# A mathematical method for constraint-based cluster analysis towards optimized constrictive diameter smoothing of saphenous vein grafts

- 96 Downloads
- 3 Citations

## Abstract

This study was concerned with the cluster analysis of saphenous vein graft data to determine a minimum number of diameters, and their values, for the constrictive smoothing of diameter irregularities of a cohort of veins. Mathematical algorithms were developed for data selection, transformation and clustering. Constrictive diameter values were identified with interactive pattern evaluation and subsequently facilitated in decision-tree algorithms for the data clustering. The novel method proved feasible for the analysis of data of 118 veins grafts, identifying the minimum of two diameter classes. The results were compared to outcome of a statistical recursive partitioning analysis of the data set. The method can easily be implemented in computer-based intelligent systems for the analysis of larger data sets using the diameter classes identified as initial cluster structure.

## Keywords

Saphenous vein Diameter irregularity Constrictive smoothing Mathematical model Pattern recognition## Abbreviations

*C*_{R,i}Constriction degree required for complete smoothing of vein

*i*, by reducing the vein’s maximum diameter*D*_{max,i, }to the minimum diameter*D*_{min,i }, where*i*= 1 to*n**C*_{A,i}Constriction degree applied to vein

*i*, to reduce the maximum diameter*D*_{max,i }to the constricted diameter*d*^{(k)}, where*i*= 1 to*n*and*k*= 1 to*l**C*_{A,i,j}Constriction degree applied to vein

*i,*to reduce the diameter*D*_{ i,j }at each measurement position*x*_{ i,j }along the length of the vein to the constriction diameter \( d_{\text{P}}^{s}\) proposed with the recursive partitioning, where*i*= 1 to*n*and*s*= 1 to*l*_{P}- \( C_{\text{A}}^{\max } \)
Maximum permissible constriction degree applied to a vein to reduce the maximum diameter

*D*_{max,i }to the constricted diameter*d*^{(k)}, where*i*= 1 to*n*and*k*= 1 to*l**C*_{A}Mean of applied constriction degree

*C*_{A,i }for all veins*i*, with*i*= 1 to*n*across all constriction diameters*d*^{(k)}, with*k*= 1 to*l*- \( C_{\text{A}}^{(k)} \)
Mean of applied constriction degree

*C*_{A,i }for all veins*z*with*z*= 1 to*n*_{ k }for one constriction diameters*d*^{(k)}, with*k*= 1 to*l**D*_{i,j}Outer diameter of vein

*i*measured at luminal pressure associated with post-harvest leak test syringe inflation at measurement position*x*_{ i,j }, where*i*= 1 to*n*and*j*= 1 to*m*_{ i }(mm)*D*_{C,i}Constriction diameter for vein

*i*, i.e. outer diameter of vein*i*after constriction, where*i*= 1 to*n*(mm)*D*_{max,i}Maximum value of the outer diameter

*D*_{ i,j }of vein*i*at luminal pressure associated with post-harvest leak test syringe inflation, where*i*= 1 to*n*(mm)*D*_{min,i}Minimum value of the outer diameter,

*D*_{ i,j }, of vein*i*at luminal pressure associated with post-harvest leak test syringe inflation, where*i*= 1 to*n*(mm)- \( D_{\text{max,z}}^{(k)} \)
Maximum outer diameter of vein

*z*of subset*k*, where*z*= 1 to*n*_{ k }and*k*= 1 to*l*(mm)- \( D_{\text{min,z}}^{(k)} \)
Minimum outer diameter of vein

*z*of subset*k*, where*z*= 1 to*n*_{ k }and*k*= 1 to*l*(mm)*d*Constrictive smoothing diameter for all veins

*i*, where*i*= 1 to*n*(mm)*d*^{(k)}Constrictive smoothing diameter for a subset

*k*of veins*z*, where*k*= 1 to*l*and*z*= 1 to*n*_{ k }(mm)- \(d_{\text{P}}^{s}\)
Partitions of

*D*_{min,i }determined with the recursive partitioning method, that represent constrictive smoothing diameters with*i*= 1 to*n*and*s*= 1 to*l*_{P}(mm)*i*Identifier of a vein in full set (

