Matt Mecham

03-08-1996, 10:00 AM

Dear All:

Following is a summary of the responses that I received regarding my

request for information about the HIC and other ways of predicting head

injury using accelerometer data. I would like to thank the individuals

who responded to my request. If anyone has any further comments or

information I would welcome your input.

Thank You

Matthew D. Mecham

matt.mecham@m.cc.utah.edu

11111111111111111111111111111111111111111111111111 11111111111111111111111111

>From MKLEINBERGER@nhtsa.dot.govMon Mar 4 18:43:02 1996

Date: Fri, 05 Jan 1996 14:51:59 -0500

From: Mike Kleinberger

To: matt.mecham@m.cc.utah.edu

Subject: HEAD INJURY CRITERION (HIC) -Reply

Matt -

I think that I can be of some help to you. Here at the National Highway Traffic

Safety Administration, we are very interested (and very involved) in predicting

head injuries from acceleration data. First of all, if you need a program to

calculate HIC, you can get it from our Web site. The URL for our ftp site is

ftp://ftp.nhtsa.dot.gov/pub/nrd/nrd10/software/tools/. There are some DOC files

available on the HIC program, and also binaries for a variety of platforms.

Regarding novel approaches to predicting head injuries, I would recommend that

you check out a paper that was just published at the 39th Stapp Car Crash

Conference (November 1995). The reference is as follows:

DiMasi FP, Eppinger RH, and Bandak FA. Computational analysis of head

impact response under car crash loadings. SAE Paper No. 952718, 39th Stapp

Car Crash Conference, November 1995.

This paper describes our current thoughts on using a finite element model of the

head in conjunction with accelometer data to predict strain in the brain tissue,

and thus brain injury.

I hope that this information is helpful.

Good Luck,

Mike Kleinberger

*****************************************

Dr. Michael Kleinberger

National Highway Traffic Safety Administration

Biomechanics Research Division

E-mail: mkleinberger@nhtsa.dot.gov

*****************************************

>>> matt mecham 01/05/96 01:28pm >>>

Dear Netters:

I am in search of a program that will calculate a HIC value (Head Injury

Criterion) for complex acceleration-time profiles. I am aware of the basics of

the HIC calculation for simple profiles, but understand that different methods

must be used for multiple peak acceleration data. If anyone has written such a

program, has access to a public domain program or specific information on how

this might be done, please contact me.

In addition, if anyone is aware of any novel approaches utilizing

acceleration-time profiles to predict head injury and set tolerance levels I would

appreciate the information. If significant interest is shown I will post a summary

of the replies.

Thanks in advance.

Matthew D. Mecham

22222222222222222222222222222222222222222222222222 22222222222222222222222

>From A.McIntosh@unsw.edu.auMon Mar 4 18:49:23 1996

Date: Tue, 9 Jan 1996 14:57:50 +1100

From: Andrew McIntosh

To: matt mecham

Subject: Re: HEAD INJURY CRITERION (HIC)

Dear Matt,

there are a number of algorythms for HIC, mostly the time period

over which HIC is calculated is defined. It would be best to consult an SAE

recommendation or FMVSS for a suitable definition. Otherwise general

information about HIC can be found in numerous papers presented over the

years at the Stapp and IRCOBI conferences. You will find many conflicting

opinions about HIC. To understand and evaluate HIC you must look at how the

Wayne State Uni. Tolerance Curve was derived.

There are more direct applications of the head's resultant acceleration time

history, e.g. maximum- or 3 ms. acceleration values. I presented a paper at

the 1993 Stapp conference where we used the logistic regression model to

develop relationships between injury predictors, e.g. maximum head

acceleration and HIC, and AIS head injury severity. This method permits one

to propose an injury tolerance threshold. The sample size is limited, as in

almost all areas of impact biomechanics, therefore the confidence limits are

wide.

Other applications of the head's acceleration are to use it as an input into

a numerical model, e.g. Stalnaker's (R.L) Translational Energy Criteria or

the Mean Strain Criterion. Head acceleration could also be used as an input

into a good FE model of the head, which then provided an output appropriate

for the establishment of injury. There is presently a great deal of

research in this area.

If you have any specific questions please write to me directly.

