Identification of new factors associated to walking impairment in patients with vascular-type claudication

Abstract

Objectives

Mechanisms of walking limitation in arterial claudication are incompletely elucidated. We aimed to identify new variables associated to walking limitation in patients with claudication.

Methods

We retrospectively analyzed data of 1120 patients referred for transcutaneous exercise oxygen pressure recordings (TcpO₂). The outcome measurement was the absolute walking time on treadmill (3.2 km/h, 10% slope). We used both: linear regression analysis and a non-linear analysis, combining support vector machines and genetic explanatory in 800 patients with the following resting variables: age, gender, body mass index, the presence of diabetes, minimal ankle to brachial index at rest, usual walking speed over 10 m (usual-pace), number of comorbid conditions, active smoking, resting heart rate, pre-test glycaemia and hemoglobin, beta-blocker use, and exercise-derived variables: minimal value of pulse oximetry, resting chest-TcpO₂, decrease in chest TcpO₂ during exercise, presence of buttock ischemia defined as a decrease from rest of oxygen pressure index ≤15 mmHg. We tested the models over 320 other patients.

Results

Independent variables associated to walking time, by decreasing importance in the models, were: age, ankle to brachial index, usual-pace; resting TcpO₂, body mass index, smoking, buttock ischemia, heart rate and beta-blockers for the linear regression analysis, and were ankle to brachial index, age, body mass index, usual-pace, decrease in chest TcpO₂, smoking, buttock ischemia, glycaemia, heart rate for the non-linear analysis. Testing of models over 320 new patients gave r = 0.509 for linear and 0.575 for non-linear analysis (both p < 0.05).

Conclusion

Buttock ischemia, heart rate and usual-pace are new variables associated to walking time.

Keywords

Peripheral artery disease claudication walking patho-physiology treadmill

Introduction

Peripheral arterial disease (PAD) affects million men and women in the United States and the European Union. It affects from 0.4 up to 14% of the population and its prevalence increases with age.¹ Ankle to brachial index (ABI) is highly accurate to detect and screen PAD in symptomatic and asymptomatic patients.² One of the first symptoms of PAD is intermittent claudication. Treadmill tests are largely used to reproduce usual symptoms in intermittent claudication and quantify walking limitation. The maximal walking time (walking time), or distance, that can be performed on treadmill is used to classify walking impairment severity and has a prognostic value of mortality.³

There have been many studies that tried to define variables associated to the severity of walking impairment. All have shown a fair to moderate relationship of ABI and walking distance,^4–11 in both man and women.¹² On the one hand, most studies lack information and objective estimation of some co-morbid conditions that are expected to impair walking ability. Among these co-morbid unstudied conditions are exercise-induced buttock ischemia and exercise induced hypoxemia. These two co-morbid conditions will remain undetectable through ABI measurements and may impair walking ability. On the other hand, some factors are clearly expected to show a non-linear relationship with walking capacity such as ABI (high and low ABI values being associated with vascular disease),¹³ hemoglobin concentration (higher values being potentially associated to chronic hypoxic conditions and lower values with anemia) or maximal heart rate at exercise (low heart rate being possibly related to chronotropic insufficiency, while high heart rate could rely on cardiac dysfunction). This underlines the potential interest of non-linear methodologies to analyze variables associated to walking limitation.

The aim of this study was to identify original variables associated to maximal walking time on treadmill in patients with vascular-type claudication.

Methods

Study design

We performed a retrospective analysis on systematically collected data over a nine years period starting July 2001 for all patients referred for vascular-type claudication and a treadmill test (3.2 km/h; 10% grade) in Laboratory for Vascular Investigations of University Hospital in Angers, France. Patients since 2011 were excluded because changes in our protocol occurred in 2011. Typical vascular-type claudication is a lower limb pain that: (i) is absent at rest, (ii) occurs at exercise, (iii) worsens until it forces the patient to stop if exercise is continued and (iv) disappears with 10 min when exercise is stopped. As a retrospective analysis of the routine database that has been administratively declared, no individual consent was required from patients, and the study conform to the declaration of Helsinki.

