Computer simulation and spatial modelling in heart surgery

Computer simulation and spatial modelling in heartsurgery

Roman Trobec a, *, BosÏ tjan Slivnika, Borut Gersakb, Tone Gabrijelcic b

aDepartment of Digital Communications and Networks, JozÏef Stefan Institute, Ljubljana, SloveniabDepartment of Cardiovascular Surgery, Medical Center, Ljubljana, Slovenia

Received 8 December 1997

Abstract

In this work, three dimensional modelling and computer simulation of heat transfer on generally-shaped nonhomogeneous bodies is proposed and described. The complexity of the calculation isestimated and the potential use of high performance parallel computers is discussed. The method isfocused on applications in medicine. As an example, a numerical algorithm for the parallel computersimulation of heart cooling procedures during surgery is presented. On the basis of simulated results,two di�erent methods of cooling are compared. # 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Parallel computer simulation; Spatial modelling; Heat transfer; Heart surgery; Topical cooling

1. Introduction

The development of high performance computing (HPC) in recent years has enabled

computer simulation to spread throughout large areas of science and technology. HPC has

opened a wide range of new applications in medicine, such as the simulation of heat transport,

blood-¯ow [1], electromagnetic ®elds [2], etc. Computer simulation in medicine have several

advantages over in vivo experiments. For example, performing in vivo experiments and

measurements is often di�cult, dangerous or even impossible, while simulation can give insight

into physiological processes in the organism without harm. Besides, computer simulations are

often less expensive and faster than experimental studies.

The fundamental part of any computer simulation is a computer model of the physical

phenomena which is to be simulated [3]. In medicine, one is usually faced with three

Computers in Biology and Medicine 28 (1998) 393±403

0010-4825/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved.PII: S0010-4825(98)00023-7

PERGAMON

* Corresponding author. Tel.: +386-61-1773-497; Fax: +386-61-219-385; E-mail: [email protected].

dimensional (3-D) bodies of a general shape and processes that can be described by partialdi�erential equations. The modelling starts with the digitalisation of sliced object pictures. Thisraw information of the object must then be further re®ned through a number of steps(segmentation, ®ltering, etc.) in order to obtain a spatial model suitable for the particularapplication [4]. Simulation is generally performed by solving the partial di�erential equationsthat describe the simulated process. In practice, these equations can rarely be solvedanalytically and the solution must, therefore, be found using numerical methods whichtransform a particular partial di�erential equation to a set of algebraic equations [5]. The formand size of the algebraic equations depend on the underlying partial di�erential equations andthe type and level of discretisation. In the case of 3-D modelling, solving the large number ofequations can become a serious problem, even on the most advanced computerworkstations [4, 6].In the rest of the paper, our attention is focused on the modelling and calculation

complexity of the proposed simulation and on the limits posed upon the experiments, evenwith the use of powerful computers. The simulation of di�erent cooling methods during heartsurgery is given as an application example. We discuss and analyse the computationalcomplexity. The paper is concluded with some further remarks on possibilities and researchdirections in the area of HPC simulation.

2. Materials and methods

2.1. Materials

For successful operations on the heart, a few basic requirements are necessary.Cardiopulmonary bypasses (heart±lung machine) enable operations on a nonbeating heart.During this period, di�erent levels of systemic-body hypothermia are used to lower metabolicrequirements. Normal (moderate) levels of hypothermia are considered to be esophagealtemperatures of 288C and rectal temperatures of 308C. The aortic cross clamping stopscoronary circulation and a cardioplegic solution (usually at 48C) is used to maximally slowdown cardiac metabolism during this ischemic period. During cardioplegic administration theheart is topically cooled to a certain level.The simulation of topical cooling during open heart surgery was performed based on the

following model. To get a spatial 3-D heart model, several (in our application 64) computer-tomography slices of a human heart on the X±Y plane were generated (see Fig. 1). The sliceswere positioned equidistantly in the Z-direction that represented the heart axis, originatingfrom the apex towards the base. The resolution on the X±Y plane was determined by ascanning procedure. Because of the integrity of the spatial model, it was necessary to guaranteethe exact overlapping of all slices. Before digitalisation, two reference points on every picturewere marked, in order to adjust all data. In addition, all ¯aws and other errors in theexposures had to be discarded. Finally, the edges between di�erent substances had to bedetermined. The environment was imitated with an isolated cube, large enough to contain thesimulated heart. The lower half of the cube represented a cooling container and was ®lled witha cooling liquid. The container walls were kept at the esophageal temperature. The upper half

R. Trobec et al. / Computers in Biology and Medicine 28 (1998) 393±403394

of the cube represented the thorax that was surrounded by air at room temperature. Due tothe air circulation, the upper wall of the cube had a constant temperature of 208C.

