Separating the complex impedance tensors into their real (X) and imaginary (Y) parts, we can write:


Thus, from eq. ( 8):


Since D is unknown the amplitude information contained in Z R can- not be recovered from Z without other independent information (e.g. Smith 1995).

However, since D is real the essential phase relation- ship between the horizontal components of the regional electric and magnetic fields must be unaffected by the distortion.

It is the nature and determination of the phase relationship in a situation where the conductivity structure is 3-D that is addressed below.


The phase of a complex number is defined from the ratio of its real and imaginary parts.

This relationship can be generalized to a complex matrix or tensor.

Hence, we define the phase tensor by the relation Φ = X −1 Y, (13) where X −1 (a tensor) is the inverse of X and Φ is real.

( Formally, eq. ( 13) is a particular contraction of the tensor product of X −1 and Y.) The relationship between the phase tensors of the observed and regional impedance tensors can be derived directly from eqs (11) and (12), i.e.


Thus the observed and regional phase tensors are identical and are independent of the distortion tensor, consistent with the behaviour expected for the phase on physical grounds.

Note, no assumption about the nature or dimensionality of the regional conductivity struc- ture is required and eq. ( 14) is applicable where the conductivity structure is 3-D.


Written in terms of the real and imaginary components of Z in a Cartesian coordinate system (x 1 , x 2 ), the phase tensor Φ can be written as the matrix


If the regional conductivity structure is both isotropic and 1-D (i.e. the conductivity varies only with depth), the impedance tensor in a Cartesian coordinate system (x 1 , x 2 ) has the form


Thus (from eq. 15) Φ 1D = Y 1D /X 1D 0 0 Y 1D /X 1D = (Y 1D / X 1D )I = tan(φ)I (18) where I is the identity matrix, and Φ is characterized by a single scalar quantity equal to the tangent of the conventional MT phase for a 1-D structure.

Note that the tensor form in eq. ( 18) is invari- ant under rotation; i.e. the regional electric and magnetic fields are perpendicular and independent of the direction of polarization.


Just as a vector is characterized by its direction and magnitude, a real second-rank, 2-D tensor is characterized (in the general case) by a direction and three independent scalar quantities that are in- dependent of the coordinate system used to express the tensor, i.e. by three coordinate invariants.

Since any function of the invariants is also coordinate invariant many different ways of expressing the invariants are possible.

In this paper we will adopt one of the repre- sentations used by Bibby (1986) in his analysis of the DC apparent resistivity tensor.

The coordinate invariants that we will use are the maximum ( max ) and minimum ( min ) tensor values, simple func- tions of the tensor components given in the Appendix, and the skew angle β given by the expression


This angle can be thought of as a rotation and is a measure of the tensor's asymmetry.

Note that β depends on the tensor's skew ( 12 − 21 ), which is invariant under rotation but changes sign if the coordinate system is reflected.


Expressed in terms of these quantities, the phase tensor can be writ- ten in the form:


where R(α + β) is the rotation matrix


R T is the transposed or inverse rotation matrix, i.e. R T (θ) = R −1 (θ) = R(−θ), and


This angle expresses the tensor's dependence on the coordinate sys- tem and with the three coordinate invariants completely defines the tensor.

Note that eq. ( 20) has the same form as the singular value de- composition (SVD) of a square matrix (e.g. Press et al. 1986).

Thus, the maximum and minimum values max and min are the principal or singular values of Φ. Since the SVD parameters have explicit forms (eqs 19, 22, A8 and A9) their standard errors can be deter- mined explicitly if the variances and covariances of the impedance tensor components are known.


The invariance of the principal values and skew angle can be demonstrated by rotating the Cartesian coordinate system used to express the tensor.

Rotating by an angle θ,


If the phase tensor is symmetric (β = 0) the principal values of the tensor are equal to its eigenvalues.

This situation occurs where the regional conductivity distribution is mirror symmetric, e.g. where the regional conductivity distribution is 1-D or 2-D. Where Φ is symmetric and has equal principal values, such as is the case if the conductivity distribution is uniform or 1-D, the electric field will be linearly polarized if the magnetic field is linearly polarized.

In the 2-D case, the principal values will usually be distinct (i.e. max = min ) and there will be two directions for which a linearly polarized magnetic field will give rise to a linearly polarized electric field.

If the 2-D response is not distorted these two directions will be orthogonal.

In all other directions the electric field will be elliptically polarized.


Any second-rank, non-symmetric 2-D tensor can be represented graphically by an ellipse (e.g. Bibby 1986) and the quantities in eq. ( 20) have simple interpretations in terms of the properties of the tensor ellipse illustrated in Fig. 1.

In particular, the major and minor axes of the ellipse depict the principal axes and values of the tensor with the orientation of the major axis specified by the angle α − β.

Note that where the phase tensor is symmetric (i.e. β = 0) the orientation of the major axis is given by α.

In the general (3-D) case, the skew angle (β) is non-zero and represents the rotation of the major axis of the phase tensor ellipse away from an identically shaped ellipse represented by a symmetric tensor.

A simple method of constructing the tensor ellipse is described in the Appendix.


The simplest example of this representation is the tensor ellipse for a uniform conductivity half-space.

In this case, a circle of unit ra- dius represents the phase tensor at all periods.

More generally, if the conductivity is both isotropic and 1-D, the radius of the circle will vary with period according to the variation of the conductivity with depth.

For example, the radius will increase if the conductivity increases with depth.

In the 1-D case, or in a situation where the conductivity distribution is cylindrically symmetric about the mea- surement point, α is undefined.

In practice this means that if the conductivity is 1-D the value of α will be unstable due to noise in the observed data.

