Introduction

For an individual, if the total serum creatinine phosphokinase (CPK) levels are very high, it often indicates that there has been stress or injury to muscle tissue, or the brain, or the heart. Higher CPK levels most likely present muscle tissue injury. If a muscle is injured, CPK leaks into the bloodstream.

During the last few decades, extreme lifestyle changes such as obesity, stress, and lack of physical activity have led to life threatening situations such as acute myocardial infarction (AMI) as a fundamental cause of death in the developing world as well as in industrialized nations [1]. Myocardial necrosis is associated with the release of structural proteins and other intracellular macromolecules. If the unity of the cellular membranes is endanger, then some biomarkers such as CPK, creatine kinase (CK) MB, cardiac specific troponin I (cTnI), and cardiac specific troponin T (cTnT) leaks into the bloodstream and are measured in serum aiding in the detection of MI [2,3]. Generally, these biomarkers are analyzed by advanced biochemistry approaches for obtaining primary observation of AMI [4-6]. The article [3] has studied the association between serum and saliva levels of CPK, and it has compared salivary CPK as a biomarker between AMI patients and healthy individuals. Using advanced statistical modeling of these biomarkers, many interesting observations can be obtained regarding AMI, which are little studied. The basic myocardial energy holds reaction for generating adenosine triphosphate (ATP) is the CK reaction that reversibly transfers high-energy phosphate between adenosine diphosphate (ADP) and phosphocreatine (PCr) [7, 8].

 The key enzyme serum CPK is considered in regulating energy metabolism. It is considered as associated with insulin resistance Type 2 diabetes mellitus (T2DM), and as a risk factor for growing low muscle mass [9-11]. The article [9] has investigated the association of CPK with low muscle mass in T2DM patients. The correlation between CPK level and plasma glucose level for T2DM patients is studied in some articles [12-15]. Pierro, et al. [16] presented a retrospective analysis of the effects of a standard mixture of some components in dyslipidemia patients. There are very few articles focused on the effects of CPK on anaemic patients [17-20].

Many earlier published articles [1-20] have studied the effects of CPK on diabetes, heart and anaemia patients, using correlation, regression, meta-analysis approaches. The earlier findings related to the effects of CPK on the different patients such as diabetes, cardiac, and anaemia invite many doubts and debates, as the outcomes are not accepted based on statistical diagnostic tests. Moreover, earlier models are not accepted based on model fitting diagnostics tests. The effects of CPK are little searched using appropriate probabilistic modeling. The current report enquires the following research hypotheses.

• Is there any association of CPK with anaemia, diabetes and heart patients along with other anatomical/ biochemical variables? For the affirmative response, what is the most suitable CPK association model?
• How do we develop the most probable CPK relationship model?
• What are the effects of CPK on the anaemia, diabetes and heart patients and the other explanatory factors?

The article searches the above research hypotheses using the following paragraphs such as materials & methods, statistical analysis & results, discussions, and conclusions. The developed CPK probabilistic model is displayed using the data set stated in the materials section. The statistical CPK mean and dispersion models are derived using joint generalized linear models (JGLMs), which is illustrated in the methods section. The developed outcomes are presented in the results paragraph, while the outcomes of the analysis are illustrated in the discussion section. On the basis of the derived CPK mean and variance models, the report has obtained some new information that are presented in the conclusions section.

Material & Statistical Methods

Materials

The CPK probabilistic model is obtained herein from a data set of 299 heart failure subjects collected from Faisalabad Cardiology Institute (under an Allied Hospital), Faisalabad, Punjab province, Pakistan. The observations were collected for nine months i.e., April, 2025 to December 2015, and the data set was clearly described in the articles [21, 22]. The data set can be found in the site (https://archive.ics.uci.edu/ml/datasets/Hea rt+failure+clinical+records). The considered research study [21] contained 194 male and 105 female study subjects, who had left ventricular systolic dysfunction. The study individuals had earlier heart failures, and they were classed in III or IV heart failure stages of New York Heart Association (NYHA) classification [23]. The subject consents and the study ethics approval are neatly illustrated in the main article [21]. The study characters such as attributes and variables in the current research are separately illustrated in the article by Chicco and Jurman [24]. Recently an article by Das et al. [25] has shortly described the data set. For more information on the data set, interested authors are requested to go through the articles [21, 22, 24, 25].

