

_{1}, X

_{2}, X

_{3}, X

_{4}, X

_{5})~MVN

_{5}(μ, Σ) with mean vector μ

^{T}= (μ

_{1}, μ

_{2}, μ

_{3}, μ

_{4}, μ

_{5}) = (μ

^{(1)}, μ

^{5}) and dispersion matrix

_{11}is the dispersion matrix of (X

_{1}, X

_{2}, X

_{3}, X

_{4}), and Σ

_{22}= (${\sigma}_{5}^{2}$). The entity μ

^{(1)}is the mean vector of (X

_{1}, X

_{2}, X

_{3}, X

_{4}). The way we have partitioned the mean vector and the dispersion matrix is influenced by the following conditional distribution. The acronym MVN stands for multivariate normal distribution.

_{i}s are equi-correlated with common correlation coefficient ρ. The dispersion matrix is positive if −1/4 < ρ < 1. We have chosen the simple model because it is a reasonable way to build a conditional profile of roughness. We can also handle the conditional probability.

_{1}− μ

_{1}≤ a, −b ≤ X

_{2}− μ

_{2}≤ b, −c ≤ X

_{3}− μ

_{3}≤ c, −d ≤ X

_{4}− μ

_{4}≤ d|X

_{5}), which will be helpful for building a prediction band. Even though we know the conditional distribution of X

_{1}, X

_{2}, X

_{3}, X

_{4}given X

_{5}, under this model, calculating the conditional probability is extremely difficult. It involves evaluating a four-dimensional integral. However, the distribution can be simulated so that the joint probability can be estimated. This is the gist of the Monte Carlo simulations.

_{5}− μ

_{5}) and

_{i}|X

_{5}is (1 − ρ

^{2}) × ${\sigma}_{i}^{2}$. The conditional variance is less now, and the correlation is also less if ρ > 0.

- Given X
_{5}, simulate the joint distribution of (X_{1}, X_{2}, X_{3}, X_{4}). This requires knowledge of the conditional mean and conditional dispersion matrix. - We need μ
_{i}s, which can be estimated using the individual data on X_{i}s. - We need σ
_{i}s, which can be estimated using the individual data on X_{i}s. - The correlation coefficient ρ glues the means, variances, and joint distribution. There was no way we can estimate the correlation coefficient using the marginal data we have. We performed simulations by assuming the value of ρ = 0.0 (0.1) 0.9.
- We conducted Monte Carlo simulations. For each choice of ρ and fluid, Steps 1 through 4 were repeated one thousand times. The average of (X
_{1}, X_{2}, X_{3}, X_{4}) s was the desired profile. The 95% band surrounding the mean was built using the following inequality:

_{1}, X

_{2}, X

_{3}, X

_{4}, X

_{5})~MVN

_{5}(μ, Σ) with mean vector μ

_{T}= (μ

_{1}, μ

_{2}, μ

_{3}, μ

_{4}, μ

_{5}) and dispersion matrix

_{i}was assumed to have the same variance. This was justified by the ANOVA procedure carried out in Section 2.2. This model was the classic equi-correlated normal distribution, which means there was the same variance and correlation (ρ) between any two X

_{i}and X

_{j}. We took the liberty in assuming equi-correlation. This assumption allowed us build a profile of roughness overtime and a 95% confidence band of the profile. We chose the simple model because this was a reasonable way to build a profile of roughness.

_{1}− μ

_{1}≤ a,−a ≤ X

_{2}− μ

_{2}≤ a,−a ≤ X

_{3}− μ

_{3}≤ a, −a ≤ X

_{4}− μ

_{4}≤ a, a ≤ X

_{5}− μ

_{5}≤ a) = 0.95.

_{1}, μ

_{2}, μ

_{3}, μ

_{4}, μ

_{5}, σ

_{2}, and ρ. We used estimates of means and common variance in the equation. We experimented with several choices of correlation for the band. We chose ρ = 0.5 for which the length of each interval 2 × a was minimum. The calculation of the probability was daunting. We resorted to Monte Carlo simulations. The multivariate normal distribution was simulated one thousand times to determine a for our choice of ρ [30].