2.2. Methods

Any physical model can be described by a set of di�erential equations which can be rarelysolved analytically. The basic physical equation which describes a heat transfer is a partialdi�erential equation called the di�usion equation. For the three space dimensions and constantdi�usion coe�cients D, it can be written as

@T

@t� D

�@2T

@x2� @

2T

@y2� @

2T

@z2

�, �1�

where T is a temperature function of space variables x, y, z and the time parameter t. It can beexpressed as T(x, y, z, t). Each substance found in the model is described by a di�usioncoe�cient D= l/cpr, where l, cp and r stand for heat conductivity, speci®c heat and speci®cweight, respectively. Therefore, the spatial model can be represented by a function D(x, y, z).The solution of Eq. (1) depends on initial values and boundary conditions. Duringphysiological modelling, temperatures at the beginning of simulation and temperatures at the

Fig. 1. Computertomography slices of a human heart on the X±Y plane, positioned along the Z axis. Slices 7, 23, 32

and 64 are shown only.

R. Trobec et al. / Computers in Biology and Medicine 28 (1998) 393±403 395

boundaries of spatial model during the simulation are determined according to theexperimentally measured values. They are used as the initial values T(x, y, z, 0) and constantboundary conditions Tb(x, y, z), respectively.

Except for the simplest models, Eq. (1) has to be solved numerically. In general, there aretwo di�erent approaches for solving PDEs, ®nite di�erence methods (FDM) [5] and ®niteelement methods (FEM) [7]. The latter are suitable for the models which can be easilydescribed in geometrical terms, particularly for homogenous bodies. However, in medicine oneis usually faced with generally shaped and nonhomogeneous spatial models and therefore theFEM are less appropriate. A discretised model is needed when a method of either type isapplied. In this paper, a combination of FDM and an equidistant 3-D orthogonal grid of npoints is used, with

�3np

points along each spatial dimension.

Using the explicit ®nite di�erence method (EFD), Eq. (1) must be discretised using theapproximations for the spatial and time derivatives and rewritten into a set of n linearequations, with each equation of the form

T�h�1�i,j,k ÿ T

�h�i,j,k

Dt� Di,j,k

�T�h�i�1,j,k ÿ 2T

�h�i,j,k � T

�h�iÿ1,j,k

Dx2� T

�h�i,j�1,k ÿ 2T

�h�i,j,k � T

�h�i,jÿ1,k

Dy2

� T�h�i,j,k�1 ÿ 2T

�h�i,j,k � T

�h�i,j,kÿ1

Dz2

�, �2�

for i, j and k from 1, 2, . . . ,�

3np

. Eq. (2) specify only the approximations of the function T(x,y,z, t) in the grid points at various time steps. A value denoted by T i,j,k

(h) is an approximation ofthe value T(iDx, jDy, kDz, hDt) in the grid point (i, j, k) in the hth time step. The di�erencesDx, Dy and Dz denote the distances between two points in each space dimension and Dt is thelength of the time interval between two consecutive approximations. The set of Eq. (2) can berewritten in a matrix form as

MT �h� � Tb � T �h�1�, �3�where the matrix M contains the coe�cients from Eq. (2), T (h) is the vector of approximationsT i,j,k

(h) , for all grid points (i, j, k) and Tb the vector of boundary conditions [8]. Starting with theinitial vector T (0), it is possible to compute T (h) for every h>0 by repeated use of Eq. (3). Ineach step this equation requires a matrix-vector multiplication and a vector addition, whichtake O(n 2) and O(n) operations, respectively. However, in this particular case the matrix M isextremely sparse and contains only seven diagonal stripes. The product MT (h) requires 7nmultiplications and 6n additions and hence the complexity of a single iteration is O(n).However, the time step Dt has an upper limit. In the 3-D model, Dt is bounded by thefollowing inequality�

maxx,y,z D�x, y, z��

Dt

�maxfDx, Dy, Dzg�2 R 1

6, �4�

which seriously slows down the EFD simulation. On the other hand, the EFD method remainsattractive because it can easily be parallelised with almost linear speed-up [9].