Whether or not α is significant within the uncer- tainties of the observed (and perhaps distorted) data may be tested using the expected properties of the phase tensor invariants as sig- nificance criteria.

In particular, if the difference in the lengths of the major and minor axes ( max − min ) is less than the correspond- ing standard error then α is insignificant and the tensor ellipse is indistinguishable from a circle.


Where a half-space has a uniform but horizontally anisotropic con- ductivity (i.e. where one of the principal axes of the conductivity tensor is aligned vertically) the conductivity is mirror symmetric about the horizontal principal axes and the phase tensor skew an- gle β = 0.

In a Cartesian coordinate system with axes aligned with the principal axes of the conductivity tensor, the impedance tensor has an antidiagonal form.

However, in contrast to an isotropic half- space (eq. 16), the magnitudes of the off-diagonal components of the impedance tensor for the anisotropic case are different.

Conse- quently, the amplitude of the electric field will vary as the direction of the magnetic field polarization changes but the phase difference between the electric and magnetic fields will not.

Thus, the phase tensor is also represented by a unit circle (Φ = I) in the case of a uniform horizontally anisotropic half-space.

This means that if the magnetic field is linearly polarized, the electric field is also lin- early polarized but will be perpendicular to the magnetic field only where the polarization directions coincide with the directions of the principal axes of the conductivity tensor.


In a horizontally layered situation where an isotropic layer (or layers) overlies an anisotropic half-space, any difference (or split) between the principal phases is a consequence of the conductivity contrast at the boundary between the isotropic and anisotropic mate- rial and is not directly indicative of the underlying anisotropy of the half-space.

Thus information pertaining directly to the anisotropy can only be obtained from the amplitude of the MT response.

In practice, the difficulty is that the amplitude response may have been distorted by structure at length scales less than the depth of the anisotropic layer.

More generally, the observation of a phase split between the principal phase values indicates that a conductivity gra- dient (lateral or vertical) exists, nothing more.


In the 2-D case (the assumption made in previous approaches to MT distortion analysis) the strike of the regional conductivity dis- tribution defines a natural orientation for the coordinate system.

In a Cartesian coordinate system (x 1 , x 2 ) aligned with x 1 parallel to the (unknown) strike direction, at say an angle θ with respect to the observation coordinate system, the regional impedance tensor The magnetotelluric phase tensor 461 has the antidiagonal form: Z R = 0 Z 12 Z 21 0 = 0 Z −Z ⊥ 0 , (24) where quantities in the rotated coordinate system are denoted by a prime ( ) and Z and Z ⊥ , are the transverse electric (TE) and trans- verse magnetic (TM) impedance respectively.

That is, Z and Z ⊥ are the impedances measured when the electric field is linearly polarized parallel or perpendicular to the strike of the conductivity distribu- tion.

The antidiagonal form of the impedance tensor expresses the orthogonality of the electric and magnetic fields where the fields are polarized parallel or perpendicular to the strike of the conductivity structure.


Rotating the observed phase tensor Φ by the unknown angle θ, the phase tensor in the rotated coordinate system is given by


Since Z R is antidiagonal so are X R and Y R .

Thus, the inverse tensor (X R ) −1 is also antidiagonal.

Hence, the matrix product (X R ) −1 Y R is diagonal and the phase tensor in the rotated coordinate system has the diagonal form:


Comparing Φ (eq. 26) with the SVD form in eq. ( 20), it can be seen that the phase tensor coordinate invariants are Y ⊥ /X ⊥ , Y /X and β = 0.

The principal values of the phase tensor for 2-D conductivity distribution are thus the tangents of the corresponding conventional TM and TE polarization phases.

Note that the condition β = 0 is a necessary but not sufficient condition for a 2-D regional conduc- tivity structure.

For a 2-D structure, β must be zero at all periods greater than the minimum value for which the galvanic approxi- mation applies.

In practice, what we seek is a range of frequencies where β is zero (within the observational errors) and the direction of the phase tensor major axis is constant.


Since β = 0 in the 2-D case, the orientation of the major axis of the phase tensor ellipse in the rotated coordinate system is given by the angle α .

Thus from eqs (22) and (26), α = 0 • and the major axis of the phase tensor ellipse is aligned parallel or perpendicu- lar to the strike of the (regional) conductivity distribution.

In the observer's frame of reference (i.e. in the unrotated coordinate sys- tem) the orientation of strike axis is α = θ or α = θ + 90 • .

The 90 • ambiguity remains since (a priori) we have no knowledge as to which of the maximum or minimum phases corresponds to the TE or TM polarization.

( In practice, the ambiguity is resolved using vertical magnetic field data.

) Note that, if the regional conductivity structure is 2-D, eq. ( 22) is an explicit expression for the direction (α) of the strike axis (or its normal), irrespective of whether or not the response has been galvanically distorted.


If the regional conductivity structure is 2-D there are two direc- tions for which a linearly polarized magnetic field will give rise to a linearly polarized electric field.

A galvanic distortion will not change the polarization state of the EM field so that there are also two directions where the observed (distorted) electric and magnetic fields are both linearly polarized.

Although the distortion will usu- ally change the direction of regional electric field polarization, the corresponding direction of the magnetic field will be virtually un- affected provided that the distortion of the regional current density does not significantly distort the magnetic field components (i.e. eq. 4 is a good approximation).

The problem of determining the 2-D strike direction from the distorted data is thus equivalent to identifying the direction of a linearly polarized magnetic field for which the electric field is also linearly polarized.

This direction de- fines the strike direction of the regional conductivity structure or its normal.


The effect of a galvanic distortion on the response of a regional 2-D conductivity distribution is illustrated in Fig. 2 using ellipses to represent all of the tensors involved.