Best wishes,

Andrew Mc. ******************************************** ANDREW McINTOSH

(PhD) Department of Safety Science The University of New South Wales

Sydney Australia 2052 ph:*61 *2 *3855413 fax: *61 *2 *3856190 EMail:

a.mcintosh@unsw.edu.au

33333333333333333333333333333333333333333333333333 333333333333333333333333

>From pbishop@healthy.uwaterloo.caMon Mar 4 18:50:34 1996

Date: Mon, 15 Jan 1996 15:47:44 -0500 (EST)

From: pbishop@healthy.uwaterloo.ca

To: matt.mecham@m.cc.utah.edu

Subject: Calculating HIC

Dear Matt

At the University of Waterloo we have been calculating the HIC for a

number of years.This has been in conjunction with our work on head

protection in hockey and football.

The program was written several years ago by John Pezzack in Quick Basic

and it runs on a PC. The calculation takes a long time (compared to

calculating the GSI) because of the need to maximize HIC.

Just out of curiosity, why are you calculating HIC anyway? You probably

know already that HIC values in helmet testing are about 18% lower than

GSI values and both are merely boundry estimates of head injury

occurrance,although GSI has proven to be useful in setting helmet standards.

Since neither index obeys a scaling law their usefullness is limited.

In any event if you want more info re a computer program you can contact

me via email or by calling me at 519 885-1211 ext 2269.

Good luck with your search and project.

Pat Bishop PhD Professor

44444444444444444444444444444444444444444444444444 44444444444444444444444

>From 75144.3653@compuserve.comTue Mar 5 10:00:03 1996

Date: 05 Mar 96 10:06:17 EST

From: Fred

To: "Matthew D. Mecham"

Subject: HIC

Hello,

Just wondering what responses you got to your 1/5/1996 HIC query.

Do you have anything you could send me?

here is somthing:

C PROGRAM **HICOP** - OPTIMIZED HIC ALGORITHM FROM DOT 9/25/85

DIMENSION R(0:8000),JL(10000),KL(10000)

COMMON X(8000,25),T(8000)

DOUBLE PRECISION RC,RM,SC,SM

CHARACTER*60 NAME,LABEL

DATA NAR/10000/

NAME='**PROGRAM HICOP**'

CALL INOUT(NAME,1,0)

c WRITE(6,1)

c 1 FORMAT(10X,'**PROGRAM HICOP**')

READ(10,2)NPT,IFILE,NCHN,LABEL

2 FORMAT(3I4,A60)

write(6,17)npt,ifile,nchn,label

17 format(3i4,1x,a60)

if(npt.gt.8000)then

write(6,30)

30 format('npt too large, must be less than 8000')

close(10)

stop

end if

if(nchn.gt.24)then

write(6,31)

31 format('nchn too large, must be less than 25')

close(10)

stop

end if

DO 3 I=1,NPT

3 READ(10,4)T(I),(X(I,J),J=1,NCHN)

4 FORMAT(F8.2,24F10.3)

close(10)

DEL=(T(2)-T(1))/1000.

C FORM RESULTANT ARRAY

DO 6 I=1,NPT

6 X(I,25)=0.0

16 WRITE(6,5)

5 FORMAT('INPUT CHANNELS TO BE USED FOR RESULTANT'/'XX XX XX')

READ(5,7)I1,I2,I3

7 FORMAT(3(I2,1X))

IF((I1.EQ.0).AND.(I2.EQ.0).AND.(I3.EQ.0))THEN

CALL STOP

stop

END IF

IF(I1.EQ.0)I1=25

IF(I2.EQ.0)I2=25

IF(I3.EQ.0)I3=25

DO 8 I=1,NPT

8 R(I)=SQRT(X(I,I1)**2+X(I,I2)**2+X(I,I3)**2)

R(0)=0.0

C COMPUTE SUM (INTEGRAL) ARRAY USING TRAPEZOID RULE

SM=0.D0

RM=ABS(R(0))

R(0)=0.0

DO 9 I=1,NPT

RC=ABS(R(I))

SC=SM+.5*(RC+RM)

R(I)=SC

SM=SC

9 RM=RC

C INITIALIZE VARIABLES

FMAX=0.0

JMAX=0

KMAX=0

JL(1)=0

KL(1)=-1

NWU=NPT

IBR=1

IER=1

INC=-1

IBL=NAR

C BEGIN A ROUND OF REGION EVALUATIONS

10 IL=IBL-INC

IEWR=0

DO 11 IR=IBR,IER,INC

JLI=JL(IR)

IF(JLI.GT.0)THEN

NJ=NWU

ELSE

JLI=ABS(JLI)

NJ=MAX0(NWU-1,0)

