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 »  Introduction
 »  Material and methods
 »  Results
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Year : 2000  |  Volume : 48  |  Issue : 1  |  Page : 43-8

A mathematical outcome prediction model in severe head injury : a pilot study.

Department of Neurosurgery, Postgraduate Institute for Medical Education and Research, Chandigarh, 160012, India.

Correspondence Address:
Department of Neurosurgery, Postgraduate Institute for Medical Education and Research, Chandigarh, 160012, India.

  »  Abstract

103 patients of head injury, with a Glasgow coma scale (GCS) score of 8 or less, were studied prospectively. GCS score, brain stem reflexes, motor score, reaction level scale, and Glasgow Liege scale were evaluated as prognostic variables. Linear logistic regression analysis was used to obtain coefficients of these variables and mathematical formulae developed to predict outcome in individual patients.

How to cite this article:
Mukherjee K K, Sharma B S, Ramanathan S M, Khandelwal N, Kak V K. A mathematical outcome prediction model in severe head injury : a pilot study. Neurol India 2000;48:43

How to cite this URL:
Mukherjee K K, Sharma B S, Ramanathan S M, Khandelwal N, Kak V K. A mathematical outcome prediction model in severe head injury : a pilot study. Neurol India [serial online] 2000 [cited 2023 Dec 11];48:43. Available from:

   »   Introduction Top

A numerical method to estimate the probability of outcome following head injury would be useful to prognosticate and direct resources effectively. Several models for this purpose have been described.[1],[2],[3],[4] These are based on the data of representative population. The conditions and factors of these patient groups are not necessarily applicable to our setting. Prognostic factors predicting the outcome following head injury in India have also been described, but formulation of a mathematical model has not been seriously attempted.[5],[6],[7] This study attempts to determine an optimum set of indicators for prognostication. It also formulates a set of coefficients, based on which a prediction model has been developed.

   »   Material and methods Top

One hundred and fifty nine consecutive patients with severe head injury were admitted to the emergency services of Post Graduate Iinsitute of Medical Education and Research, Chandigarh, during a 3 months period. Of these, one hundred and three were selected prospectively for this study according to the coma data bank criteria, viz. any patient of any age or sex presenting within 24 hours of trauma, Glasgow coma scale(GCS) score of 8 or less after resuscitation and who have stayed within the above score for at least 6 hours following resuscitation.[8] Their haemoglobin was greater than 9 gm dl-1 with normal serum electrolytes and renal functions on admission. Patients with a history of alcohol intake, gun shot wound, associated significant soft tissue, bony, spinal, or multiple organ injuries causing hypovolaemic shock, were excluded from the study. Two infants in the series were not given the GCS score.
Variables used in the model: Besides detailed neurological examination, motor response (M) as per the motor score in GCS,[9],[10] brain stem reflexes (BSR)[11] and reaction level scale (RLS)[12] were determined at admission, at the end of 24 and 48 hours, and 1st and 2nd week. GCS and BSR were done as a part of the Glasgow Liege Scale (GLS)[1],[13] The RLS was given a score inversely related to its grade, i.e. grade 1(best response) was given a score of 8, grade 2 was given a score of 7, and so on upto a score of 1 for grade 8. Outcome of patients was assessed by Glasgow outcome scale (GOS) at discharge,[14] but collapsed into 3 categories: (a) good recovery and moderate disability (G-MD), (b) severe disability and persistent vegetative state (PVS-SD), and (c) death(D).
Statistical Methods: Linear logistic regression has been used to calculate the coefficients of the above mentioned variables.[15],[16],[17] The probability of attaining any of the above three outcomes has been evaluated.
A curve fitting conformal mapping, employing least square deviation, provided a linear regressional analysis for each set of prognostic indicator values and combinations of these indicators. These were further assessed for evaluating the respective probabilities for each outcome. The expected outcome values once obtained were between 0 and 2, with 0 denoting death, 1 denoting PVS-SD, and 2 denoting G-MD. Probabilities of predicted outcomes at extremes of the scale were higher than those at the middle. The accuracy of prediction varied with an estimated probability of outcome.
Information was processed on database software Oracle, version 6.0, and regressional correlation was established using the spreadsheet Lotus, version 3.4 and 4.0.

