Phys. Med. Biol., 1986, Vol. 31, No 11, 1247-1256. Printed in Great Britain

The modelling of biological systems in three dimensions using the time domain finite-difference method: I. The implementation of the model

R W M Lau and R J Sheppard Department of Physics, King’s College, Strand, London WC2, England

Received 18 February 1986, in final form 13 May 1986

Abstract. A computer method has been developed which uses the time domain finite- difference (TDFD) algorithm to calculate the deposition of the electromagnetic (EM) field in three-dimensional biological models. This, the first of two papers, describes the algorithm and the computer programs developed. The method is demonstrated by calculating the penetration of the EM field from a rectangular waveguide radiating into a homogeneous model, the calculation being carried out in two dimensions for simplicity in this paper.

1. Introduction

RF- or microwave-induced hyperthermia is now accepted as a useful form of cancer therapy. However, one major problem is to ensure that the electromagnetic heating is concentrated in the tumour. Unfortunately it is very difficult to monitor either the electromagnetic field or the temperature rise in a clinical situation and the value of calculating the energy deposition is realised.

In particular it is necessary to solve the Maxwell equations of electromagnetic radiation, subject to the appropriate boundary conditions, and this can only be done using numerical methods when a realistic biological model is being considered. The state of the art of such computer modelling has been summarised in recent articles (Spiegel 1984, Strohbehn and Roemer 1984, Durney 1980). In electromagnetic-wave- induced hyperthermia, the employment of such computer models gives useful informa- tion on the energy deposition pattern inside the body and, as a result, various aspects of the treatment such as the design of the applicators and their positioning can be optimised.

Among other methods, one algorithm to solve Maxwell’s equations using a time- stepping finite-difference scheme was first introduced by Yee (1966). Since then it has been applied in solving problems such as the response of a military aircraft to a nuclear electromagnetic pulse (Holland 1977, Kunz and Lee 1978), the penetration of a plane wave into a guided missile cone (Taflove 1980) and the scattering from lossy dielectric spheres (Holland et a1 1980). The time domain finite-difference (TDFD) algorithm has been compared favourably with other analytical and numerical techniques (Yee 1966, Holland er a1 1980, Umashankar and Taflove 1982) as well as with experimental data (Holland 1977, Taflove 1980). The accuracy of the method has been established to be between 10 and 25% in the examples quoted above. The inherent advantages of the time domain finite-difference method over other modelling techniques such as the moment method have been discussed by Spiegel (1984) and Taflove and Brodwin

0031-9155/86/111247+ 10$02.5U @ 1986 The Institute of Physics 1247

1248 R W M Lau and R J Sheppard

(1975a). Essentially, the algorithm does not require the formulation of integral equations and relatively detailed models can be treated without the inversion of large matrices. Although the method has been used for computation of electromagnetic energy deposition in a human eye (Taflove and Brodwin 1975b), the potential applica- tion of such a model in bioelectromagnetic dosimetry has only been realised recently (Spiegel 1984) and it appears that previous modelling of biological systems particularly at high frequencies has been in two dimensions. The desirability of a full three- dimensional analysis is certainly realised, for example, Turner (1984) in his Daper concerning the BSD annular phased array system concluded 'the demonstrated effect of variations in anatomical configurations suggests that three-dimensional numerical methods which take into account the overall interaction of the body with electromag- netic fields are required to model and accurately predict heating patterns'. Turner also suggests that such an exact solution may be impossible at present.

The purpose of this, the first of two papers, is to describe the implementation of the time domain finite-difference method (TDFD) in three dimensions. The second paper demonstrates the use of the method in analysing the performance of a single hyperthermia applicator and then computes the fields in a realistic three-dimensional model. This paper also demonstrates the validity of the method by testing experi- mentally a simple homogeneous model.