END IF

KLI=KL(IR)

IF(KLI.GT.0)THEN

NK=NWU

ELSE

KLI=ABS(KLI)

NK=MAX0(NWU-1,0)

END IF

KUI=KLI+NK

C BLANK LINE

IF((NJ.EQ.0).AND.(NK.EQ.0))THEN

C THIS IS A SINGLE POINT REGION - EVALUATE

F=(R(KUI)-R(JLI))*(KUI-JLI)**(-.6)

IF(F.GT.FMAX)THEN

FMAX=F

JMAX=JLI

KMAX=KUI

END IF

ELSE

C THIS IS A FINITE REGION WITH FEASIBLE POINTS - BOUND

JUI=JLI+NJ

FS=R(KUI)-R(JLI)

IF(FS*MAX0(KLI-JUI,1)**(-.6).GT.FMAX)THEN

C REGION IS POSSIBLY OPTIMAL - EVALUATE AND PARTITION

F=FS*(KUI-JLI)**(-.6)

IF(F.GT.FMAX)THEN

FMAX=F

JMAX=JLI

KMAX=KUI

END IF

IF((IL+4*INC)*INC.GT.IR*INC)THEN

C SIZE LIMITS REQUIRE EXHAUSTIVE ASSESSMENT TO FINISH

IF(IEWR.EQ.0)WRITE(6,12)

12 FORMAT(' **HICOP ARRAY SIZE EXCEEDED - EXHAUSTIVE EVAL. NEC.')

IEWR=1

DO 13 KJD=KUI-JLI,MAX0(KLI-JUI,1),-1

SKJM=0.0

DO 14 J=MAX0(JLI,KLI-KJD),MIN0(JUI,KUI-KJD)

SKJ=R(J+KJD)-R(J)

IF(SKJ.GT.SKJM)THEN

SKJM=SKJ

JM=J

END IF

14 CONTINUE

FB=SKJM*KJD**(-.6)

IF(FB.GT.FMAX)THEN

FMAX=FB

JMAX=JM

KMAX=JM+KJD

END IF

13 CONTINUE

ELSE

CALL PART(JLI,JUI,NJ,NWU-NJ,KLI,KUI,NK,NWU-NK,IL,INC,JL,KL)

END IF

END IF

END IF

11 CONTINUE

IF(ABS(IBL-IL-INC).GT.0)THEN

C SOME REGIONS REMAIN - SET UP BOUNDARY INDEX ARRAYS

ITMP=IBL

IBL=IER

IER=ITMP

IBR=IL

INC=-INC

NWU=NWU/2

GO TO 10

ELSE

C DONE - NO REGIONS REMAIN - REPORT RESULTS

HIC=DEL*FMAX**2.5

T1=1000*JMAX*DEL

T2=1000*KMAX*DEL

WRITE(6,15)HIC,T1,T2

15 FORMAT(' HIC= ',F15.2,' T1= ',F10.2,' T2= ',F10.2)

GO TO 16

END IF

CALL STOP

stop

END

SUBROUTINE PART(JLI,JUI,NJ,NJD,KLI,KUI,NK,NKD,IL,INC,JL,KL)

C PERFORMS PARTITIONING OF POSSIBLY OPTIMAL REGIONS

DIMENSION JL(*),KL(*)

J2M=2*MOD(NJ,2)

NJ2=NJD*J2M

JMU=(JUI-NJ/2)*(1-NJ2)

K2M=2*MOD(NK,2)

NK2=NKD*K2M

KMU=(KUI-NK/2)*(1-NK2)

C FIRST REGION

IL=IL+INC

JL(IL)=JMU

KL(IL)=KMU

C SECOND REGION IF FINITE WIDTH ALONG J AXIS

IF(NJ.NE.0)THEN

JLI=JLI*(J2M-NJ2-1)

IL=IL+INC

JL(IL)=JLI

KL(IL)=KMU

END IF

IF(NK.NE.0)THEN

KLI=KLI*(K2M-NK2-1)

C THIRD REGION IF FINITE WIDTH ALONG J AND K AXES

IF(NJ.NE.0)THEN

IL=IL+INC

JL(IL)=JLI

KL(IL)=KLI

END IF

C FOURTH REGION IF FEASIBLE AND FINITE WIDTH ALONG K AXIS

IF(ABS(KMU)-1.GT.ABS(JMU))THEN

IL=IL+INC

JL(IL)=JMU

KL(IL)=KLI

END IF

END IF

RETURN

END

********************************************

Following is a summary of the responses that I received regarding my

request for information about the HIC and other ways of predicting head

injury using accelerometer data. I would like to thank the individuals

who responded to my request. If anyone has any further comments or

information I would welcome your input.