   »   Results Top

The average time between the injury and first examination was 9 hours (range 15 minutes to 24 hours). The median age was 30 years. There were 87 males and 16 females in the series. Road traffic accident was the mode of injury in 71 (68.9%), fall from height in 26 (25.2%), and assault in 3 (2.9%) patients. The mode of injury could not be determined in 3(2.9%) patients.
Seventy six patients had a solitary or multiple mass lesions, 37 of whom required an operative procedure. The post- resuscitation GCS, RLS, motor and BSR scores of the patients at admission are summarized in [Table I]. Pupillary inequality was observed in 30.1% of patients. 40.8% had an abnormal pupillary reaction at admission. Thirty six (34.9%) patients had good recovery/moderate disability, 24(23.3%) patients were in persistent vegetative state/ severe disability, and 43(41.7%) patients died. The coefficients of each prognostic variable and the probability of correctness of the prediction based on admission and 24 hour scores are given in [Table II].
The decreasing sample size with progression of time reduces the predictive efficacy of this methodology. The nature of progress of the patient in the course of time has not been taken into account as a factor determining the final outcome. The predictive efficacy of the scores can be graded by comparing the closeness of values predicted on the basis of fit to the actual values of outcome, after quantification. The combination of BSR and GCS, i.e GLS, was the most accurate predictor of outcome with a correct predictability of 84 per cent.
Mathematical Formulation: Expected value of outcome score = K + (G x C1) + (M x C2) + (B x C3)
+ (R x C4) = a value between 0 and 2. (K-constant, G-GCS score, M- Motor score, B- BSR score, R-RLS score, and C1,C2,C3 and C4 the coefficients of the respective scores). Constants have been given for individual coma score, its combination with BSR and all of them together, so that any scale could be used depending on the protocol used at a centre.
The expected numerical value of outcome score has a central bias. Thus domain of the central value (PVS-SD), must be suitably reduced to put it on equal footing with the extreme values (G/MD and D). The total domain is from '0' to '2', and can be divided into three sub-domains, i.e. 0 to 0.666 for outcome D, 0.667 to 1.332 for outcome PVS-SD and 1.333 to 2 for outcome G/MD. The net sum of mathematical probabilities of outcome, for any calculated outcome score, for a given patient has to be 1. One can use a quadratic function, or better still assume a cubic spline, to model the variation of probability of outcome with the calculated outcome score.
A few examples based on the above formulae, and constants and coefficients from [Table II] are described below:
1. Predicted mathematical outcome based on GCS score of 8 at admission is calculated as -0.6550 (constant for GCS) + 8 x 0.244 (coeff. when only Predicition Model in Head Injury GCS has been evaluated) = 1.297. Probability of PVS/SD is greatest because it lies in the domain (0.667 - 1.332), and the actual outcome was PVS/SD.
2. Predicted mathematical outcome based on GCS of 8, M of 5, BSR of 5, and RLS score of 5 at admission is calculated as -0.6027 (constant for GCS) + 8 x 0.2047 (coeff of GCS) + 5 x -0.1992 (coeff of M) + 5 x 0.0794 (coeff of BSR) + 5 x 0.1918 (coeff of RLS) = 1.3949. Probability of G/MD is greatest because it lies in the domain (1.33-2.0), and the actual outcome was G/MD.
3. Predicted mathematical outcome based on GCS of 7 + BSR of 3 (i.e GLS of 10) at admission is calculated as -0.2980 (constant of GLS) + 10 x 0.131 (coeff of GLS) = 1.012. Probability of PVS-SD is greatest, and the actual outcome was PVS-SD.