## 3. Results

#### 3.1. Conditional Profile

_{5}to equal the average of observed X

_{5}. For AF, X

_{5}= 291 and for PBS, X

_{5}= 217. The Monte Carlo average profile remained more or less the same across a whole range of ρ s. We calculated the average profile at ρ = 0.6 for each fluid.

#### 3.2. Unconditional Profile

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Iskandar, B.J.; Finnell, R.H. Spina Bifida. N. Engl. J. Med.
**2022**, 387, 444–450. [Google Scholar] [CrossRef] [PubMed] - Avagliano, L.; Massa, V.; George, T.M.; Qureshy, S.; Bulfamante, G.; Finnell, R.H. Overview on Neural Tube Defects: From Development to Physical Characteristics. Birth Defects Res.
**2018**, 111, 1455–1467. [Google Scholar] [CrossRef] [PubMed] - Hassan, A.-E.S.; Du, Y.; Lee, S.Y.; Wang, A.; Farmer, D.L. Spina Bifida: A Review of the Genetics, Pathophysiology and Emerging Cellular Therapies. J. Dev. Biol.
**2022**, 10, 22. [Google Scholar] [CrossRef] [PubMed] - About Spina Bifida. Available online: https://www.nichd.nih.gov/health/topics/spinabifida/conditioninfo (accessed on 23 January 2024).
- How Do Healthcare Providers Diagnose Spina Bifida? Available online: https://www.nichd.nih.gov/health/topics/spinabifida/conditioninfo/diagnose (accessed on 23 January 2024).
- Song, R.B.; Glass, E.N.; Kent, M. Spina Bifida, Meningomyelocele, and Meningocele. Vet. Clin. N. Am. Small Anim. Pract.
**2016**, 46, 327–345. [Google Scholar] [CrossRef] [PubMed] - Muñiz, L.M.; Del Magno, S.; Gandini, G.; Pisoni, L.; Menchetti, M.; Foglia, A.; Ródenas, S. Surgical Outcomes of Six Bulldogs with Spinal Lumbosacral Meningomyelocele or Meningocele. Vet. Surg.
**2019**, 49, 200–206. [Google Scholar] [CrossRef] [PubMed] - Piatt, J.H. Treatment of Myelomeningocele: A Review of Outcomes and Continuing Neurosurgical Considerations among Adults. J. Neurosurg.
**2010**, 6, 515–525. [Google Scholar] [CrossRef] [PubMed] - Copp, A.J.; Adzick, N.S.; Chitty, L.S.; Fletcher, J.Μ.; Holmbeck, G.N.; Shaw, G.M. Spina Bifida. Nat. Rev. Dis. Primers
**2015**, 1, 15007. [Google Scholar] [CrossRef] - Bibbins-Domingo, K.; Grossman, D.C.; Curry, S.J.; Davidson, K.W.; Epling, J.W.; García, F.; Kemper, A.R.; Krist, A.H.; Kurth, A.; Landefeld, C.S.; et al. Folic Acid Supplementation for the Prevention of Neural Tube Defects. JAMA
**2017**, 317, 183. [Google Scholar] [CrossRef] - Spina Bifida Data and Statistics|CDC. Centers for Disease Control and Prevention. Available online: https://www.cdc.gov/ncbddd/spinabifida/data.html (accessed on 23 January 2024).
- Spina Bifida—Diagnosis and Treatment—Mayo Clinic. Available online: https://www.mayoclinic.org/diseases-conditions/spina-bifida/diagnosis-treatment/drc-20377865 (accessed on 23 January 2024).
- Adzick, N.S.; Thom, E.; Spong, C.Y.; Brock, J.W.; Burrows, P.K.; Johnson, M.P.; Howell, L.J.; Farrell, J.A.; Dabrowiak, M.E.; Sutton, L.N.; et al. A Randomized Trial of Prenatal versus Postnatal Repair of Myelomeningocele. N. Engl. J. Med.