Data collection

As a routine, we recorded age, body weight and height to calculate body mass index, heart rate at rest, gender, treatments and specifically the use of beta blockers, the presence or absence of diabetes, the number of comorbidities that may impair walking ability such as cardiac, pulmonary or osteo-articular diseases, active smoking, the time required to walk 10 m at a usual-pace in the laboratory corridor to estimate usual walking speed (usual-pace), capillary glycemia and hemoglobin were systematically measured before exercise using a Freestyle Opitum-H glucometer (Abbott Laboratory Ltd, Illinois, USA) and Hemocue Hb201 (Hemocue® AB, Meaux, France). ABI was measured with a hand hold Doppler at rest manually on both sides and we used the lowest ankle systolic pressure value as the numerator and highest arm systolic pressure value as the denominator.

The exercise test procedure has been extensively described elsewhere.^14,15 Note that patients that were unable to walk 10 m in less than 15 min underwent a treadmill test at a low (non-standard speed). As a standard for patients that were expected to be able to walk on the treadmill, the standard exercise test was performed at a constant load (3.2 km/h 10% grade test) for 15 min and was changed to an incremental load test if patients are not stopped in the initial 15 min of the test.¹⁶ The interest of such an approach is to have all patients reaching their maximal capacity, and avoid the ceiling effect of constant load procedures. In brief, we record transcutaneous oxygen pressures (TcpO₂) with a 5 probe TCM400 (Radiometer, DK) at the buttocks, calves and chest to calculate a “decrease from rest of oxygen pressure” (DROP) index.¹⁷ DROP is defined as limb changes minus chest changes from rest and the minimal value of this index is used for the analysis. As previously validated, a minimal DROP ≤15 mmHg is highly predictive for the presence of ischemia at the buttock level¹⁷ and shown excellent reliability.¹⁸ Further we recorded oxygen saturation with a pulse oximeter throughout exercise and recovery using Radical7 (Masimo®, Neuchatel, Switzerland).

From the exercise test, we recorded the lowest saturation (Sat-min) recorded from a pulse oximeter, the absolute value of chest TcpO₂ at rest as the mean of the values recorded over the 2 min resting period (TcpO₂-rest), the minimal value of chest TcpO₂ observed during exercise from which we calculated the decrease from rest in chest TcpO₂, the presence of buttock ischemia.

Outcome measure

The outcome measure was the absolute walking time on treadmill. The maximal walking time was recorded as the time at which pain forces to stop and not at pain occurrence.

Data analysis

The database has been randomly divided into two parts: A first group of patients to build the model (Group A) and another group of different patients (Group B) to test and validate the models defined in Group A. Explanatory variables were defined from Group A and applied on patients of Group B for both linear regression analysis and non-linear analysis. We used using two different statistical methods: a classical stepwise linear regression analysis and a non-linear analysis. The former analysis is easy and provides the direction of the relationship (positive or negative) between a variable and walking time but may exclude some variables due to non-linearity. The latter is independent from linearity but cannot define the direction and absolute weight of the relationship. Both classify the variables from the one having the most important relative “weight” as an explanatory variable, to the one having the smallest importance. The explanatory variables introduced in the model were resting variables (age, gender, body mass index, diabetes, ABI, usual-pace, comorbidities, smoking, heart rate, glycemia, hemoglobin, beta-blockers) and exercise-derived variables (saturation, TcpO₂-rest, minimal saturation, buttock ischemia).

Linear regression analysis

We used a stepwise multivariate linear regression analysis used to assess the influence of the selected explanatory variables on walking time from Group A. In brief, in a stepwise linear regression analysis, the program determines which variables among the set of explanatory variables included in the analysis will be used for the model, and in which order each independent explanatory variable will be introduced in the models. The models aim at fitting a linear equation to observed data. For each model, the program calculates a “residual term” using a least square calculation (estimation of variance). The least-squares process, aims at minimizing the sum of the squares of the vertical deviations from each data point to the regression line. After each step the programs selects from the explanatory variables the variable or variables, from which a linear regression model results in the largest reduction in the residual variance of walking time. The steps are continued until the introduction of an additional variable shows no significant contribution to the total F-ratio. The F-ratio chosen for the present analysis was 0.05 and the model was stopped at nine factors to compare with the number of factors observed with the non-linear analysis.