*i*= 1 to*n*)*j*Identifier of the measurement position along a vein (

*j*= 1 to*m*_{ i })*k*Identifier of a vein set (

*k*= 1 to*l*)*L*_{i}Harvested length of vein

*i,*where*i*= 1 to*n*(mm)*l*Number of vein sets:

*l*= 1 for the entire set of veins*i*with*i*= 1 to*n*and*l*> 1 if it is required to divide the set of*n*veins into subsets*k*of veins*l*_{P}Number of partitions \(d_{\text{P}}^{s}\) determined with the recursive partitioning method

*m*_{i}Number of measurement points for the outer diameter

*D*_{ i,j }along vein*i*with the length*L*_{ i }, where*m*_{ i }= int (*L*_{ i }/20)*n*Number of veins in analysis

*n*_{e}Number of veins excluded from analysis

*n*_{k}Number of veins in subset

*k*where*k*= 1 to*l*- \( \hat{n}_{k} \)
Normalized number of veins in subset k: \( \hat{n}_{k} = {{n_{k} } \mathord{\left/ {\vphantom {{n_{k} } {\sum\nolimits_{k = 1}^{l} {n_{k} } }}} \right. \kern-\nulldelimiterspace} {\sum\nolimits_{k = 1}^{l} {n_{k} } }}, \) where

*k*= 1 to*l*(%)*s*Identifier of partitions of

*D*_{min,i }, \(d_{\text{P}}^{s} ,\) determined with the recursive partitioning method, where*s*= 1 to*l*_{P}*x*_{i,j}Position of

*D*_{ i,j }measurement along vein*i*, where*i*= 1 to*n*and*j*= 1 to*m*_{ i }*z*Identifier of a vein in subset

*k*, where*z*= 1 to*n*_{ k }and*k*= 1 to*l*

## Notes

### Acknowledgements

The authors wish to acknowledge Dr. L. Moodley, Dr. W. Lichtenberg, Dr. J. Scherman, Dr. H. Legodi, Dr. G. Mphahlele and Dr. I. Taunyane of the Chris Barnard Division of Cardiothoracic Surgery, Groote Schuur Hospital, Cape Town, for performing the saphenous vein dimensional measurements. This research was mainly funded through a research collaboration grant by Medtronic Inc. (Minneapolis, MN, USA) to the University of Cape Town and a THRIP grant of the National Research Foundation (South Africa). The salary of TF was fully funded and that of PH was partially funded from the Medtronic research grant.