The case illustrated is a synthetic example taken from McNeice & Jones (2001).

Note that rotating Fig. 2 leaves the configuration of the ellipses unchanged; only the angle specifying the orientation of the diagram with respect to the observer changes.

The lengths of the ellipse axes, their relative orientation with respect to each other and their orientation with respect to the conductivity distribution (i.e. with respect to the Earth) do not depend on the particular frame of reference chosen.

It is these quantities (the coordinate invariants) that express the essential physical relationship between the conductivity distribution and the EM field observations.


Fig. 2(a) shows the phase tensor derived from the real (solid) and imaginary (dotted) parts of distorted impedance tensor (Fig. 2b).

Note that the orientations of the principal axes of the phase tensor are the same as the major and minor axes of the ellipses that represent the real and imaginary parts of the regional impedance tensor (Fig. 2d).

This set of axes defines the strike axes of the regional conductivity structure, i.e. the directions in which a linearly polarized regional magnetic field give rise to a linearly polarized electric field.


These ellipses, and the ones used in Fig. 2(b) to represent the observed impedance, are (polar) plots of the electric field magnitudes | E| and | E| produced by a unit-magnitude linearly polarized magnetic field drawn in the direction of the corresponding electric field vector (e.g. Bibby 1986; Caldwell et al. 2002; Appendix A).


This method of graphically representing the impedance tensor has the advantage that the two sets of principal axes that characterize the (complex) impedance tensor can be simply and directly depicted.

For example, if the regional conductivity structure is 2-D the major axes of both X R and Y R must be parallel or perpendicular to the conductivity structure as is illustrated in Fig. 2(d).

This requirement can also be expressed as a condition on the skew angles (β X and β Y ) and on the angle between the major axes of the two ellipses representing the regional impedance tensor.

All of these quantities are coordinate invariant.


Since the traces (another coordinate invari- ant form) of X R and Y R are zero in 2-D (eq. 24), the skew angles β X and β Y have values of ±45 • .

In the Cartesian coordinate sys- tem with axes parallel to the strike axes, the diagonal terms of the impedance tensor are both zero and α X = ± 45 • and α Y = ± 45 • .

Thus the condition on the angle between the major axes of the real and imaginary parts reduces to the condition that α X = α Y or α X = α Y ± 90 • .


The ellipse representing the distortion tensor in Fig. 2(c) is a polar plot of the magnitude ratio |E|/|E R | in the direction of the observed electric field E. The effect of the distortion tensor on the regional impedance tensor (Fig. 2d) is to amplify or attenuate the principal values of X R and Y R by different amounts and to reorient the prin- cipal axes.

Thus two distinct sets of principal axes are needed to characterize the observed impedance tensor as shown in Fig. 2(b).

This characteristic is indicative of a 3-D conductivity distribution, either in the neighbourhood of the measurement site or regionally.

Only where further analysis shows that the phase tensor skew an- gle (β) is insignificant for all periods greater than some minimum value can we infer that the regional conductivity structure is 2-D, and thus that the observed response has been distorted by a localized conductivity heterogeneity.


We will demonstrate the properties of the phase tensor in 3-D us- ing synthetic data calculated with the modelling code described in Xiong (1992) and Xiong & Tripp (1995).

The model used for this demonstration (Fig. 3) consists of a small conductive (10 m) cube situated near the surface and a much larger (regional) conductive (1 m) body at depth both embedded in a 100 m half-space.

The cube is situated near the corner of the rectangular body in the area where 3-D effects are expected to be large.

Maps of the phase tensor ellipses at three different periods (0.0316, 0.316 and 3.16 s) with contours showing the direction of the major axes and skew angle (β) are shown in Figs 4 and 5, respectively.


Where the difference in the lengths of ellipse axes ( max and min ) is less than 2 per cent of their corresponding geometric mean (i.e. of the radius of a circle with the same area as the ellipse), the ellipses in Fig. 4 have been left unfilled.

At these locations, where the phase tensor is indistinguishable from a circle, the azimuth calculation is potentially unreliable (unstable) because of the limitations in the accuracy of the forward modelling code.


At short periods the MT response will be unaffected by the deep conductor and the observed response will reflect the influence of the near-surface body alone.

This can be seen in the phase tensor response (Fig. 4a) as a radial alignment of the major axes around the conductive cube.

If the cube is resistive (not shown) rather than conductive the phase tensor ellipses are oriented tangentially.

This change in the orientation of the principal axes suggests that the direction of the major axes indicates the preferred flow direction of the induction current.


Support for this interpretation is provided by the induction arrows (Parkinson 1962) also shown in Fig. 4.

These arrows are graphical representations of the real part of the vertical magnetic field transfer function or tipper vector (K) defined by the equation H z = −K·H, (27) where H z is the vertical component of the total magnetic field and H is the horizontal magnetic field vector.

In situations where the conductivity distribution varies laterally, the real part of the tipper vector will point towards the region of highest conductance.

At 10 Ωm 1 Ωm 100 Ωm short periods (Fig. 4a), outside the immediate vicinity of the cube, where the conductance sensed by the EM field will be approximately cylindrically symmetric, the induction arrows and major axes of the phase tensor are parallel, consistent with the interpretation of the principal axes of the phase tensor suggested above.


At intermediate periods (Fig. 4b) the alignments of the major axes reflect a mixture of the effects of both the near-surface and deep conductors.

At long periods (Fig. 4c), inductive effects produced by the cube are insignificant and the deep conductor dominates the phase response, the alignment of the major axes indicating the preferred flow direction of the inductive current associated with the deep conductor.

Note that the direction of the ellipse major axes at long periods (Fig. 4c) changes rapidly (approximately 10 • in 0.5 km) in a northwest-southeast direction.