Nonlinear analysis

We developed a tool that combines support vector machines (SVM), first described by Vapnik¹⁹ and genetic algorithms that were developed by Holland.²⁰

SVM have been developed in the 90s, in the wide area of machine learning. Due to their ability to work on high-dimensional data, their theoretical guarantees and their reliable results in practice, SVM have been rapidly adopted in many areas, among them is medical decision support systems.²¹ SVM are based on two basic ideas: a wide margin and a kernel notion. As will be shown, while the first idea gives to SVM a remarkable resistance to over fitting, the second idea permits to deal with nonlinear problems. To explain the notion of wide margin, one must suppose that we want a machine to learn how to separate plus versus minus items (Figure 1 upper panel). We could find an infinity number of lines. But, as shown in Figure 1 (middle panel), intuitively the best separation is achieved by the line that has the largest distance to the nearest training data points of any class.²² Nevertheless, the problem becomes intractable when the data are mixed (Figure 1 lower panel). In this case, data are mapped into a hyperspace (for example ℜ¹⁰), then a separating hyperplane is computed in this hyperspace. This mapping in the hyperspace is achieved through kernel functions.²³ In practice, the more used kernels are linear kernels, polynomial kernels and Gaussian kernels.²³ Initially, SVMs were only designed to work on classification problems (typically binaries issues: 0 versus 1). Later Drucker et al.²⁴ proposed an extension of SVMs for addressing regression problems. In the same way as with classification problems, the regression functions built by SVMs optimize the generalization performance by enforcing flatness of the function. Therefore, during the “learning” step, slack variables are used in order to reduce the constraints and authorize some errors.¹⁹ On our side, we used the toolbox least-square SVM provided by Suykens and Vandewalle²⁵ this software works under Matlab©.

Figure 1.

Schematic representation of the three steps of the SVM approach: Step 1 (upper panel), an infinity number of separating lines. Step 2 (middle panel) optimal line and maximum margin for a SVM trained with samples from two classes. Step 3 (lower panel) A mixture of plus versus minus.

Genetic algorithms have proven the ability to explore wide spaces as result from combinatorial explosion.²⁰ They have been applied to numerous medical problems.²⁶ Genetic algorithms function in a manner similar to Darwinian natural selection. They work in three steps (detailed below this paragraph): (a) the algorithm encodes solutions in arrays forms and begins with a random population of solutions. Traditionally, solutions are represented in binary as arrays of 0s and 1s. (b) Each pattern is evaluated to see its fitness. (c) Iterations of mutation and crossover operations on selected individuals conduce to new individuals. For the first step, 64 random arrays were built. The length of each array is equal to 16 (because they are 16 variables that can be selected). If the ith value is equal to 1, then the ith variable is selected, otherwise the variable is not selected. The fitness is computed by a five-fold cross validation (Figure 2). A SVM is trained on 640 patients and its performances are evaluated on the 160 remaining patients of the first group of patients. The performance of the SVM is equal to the correlation that is predicted by the learning machine and the real value.

c o r r = \frac{\sum_{i = 1}^{N} (x_{i} - \bar{x}) . (y_{i} - \bar{y})}{\sqrt{\sum_{i = 1}^{N} {(x_{i} - \bar{x})}^{2} . \sqrt{\sum_{i = 1}^{N} {(y_{i} - \bar{y})}^{2}}}}

Figure 2.

Schematic representation of the five cross validations.

This is achieved five times and the mean value of the five performances values is computed; this value gives the fitness of a single individual. Finally, step 3 (mutations and crossover) works in the following way: (a) Selection of the 32 best individual (with the highest fitness value). (b) Crossover is defined so that two individuals (the parents) combine to produce two new individuals (the children). Thus a crossover between parent 1 and parent 2 is achieved; a crossover between parent 3 and parent 4 and so on. Crossover is performed by selecting a random position along the length of the 2 parents and swapping all the values after that point (Figure 3). This inversion gives two new children. (c) Finally, the mutation is the chance that a bit within an individual will be flipped (0 becomes 1, 1 becomes 0). A mutation can occur with a probability of 0.05.

Figure 3.

Schematic representation of the crossover operation.