2. The time domain finite-difference (TDFD) model (Yee 1966)

In order to study the interactions of electromagnetic waves with scattering objects, the time domain finite-difference formalism requires that a three-dimensional mesh be imposed on the volume of interest usin'g a rectangular coordinate system. The scattering object is composed of a collection of dielectric cells, each of which is specified by a pair of dielectric parameters (permittivity E ' and conductivity a ) and has associated with it a set of electric- and magnetic-field components. At the beginning of the simulation, a source of electromagnetic field is excited within the volume. The propaga- tion and scattering of the field is then simulated by time stepping a set of finite-difference approximations of Maxwell's equations of electromagnetic radiation over all the lattice points. Either the transient response of the whole system can be followed in time or the steady-state solutions can be obtained by allowing the time stepping to continue over a number of complete cycles of the continuous-wave source.

Using the usual notation, Maxwell's equations can be written as

d H V x E = - p -

a t

V X H = E ' - - + U E . dE a t

The vector equations (1) and (2) represent a system of six Scalar equations, two of which can be expressed as

Modelling of biological systems in 3 D: l 1249

A grid point of the problem space is defined as

(i, j, k ) = (iAx, jAy, kAz)

and any function of space and time as

F ” ( i, j, k ) = F ( iAx, jAy, kAz, n A t )

where Ax, Ay and Az are the spatial increments and Af is the temporal increment. By placing the electric- and the magnetic-field components on a unit cell according to the Yee grid (Yee 1966) illustrated in figure 1, a two-point central finite-difference approxi- mation to the scalar equations can be formulated.

In particular from equation ( 3 a )

H:+’/’( i, J + f , k + f )

+ Ey(i , j , k+f) -E:( i , j+ l ,k+f)

AY (6)

and from equation (3b)

E;+’(i,j,+f, k )

= ( l - u(i,J+i, k ) A t

E;( i, j +f , k ) + A t E ‘ ( i , j + f , k ) E’(i, j+f, k )

Similar difference equations can be constructed for the rest of the scalar equations. The value of any field component can be computed from its previous value and the previous values of the field components in its neighbourhood. The electric and magnetic

Figure 1. Field components in a unit cell.

1250 R W M Lau and R J Sheppard

components are interlaced within the unit cell and are evaluated at alternate half time steps. When translated into a computer code then, within the same time loop, one type of field component is calculated first and the results obtained are then used in the computation of the other field values. The spatial increments must be set to be a small fraction of either the minimum wavelength or minimum dimension of the scattering object in the modelled domain. The temporal increment is determined by the spatial increments and must satisfy the following stability condition (Taflove and Brodwin 1975a):

where v,,, is the maximum wave phase velocity within the model.

stability criterion then becomes In the present implementation a cubic cell is used and A x = A y = A z = 6. The

A t S / & (in two dimensions) (9)

and

A t 6 / c 8 (in three dimensions). (10)

If the electromagnetic field within the modelled space can be separated into its incident and scattered components, i.e.

total - - Einc ident + Escattered

H t o t a l = Hinc ident + Hrcattered

then the finite-difference approximation to Maxwell's equations can either be applied to the total field or the scattered field alone, resulting in two different formulations of the algorithm. The choice between the total-field and the scattered-field formulations is largely determined by the type of scattering problem being studied and the degree of accuracy required (Taflove 1980, Holland and Williams 1983). One advantage of the scattered-field formalism is the possibility of using a radiation boundary condition at the problem space lattice, as discussed below. In the current study, localised regions of biological tissues are modelled and the scattered field arriving at the lattice boun- daries is most likely to be in the near-field region, hence violating the assumption required by the radiation boundary condition. Furthermore, the total-field formalism is more accurate particularly in regions where the total field diminishes to levels far below the incident (Taflove 1980). As a result, the total-field formalism has been adopted in the current study.