## References

- 1.Han J, Kamber M (2006) Data mining: concepts and techniques. In: Gray J (ed) The Morgan Kaufmann series in data management systems, 2nd edn. Elsevier, Amsterdam, p 770Google Scholar
- 2.Durbin B, Dudoit S, van der Laan MJ (2008) A deletion/substitution/addition algorithm for classification neural networks, with applications to biomedical data. J Stat Plan Inference 138:464–488CrossRefGoogle Scholar
- 3.Hu Y-C, Chen R-S, Tzeng G-H (2003) Finding fuzzy classification rules using data mining techniques. Pattern Recognit Lett 24:509–519CrossRefGoogle Scholar
- 4.Paetz J (2006) Supervised neuro-fuzzy clustering for life science applications. In: Maglaveras N, Chouvarda I, Koutkias V, Brause R (eds) Biological and medical data analysis. Springer, Berlin, pp 378–389CrossRefGoogle Scholar
- 5.Zhang H, Ishikawa M (2007) Bagging using hybrid real-coded genetic algorithm with pruning and its applications to data classification. Int Congr Ser 1301:184–187CrossRefGoogle Scholar
- 6.Zhang Y, Bhattacharyya S (2004) Genetic programming in classifying large-scale data: an ensemble method. Inf Sci 163:85–101CrossRefGoogle Scholar
- 7.Vieira A, Barradas N (2003) A training algorithm for classification of high-dimensional data. Neurocomputing 50:461–472CrossRefGoogle Scholar
- 8.Xu P, Brock GN, Parrish RS (2009) Modified linear discriminant analysis approaches for classification of high-dimensional microarray data. Comput Stat Data Anal 53:1674–1687CrossRefGoogle Scholar
- 9.Zilla P, Human P, Wolf M, Lichtenberg W, Rafiee N, Bezuidenhout D, Samodien N, Schmidt C, Franz T (2008) Constrictive external nitinol meshes inhibit vein graft intimal hyperplasia in nonhuman primates. J Thorac Cardiovasc Surg 136:717–725CrossRefPubMedGoogle Scholar
- 10.Zilla P, Wolf M, Rafiee N, Moodley L, Bezuidenhout D, Black M, Human P, Franz T (2009) Utilization of shape memory in external vein-graft meshes allows extreme diameter constriction for suppressing intimal hyperplasia: a non-human primate study. J Vasc Surg 49:1532–1542CrossRefPubMedGoogle Scholar
- 11.Sarkar S, Schmitz-Rixen T, Hamilton G, Seifalian A (2007) Achieving the ideal properties for vascular bypass grafts using a tissue engineered approach: a review. Med Biol Eng Comput 45:327–336CrossRefPubMedGoogle Scholar
- 12.Yang C, Tang D, Liu SQ (2003) A multi-physics growth model with fluid-structure interactions for blood flow and re-stenosis in rat vein grafts: a growth model for blood flow and re-stenosis in grafts. Comput Struct 81:1041–1058CrossRefGoogle Scholar
- 13.Liu SQ, Moore MM, Yap C (2000) Prevention of mechanical stretch-induced endothelial and smooth muscle cell injury in experimental vein grafts. J Biomech Eng 122:31–38CrossRefPubMedGoogle Scholar
- 14.Mills JL, Bandyk DF, Gahtan V, Esses GE (1995) The origin of infrainguinal vein graft stenosis: a prospective study based on duplex surveillance. J Vasc Surg 21:16–22 (discussion 22–25)CrossRefPubMedGoogle Scholar
- 15.Zhang JM, Chua LP, Ghista DN, Yu SC, Tan YS (2008) Numerical investigation and identification of susceptible sites of atherosclerotic lesion formation in a complete coronary artery bypass model. Med Biol Eng Comput 46:689–699CrossRefPubMedGoogle Scholar
- 16.Cacho F, Doblare M, Holzapfel GA (2007) A procedure to simulate coronary artery bypass graft surgery. Med Biol Eng Comput 45:819–827CrossRefPubMedGoogle Scholar
- 17.Tai NR, Salacinski HJ, Edwards A, Hamilton G, Seifalian AM (2000) Compliance properties of conduits used in vascular reconstruction. Br J Surg 87:1516–1524CrossRefPubMedGoogle Scholar
- 18.Schanzer A, Hevelone N, Owens CD, Belkin M, Bandyk DF, Clowes AW, Moneta GL, Conte MS (2007) Technical factors affecting autogenous vein graft failure: observations from a large multicenter trial. J Vasc Surg 46:1180–1190CrossRefPubMedGoogle Scholar
- 19.Mardia KV, Kent JT, Bibby JM (1980) Multivariate analysis. Academic Press, San Diego, p 521Google Scholar
- 20.Armand S, Watelain E, Mercier M, Lensel G, Lepoutre F-X (2006) Identification and classification of toe-walkers based on ankle kinematics, using a data-mining method. Gait Posture 23:240–248CrossRefPubMedGoogle Scholar
- 21.Moon H, Ahn H, Kodell RL, Baek S, Lin C-J, Chen JJ (2007) Ensemble methods for classification of patients for personalized medicine with high-dimensional data. Artif Intell Med 41:197–207CrossRefPubMedGoogle Scholar
- 22.Nadkarni NM, McIntosh ACS, Cushing JB (2008) A framework to categorize forest structure concepts. For Ecol Manag 256:872–882CrossRefGoogle Scholar
- 23.Papadopoulos NJ, Sherif MF, Albert EN (1981) A fascial canal for the great saphenous vein: Gross and microanatomical observations. J Anat 132:321–329PubMedGoogle Scholar
- 24.Human P, Franz T, Scherman J, Moodley L, Zilla P (2009) Dimensional analysis of human saphenous vein grafts: implications for external mesh support. J Thorac Cardiovasc Surg 137:1101–1108CrossRefPubMedGoogle Scholar