Statistical analysis

Values are reported as mean ± standard deviation (SD) or numbers and percentages (%). Comparison of the patients of Group A (building group) and Group B (testing group) were performed with unpaired t-test and Chi-square tests. We analyzed the correlation of the model derived from the population used to build the models (Group A) with the results observed on the new population (Group B) with both linear regression analysis and non-linear analysis. For all tests, a p≤0.05 was considered to be statistically significant. The data were presented as mean ± SD for continuous variable and number of patients for categorical variables.

Results

Over the study period, 3326 tests were performed in 2371 patients. For the study, only the first test of each patient was used for the analysis. We excluded patients with non-available Hemocue® or Glucometer® results (mainly due to non-available reactive strips, n = 1043), missing or non-feasible ABI (n = 133), non-standard treadmill procedures (n = 49), lack of information on smoking status (n = 13), missing maximal heart rate (n = 8), missing BMI or sex (n = 5). Thereafter, the remaining 1120 patients were arbitrarily divided in the 800 first patients (building group: Group A) and the 320 last patients (testing group: Group B).

Characteristics of participants in these two groups are reported in Table 1. As shown, no difference exists between the two populations. Specifically approximately one-fifth of the patients were diabetic patients. Further, most patients had a regional blood flow impairment at the buttock level as a result of its systematic detection with oximetry and possibly of the specific interest of our laboratory to buttock claudication.

Table 1.

Characteristics of populations in the training set (n = 800) and test set (n = 320): Results are mean ± SD for continuous variables and number of patients for categorical variables.

	Training set	Test set	p
Age (year)	63.3 ± 11.6	62.7 ± 12.2	0.414
Gender: Females/males	153/647	64/256	0.738
Diabetes: no/yes	619 /181	258/62	0.233
Number of co morbid conditions 0/1/2/3 or more	396/323/78/3	169/127/22/2	0.398
Active smoker no/yes	253/547	110/210	0.374
Body mass index (kg/m²)	26.7 ± 4.6	26.9 ± 4.8	0.494
Ongoing beta blocking treatment : no/yes	564/236	238/82	0.194
Time to walk 10 m (s)	9.4 ± 1.5	9.3 ± 1.6	0.627
Glycemia (mmol/l)	115 ± 37	120 ± 41.4	0.074
Hemoglobin (g.% ml)	14.2 ± 1.7	14.1 ± 1.7	0.824
Ankle to brachial index	0.76 ± 0.24	0.75 ± 0.23	0.703
Resting heart rate (beats per min)	80 ± 16	81.9 ± 16.8	0.167
Minimal value of pulse oximetry (%)	95 ± 5	95.3 ± 4.9	0.647
Resting chest TcpO₂ (mm Hg)	69 ± 12	68.8 ± 12.3	0.641
Decrease in chest TcpO₂ (mm Hg)	−5 ± 5	−4.6 ± 5.4	0.965
Buttock regional blood flow impairment: no/yes	353/447	126/194	0.147
Maximal walking time on treadmill (m)	417 ± 409	388 ± 386	0.259

TcpO₂: transcutaneous oxygen pressure.

Linear regression analysis

As expected, the two most important explanatory factors of walking time were age and ABI. Interestingly, the third determinant was usual-pace. Among the following factors of less impact are body mass index, active smoking. Among original variables that appear significant in the linear regression analysis model are buttock ischemia and heart rate (r = 0.518; p < 0.01).

Of interest is to note that, from the linear regression analysis, none of the following variables seemed to have an impact on treadmill performance: gender, sat-min, diabetes, comorbidities, hemoglobin, glycemia and decrease in chest TcpO₂.

Classification of the variables in decreasing importance in the two models is shown in Table 2. Coefficients observed in the linear regression analysis are reported in Table 3.

Table 2.

Classification of the variables in decreasing importance, in the two models.

	Nonlinear analysis	Linear regression analysis
Parameters with the highest impact on walking timeParameters with the lowest impact on walking time	Ankle brachial index	Age
	AgeBody mass indexTime to walk 10 mDecrease in chest TcpO₂Active smokingPresence of buttock ischemiaGlycemiaResting heart rate	Ankle brachial indexTime to walk 10 mResting chest TcpO₂Body mass indexActive smokingPresence of buttock ischemiaResting heart ratebeta blocker use

In bold characters are variables present in both linear regression analysis and non-linear analysis. Note that coefficients for the linear regressions are provided in Table 3 whereas, by construction, non-linear analysis does not result is coefficients.