Since the problem space being modelled must be finite it is necessary when dealing with open systems to apply certain conditions at the volume boundaries such that any outgoing waves will not be reflected back into the model in order that an effective infinite space is simulated. In a scattered-field formulation when the boundaries are in the far-field region the scattered field can be assumed to behave like an outgoing radial field and first- or higher-order approximations can be enforced at the boundaries (Holland 1977, Kunz and Lee 1978, Mur 1981). However, these techniques cannot be applied to the total-field formulation and other absorbing boundary conditions have to be used. For the present TDFD model wave propagation delays and a field-average process are used to simulate the outgoing waves from the lattice (Taflove and Brodwin

Modelling of biological systems in 30: I 1251

1975a). If the number of time steps required for a wave to propagate through a unit cell at ( i , j , k ) with a complex dielectric constant E* is

L(i, j , k ) = 2AdA;(i.j,k) ( 1 1 )

then the boundary conditions for a I,,, X J,,, X K,,, lattice point are (for clarity, the indices of L will be omitted in the following)

(i) plane i = f H," ( f , j , k + f ) = f ( H : - L ( i , j , k - i ) + H ! - L ( $ , j , k + $ ) + H , n - L ( I , j , 3 k+g) ) (12 )

H:( i , j+$, k ) = f ( H : - L ( $ , j + & k - l ) + H : - L ( $ , j + $ , k ) + H : - L ( i , j + f , k + l ) ) ( 1 3 )

These boundary conditions minimise the reflection of outgoing waves whilst the natural dielectric losses of biological tissues also provide damping of reflected waves from the problem space boundaries.

1252 R W M Lau and R J Sheppard

Experience in obtaining steady-state solutions from the model suggests that the imperfection of the boundary conditions causes the model to develop instability if it is left to run for too long. The instability is seen by the positions of the null or minimum field shifting and the possible appearance of artificial peaks. It is therefore important to choose an appropriate maximum number of time steps. In particular the incident waves must be allowed sufficient time to penetrate the whole of the problem space and yet the computation should be terminated at soon as a steady state is reached.

Another boundary condition within the modelling volume to be considered is that of a conductor. When any metallic structure exists it is possible to assign a high loss to the appropriate cell; however, this can cause a large transient field to be induced near the metallic surface. One possible solution is to add a non-physical loss factor in the cells adjacent to metal surfaces (Taflove 1980) which will shorten the decay time of the transient field. In the present application the problem has been solved by applying a boundary condition to any conductor surface which forces the tangential electric field to vanish.

3. A general time domain finite-difference modelling system

Based on the theory described above, a set of FORTRAN 77 programs have been developed and to ensure that the implementation is both general and flexible the folowing features have been incorporated into the software.

(1) The number of grid points, the modelling frequency and the spatial resolution are specified as input parameters to the main program rather than predefined.

(2) A large class of scattering objects with different geometric shapes can be constructed with ease.

(3 ) The configuration of the excitation source in the model is defined by a user- written subroutine. Either the resultant field or field components from any plane within the volume can be printed or plotted on completion of the run.

The overall architecture of the package is depicted in figure 2, which contains three main types of programs, the model generators, the numeric processors and the output analyser.

Within a defined grid (either 2~ or 3 ~ ) , the model generators allow a class of primitive structures such as lines, planes, blocks, spheres and cylinders to be placed anywhere in the grid. Each primitive structure can then be assigned a permittivity and a conductivity, neither of which needs to be isotropic. The reason for allowing anisotropic dielectric parameters is to facilitate the modelling of conductor surfaces according to the boundary conditions mentioned before. A negative value of conduc- tivity value along an axis, say x, would signify to the numeric processors that the electric-field component E, is tangential to a metallic surface and should be set to zero rather than computed in the normal way. Obviously, within a conductor, all three orthogonal conductivity components should be negative and all the electric-field components would be set to zero.

Using the model generators, a complicated problem space can be built up gradually by selecting the appropriate primitive structures. The dielectric parameter map is then submitted as a data input file to a numeric processor.