Using the same notation and discretisation as above, implicit ®nite di�erence method (IFD)transforms (1) into a system of n linear equations of the form

T�h�1�i,j,k ÿ T

�h�i,j,k

Dt� Di,j,k

�T�h�1�i�1,j,k ÿ 2T

�h�1�i,j,k � T

�h�1�iÿ1,j,k

Dx2� T

�h�1�i,j�1,k ÿ 2T

�h�1�i,j,k � T

�h�1�i,jÿ1,k

Dy2

� T �h�1�i,j,k�1 ÿ 2T �h�1�i,j,k � T �h�1�i,j,kÿ1Dz2

�, �5�

which can be written in the matrix form as

M 0T �h�1� � Tb0 � T �h�, �6�

where the matrix M 0 contains the coe�cients from Eq. (5) and Tb0 represents boundary

conditions. It takes O(n 3) operations to solve a system of linear equations using a direct solver,but if large systems are considered, only iterative solvers can be used. Even if the number ofiterations within any time step remains bounded, the upper bound can usually be estimatedonly and depends heavily on the matrix M 0 and on the time step Dt. No exact limit of practicalimportance as in the inequality Eq. (4) can be imposed upon Dt if the IFD method is used.However, the experiments show that Dt suitable for the IFD method can be of some orders ofmagnitude larger that the one used with the EFD method. Although solving a system of linearequations is much harder problem than mere matrix-vector multiplication, the IFD methodproves to be often more e�cient than the EFD method due to the larger Dt.More sophisticated implicit methods allows even larger time steps. For example, s-stage

multi-implicit Runge±Kutta method (MIRK) [10, 11], which has been used for the ®rst time insuch a kind of application, permits even 1000� longer time intervals Dt than the EFD methodwhile still providing the accuracy of results suitable for medical use [12]. The comparison withthe EFD method is shown in Fig. 2.Before the MIRK method can be applied, Eq. (1) must be semidiscretised in order to get a

set of ordinary di�erential equations (ODE) [8] and thus transformed into an initial-valueproblem

dT

dt�M0T�t� � Tb0�t�, �7�

where the matrix M0 and the vector T0 represent the system of ODE and boundary conditions,respectively. The MIRK method is based on the s-stage implicit Runge±Kutta method(IRK) [13], which can be expressed in the matrix form as

Yi � T �hÿ1� � DtXsj�1

aij�M0Yj � Tb0�, i � 1, 2, . . . , s, �8�

and

T �h� � T �hÿ1� � DtXsi�1

bi�M0Yi � Tb0�: �9�


In order to calculate intermediate approximations Y i, Eq. (8) of sn equations has to be solvedin each time step. The s-stage IRK can be speci®ed by the Butcher tableau

�10�

If the eigenvalues g i of the matrix A are all real and distinct, the corresponding eigenvectorsform the matrix P so that the matrix D= Pÿ1 AP is diagonal with eigenvalues of A on thediagonal. By introducing transformed vectors

~Y � �Pÿ1 In�Y and ~Tb � �Pÿ1 In��e Tb0�, �11�where Y=[Y 1

T, Y 2T, . . . , Ys

T ]T and in general means A B=[a ijB], Eq. (8) can betransformed into s independent systems

�Id ÿ DtgiM0� ~Yi �Xsj�1�Pÿ1�ijT �hÿ1� � Dtgi ~Tb,i, i � 1, 2, . . . , s, �12�

Fig. 2. The maximal absolute error in 8C of the MIRK method (s=3) for various time steps (h=0.03� 101,0.03�102, 0.03�103, 0.03�104 s) during the simulation which lasted 3600 s. The solution obtained by the EFDmethod with the time step h=0.03 s was taken as a reference.


where TÄ b,i is the ith component of TÄb= [TÄ b,1T , TÄ b,2

T , . . . , TÄ b,sT ]T. The systems of Eq. (12), each

consisting of n equations, can be solved independently what reduces the complexity ofcalculation from O((sn)3) to O(sn 3) and makes the MIRK method attractive for parallelimplementation.