The current study considers 13 interested characters out of which 7 are continuous variables, and the rest 6 are attributes. The 7 continuous variables are age (c1), creatinine phosphokinase (CPK) (c3), ejection fraction (EFT) (c5), platelets count (PLC) (c7), serum creatinine (SCT) (c8), serum sodium (SNa) (c9), total time to follow-up period (TTP) (c12), while the 6 attribute characters are the subject’s anaemia disease status (ADS) (0= no anaemia, 1= anaemia) (Fa2), diabetes mellitus status (DMS) (0= no diabetes, 1= diabetes) (Fa4), blood pressure status (BPS) (0= normal BP, 1=high BP) (Fa6), sex (0=female, 1=male) (Fa10), smoking status (SMS) (0=no smoking, 1= smoking) (Fa11), death event (DEE) (0=alive, 1=death) (Fa13). A study individual was classed as anaemia subject (Fa2) if her/his haematocrit level was lower than 36% [21]. The main article [21] of this data source does not give any information regarding the grouping of high BPS (Fa6) of the subjects. The death event (Fa13) presents that the patient survived (=0), or died (=1) up to the study termination time of 130 days on average [21].

Serum CPK levels gradually rise up early in the process of an acute myocardial infarction. Generally, CPK levels show its enzyme levels in blood. Enzyme elevations are encountered in the presence of brain damage and skeletal muscle. CPK flows into the blood, if a muscle tissue gets damaged. In contrast, individuals with diseases of the lung and liver consistently have normal CPK levels. Therefore, high blood’s CPK enzyme levels of an individual might show an injury or a heart failure [26]. A waste product serum creatinine (SCT) is produced by creatine if a muscle breaks down. Higher SCT blood levels may show renal dysfunction [27]. Therefore, kidney failure function is examined based on blood SCT levels. The original data source article [21] does not give any information regarding the subject’s kidney disease status/. The ejection fraction (EFT) represents the percentage blood amount the left ventricle pumps out per contraction. A mineral element, serum sodium (SNa) that is responsible for the accurate functioning of muscles and nerves. An abnormally very low SNa blood levels might be caused by heart failure [28]. More information about the data set is available in the original data source article [21].

Statistical Methods

The article considers CPK is the aimed continuous response random variable that is to be modeled with the left diabetes/cardiac/ anaemia status, physiological and biochemical factors. It is found that the response CPK is continuous, heteroscedastic, and nonnormally distributed random variable. The variance of CPK can’t be stabilized using any suitable transformation, so it is modeled in the article adopting joint generalized linear models (JGLMs) under both the log-normal and gamma distribution, which is clearly illustrated in [29-32]. Interested readers may go through JGLMs from the book by Lee, Nelder and Pawitan [29]. Log-normal and gamma JGLMs are shortly presented herein.

JGLMs for log-normal distribution

For the positive response Yi (=CPK) with E (Y_i=CPK) = µ_i (mean) and Var (Y_i=CPK) = ^σ_i² µ_i² = ^σ_i² V(^µ_i ) say, where, ^σ_i² are dispersion parameters and V() reveals the variance function. Generally, log transformation Z_i = log (Y_i=CPK) is adopted to stabilize the variance Var (Z_i) ≈ ^σ_i² , but the variance may not always be stabilized [33]. For developing a CPK improved model, JGLMs for the mean and dispersion are considered. For the response CPK, assuming log-normal distribution, JGL mean and dispersion models (with Z_i = log(Y_i=CPK)) are as follows:

E(Z_i)= µ_zi  and  Var(Z_i) = σ_zi², µ_zi=x_i^t β    and   log (σ_zi²)=  g_i^t γ,

Where ^x_i^t and ^g_i^t are the explanatory factors/variables vectors of CPK associated with the mean regression coefficients β and dispersion regression coefficients γ, respectively.

JGLMs for gamma distribution

In the above stated Y_i’s (=CPK), the variancehas twoportions such as V(^µ_i ) (based on the mean parametersµ_i’s) and ^σ_i²

(free of µ_i’s). The variance functionV ()shows the GLM family distributions. For example, if V(µ)= 1 , it is normal, Poisson if

V(µ) =µ, and gamma if V(µ) = µ²etc. Gamma JGLMs mean and dispersion models of CPK are as follows:

η_i = g(µ_i ) = x_i^tβandε_i =h(σ_i²) =w_i^tγ,

Where g(⋅) and h(⋅)are the GLM link functions attached with the mean and dispersion linear predictors respectively, and ^x_i^t , w_i^t , are the explanatory factor vectors of CPK attached with the mean and dispersion parameters respectively. Maximum likelihood (ML) method is used for estimating the mean parameters, while the restricted ML (REML) method is applied for estimating the dispersion parameters, which are explicitly stated in the book by Lee, Nelder and Pawitan [29].