TcpO₂: transcutaneous oxygen pressure.

Table 3.

Coefficients resulting from the model of linear regression analysis in the training set.

	Beta ± SD of beta	t	p
Constant	1558.0 ± 183.6	+8.48	<0.0001
Age	−8.0 ± 1.3	−6.30	<0.0001
Ankle to brachial index	308.0 ± 55.1	5.59	<0.0001
Time to walk 10 m	−45.9 ± 9.1	−5.03	<0.0001
TcpO₂ at rest	3.9 ± 1.1	+3.61	<0.001
Body mass index	−13.5 ± 2.9	−4.72	<0.0001
Active smoking	−123.6 ± 29.4	−4.21	<0.0001
Presence of buttock ischemia	−98.1 ± 26.2	−3.74	<0.001
Resting heart rate	−2.9 ± 0.8	−3.44	<0.001
Ongoing use of beta blockers	−102.2 ± 29.8	−3.43	<0.001

TcpO₂: transcutaneous oxygen pressure.

Nonlinear analysis

The best results were obtained through nine indexes. The first three include ABI, age and body mass index. In the following factors notably are usual-pace, smoking, buttock ischemia and heart rate that were also found in the linear regression analysis model.

Of interest is to note that, from the non-linear analysis, none of the following variables seemed to have an impact on treadmill performance: gender, Sat-min, diabetes, the number of reported co-morbidities, hemoglobin, beta-blockers intake, absolute chest TcpO₂ at rest.

Application of the models

Application of the models to the second group of 320 patients (Group B) confirms that the models were robust with a coefficient of correlation of 0.509 and 0.575 for the linear regression analysis and non-linear analysis respectively with the whole nine variables (both p < 0.05). Of interest is to note that when the number of variables introduced in the model are reduced to keep only the variables that show the highest impact over the model. The correlation on the results issued from the model to the 320 patients of Group B, decreases when less than five variables are used (Figure 4).

Figure 4.

Correlations with the number of variable (Nb) variables for the linear regression analysis (black bars) and non-linear analysis (gray bars) in Group B.

Discussion

Walking impairment in claudication is basically due to the misfit between oxygen supply and oxygen consumption of the exercising muscle. Thereby, beyond indices of hemodynamic impairment, variables that may indicate factors able to worsen exercise ability (walking time) are of interest.

In the estimation of walking disability, the most largely used and readily available hemodynamic and clinical variables studied in patients with vascular-type claudication are ABI and age. Lower ABI values were shown to be associated with shorter distance achieved in the 6 min walk test and walking distance on treadmill,¹² but the associations between walking distance and absolute ABI value ranges non-significant associations r = 0.09 to 0.46 at best, and generally results from the analysis of small groups.^8–11 Consistently, repeated evidence exists for the decrease in performance with aging in PAD patients.²⁷ The fact that Age and ABI are the two most important factors of walking time in both the linear regression analysis and non-linear analysis approaches are only confirmatory of previous evidence. The specific interest here is the fact that when reducing the number of factors in Group B, the correlation dramatically decreases for both linear regression analysis and non-linear analysis. We assume that fact that the final and very first factor is ABI for the non-linear analysis and Age for the linear regression analysis rely on the expected non-linear relationship between ABI and the severity of walking impairment vs. the linear relationship between walking impairment and age. This is consistent with the non-linear relationship between ABI and morbid-mortality in population studies.²⁸

Our study confirms the results reported in the literature that obesity (body mass index) is associated with functional decline in persons with PAD.²⁹ This might rely in higher calf muscle percentage of fat and greater declines in calf muscle in obese PAD patients³⁰ PAD patients have a reduced walking speed than patients without PAD,⁵ nevertheless, previous studies failed to prove a relationship between usual walking speed and exercise performance in PAD patients.³¹ The fact that usual-pace was found in both models as one of the most important factor is thus yet an original, and quite logical, result. Recently, Ritti-Dias et al.³² showed that gender (female), race (non-Caucasian), and hypertension participated to walking limitations but the statistical analyses of the data were done using univariate analyses. Several prior studies reported an association between gender and treadmill walking performance.³³ Whether the high prevalence of males in our population and the small number of females could explain our negative result is a possibility. Specific distribution of race of our patients as compared to the American population is the specific reason why we did not include race as a factor. Almost all our patients were Caucasians.