**2011**, 364, 993–1004. [Google Scholar] [CrossRef] - Moldenhauer, J.S.; Soni, S.; Rintoul, N.E.; Spinner, S.S.; Khalek, N.; Martinez-Poyer, J.; Flake, A.W.; Hedrick, H.L.; Peranteau, W.H.; Rendon, N.; et al. Fetal Myelomeningocele Repair: The Post-MOMS Experience at the Children’s Hospital of Philadelphia. Fetal Diagn. Ther.
**2014**, 37, 235–240. [Google Scholar] [CrossRef] - Cortés, M.S.; Chmait, R.H.; Lapa, D.A.; Belfort, M.A.; Carreras, E.; Miller, J.L.; Brawura-Biskupski-Samaha, R.; González, G.S.; Gielchinsky, Y.; Yamamoto, M.; et al. Experience of 300 Cases of Prenatal Fetoscopic Open Spina Bifida Repair: Report of the International Fetoscopic Neural Tube Defect Repair Consortium. Am. J. Obstet. Gynecol.
**2021**, 225, 678.e1–678.e11. [Google Scholar] [CrossRef] - Tatu, R.; Oria, M.; Pulliam, S.; Signey, L.; Rao, M.B.; Peiró, J.L.; Lin, C. Using Poly(L-lactic Acid) and Poly(Ɛ-caprolactone) Blends to Fabricate Self-expanding, Watertight and Biodegradable Surgical Patches for Potential Fetoscopic Myelomeningocele Repair. J. Biomed. Mater. Res. Part B Appl. Biomater.
**2018**, 107, 295–305. [Google Scholar] [CrossRef] - Oria, M.; Tatu, R.; Lin, C.; Peiró, J.L. In Vivo Evaluation of Novel PLA/PCL Polymeric Patch in Rats for Potential Spina Bifida Coverage. J. Surg. Res.
**2019**, 242, 62–69. [Google Scholar] [CrossRef] - Tatu, R.; Oria, M.; Rao, M.B.; Peiró, J.L.; Lin, C. Biodegradation of Poly(l-Lactic Acid) and Poly(ε-Caprolactone) Patches by Human Amniotic Fluid in an in-Vitro Simulated Fetal Environment. Sci. Rep.
**2022**, 12, 3950. [Google Scholar] [CrossRef] - Bonate, P.L. A Brief Introduction to Monte Carlo Simulation. Clin. Pharmacokinet.
**2001**, 40, 15–22. [Google Scholar] [CrossRef] - Martins, M.T.; Lourenço, F.R. Measurement Uncertainty for <905> Uniformity of Dosage Units Tests Using Monte Carlo and Bootstrapping Methods—Uncertainties Arising from Sampling and Analytical Steps. J. Pharm. Biomed. Anal.
**2024**, 238, 115857. [Google Scholar] [CrossRef] - Lecina, D.; Gilabert, J.F.; Guallar, V. Adaptive Simulations, towards Interactive Protein-Ligand Modeling. Sci. Rep.
**2017**, 7, 8466. [Google Scholar] [CrossRef] - Bailer, A.J.; Ruberg, S.J. Randomization tests for assessing the equality of area under curves for studies using destructive sampling. J. Appl. Toxicol.
**1996**, 16, 391–395. [Google Scholar] [CrossRef] - Holder, D.J.; Hsuan, F.; Dixit, R.; Soper, K. A method for estimating and testing area under the curve in serial sacrifice, batch, and complete data designs. J. Biopharm. Stat.
**1999**, 9, 451–464. [Google Scholar] [CrossRef] [PubMed] - Wolfsegger, M.J.; Jaki, T. Estimation of AUC from 0 to infinity in serial sacrifice designs. J. Pharmacokinet. Pharmacodyn.
**2005**, 32, 757–766. [Google Scholar] [CrossRef] [PubMed] - Rabbee, N. Biomarker Analysis in Clinical Trials with R, 1st ed.; CRC Press: Boca Raton, FL, USA; Taylor & Francis Group: Abingdon, UK, 2020. [Google Scholar] [CrossRef]
- Rubinstein, R.Y.; Kroese, D.P. Simulation and the Monte Carlo Method, 3rd ed.; John Wiley & Sons: Hoboken, NJ, USA, 2016. [Google Scholar] [CrossRef]
- Dykstra, R.L. Product Inequalities Involving the Multivariate Normal Distribution. J. Am. Stat. Assoc.
**1980**, 75, 646–650. [Google Scholar] [CrossRef] - Tong, Y.L. Some Probability Inequalities of Multivariate Normal and Multivariate t. J. Am. Stat. Assoc.
**1970**, 65, 1243–1247. [Google Scholar] [CrossRef] - Tong, Y.L. Probability Inequalities in Multivariate Distributions. J. Am. Stat. Assoc.
**1982**, 77, 690. [Google Scholar] [CrossRef] - Ripley, B.D. Ohio Library and Information Network. In Stochastic Simulation; Wiley: New York, NY, USA, 1987; p. 28. [Google Scholar]