Statistical Analysis & Results

Statistical Analysis

The response CPK is modeled herein by JGLMs with both the log-normal and gamma distributions. In the present statistical analysis, CPK is considered as the dependent variable, and the other 12 factors/ variables are considered as the CPK’s explanatory variables. Final CPK model has been taken based on the smallest Akaike information criterion (AIC) value (within each class), that reduces both the squared error loss and predicted additive errors [33, p. 203--204]. Based on the AIC rule, JGLMs Log-normal fit (AIC= 4197) is better than gamma fit (AIC=4249.731).

In the selected Log-normal mean model, three effects such as SMS^*TTP (P=0.1590) (for gamma fit P=0.0298), PLC^*DEE (P=0.1319) (for gamma fit P=0.0103), ADS^*PLC (0.5516) (for gamma fit P=0.0184) show partially significant, while they are significant in the gamma model (Table 1). In addition, ADS^*PLC (P=0.5516) is insignificant in the mean Log-normal fitted model, while it is highly significant (ADS^*PLC; P=0.0184) in the gamma fitted mean model. Without ADS^*PLC, the Log-normal model can’t be fitted. In the Log-normal dispersion model, three effects such as SEX^*TTP (P=0.0773), SCT^*TTP (P=0.1509) and SNa (P=0.1437) are partially significant. In Epidemiology, the partially significant effects (<16%) are called confounders, which are important in Epidemiology study. Also some partial or insignificant effects are sometimes very important for better model fitting [33].

Due to the marginality rule by Nelder [34], that isif an interaction effect (as an example in Table 1, SMS^*TTP) is significant, or partially significant, all its associated lower-order effects (as an example in Table 1, SMS and TTP) should be included in the model even insignificant. Based on the marginality rule, some partially significant, or insignificant effects such as SMS (P=0.1350), PLC (P=0.9045) and ADS (P=0.5456) are included in the mean model. Similarly, in the dispersion model, some partially significant, or insignificant effects such as AGE (P=0.2182), SCT (=0.6924), DMS (P=0.3861), SMS (P=0.2815) and PLC (P=0.8043) are included based on the marginality rule by Nelder [34].

Figure 1: For the joint Log-normal fitted models of Creatinine Phosphokinase (Table 1), the (a) absolute residuals plot with the fitted values, and (b) the normal probability plot for the mean model

The derived CPK Log-normal fitted probabilistic JGLM (Table 1) is a data derived model that is to be examined by model checking diagnostic plots. The valid conclusions regarding CPK are considered from the data derived Log-normal fitted CPK probabilistic model (Table 1) that should be admitted using model diagnostic checking plots, which are revealed in Figure 1. Figure 1(a) presents the absolute residuals for the Log-normal fitted CPK model (Table 1) with respect to the fitted values that is approximately a flat line, implying that the variance is constant with the running means. The normal probability plot for the Lognormal fitted CPK mean model (Table 1) is presented by Figure 1(b), which does not show any lack of fit. These two figures 1(a) and 1(b) show that the Log-normal CPK model fit is appropriate, which can be accepted as an approximate model of the unknown CPK model.

Results

Based on the AIC criterion, the Log-normal fitted model is better than the gamma fit. Here the outcomes of the Log-normal CPK fitted model are presented based on the comparison of the gamma CPK fitted outcomes.

The derived CPK mean model shows that mean CPK is positively associated with the AGE (P=0.0335) and EFT (P=0.0417), while it is negatively associated with their joint interaction effect (JIE) of AGE^* EFT (P=0.0058). Mean CPK is negatively associated with the SCT (P<0.0001), while it is positively associated with the EFT (P=0.0417), and their JIE of EFT^*SCT (P<0.0001). Mean CPK is indifferent to both the marginal effects of the subject’s ADS (P=0.5456) and PLC (P=0.9045), while it is negatively associated with their JIE of ADS^*PLC (P=0.0184, from gamma fit) (or P=0.5516, from Log-normal fit). Mean CPK is negatively associated with SCT (P< 0.0001) and the subject’s DMS (P=0.0103), while it is positively associated with their JIE of DMS^*SCT (P<0.0001).