A rapid lateral change in direction of the principal axes of the phase tensor is a clear indication of a 3-D (regional) conductivity structure.


The corresponding maps of the phase tensor skew angle (β) are shown in Fig. 5.

At short periods (Fig. 5a) the skew angle is zero on the cube's axes of symmetry and greatest (although still small, about 0.25 • ) near the corners of the cube where the conductivity distribution is asymmetrically distributed around the measurement point.

At intermediate periods, where the combined effects of both bodies are significant, the variation of the skew angle is more com- plicated (Fig. 5b) reflecting the asymmetric configuration of both bodies with respect to the measurement point.

At longer periods (Fig. 5c) the influence of the near-surface body is insignificant and the skew angle reflects the effect of the deep conductor alone.


Note that at longer periods the ellipse major axis and the induction arrows (Figs 4b and c) are not as closely aligned as they are at short periods (Fig. 4a).

This misalignment reflects the asymmetry of the 3-D response at longer periods, which is expressed in the phase tensor as a non-zero skew angle (Figs 5b and c).


Fig. 6 shows graphs of the phase tensor properties as a function of period at the location marked by the cross in Fig. 4.

To simulate the effect of noise, an ensemble of 1000 impedance tensors was created by adding 2 per cent Gaussian noise to each component of the calculated impedance tensor.

The error bars in Fig. 6 show the standard deviation from the mean value (points) of the corre- sponding tensor property derived from this ensemble.


For periods less than 0.1 s the maximum and minimum phases (Fig. 6a), i.e. tan −1 ( max ) and tan −1 ( min ), are close to 45 • indicating that the effect of the conductive cube at this location is small.

For periods between about 0.1 and 3 s, the phase difference or split between the principal phases increases rapidly as the influence of the inductive currents in the deep conductor becomes dominant.

At periods >3 s, the phase split decreases as the induction currents associated with the deep conductor weaken.


Note that the magnitude of the phase difference between the prin- cipal phase values is reflected in the size of the error bars for the direction of the major axis (Fig. 6b).

In particular the uncertainties are large for periods <0.3 s where the phase difference is small (<3 • ).

Despite the large uncertainty, the direction of maximum in- ductive current flow associated with each of the conducting bodies at short (Fig. 4a) and long periods (Fig. 4c) can clearly be distin- guished in Fig. 6(b).


In particular, for periods between 0.3 s and 30 s (where the difference between the principal phases is large) the azimuth of the major axis can be determined within ±5 • .

At points where the error bars of the principal phases overlap (enclosed by a small grey boxes in Fig. 6(a)) the tensor ellipse is indistinguishable from a circle.

At these points, omitted from Fig. 6(b), the azimuth is ill-defined.


Note that the large uncertainty in the direction (α − β) of the major axis in Fig 6(b) at periods <0.1 s is a consequence of the small phase difference and thus the uncertainty in the angle α rather than the uncertainty in the skew angle (β), Fig. 6(c).


For the model used in this example β is small (< 3 • ) at all periods and at all locations (Fig. 5).

Although small, β is well-determined as shown by the small error bars in Fig. 6(c).


Between 0.3 s and 1 s, where β (Fig. 6c) begins to increase and the direction (Fig. 6b) of the major axis varies slightly with period, inductive effects pro- duced by the deep conductor start to become significant.


Note that at longer periods (>3 s), where the skew angle begins to decrease, the direction of the major axis (Fig. 6b) remains nearly constant.


If the regional conductivity structure is 3-D and the phase difference between the principal values is large enough so that the orientation of the phase tensor principal axes can be determined reliably, a sin- gle (well-defined) direction can be associated with the MT phase response at each period.

As is illustrated in Fig. 4, the orientations of the major axes of the phase tensor indicate the preferred flow di- rection of the induction current in a similar fashion to the induction arrows.

In particular, if the regional conductivity structure is 2-D, the principal axes of the phase tensor are aligned with the strike axis of the regional conductivity structure.

The orientation of the principal axes of the phase tensor can thus be considered to be a generalization of the concept of strike direction (the so-called geo- electric strike) to a 3-D situation.

In essence, the principal axes of the phase tensor indicate the (horizontal) directions of the maxi- mum and minimum induction current, which in turn reflect lateral variations in the underlying regional conductivity.


In the example illustrated in Fig. 4(c), the direction of the principal axes of the phase tensor at periods >3 s is nearly constant along lines running northwest-southeast.

For a single northwest-southeast line of measurements, the MT response will appear to be nearly 2-D. That is, two nearly perpendicular polarization directions of the magnetic field can be found where the ellipticity of the resulting electric field is small.

Although the phase tensor skew angle β tells us that the situation is 3-D, the value of β is small (< 3 • ).

This suggests that a small value of β by itself is not necessarily a good indication of the nearness of the regional conductivity structures to 2-D. A much better indication of the inappropriateness of a 2-D analysis in this example is the rapid (10 • in 0.5 km) northeast-southwest variation in the direction of the major axis of the tensor ellipse.


If the regional structure is 2-D, the direction of the principal axes of the phase tensor (α) will be the same as the strike direction given by the two most commonly used approaches to distortion analysis, Bahr (1988, 1991) and Groom & Bailey (1989, 1991).

In particular, the formula for the strike direction given by Bahr can be reduced to eq. ( 22).

Thus, in 3-D situations where β is small, Bahr's expression for the regional strike direction will be close to one of the principal axes of the phase tensor.

Bahr (1988, 1991) also proposed a dimensionless, coordinate- invariant measure of the dimensionality of the regional conductivity structure similar to β which he called the phase-sensitive skew (η).