3. Results

Cardiac surgery conditions were imitated as far as possible. At the start of the simulation,moderate hypothermia was assumed. At the end of the cardioplegia [14] the heart septaltemperature was approximately 118C. Further more, it was supposed that the aortic crossclamping lasted 60 min. We performed two simulations of topical heart cooling usingdi�erent conditions. In simulation A, a topical cooling liquid (TCL) with a constanttemperature of 0.28C was used; in simulation B, the initial TCL temperature was set to 0.28Cwithout forced cooling. The initial phase of the heat transfer is shown in Fig. 3. Thetemperature distribution after 50 min is represented in Fig. 4. One can clearly see the rise ofthe heart septal and especially, the ventricular wall temperature in simulation B, in contrastwith simulation A, where the situation is just the opposite. This is illustrated in more detail inFig. 5.The heat transfer was actually computed for the model of the human heart described in

Section 2.2. A standard monochrome CCD-camera with a resolution of 256�256 pixels was

Fig. 3. The initial state (time=0) of the slice 32 on the X±Y plane (left) and on the Z±X plane (right), with aresolution of 64�64 grid points. The horizontal bar shows the temperature values in the range from 0 to 288C.


used, similar as in [15, 16]. The digitalised values, in groups of the sixteen nearest neighbouringpixels, were averaged in order to get the ®nal resolution of 64� 64 grid points in each slice.The calculation was performed in the resolution of 64� 64� 64 grid points. The calculationswere performed on several sequential computers as well as on the parallel computer with 16ring-connected processors. The computation results are given in Table 1.

Fig. 4. Simulations A (left) and B (right): the temperature distribution on the X±Y plane after 50 simulated min. All

parameters are the same as in Fig. 3.

Fig. 5. Left ventricular free wall temperature (LVTemp) and cardiac septal temperature (STemp) during the

simulated period for the simulation with continuous topical cardiac cooling Ð simulation A (left) and without it Ðsimulation B (right).


4. Discussion

For a reliable estimation of heat transport even on ®ner and more sensitive structures of theheart, one would usually like the simulation procedure to be faster than in nature. This isparticularly true if several consecutive trials, simulating di�erent methods in di�erentconditions, must be performed. With ®ner time intervals and increased number of points, thenumber of equations and storage requirements increase signi®cantly. Such time and spacecomplexity can only be e�ciently overcame by the use of parallel computers. On the basis ofthe previous work, it is believed that in medicine, the 3-D simulations, supported by parallelHPC, can give adequate results. In order to produce more realistic simulation the di�usionequations should be replaced by another partial di�erential equation describing the di�usion aswell as the heat production as consequence of metabolism and the convectional transportinitiated by surrounding air. The resulting equation can be solved in the same way, with thesame methods as described in this paper with the complexity of the same magnitude.With slight modi®cations of mathematical formulation, the proposed methods of modelling

and simulation can also be implemented in other areas of interest i.e. in exploring the processesof blood ¯ow at di�erent vessel anastomoses in the heart or other parts of the arterial andvascular system, in searching for the correlation between the ¯ow and symptoms of disease inthe vessels [17], or in calculating forces and stresses as in the patellofemoral joint [18].

5. Conclusion

We proposed a procedure for 3-D modelling of generally-shaped bodies. The procedure issimple and requires only a little knowledge of computers. The resulting model can serve as astarting point for any other kind of computer simulation, or can be transformed further toreach the demands of even the most demanding simulation. In our example, the simpli®edmodel has been used because we focused our attention to the numerical methods andcalculation complexity. We have shown that the HPC computer simulation is powerful enoughto deal with 3-D models in the resolution of 64� 64� 64 domains. To get more precise resultsthe re®nement of physical and spatial models has to be done in the future. Consequently,parallel computer o�er one of the best choices for the future work. It is believed that thepresented methodology can be a valuable alternative in the development of new medicalknowledge.