Thank You

Matthew D. Mecham

matt.mecham@m.cc.utah.edu

11111111111111111111111111111111111111111111111111 11111111111111111111111111

>From MKLEINBERGER@nhtsa.dot.govMon Mar 4 18:43:02 1996

Date: Fri, 05 Jan 1996 14:51:59 -0500

From: Mike Kleinberger

To: matt.mecham@m.cc.utah.edu

Subject: HEAD INJURY CRITERION (HIC) -Reply

Matt -

I think that I can be of some help to you. Here at the National Highway Traffic

Safety Administration, we are very interested (and very involved) in predicting

head injuries from acceleration data. First of all, if you need a program to

calculate HIC, you can get it from our Web site. The URL for our ftp site is

ftp://ftp.nhtsa.dot.gov/pub/nrd/nrd10/software/tools/. There are some DOC files

available on the HIC program, and also binaries for a variety of platforms.

Regarding novel approaches to predicting head injuries, I would recommend that

you check out a paper that was just published at the 39th Stapp Car Crash

Conference (November 1995). The reference is as follows:

DiMasi FP, Eppinger RH, and Bandak FA. Computational analysis of head

impact response under car crash loadings. SAE Paper No. 952718, 39th Stapp

Car Crash Conference, November 1995.

This paper describes our current thoughts on using a finite element model of the

head in conjunction with accelometer data to predict strain in the brain tissue,

and thus brain injury.

I hope that this information is helpful.

Good Luck,

Mike Kleinberger

*****************************************

Dr. Michael Kleinberger

National Highway Traffic Safety Administration

Biomechanics Research Division

E-mail: mkleinberger@nhtsa.dot.gov

*****************************************

>>> matt mecham 01/05/96 01:28pm >>>

Dear Netters:

I am in search of a program that will calculate a HIC value (Head Injury

Criterion) for complex acceleration-time profiles. I am aware of the basics of

the HIC calculation for simple profiles, but understand that different methods

must be used for multiple peak acceleration data. If anyone has written such a

program, has access to a public domain program or specific information on how

this might be done, please contact me.

In addition, if anyone is aware of any novel approaches utilizing

acceleration-time profiles to predict head injury and set tolerance levels I would

appreciate the information. If significant interest is shown I will post a summary

of the replies.

Thanks in advance.

Matthew D. Mecham

22222222222222222222222222222222222222222222222222 22222222222222222222222

>From A.McIntosh@unsw.edu.auMon Mar 4 18:49:23 1996

Date: Tue, 9 Jan 1996 14:57:50 +1100

From: Andrew McIntosh

To: matt mecham

Subject: Re: HEAD INJURY CRITERION (HIC)

Dear Matt,

there are a number of algorythms for HIC, mostly the time period

over which HIC is calculated is defined. It would be best to consult an SAE

recommendation or FMVSS for a suitable definition. Otherwise general

information about HIC can be found in numerous papers presented over the

years at the Stapp and IRCOBI conferences. You will find many conflicting

opinions about HIC. To understand and evaluate HIC you must look at how the

Wayne State Uni. Tolerance Curve was derived.

There are more direct applications of the head's resultant acceleration time

history, e.g. maximum- or 3 ms. acceleration values. I presented a paper at

the 1993 Stapp conference where we used the logistic regression model to

develop relationships between injury predictors, e.g. maximum head

acceleration and HIC, and AIS head injury severity. This method permits one

to propose an injury tolerance threshold. The sample size is limited, as in

almost all areas of impact biomechanics, therefore the confidence limits are

wide.

Other applications of the head's acceleration are to use it as an input into

a numerical model, e.g. Stalnaker's (R.L) Translational Energy Criteria or

the Mean Strain Criterion. Head acceleration could also be used as an input

into a good FE model of the head, which then provided an output appropriate

for the establishment of injury. There is presently a great deal of

research in this area.

If you have any specific questions please write to me directly.