In terms of the quantities used in this paper, η is given by the expression


Note that unlike β (eq. 19), η is a function of the coordinate invari- ants of the observed impedance tensor and the phase tensor skew ( 12 − 21 ).


The condition η = 0 (or equivalently 12 − 21 = 0), represents the requirement that the component phases of the impedance tensor in each column of the impedance tensor matrix are equal, which is true if the regional conductivity structure is 2-D (Bahr 1988, 1991).

However, if | 12 − 21 | = 0 the effect of a galvanic distortion on eq. ( 28), contained in the det(X) and |Z 12 − Z 21 | 2 terms, will The magnetotelluric phase tensor 465 not cancel out.

Thus, if the regional conductivity structure is not 2-D, the value of η will depend on whether or not the observed response is distorted.

For this reason it is questionable whether the magnitude of η is a good indication of the dimensionality of the regional conductivity structure.


Any distortion analysis based on the assumption that the regional conductivity structure is 2-D cannot include the asymmetry in the MT response represented by the phase tensor skew angle (β).

In what is, in essence, an attempt to represent this asymmetry, Bahr (1991) introduced a complex parameter (the phase deviation) into the distortion tensor while retaining the 2-D form for the regional impedance tensor.

This parametrization is inconsistent with the 3-D source of the asymmetry and violates the basic physical assumption that the distortion tensor is real for a galvanic distortion.

One of the advantages of the phase tensor analysis is that the asymmetry in the phase response is represented explicitly and no assumption about the form of regional impedance tensor is required.


In this (Groom & Bailey 1989, 1991) and other similar approaches (Zhang et al. 1987; Chakridi et al. 1992; Chave & Smith 1994; Smith 1995) to distortion analysis the observed impedance tensor is decomposed into a 2-D regional impedance tensor Z R and a (real) distortion tensor D , which is parametrized in different ways by different authors.

The observed impedance tensor Z can therefore be expressed by the relation Z = R(θ)[D Z R ]R T (θ ), (29) where Z R is the 2-D (anti-diagonal) impedance tensor (eq. 24) and θ is the (unknown) strike of the conductivity structure.

Eq. ( 29) has a total of nine unknowns θ, X ⊥ , Y ⊥ , X , Y and the four components of D .

The number of measured quantities in Z is eight (i.e. the components of X and Y) so eq. ( 29) is underdetermined.

Groom & Bailey (1989, 1991) factorize the distortion tensor into three, linearly independent, 2 × 2 matrices and a scalar, i.e.


and g, the 'site gain', is a scalar.

That is Groom & Bailey rewrite eq. ( 29) as


In essence, the four unknown components of D have been rewritten in terms of the (unknown) parameters: g, t, e and s. These parameters cannot be determined from the observed impedance tensor without independent information or assumptions.


Since the operation of g and A on Z R is equivalent to a non- determinable scale change or gain (sayg ⊥ andg ) for each of the off-diagonal (non-zero) components of Z R , eq. ( 32) can be reduced to the form Z = R(θ)[TSZ R ]R T (θ ), (33) whereZ R is the scaled version of Z R (Groom & Bailey 1989, 1991).

This reduction implicitly introduces a pair of constraint equations (g = 1 and s = 0) for the components of D, which reduce the num- ber of unknowns by two.

The seven unknown parameters in eq. ( 33) (i.e.g ⊥ Y ⊥ ,g Y , t, e, and θ) and thus the regional phases (Y ⊥ /X ⊥ and Y /X ) can then be determined at each period from what is now an overdetermined problem.


As emphasized by McNeice & Jones (2001) one of the attractive features of the Groom-Bailey ap- proach is that it allows a stable estimate of the strike direction to be recovered from a distorted 2-D impedance tensor that contains noise.

However, as was first shown by Bahr (1988) and by the phase tensor analysis, the strike direction of a 2-D conductivity distribu- tion can be obtained directly from the observed impedance tensor (eq. 22) without decomposition, i.e. without attempting to recover information about the structure of the distortion tensor.

In practice, where the situation may only be approximately 2-D, the strike angle recovered by Groom-Bailey decomposition will depend on how the asymmetry in the regional response is distributed into the misfit.


Other parametrizations of the distortion tensor are equally plau- sible.

For example, the distortion tensor could be represented by its SVD i.e. D = R T (α D − β D ) d max 0 0 d min R(α D + β D ), (34) where β D and the principal values (d max and d min ) are coordinate invariant and α D is the coordinate dependent angle corresponding to eq. ( 22).

Like the Groom-Bailey parameters, the SVD parameters of D are functions of the distortion tensor components and can- not be determined from the observed impedance without auxiliary information.

By assuming a 2-D form for the regional impedance tensor (with strike direction determined from the principal axes of the phase tensor) and introducing constraints on the distortion ten- sor we can solve for the SVD parameters of D in a similar way to how Groom & Bailey (1989, 1991) solve for T and S.


One of the less satisfactory features of the Groom-Bailey ap- proach is that only one of the two constraints used for the decompo- sition is coordinate invariant.

Other choices of constraint are possible and equally plausible.

In particular both constraints can be chosen to be coordinate invariant.

However, it is important to stress that the choice of constraint partially determines the values of the distortion parameters recovered and thus limits their physical significance.

We will discuss the analysis of the distortion tensor in more detail elsewhere.


To demonstrate the application of the phase tensor to a real data set we have analysed data from a MT survey conducted in the Taupo Volcanic Zone (TVZ) in the North Island of New Zealand.

Phase tensor ellipses from this survey are shown in Fig. 7 superimposed on a gravity map of the region.

Details of the collection and interpre- tation of these data using conventional techniques (i.e. techniques based on a 2-D model of the regional conductivity structure) are given in Ogawa et al. ( 1999).