Mean CPK is partially negatively associated with SMS (P=0.1350), while it is positively associated with TTP (P<0.0001), and the JIE of SMS^*TTP (P=0.1590) (partially). It is positively associated with SEX (P<0.0001) and TTP (P<0.0001), while it is negatively associated with their JIE of SEX^*TTP (P=0.0003). Mean CPK is negatively associated with SNa (P=0.0022) and DEE (P=0.0092), while it is positively associated with their JIE of SNa^*DEE (P=0.0121). Mean CPK is negatively associated with DEE (P=0.0092), and it is indifferent to PLC (P=0.9045), while it is positively associated with their JIE of PLC^*DEE (P=0.1319) (partially). Mean CPK is positively associated with TTP (P<0.0001), while it is negatively associated with SCT (P<0.0001) and the JIE of SCT^*TTP (P=0.0125).

Variance of CPK is indifferent to DMS (P=0.3861) and SMS (P=0.2815), while it is negatively associated with their JIE of DMS^*SMS (P=0.0023). Variance of CPK is positively associated with ADS (P=0.0229) and SEX (P= 0.0005), while it is negatively associated with their JIE of ADS^*SEX (P=0.0212). CPK’s variance is indifferent to PLC (P=0.8043) and positively associated with ADS (P=0.0229), while it is negatively associated with their JIE of ADS^*PLC (P=0.0400).

Variance of CPK is partially positively associated with SNa (P=0.1437) and AGE (P=0.2182). CPK’s variance is positively associated with SEX (P=0.0005) and TTP (P=0.0205), while it is negatively associated with their JIE of SEX^*TTP (P=0.0773). Variance of CPK is indifferent to SCT (P=0.6924), and positively associated with TTP (P=0.0205), while it is negatively associated with their JIE of SCT^*TTP (P=0.1509) (partially).

Log-normal fitted CPK mean (µ^ˆ _z) model (Table 1) is µˆ _z =

9.7874 + 0.0063 TTP – 0.3558 SMS + 0.0024 SMS^*TTP + 0.9725 SEX – 0.0060 SEX^*TTP - 0.0439 SNa -7.1459DEE + 0.0512 SNa^*DEE – 0.6232 SCT – 0.4787 DMS + 0.4091 DMS^*SCT + 0.0001 PLC+ 0.0001 PLC^*DEE – 0.2058 ADS – 0.0001 ADS^*PLC

+ 0.0332 AGE + 0.0474 EFT – 0.0010 AGE^*EFT + 0.0131

2 EFT^*SCT – 0.0015 SCT^*TTP, and the fitted CPK variance (σˆ z) model is 2

σˆ = exp.(-7.0836 + 0.0098 AGE + 0.0068TTP + 1.7261 SEX z

– 0.0050 SEX^*TTP+ 0.0698 SCT - 0.0023 SCT^*TTP+ 0.1872 DMS + 0.2753SMS –1.2557 DMS^*SMS + 1.5743 ADS – 0.9375 ADS^*SEX + 0.0001 PLC – 0.0001 ADS^*PLC + 0.0373 SNa).

From the above CPK mean ( µˆ ) and variance (σˆ² ) models, it z        z

is noted that mean CPK is expressed by TTP, SMS, SMS^*TTP, SEX, SEX^*TTP, SNa, DEE, SNa^*DEE, SCT, DMS, DMS^*SCT, PLC, PLC^*DEE, ADS, ADS^*PLC, AGE, EFT, AGE^*EFT, EFT^*SCT, and SCT^*TTP, while the CPK’s variance is expressed by AGE, TTP, SEX, SEX^*TTP, SCT, SCT^*TTP, DMS, SMS, DMS^*SMS, ADS, ADS^*SEX, PLC, ADS^*PLC, and SNa.

Discussion

The CPK analysis outcomes for Log-normal and gamma fitted models are displayed in Table 1. The Log-normal fitted CPK mean and variance models are displayed above. There are some discrepancies between the fitted Log-normal and gamma models, which are well discussed in [35]. Note that the considered data set contains two cardiac disease factors/parameters such as ejection fraction (c5) and high blood pressure status (0= normal BP, 1=high BP) (Fa6). It contains only one diabetes disease factor, namely the subject’s diabetes mellitus status (0= no diabetes, 1= diabetes) (Fa4), and one anaemia disease factors, namely the subject’s anaemia disease status (0= no anaemia, 1= anaemia) (Fa2). The given data set is a multivariate form. So, simple correlation study is not a proper method of finding the relationship. Again, multiple regression analysis is not a suitable method as the data set is heteroscedastic. Herein, appropriate data analysis models (Lognormal and gamma) are derived. Based on the derived models, the effects of CPK on heart, anaemia and diabetes patients are discussed.

Table 1: Results for mean and dispersion models for serum Phosphokinase from Log-normal & Gamma fit.