Best wishes,

Andrew Mc. ******************************************** ANDREW McINTOSH

(PhD) Department of Safety Science The University of New South Wales

Sydney Australia 2052 ph:*61 *2 *3855413 fax: *61 *2 *3856190 EMail:

a.mcintosh@unsw.edu.au

33333333333333333333333333333333333333333333333333 333333333333333333333333

>From pbishop@healthy.uwaterloo.caMon Mar 4 18:50:34 1996

Date: Mon, 15 Jan 1996 15:47:44 -0500 (EST)

From: pbishop@healthy.uwaterloo.ca

To: matt.mecham@m.cc.utah.edu

Subject: Calculating HIC

Dear Matt

At the University of Waterloo we have been calculating the HIC for a

number of years.This has been in conjunction with our work on head

protection in hockey and football.

The program was written several years ago by John Pezzack in Quick Basic

and it runs on a PC. The calculation takes a long time (compared to

calculating the GSI) because of the need to maximize HIC.

Just out of curiosity, why are you calculating HIC anyway? You probably

know already that HIC values in helmet testing are about 18% lower than

GSI values and both are merely boundry estimates of head injury

occurrance,although GSI has proven to be useful in setting helmet standards.

Since neither index obeys a scaling law their usefullness is limited.

In any event if you want more info re a computer program you can contact

me via email or by calling me at 519 885-1211 ext 2269.

Good luck with your search and project.

Pat Bishop PhD Professor

44444444444444444444444444444444444444444444444444 44444444444444444444444

>From 75144.3653@compuserve.comTue Mar 5 10:00:03 1996

Date: 05 Mar 96 10:06:17 EST

From: Fred

To: "Matthew D. Mecham"

Subject: HIC

Hello,

Just wondering what responses you got to your 1/5/1996 HIC query.

Do you have anything you could send me?

here is somthing:

C PROGRAM **HICOP** - OPTIMIZED HIC ALGORITHM FROM DOT 9/25/85

DIMENSION R(0:8000),JL(10000),KL(10000)

COMMON X(8000,25),T(8000)

DOUBLE PRECISION RC,RM,SC,SM

CHARACTER*60 NAME,LABEL

DATA NAR/10000/

NAME='**PROGRAM HICOP**'

CALL INOUT(NAME,1,0)

c WRITE(6,1)

c 1 FORMAT(10X,'**PROGRAM HICOP**')

READ(10,2)NPT,IFILE,NCHN,LABEL

2 FORMAT(3I4,A60)

write(6,17)npt,ifile,nchn,label

17 format(3i4,1x,a60)

if(npt.gt.8000)then

write(6,30)

30 format('npt too large, must be less than 8000')

close(10)

stop

end if

if(nchn.gt.24)then

write(6,31)

31 format('nchn too large, must be less than 25')

close(10)

stop

end if

DO 3 I=1,NPT

3 READ(10,4)T(I),(X(I,J),J=1,NCHN)

4 FORMAT(F8.2,24F10.3)

close(10)

DEL=(T(2)-T(1))/1000.

C FORM RESULTANT ARRAY

DO 6 I=1,NPT

6 X(I,25)=0.0

16 WRITE(6,5)

5 FORMAT('INPUT CHANNELS TO BE USED FOR RESULTANT'/'XX XX XX')

READ(5,7)I1,I2,I3

7 FORMAT(3(I2,1X))

IF((I1.EQ.0).AND.(I2.EQ.0).AND.(I3.EQ.0))THEN

CALL STOP

stop

END IF

IF(I1.EQ.0)I1=25

IF(I2.EQ.0)I2=25

IF(I3.EQ.0)I3=25

DO 8 I=1,NPT

8 R(I)=SQRT(X(I,I1)**2+X(I,I2)**2+X(I,I3)**2)

R(0)=0.0

C COMPUTE SUM (INTEGRAL) ARRAY USING TRAPEZOID RULE

SM=0.D0

RM=ABS(R(0))

R(0)=0.0

DO 9 I=1,NPT

RC=ABS(R(I))

SC=SM+.5*(RC+RM)

R(I)=SC

SM=SC

9 RM=RC

C INITIALIZE VARIABLES

FMAX=0.0

JMAX=0

KMAX=0

JL(1)=0

KL(1)=-1

NWU=NPT

IBR=1

IER=1

INC=-1

IBL=NAR

C BEGIN A ROUND OF REGION EVALUATIONS

10 IL=IBL-INC

IEWR=0

DO 11 IR=IBR,IER,INC

JLI=JL(IR)

IF(JLI.GT.0)THEN

NJ=NWU

ELSE

JLI=ABS(JLI)

NJ=MAX0(NWU-1,0)

END IF

KLI=KL(IR)