   »   Discussion Top

The use of a mathematical prognostic model would be helpful for counselling of the family regarding the chances and quality of survival following acute craniocerebral trauma. A good model can be utilized for stratification in clinical trials. If the expected probability of prediction changes significantly after the institution of a newer procedure, it can help us in accepting or rejecting it. It would be useful for audit too. However, one should not expect any system to be unfailingly accurate in a clinical setting. All prediction models contain simple clinical indicators like age, level of consciousness, reflexes, etc., which do not vary between centres. However caution should be exercised when directly applying western experience to the Indian setting because of differences in injury to treatment delay, treatment techniques, resources, and other support systems. These differences lead to variability in the numerical values attached to different indicators, and these may just as well vary between centres in the same country as between different countries.
An ideal prediction model should be based on only those criteria which have a high predictive value. It should be based on a large sample size, with specific inclusion and exclusion criteria. Various attempts for predicting outcome following head injury have been made earlier using different statistical techniques, like contingency tables, logistic models,[2-4] prognostic scores[1] and regression tree models.[18] A contingency table can easily estimate the probability of a good outcome, provided the predictor has a high value. The outcome is 'contingent' on the value of the predictor variable. Chi-square test is usually used for these tables. The problem with a contingency table is its inability to adjust for a potentially important covariable.
A discriminant function method is built to discriminate between two or more groups of patients with the use of a set of data. A linear discriminant makes several assumptions on the distribution of the predictors, which are often violated as the data becomes more complex. An alternative to this is a logistic regression model. Narayan et al described a model based on this method,[4] using age, GCS, pupillary reaction, eye movements and surgical mass as variables. Their equation for predicting poor outcome was 1/(1+reciprocal exponential of intercept + summation of the product of predictors and theircoefficients). The correct predictability rate with this model was 82 percent. However, it requires a scientific calculator with exponential function.
Choi et al used the linear discriminant analysis which yields a model with only three variables, viz. age, motor component of GCS, and pupillary response.[2] The final model is: log (7-motor component), age and the number of non-reactive pupils. This model is useful for modeling multiple outcomes, but application of the model to external data is complicated. The authors chose to present graphs that allow prediction based on the three factors. The correct prediction rate with this model was 78.4 percent. A stepwise logistic regression model was fit to select the best set of predictors in Klauber's model.[3] It used six variables and utilized the deciles of risk approach. Their equation to predict survival is 1/ (1+ exponential summation of the predictor variables times coefficients for the predictors). The overall goodness of this model is excellent and unlike other models is designed to be used with GCS scores of 3 to 15. The Glasgow Liege scale (GLS) is a combination of GCS and the brain stem reflexes.[1],[13] A multiple group multiple logistic discriminant analysis has been used in this model. This produces two equations which correctly classify 79.3% of the patients.
Choi et al reported a new method of prediction using the classification and regression tree model.[2],[18] It performs a series of analyses that divide the sample into smaller and smaller subsets. At each step the variable that best divides the sample into two groups is found, and the sample divided. The process is repeated on each subset until a predetermined set of groups or sample size in each group is reached. Four variables have been used in the final model with an overall correct prediction rate of 77.7 percent.
Several other prognostic indicators are undergoing intense validation studies. They include brain tissue oxygenation,[19] intracranial pressure plateau waves,[20] cerebral perfusion pressure, electroencephalography, blood flow velocity on transcranial doppler, somatosensory evoked potential, and brainstem auditory responses.[21],[22] Factors which promote secondary brain injury like hypoxia, hypotension, and hyperglycaemia too, are being stressed upon.[23],[24]
From the statistical stand point, it is impossible to compare the efficacy of these models, as they were derived on different patients using different variables. Titterington et al demonstrated that it was the choice of variables and the setting in which they were applied which was more important rather than the formulae per se.[25]An implication of the analysis of Choi et al is that the importance of specific predictors may vary among different centres. The inclusion and exclusion criteria are also variable. As most of the models are based on western population surveys, it may not be well suited for our country. Consequently we have attempted to derive a set of variables suited to our setting to establish the relevent coefficients and constants. Ours is a simple linear relationship, relatively easier for a doctor to apply, as it utilizes simple addition and multiplication. It does not require even a calculator to derive the outcome value. Moreover we have reduced the list of potential predictors to a minimal one, that is not likely to be affected by the sophistication of care. Therefore, such variables as GCS, brain stem reflexes, etc., were included, while those obtainable by CT scan and intracranial pressure monitoring, were excluded from this report. This has not reduced its prediction efficiency, which at 84% for GLS is comparable to other models. However, addition of these excluded variable might enhance the efficacy further. The findings of CT scan in our study is being analysed separately and a prediciton model based on it is being developed on the lines of the abbreviated injury scale.[26] This will be published as a subsequent report. As with other models, this too tries to give a numeric value to the outcome and hence predicts a probability of a particular outcome. Addition of data from different centres will further improve the reliability of its prediction.
The limitation of this mathematical model should be kept in mind during application. The attempt to utilize Glasgow outcome scale at discharge is inappropriate, but being a pilot study it was chosen due to logistical ease. The subsequent plan is to attempt this model on an independent set of patients and Glasgow outcome scale at six months. Other patient variables like alcohol, drugs and hypotension, though excluded, would influence the outcome in an individual patient. It is of utmost importance that the calculated prediction should not cloud the clinical judgement of the surgeon in deciding appropriate management of individual patients.