The central part of the TVZ shown in Fig. 7 is characterized by northwest-southeast extension and late Pliocene to Quaternary rhyolitic volcanism (Wilson et al. 1995).

This has resulted in the formation of a large volcano-tectonic depression filled with low- density volcaniclastic sediments and the central part of the TVZ can be identified with the area of low gravity (say <20 mGal) in Fig 7 (Bibby et al. 1995).

The volcaniclastic sediments are much more conductive (∼3-30 m) than the underlying basement rocks (∼1000 m) or the layer of recent volcanics (100-1000 m) that covers the entire region.

Thus where the basement is close to the surface (in the west and southeast) the conductivity structure of the upper part of the crust will reflect the structure shown by the gravity map (Fig. 7).


Estimates of the Groom-Bailey strike direction for the measure- ment sites shown in Fig. 7 are concentrated between N40 • E and N50 • E suggesting that the overall conductivity structure is approx- imately 2-D (Ogawa et al. 1999).

We have also analysed these data using the multisite Groom-Bailey decomposition technique pub- lished by McNeice & Jones (2001).

This analysis gives an overall apparent strike direction of N43.5 • E ± 1 • in good agreement with the N45 • E strike direction used by Ogawa et al. ( 1999).


As can be seen in Fig. 7, the southeastern margin of the TVZ is marked by an almost linear northeast-southwest gravity gradient.


The major axes of the phase tensor ellipses near this margin are ori- ented parallel and perpendicular to the strike of the gravity gradient similar to the behaviour expected for a 2-D conductivity distribu- tion.

Independent evidence for the 2-D nature of the conductivity structure along this margin is provided by detailed long-offset DC resistivity surveys (Bibby et al. 1998) and by the behaviour of the induction arrows shown in Fig. 7.

In a 2-D situation, the induction arrows will be perpendicular to the strike of the conductivity dis- tribution and the agreement between the directions of the induction arrows and ellipse axes near the southeastern margin of the TVZ is good.


In Fig. 8 we have plotted the phase tensor properties as functions of period for four of the measurement sites shown in Fig. 7. shown (Fig. 8a) are the ρ xy and ρ yx apparent resistivities (propor- tional to |Z xy | 2 and |Z yx | 2 ) in a Cartesian coordinate system with its x-axis aligned N45 • E. That is ρ xy and ρ yx are (respectively) the nominal TE and TM apparent resistivities with respect to the southeastern margin of the TVZ.


At the three sites within the TVZ (i.e. sites 300, 400 and 550) the apparent resistivity curves have a well-developed minimum reflecting the underlying conductive sed- iments.

This minimum is absent at site 250 outside the TVZ where the resistive basement rocks are close to the surface.


For periods <0.3 s, the TM and TE apparent resistivity curves (Fig. 8a) at sites 300 and 400 appear to be displaced by a constant factor with respect to each other.


This suggests that the data at these two sites have been distorted.


In Fig. 8(b) we have superimposed the principal values of the phase tensor (i.e. tan −1 min and tan −1 min ) on plots of the (nominal) TE and TM phases.

For the two southeastern sites (250 and 300) the differences between the principal phase values and the TE and TM phases at periods <100 s are small, indicating that the conductivity structure in this area is approximately 2-D with a strike of N45 • E. However, significant differences between the principal phase values and the TE and TM phases are present at sites 400 and 550.

This difference suggests that either 3-D effects are present or that the strike direction has been chosen incorrectly.

At sites 400 and 550, the directions of the principal axes change with period, implying that the regional conductivity structure in the neighbourhood of these two sites is 3-D. Plots of the skew angle β (Fig. 8d) for these sites support this inference.


For the two southeastern sites (250 and 300), the variation in the direction of the major axis of the phase tensor (Fig. 8c) is <15 • in the period range between 1 and 100 s, consistent with a quasi 2-D regional structure near the southeastern margin of the TVZ.

The variation is larger at site 300 where the skew angle also changes by a total of about 7 • over the same period range.

At greater periods >100 s, the principal direction at site 300 changes by about 45 • so that one interpretation of the response at this site is that the effect of a distant, off-strike structure becomes significant at periods >100 s. Indeed, the directions of the principal axes from all the sites within the TVZ (not all shown in Fig. 8) vary significantly in this period range.


At the sites southeast of the TVZ (of which site 250 is one example), the principal directions show much less variation and are coherent between sites.

It is clear from the phase tensor analysis that the conductivity structure within the TVZ is 3-D and a 3-D approach to data collection and analysis will be required to more clearly resolve the conductivity structure.


One of the difficulties of interpreting MT data is that the information from deeper levels is distorted by the effect of near-surface conduc- tivity heterogeneities.

The traditional approach to this difficulty has been to assume a simplified 2-D structure for the underlying regional conductivity distribution.

This assumption allows the regional phase information and strike of a 2-D conductivity distribution to be de- rived from a distorted set of observations, provided the distortion is galvanic.


We have shown that the 2-D assumption is unnecessary and that the phase relationship between the horizontal components of the electric and magnetic fields may be derived directly from the observed impedance tensor in situations where the regional conduc- tivity structure is 3-D. In essence we have generalized the represen- tation of the MT phase so that it applies in the 3-D case, where the electric field response to a linearly polarized magnetic field is (in general) elliptically polarized.

This representation takes the form of a (second-rank) tensor that is unaffected by a galvanic distortion of the electric field, consistent with the physical behaviour expected for the MT phase.


The phase tensor components and their associated three coor- dinate invariants are simple functions of the observed impedance tensor components.

Graphically, the tensor can be represented as an ellipse where the principal axes of the tensor are represented by the major and minor axes of the ellipse.