**Figure 1.**Conditional profile of roughness at zero, four, eight, and twelve weeks given roughness at sixteen weeks + 95% prediction band.

**Figure 2.**Unconditional profile of roughness at zero, four, eight, twelve and sixteen weeks + 95% prediction band for AF.

**Figure 3.**Unconditional profile of roughness at zero, four, eight, twelve and sixteen weeks + 95% prediction band for PBS.

Roughness | |||
---|---|---|---|

Week | Baseline | AF | PBS |

0 | 139 | ||

0 | 122 | ||

0 | 132 | ||

4 | 223 | 177 | |

4 | 267 | 202 | |

4 | 217 | 212 | |

8 | 245 | 185 | |

8 | 269 | 198 | |

8 | 257 | 205 | |

12 | 265 | 167 | |

12 | 283 | 217 | |

12 | 285 | 248 | |

16 | 306 | 224 | |

16 | 247 | 198 | |

16 | 320 | 229 |

Week | Baseline | AF | PBS | |||
---|---|---|---|---|---|---|

Mean | SD | Mean | SD | Mean | SD | |

0 | 131 | 8.54 | ||||

4 | 235.67 | 27.3 | 197 | 18.03 | ||

8 | 257 | 12 | 196 | 10.15 | ||

12 | 277.67 | 11.02 | 210.67 | 40.87 | ||

16 | 291 | 38.74 | 217 | 16.64 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Guan, T.; Tatu, R.; Wima, K.; Oria, M.; Peiro, J.L.; Lin, C.-Y.; Rao, M.B.
Profile of a Multivariate Observation under Destructive Sampling—A Monte Carlo Approach to a Case of Spina Bifida. *Bioengineering* **2024**, *11*, 249.
https://doi.org/10.3390/bioengineering11030249

**AMA Style**

Guan T, Tatu R, Wima K, Oria M, Peiro JL, Lin C-Y, Rao MB.
Profile of a Multivariate Observation under Destructive Sampling—A Monte Carlo Approach to a Case of Spina Bifida. *Bioengineering*. 2024; 11(3):249.
https://doi.org/10.3390/bioengineering11030249

**Chicago/Turabian Style**

Guan, Tianyuan, Rigwed Tatu, Koffi Wima, Marc Oria, Jose L. Peiro, Chia-Ying Lin, and Marepalli. B. Rao.
2024. "Profile of a Multivariate Observation under Destructive Sampling—A Monte Carlo Approach to a Case of Spina Bifida" *Bioengineering* 11, no. 3: 249.
https://doi.org/10.3390/bioengineering11030249