A systematic review of studies that measured the magnitude of the effect of smoking on developing PAD reported an average risk for developing symptomatic PAD in smokers to over twice that of non-smokers, and supported a dose-response association between smoking and PAD.³⁴ In a recent work, Fritschi et al. showed that patients with PAD who smoke have earlier onset of claudication pain than patients with PAD who do not smoke.³⁵ The present work confirms the smoking behavior as an important factor affecting walking time in the both linear regression analysis and non-linear statistical analyses. This underlines that smoking cessation is of major importance in the treatment strategy for patients with PAD and claudication.

A recent Cochrane review included six randomized controlled trials to evaluate the role of beta blockers compared with placebo in patient with PAD.³⁶ This review showed that currently, there is no evidence to suggest that beta blockers adversely affect walking distance.³⁶ Our study suggests that, if any effect of beta-blockers use is to be found, it is extremely limited. Indeed beta-blocker use was not found significant in the non-linear analysis and was found the less important (although significant) factor in the linear regression analysis approach.

There is a high rate of patients with co-morbidities such as osteo-articular, cardiac-vascular or pulmonary diseases. Few studies have tried to approach indirect estimations of cardiac or pulmonary function as limiting factors of exercise.³⁷ Because patients may be unaware of their co-morbid conditions, it seemed of interest to include objective estimators of pulmonary or cardiac unknown or unreported diseases. Because higher heart rates are associated with increased risk of cardiomyopathy,³⁸ we used resting heart rate, in addition to self-reported known co-morbid conditions, as a potential indirect estimator of unknown cardiac dysfunction. Although heart rate had one of the lowest impacts on walking time is was found significant in both linear regression analysis and non-linear analysis. Resting heart rate likely remains a rarely studied but interesting parameter, which deserves further investigations in PAD patients. Consistently, beyond self-reported pulmonary diseases, to our knowledge, no prior studies have ever analyzed potential indirect estimator of hypoxemia as a factor of walking impairment in PAD patients. Of interest is the fact that minimal saturation was not observed as a significant factor for walking time. It must be kept in mind that the relationship of oxygen saturation to oxygen pressure is sigmoid. Thereby, if starting oxygen pressure value is sufficiently high, a decrease of TcpO₂ during exercise (assumed to reflect arterial oxygen pressure decrease) may occur without apparent (or at least measurable) saturation decrease.

The role of hemoglobin in the functional capacity of patients with PAD remains unclear. In a cohort of patients with chronic obstructive pulmonary disease, anemia independently predicted dyspnea and reduced exercise capacity.³⁹ In theory, hemoglobin is the support for oxygen content and an increase in hemoglobin should show a favorable effect on walking time for a given flow. Amazingly, on the contrary hemodilution (decrease hemoglobin content) has been proposed as a treatment method of chronic limb ischemia.⁴⁰ Very few data on hemodilution are available in claudication. A small sample study, showed no clinically or physiologically beneficial effect of hemodilution in patients with severe intermittent claudication.⁴¹ Neither linear regression analysis nor non-linear analysis found hemoglobin as a factor associated to walking time in our group.

Self-reported walking capacity is comparable between patient with and without diabetes but patients with diabetes show more severe limitation on the treadmill.⁴² Beyond the presence of diabetes itself, the question remains whether what impairs walking ability is diabetes or the equilibrium of glycaemia at the time of the test. From our results, at least on the non-linear analysis, it seems that the glycaemia at the time of the test is still a significant independent factor, although diabetes was included in the number of co-morbid conditions. Further studies are needed to clarify this interesting point.