The third coordinate invariant, the skew angle, is a measure of the asymmetry in the regional MT response.

In 2-D the phase tensor is symmetric and the skew angle is zero.

While a large value of the skew angle necessarily implies a 3-D regional conductivity structure, our results suggest that a small value of the skew angle is not necessarily a good indication of the nearness of the conductivity structure to 2-D. Much more reliable criteria for two-dimensionality are the constancy of the direction of the principal axes of the phase tensor with period and with location along strike.


Our modelling results suggest that the direction of the phase ten- sor major axis indicates the preferred flow direction of the regional induction current.

Maps of the phase tensor ellipses at different periods provide a simple way of depicting this direction and thus (indirectly) of visualizing lateral changes in the regional conductiv- ity structure at different depths.

Such maps will not be influenced by near-surface galvanic effects.


Spatial and temporal variations in the near-surface conductivity distribution dominate the measured amplitudes in MT surveys de- signed to monitor temporal conductivity changes at deeper levels.

Provided the effect of any temporal change in the near-surface con- ductivity can be represented by a galvanic distortion of the electric field, any change in the phase tensor will reflect only the conduc- tivity variations at deeper levels.

Detecting conductivity changes at depth is potentially an important future application of this method of analysis.


The calculation of the phase tensor requires no assumption about the dimensionality of the underlying conductivity distribution and is applicable where both the heterogeneity and the regional conduc- tivity structures are 3-D. Although we have introduced the phase tensor in the context of the galvanic distortion problem, the rep- resentation of the MT phase as a tensor is not restricted to this context.

Rather, the phase tensor should be considered to be a fundamental property of the impedance tensor expressing how the phase relationship between the electric and magnetic field changes with polarization in situations where the conductivity structure is 3-D.


Much of this work would not have been possible without access to the 3-D EM modelling code developed by Zonghou Xiong.

We would also like to thank Gary McNeice and Alan Jones for making their multisite Groom-Bailey code available.

Reviews by John Booker, Wolfgang Soyer and Martyn Unsworth helped us to clarify a num- ber of ideas in the paper.

Mark Berdichevsky and John Weaver also commented on an earlier version of this paper.

The MT data from the Taupo Volcanic Zone were collected in a joint project between the Institute of Geological and Nuclear Sciences in New Zealand and the Geological Survey of Japan.

We would especially like to acknowl- edge the contribution of our Japanese colleagues Yasuo Ogawa (now at the Tokyo Institute of Technology), Shinichi Takakura, Toshihiro Uchida, Nobuo Matsushima, Toshiyuki Tosha and Yuji Nishi for making that project and on going work in the Taupo Volcanic Zone so successful.

This work was supported by the Marsden Fund admin- istered by the Royal Society of New Zealand (Institute of Geological and Nuclear Sciences contribution 2902) and by the Enterprise Ire- land Basic Research Grants Scheme (project SC-1998-501).


Expressions for the coordinate invariants of a (non-symmetric) second-rank 2-D tensor can be found in Bibby (1986) and are sum- marized here for completeness.

Writing the phase tensor as the ma- trix Φ = 11 12 21 22 (A1) the simplest algebraic representations of the tensor invariants are the trace tr(Φ) = 11 + 22 , (A2) the skew sk(Φ) = 12 − 21 (A3) and the determinant det(Φ) = 11 22 − 12 21 (A4) of the matrix.

Note that the determinant is a quadratic function of the tensor components.


It is convenient to re-express all the invariants as the first order functions: In terms of these quantities the maximum, minimum and skew angle (eq. 19) are given by the expressions:


The fourth SVD parameter α (eq. 22), which is needed to com- plete the specification of the tensor, is not coordinate invariant and cannot be expressed as a function of the coordinate invariants alone.


Note that we have tacitly assumed in eq. ( A6) that det(Φ) ≥ 0.

If the conductivity distribution is very unusual and 3-D it is pos- sible that the det(Φ) may vanish or become negative.

In the singu- lar case (det(Φ) = 0) the expressions (eqs A8 and A9) for max and min are well-defined and min = 0.

That is the phase ten- sor ellipse has zero area and the tensor is represented by a line of length 2 max .

If det(Φ) < 0 then we can retain eq. ( A8) by defining 2 = |det(Φ)| 1/2 and assign a negative sign to the value of min .


The behaviour of the phase tensor in this unusual situation can be envisaged graphically.

Assume that the effect of the conductivity distribution is insignificant at some period and that max and min are >0.

As the unusual 3-D effects become important, the minor axis of the tensor ellipse shrinks.

Eventually, the effect of the con- ductivity structure causes the ellipse to shrink down to a line (length 2 max ) where min and det(Φ) are both zero.

As these effects in- crease the ellipse will re-expand, although now min and det(Φ) are negative.


The angle α − β defines direction of the major axis of the tensor ellipse in the Cartesian coordinate system used to express the tensor.

Knowing the orientation of the major axis the ellipse may then be constructed from knowledge of the lengths of the major and minor axes (i.e. from max and min ).

A more direct and simpler way of drawing the tensor ellipse is to recall that the matrix representing the tensor can be thought of as mapping of one vector into another.

By generating a set of unit vectors at say 1 • intervals around a circle and transforming each unit vector with the tensor matrix the resultant (transformed) set of vectors define the tensor ellipse.

This is easily and compactly implemented in modern computer languages such as MATLAB.


We demonstrate a minor inconsistency in the appendix of ‘The magnetotelluric phase tensor’ by Caldwell et al. that can lead to incorrect results, and provide a simple solution for this problem.

Since its publication the phase tensor has become an important tool to analyse magnetotelluric data.