Table 1CPU time for a single time step on di�erent computer platforms

Computer platform EFD (s) 3-MIRK (s) 5-MIRK (s)

Power Challenge, R8000/75 0.096 4.656 9.376DEC Alpha/233 0.128 7.992 14.000HP J210C, PA 7200/120 0.160 9.424 17.640HP 735/125, PA 7200/125 0.544 13.936 35.104

Pentium/133, Linux 1.008 21.000 43.97616�T800/20 0.330


6. Summary

The development of high performance computing (HPC) in recent years has enabled computersimulation to spread throughout large areas of science and technology. HPC has opened a widerange of new applications in medicine, such as the simulation of heat transport, blood ¯ow,electromagnetic ®elds, etc. Computer simulation in medicine has many advantages over in vivoexperiments. The fundamental part of any computer simulation is a computer model of thephysical phenomena, which are to be simulated. In medicine, one is usually faced with threedimensional (3-D) bodies of a general shape. Simulation is generally performed by solving thepartial di�erential equations that describe the simulated process. In practice, these equations canrarely be solved analytically. In the case of 3-D modelling, solving the large number of equationscan become a serious problem, even on the most advanced computer workstations.In the paper, our attention was focused on the modelling and calculation complexity of the

proposed simulation and on the limits posed upon the experiments, even with the use ofpowerful computers. A three dimensional modelling and heat transfer computer simulation ofgenerally-shaped, nonhomogeneous bodies, is proposed and described in this work. Thecomplexity of the calculation is estimated and the potential use of high performance parallelcomputers is discussed. The method is focused on applications in medicine. As an example, anumerical algorithm for the parallel computer simulation of heart cooling procedures duringsurgery is presented and, on the basis of simulated results, di�erent methods of cooling arecompared.In our example, the simpli®ed model has been used because we focused our attention on the

numerical methods and calculation complexity. We prove that the parallel computer simulationis powerful enough to deal with 3-D models in the resolution of 64�64� 64 domains. To getmore precise results the re®nement of physical model has to be done in the future. In order toproduce more realistic simulation the di�usion equations must be replaced by another partialdi�erential equation describing the di�usion, as well as the heat production as consequence ofmetabolism and the convectional transport initiated by surrounding air. The resulting equationcan be solved in the same way, using the same methods as described in this paper and thecomplexity of the same magnitude.

Acknowledgement

This work was supported in part by the Ministry of Science and Technology of the Republicof Slovenia, grant J2-5092-106-95.

References

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Roman Trobec received his BSc., MSc. and Ph.D. in electrical engineering from the University of Ljubljana. Since 1979 he has been

with the JozÏ ef Stefan Institute, where he is a research member in the department for communications and computer networks. His

group is involved in the design and development of digital transmission systems. His research interests are digital communication

systems, VLSI design, parallel computing and biomedical research.

BosÏ tjan Slivnik received his BSc. and MSc. degrees in Computer Science from the University of Ljubljana. Since joining the

department of digital communications and networks at JozÏ ef Stefan Institute in 1992, he has been involved in the research of

programming techniques for distributed memory computers and high performance computing applications. He is currently working

towards the Ph.D. degree in compilers.

Borut Gersak received his BSc., MSc. and Ph.D. from the University of Ljubljana, School of Medicine. Since 1986 he has been

with the department of cardiovascular surgery, University Medical Center, where he is an associate Professor of cardiovascular

surgery. His research interests extend beyond pure cardio Ð surgery to interdisciplinary approaches in surgery, biomedical

engineering and computer simulations of procedures used in cardiovascular surgery.

Tone GabrijelcÏ icÏ received his BSc., MSc. and Ph.D. from the University of Ljubljana, School of Medicine. Since 1979 he has been

working in the department for cardiovascular surgery of the University Medical Center, where he works as an assistant Professor

in surgery from 1995. He started heart transplant program in Clinical Center Ljubljana in 1990. This ®eld of surgery is his major

point of interest. He is the head of the department for cardiovascular surgery of Clinical Center Ljubljana, president of Slovenian

Society for Cardiac Surgery, and member of many medical associations in Slovenia and abroad.


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Computer simulation and spatial modelling in heart surgery