IF(KLI.GT.0)THEN

NK=NWU

ELSE

KLI=ABS(KLI)

NK=MAX0(NWU-1,0)

END IF

KUI=KLI+NK

C BLANK LINE

IF((NJ.EQ.0).AND.(NK.EQ.0))THEN

C THIS IS A SINGLE POINT REGION - EVALUATE

F=(R(KUI)-R(JLI))*(KUI-JLI)**(-.6)

IF(F.GT.FMAX)THEN

FMAX=F

JMAX=JLI

KMAX=KUI

END IF

ELSE

C THIS IS A FINITE REGION WITH FEASIBLE POINTS - BOUND

JUI=JLI+NJ

FS=R(KUI)-R(JLI)

IF(FS*MAX0(KLI-JUI,1)**(-.6).GT.FMAX)THEN

C REGION IS POSSIBLY OPTIMAL - EVALUATE AND PARTITION

F=FS*(KUI-JLI)**(-.6)

IF(F.GT.FMAX)THEN

FMAX=F

JMAX=JLI

KMAX=KUI

END IF

IF((IL+4*INC)*INC.GT.IR*INC)THEN

C SIZE LIMITS REQUIRE EXHAUSTIVE ASSESSMENT TO FINISH

IF(IEWR.EQ.0)WRITE(6,12)

12 FORMAT(' **HICOP ARRAY SIZE EXCEEDED - EXHAUSTIVE EVAL. NEC.')

IEWR=1

DO 13 KJD=KUI-JLI,MAX0(KLI-JUI,1),-1

SKJM=0.0

DO 14 J=MAX0(JLI,KLI-KJD),MIN0(JUI,KUI-KJD)

SKJ=R(J+KJD)-R(J)

IF(SKJ.GT.SKJM)THEN

SKJM=SKJ

JM=J

END IF

14 CONTINUE

FB=SKJM*KJD**(-.6)

IF(FB.GT.FMAX)THEN

FMAX=FB

JMAX=JM

KMAX=JM+KJD

END IF

13 CONTINUE

ELSE

CALL PART(JLI,JUI,NJ,NWU-NJ,KLI,KUI,NK,NWU-NK,IL,INC,JL,KL)

END IF

END IF

END IF

11 CONTINUE

IF(ABS(IBL-IL-INC).GT.0)THEN

C SOME REGIONS REMAIN - SET UP BOUNDARY INDEX ARRAYS

ITMP=IBL

IBL=IER

IER=ITMP

IBR=IL

INC=-INC

NWU=NWU/2

GO TO 10

ELSE

C DONE - NO REGIONS REMAIN - REPORT RESULTS

HIC=DEL*FMAX**2.5

T1=1000*JMAX*DEL

T2=1000*KMAX*DEL

WRITE(6,15)HIC,T1,T2

15 FORMAT(' HIC= ',F15.2,' T1= ',F10.2,' T2= ',F10.2)

GO TO 16

END IF

CALL STOP

stop

END

SUBROUTINE PART(JLI,JUI,NJ,NJD,KLI,KUI,NK,NKD,IL,INC,JL,KL)

C PERFORMS PARTITIONING OF POSSIBLY OPTIMAL REGIONS

DIMENSION JL(*),KL(*)

J2M=2*MOD(NJ,2)

NJ2=NJD*J2M

JMU=(JUI-NJ/2)*(1-NJ2)

K2M=2*MOD(NK,2)

NK2=NKD*K2M

KMU=(KUI-NK/2)*(1-NK2)

C FIRST REGION

IL=IL+INC

JL(IL)=JMU

KL(IL)=KMU

C SECOND REGION IF FINITE WIDTH ALONG J AXIS

IF(NJ.NE.0)THEN

JLI=JLI*(J2M-NJ2-1)

IL=IL+INC

JL(IL)=JLI

KL(IL)=KMU

END IF

IF(NK.NE.0)THEN

KLI=KLI*(K2M-NK2-1)

C THIRD REGION IF FINITE WIDTH ALONG J AND K AXES

IF(NJ.NE.0)THEN

IL=IL+INC

JL(IL)=JLI

KL(IL)=KLI

END IF

C FOURTH REGION IF FEASIBLE AND FINITE WIDTH ALONG K AXIS

IF(ABS(KMU)-1.GT.ABS(JMU))THEN

IL=IL+INC

JL(IL)=JMU

KL(IL)=KLI

END IF

END IF

RETURN

END

********************************************