One original parameter analyzed in the present study is the estimation of the role of buttock ischemia as a confounding factor in the analysis of the relationship between ABI and walking impairment. Proximal ischemia is a frequent cause of exercise-induced pain in patients with a normal ABI at rest.¹⁵ The association of buttock ischemia to walking time observed here seems logical to us, but had never been shown previously. Indeed ABI is likely not the optimal hemodynamic factor to assess proximal ischemia. A bias may result from the recruitement of our population, due to the specific interest of our laboratory in proximal claudication.¹⁷ Although exercise TcpO₂ is not a routine technique, it is rapidly spreading in France⁴³ and abroad⁴⁴ and is more accurate than ultrasound or segmental pressures for the detection of proximal ischemia.^45,46

|Methodologies used in the study can also be discussed. Several points of criticism have been made against stepwise linear regression analyses.⁴⁷ Of interest here is the fact that the model issued from Group A was verified on a second group of 320 patients (Group B), although this new group is issued from the same laboratory and may suffer the same selection bias. The limits of the linear regression analysis, as well as non-linear associations expected for some of the studied factors, led us use an Non linear analysis (NLA). Compared to other non-linear methods, we chose SVM for many reasons: they are relatively new in the scope of machine learning methods and they are deterministic and robust for managing outliers.¹⁹ Furthermore, they resist to overfitting and are very efficient in generalization.²⁵ Moreover, SVM lead to small sets of selected features^22,23; finally, compared to other methods, the performances of SVM are often very high.²² If we focus specifically on the choice between SVM and neural networks, we chose SVM rather than neural networks because the former works much faster than the latter. This fact results from the SVM learning process that finds a unique optimal hyperplane. On the contrary, neural networks need more and more iterations.⁴⁸ Nevertheless one K-cross validation still takes 5 min time to be achieved on an Intel Quad-core 2.66 Mhz with 4 Mo of RAM, under Windows 7. This lapse of time is explained by the fact that, even if SVM are very fast to learn classification problems, they need more time to achieve regression prediction tasks.²⁵ Thus, the test of all the 2¹⁶ possibilities would have taken 228 days; therefore, genetic algorithms were necessary. In Group A, we obtain a correlation equal to 0.53 (±0.03). In the final Group B, the correlation reached 0.57. It could be surprising that the performances in this group are higher than the performances of the K mean correlations in Group A. Therefore, we could be worried about the generalization aspects of the learning machine. This is due to the fact that the final learning machine is trained on 800 patients, while the cross validations could only take 640 patients for each of the K learning processes. We could be interested in determining the impact of each variable in the final best computed SVM. Unfortunately, the SVM, we used, are based on Gaussian kernels which are highly nonlinear. Therefore, unlike the multiple linear regression method,⁴⁹ for example, we cannot determine precisely the percentage of the contributions (relative weight) of each given variable to the final learning machine. The results for Table 3 are not available for the non-linear approach. Nevertheless, we can compute the K-mean correlation coefficients while removing each time one of the nine variables as shown in Table 2. Thus, the variables can be ranked in order of their decreasing impact on the performances.

Limitation

First, this is a retrospective monocentric study with a relatively high prevalence of proximal and atypical claudication. Second, we did not study the impact of each comorbid condition separately and did not analyze the influence of clustered comorbid conditions on walking capacity. Third, we did not analyze the effects of anti-platelet agents, lipid-lowering agents, phosphordiesterase inhibitors on walking capacity although some were previously shown to influence walking capacity.⁵⁰ Last, the results observed may be influenced by the treadmill testing procedure used and may not be applicable to other methodologies.

Conclusion

As expected, we found that ABI and age have the highest impacts on the walking ability on treadmill in patients with PAD, but in an order that likely depends on the linear vs. non-linear approach used. Specifically, ABI is the main determinant of walking capacity in the non-linear approach (but not in the linear regression) consistent with the U-shape relationship between ABI and the severity and prognosis of the cardiovascular disease.¹³ Body mass index, walking speed (over 10 m in the corridor), active smoking, are also significant (in both the linear regression analysis and non-linear analysis models) and previously observed factor of walking impairment. Yet unknown factors observed in both the linear regression analysis and non-linear analysis models are the presence of Buttock regional blood ischemia at exercise, glycaemia and resting heart rate at the time of the test. These factors greatly improve the prediction of walking ability in the non-linear model (with “r” being almost doubled as compared to the sole use of ABI) and explain why intends to determine walking ability on the sole estimation of ABI has shown fair to low correlation in the past.

Footnotes

Acknowledgments

The authors thank I Laporte for technical help. The work is presented with collaboration of the SOCOS group.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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