  »   References Top

1.Born JD, Albert A, Hans P et al: Relative prognostic value of best motor response and brain stem reflexes in patients with severe head injury. Neurosurgery 1985; 16: 595-601.   Back to cited text no. 1    
2.Choi SC, Narayan RK, Anderson RL et al: Enhanced specificity of prognosis in severe head injury. J Neurosurg 1988; 69: 381-385.   Back to cited text no. 2    
3.Klauber MR, Marshall LF, Luerssen TG et al: Determinants of head injury mortality: Importance of the low risk patient. Neurosurgery 1989; 24: 31-36.   Back to cited text no. 3    
4.Narayan RK, Greenberg RP, Miller JD et al: Improved confidence of outcome prediction in severe head injury. J Neurosurg 1981; 54: 751-762.   Back to cited text no. 4    
5.Mahapatra AK, Tandon PN, Bhatia R et al: Bilateral decerebration in head injury patients. An analysis of sixty two cases. Surg Neurol 1985; 23: 536 - 540.   Back to cited text no. 5    
6.Roy Choudhury S, Sharma BS, Gupta VK et al: Risk factors in severe head injuries. Neurol India 1996; 44: 187-194.   Back to cited text no. 6    
7.Tandon PN: Management of head injury. Fads, fashion and facts. Neurol India 1986; 34: 1-30.   Back to cited text no. 7    
8.Marshall LF, Baker DP, Bowers SA: The national coma data bank. J Neurosurg 1983; 59: 276-284.   Back to cited text no. 8    
9.Jennett B, Teasdale G: Aspect of coma after severe head injury. Lancet 1977; 1: 878-881.   Back to cited text no. 9    
10.Selhorst JB: Neurological examination of head injured patients. In: Becker DP, Gudeman SK, (Eds.) Textbook of Head Injury. WB Saunders, Philadelphia 1989: 85-92.   Back to cited text no. 10    
11.Fisher CM: Neurological examination of the comatose patient. Acta Neurol (Scand) 1969; 45: 5-56.   Back to cited text no. 11    
12.Starmark JE, Stalhammar D, Holmgren E: The reaction level scale (RLS 85) manual & guidelines. Acta Neurochir (Wien) 1988; 91: 12-20.   Back to cited text no. 12    
13.Born JD: The Glasgow Liege Scale. Acta Neurochir (Wien) 1988; 91: 1-11.   Back to cited text no. 13    
14.Jennett B, Bond M: Assessment of outcome after severe brain damage. Lancet 1975; 1:480-484.   Back to cited text no. 14    
15.Experienced professors. Statistics simplified. Naba Bharat Publishers, Calcutta. 1966.   Back to cited text no. 15    
16.Mounsey J: An introduction to statistical calculations. The English Universities Press Ltd. London. 1964.   Back to cited text no. 16    
17.Shukla SM, Sahay SP: Principles of Statistics. Sahitya Bhavan Publications, Agra. 1995: 484-488.   Back to cited text no. 17    
18.Choi SC, Muizelaar JP, Barnes TY et al: Prediction tree for severely head injured patients. J Neurosurg 1991; 75: 251-255.   Back to cited text no. 18    
19.Menzel M, Doppenberg EMR, Zauner A et al: Increased inspired oxygen concentration as a factor in improved brain tissue oxygenation and tissue lactate levels. J Neurosurg 1999; 91: 1-10.   Back to cited text no. 19    
20.Czosnyka M, Smielewski R, Piechnik S et al: Hemodynamic characterization of intra cranial pressure plateau waves in head injured patients. J Neurosurg 1999; 91: 11-19.  Back to cited text no. 20    
21.Guerra WKW, Gaab MR, Dietz H et al: Surgical decompression for traumatic brain swelling; indicators and results. J Neurosurg 1999; 90: 187-196.   Back to cited text no. 21    
22.Moulton RJ, Shedden PM, Tucker WS et al: Somatosensory evoked potential monitoring following severe closed head injury. Clin Invest Med 1994; 17(3): 187-195.   Back to cited text no. 22    
23.Yang SY, Zhang S, Wang ML: Clinical significance of admission hyperglycemia and factors related to it in patients with acute severe head injury. Surg Neurol 1995; 44(4): 373-377.  Back to cited text no. 23    
24.Chesnut RM: Secondary brain insults after head injury: clinical perspectives. New Horiz 1995; 3(3): 366-375. 25.   Back to cited text no. 24    
25.Titterington DM, Murray GD, Murray LS et al: Comparison of discrimination techniques applied to a complex data set of head patients. J Royal Statist Soc Assoc 1981, 144: 145-175.   Back to cited text no. 25    
26.Walder AD, Yeoman PM, Turnbull A: The abbreviated injury scale as a predictor of outcome of severe head injury. Intensive Care Med 1995; 21(7): 606-609   Back to cited text no. 26    


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