The numeric processors, consisting of a 2~ and a 3~ version of the total-field computation codes, compute all the field values by time stepping for a prescribed number of steps. This number is chosen so as to give an expected steady-state solution. The excitation of the model is performed by calling a specific subroutine which can

Modelling of biological systems in 3 D: I 1253

DEFINE ser les

Generatlon of problem

spoce In 20 or 30

parameters subrout lne

SIMUL2. SIMUL 3 1 To ta l - f l e ld computat lon

Computed

ANALYSER

Comparatlve

array processor

Graphlcal o u t p u t

Figure 2. Time domain finite-difference simulation system.

be tailored to a particular form of electromagnetic excitation. On the completion of a run the arrays of field values can be extracted for analysis.

The output processor is an interactive program suite for handling arrays of numbers. Basic arithmetic operations such as normalisation, scaling, sampling and filtering are available to preprocess an array before displaying the data in either two- or three- dimensional formats. Field arrays from the numeric processors can be efficiently viewed and compared with the aid of this output processor.

At present the programs have been implemented for a Digital VAX 11/780 micro- computer. The output is transferred to a Research Machine 3802 microcomputer for output to either a VDU, a printer or a Calcomp plotter. However, it is envisaged that in the near future the main numerical computation will be done by the University of London Cray computer.

4. On-axis attenuation of waveguide applicators

To demonstrate how the general TDFD model can be used to yield information on the performance of applicators numerical results are now presented which show the familiar problems associated with the miniaturisation waveguide applicators, namely the reduc- tion of penetration depth of EM field into tissues as compared with the plane-wave

1254 R W M Lau and R J Sheppard

irradiation condition (Audet et a1 1980, Cheung 1981). The analysis also enables the profile of the beam to be studied as it propagates into the model.

Since the aperture field of a rectangular waveguide propagating in the T E ] ~ mode varies only in one dimension across the aperture, provided that the tissue being irradiated is also homogeneous, the whole system can be approximated by a two- dimensional parallel-plate waveguide with a lossless dielectric sandwiched in between as shown in figure 3. However, it is stressed that such a simple two-dimensional model is being used for a preliminary demonstration of the method; a rigorous analysis of a realistic three-dimensional biological model is presented in the second paper. Only three field components E?, H, and H , are required for this problem. For the following numerical experiments, a two-dimensional domain consisting of 55 x 80 unit cells was used, the unit cell size was 0.387 cm and the incident wave source was operating at 433 MHz. The unit cell size was chosen to be 0.05 of the minimum wavelength in the models (i.e. the wavelength in deionised water with E ‘ = 80). The source field distribu- tion, placed at one end of the domain, was either that of a plane wave or a wave of the form E? = Eo sin( v x / a ) where a is the separation of the parallel-plate waveguide. Steady-state solutions were obtained after time stepping to four cycles (equivalent to 1440 time steps) of the incident wave. The choice of four cycles was to ensure the coverage of the whole model by the propagating waves.

Muscle t lssue f ’ = 53 c ” = 59

Lossless dlelectrlc loadlng

Figure 3. Parallel-plate waveguide.

A muscle-equivalent dielectric with E ’ = 53, (T = 1.43 S m” (Johnson and Guy 1972) was first irradiated with a plane wave, then with a parallel-plate waveguide (plate separation a = 11.6 cm) loaded with a lossless dielectric ( E ’ = 9). Finally the dielectric was irradiated with a waveguide (a = 3.9 cm) loaded with the equivalent of de-ionised water ( E ‘ = 80).

The attenuation of a 433 MHz plane wave by the muscle is shown in figure 4. The penetration depth D (where the E field drops to l / e of its original value) is 2.90 cm. This can be compared with the value of 3.02 cm obtained from the usual equation D = l /a where

cy = ( 2 7 r / ~ 0 ) { f [ - ~ ‘ + ( ~ ’ 2 + ~ ’ ’ 2 ) 1 ’ 2 ] } 1 ’ 2 . (22)

The on-axis attenuations of the two dielectric-loaded guides are shown in figure 5. It can be seen that the guide loaded with the dielectric E ‘ = 9 gives similar on-axis penetration performance to that of a plane wave whereas the guide loaded with the dielectric E ’ = 80 gives a reduced penetration depth of 2.05 cm. The field distributions inside the muscle in both cases are illustrated in figure 6. The shape of the TElO aperture field is mostly retained but progressively attenuated; the spreading of the beam width can also be noted.

Modelling of biological systems in 3 D: l 1255

Depth ( c m 1 0 1 2 3 4 5

Dlstonce from surface (cm1

0 4 8 12 16 20 0 4 8 12 16 20 x ( c m )

Figure 6. Field inside muscle. Guide loaded with ( a ) E ' = 9 and ( b ) E ' = 80.

5. Conclusions and discussions

A general three-dimensional computer model based on the time domain finite-difference formalism of Maxwell's equations has been implemented. The model can be used to predict the electromagnetic-field deposition within biological tissues when irradiated with a known source. Due to the virtual-memory-addressing capability of the VAX/VMS operating system, a problem space of 38 x 45 x 38 unit cells is just within the capacity of the current computer configuration; however, the number of cells will be increased when it becomes possible to use the Cray computer. To compute the field values of a typical domain consisting of 65 000 cells and 560 time steps, about 3.5 h of CPU time is required. The number of time steps to be followed for a particular simulation depends on the content of the problem space; sufficient time must be allowed for incident waves to penetrate the whole of the problem space and for a steady state to be established. The accuracy of the computed field values depends on the validity of all the approximations made in the problem space, namely, the approximation of the geometric shape of the scatterers, the approximation of the known excitation field and the approximation of the invisible domain boundaries. Although it has been established by other workers that the algorithm is accurate to between 10 and 25% in a number of scattering problems, it would be unsatisfactory to use the model in a new application liberally without any validation. In the companion paper a simple three- dimensional structure model is physically realised and experimental data are obtained for comparison with the output from the TDFD modelling system. A circular applicator

1256 R W M Lau and R J Sheppard

is also described and its performance evaluated when radiating into a real biological model.

Acknowledgments

Thanks are due to the Cancer Research Campaign for an equipment grant and for providing a PhD studentship for RWML. The authors would also like to thank Professor R E Burge of the Physics Department, for providing the facilities on the VAX computer.

Resume

Modtlisation tridimensionnelle des systkmes biologiques B l’aide de la mtthode des differences finies dans le domaine temporel: I. Mise en oeuvre du modkle.

Les auteurs ont dtveloppt une methode numirique reposant sur un algorithme de calcul utilisant les difftrences finies dans le domaine temporel (TDFD) pour determiner, B partir d’un modele biologique tridimensionnel, I’tnergie diposte par un champ Blectromagnitique (EM). Ce travail publie en deux articles dont celui-ci est le premier, decrit I’algorithme et les programmes de calcul dtveloppes. La methode est illustrte par le calcul de la ptnttration d’un champ E M dans un milieu homogene irradie i I’aide d’un guide d’aide B section rectangulaire; par souci de simplification, les calculs presentes ici ont et6 effectuts en deux dimensions.

Zusammenfassung

Dreidimensionale Rechnermodelle biologischer Systeme mit Hilfe des Differenzenverfahrens iiber einen Zeitbereich: I. Die Einfiihrung des Modells.

Auf der Grundlage von Zeitbereichs-Differenzenalgorithmen wurde ein Computerverfahren entwickelt, um die Ausbreitung elektromagnetischer Felder in dreidimensionalen biologischen Modellen zu berechnen. In diesem ersten von zwei Berichten werden die Algorithmen und Rechner-programme beschrieben. Die Funktionsweise des Verfahrens wird demonstriert anhand der Berechnung der Penetration elektromagnetis- cher Felder, die aus einem rechteckigen Wellenleiter in ein homogenes Modellsystem eindringen, wobei die Berechnung aus Griinden der Vereinfachung hier zunachst zweidimensional erfolgt.

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