The reason for this success is, that it is not affected by galvanic distortion and does not make any assumptions about the conductivity structure in the subsurface [1].

However the discussion in the appendix on how to treat the problematic case of a negative determinant is misleading and leads to incorrect expressions when used to calculate the two invariants Φmax and Φmin.


Given the magnetotelluric transfer function Z = X + iY the phase tensor is given by Φ = X−1Y = 1/det X [X22Y11 − X12Y21, X22Y12 − X12Y22; X11Y21 − X21Y11, X11Y22 − X21Y12].

This real second rank tensor can also be expressed in terms of three rotational invariants and one rotation angle.

In the appendix of Caldwell et al. ( 2004) the formulae to calculate the invariants Φmax, Φmin and β are given as Φ1 = tr(Φ)/2, Φ2 = det(Φ), Φ3 = sk(Φ)/2, Φmax = (Φ1^2 + Φ3^2)^(1/2) + (Φ1^2 + Φ3^2 − Φ2^2)^(1/2), Φmin = (Φ1^2 + Φ3^2)^(1/2) − (Φ1^2 + Φ3^2 − Φ2^2)^(1/2), β = 1/2 tan−1(Φ3/Φ1).


In certain cases the determinant of the phase-tensor det(Φ) can become negative.

The recommendation by Caldwell et al. is to use the absolute value |det(Φ)| in these cases instead to avoid problems while calculating the intermediate quantity Φ2 which should be real.

Through this manipulation the expressions for Φmax and Φmin become incorrect for det(Φ) < 0.


Short algebraic manipulation of eq. ( 5) shows that in this case Φmax = (1/4 + Φ1^2 + Φ3^2)^(1/2) + (Φ2^2 + 3/2)^(1/2) and similarly for Φmin.

There are two simple ways to avoid problems with negative determinants: Either use the formulae (10) and (11) given by Bibby (2005) that avoid the calculation of the determinant altogether [2], or circumvent the calculation of Φ2 by inserting (3) into (5) and (6).

The resulting expressions are an equivalent definition for Φmax and Φmin is given by Bibby (1986, 2005) [2][3].


In the appendix of Caldwell et al. ( 2002), summarising results from Bibby (1986), we discuss the difficulty that arises from the definition (A6 in Caldwell et al.) of (cid:2)2 = det((cid:2)) in rare 3-D situations (for good quality MT data) where the determinant of the phase tensor Φ is negative.


Unfortunately, a sentence in our discussion, interpreted literally, introduces an ambiguity into the expressions (A8 and A9 in Caldwell et al.) for the maximum and minimum phase.


As Moorkamp (2007) points out, any ambiguity in these two expressions is removed by writing det(Φ) for (cid:2)2 2 in equations A8 and A9.


Alternatively, other equivalent expressions contained in Bibby (1986), but omitted from the summary given in the appendix, may be used.


A rise in core temperature reportedly leads to increased minute ventilation (V E) (hyperthermic hyperventilation) at rest (10, 17, 20, 40) and during exercise (18, 33, 45, 48).

Given that oxygen uptake remains largely unchanged from normothermic levels during hyperthermia in passively heated resting subjects (10, 17, 40), the influence of metabolic factors on hyperthermic hyperventilation appears to be trivial.

Consequently, this hy- perventilation results in excessive elimination of CO 2 from the body, decreasing arterial CO 2 partial pressure (Pa CO 2 ) (hypo- capnia) (20, 33, 40).

This hypocapnia likely contributes to the cerebral hypoperfusion observed during hyperthermia (9, 16, 31, 33).

By limiting heat exchange in the brain, cerebral hypoperfusion would eventually lead to elevation in brain temperature (34), which may ultimately underlie central fa- tigue during hyperthermia (39).

Therefore, preventing the ce- rebral hypoperfusion associated with hyperthermic hyperven- tilation may be an effective strategy for minimizing central fatigue during exercise in the heat.

Before implementing this strategy, however, it will be important to better understand the characteristics and underlying mechanisms of hyperthermic hyperventilation, most of which are currently unclear.


When V E is expressed as a function of esophageal temper- ature (T es ; an index of core temperature) during passive heat- ing, V E remains nearly unchanged until T es surpasses~37.8 - 38.5°C (10, 17, 44).

Above this threshold temperature, V E increases linearly with rising T es such that V E increases by 20 -32 l·min Ϫ1 ·°C Ϫ1 rise in T es (10, 17, 44).

The slope relating T es to V E would reflect ventilatory sensitivity to rising core temperature (5, 17, 25), as with the thermosensitivity of sweat- ing and cutaneous vasodilation in those relations (7).

The hyperthermic hyperventilation-induced hypocapnia potentially suppresses ventilatory sensitivity, given that ventilatory re- sponses are highly sensitive to Pa CO 2 (14).

Along these lines, we previously found that during prolonged moderate exercise in the heat, continuous restoration of Pa CO 2 to eucapnia in- creased the ventilatory sensitivity (24).

However, it remains to be seen whether this is also true during passive heating at rest, as the characteristics of hyperthermic hyperventilation differ depending on whether one is resting or exercising (17, 44).


Previous reports using passive heating at rest showed that acute restoration of end-tidal CO 2 partial pressure (PET CO 2 , an index of Pa CO 2 ) to normothermic levels did not change V E at moderate hyperthermia (core temperature of~37.6 -38.0°C) (16), but it did increase V E at severe hyperthermia (38.8°C) (31).

The latter result lends support to the possibility that continuous maintenance of Pa CO 2 at eucapnia augments ventilatory sensi- tivity to increasing core temperature at rest.

It should be emphasized that to precisely evaluate the influence of hypo- capnia on the ventilatory sensitivity to rising core temperature, hypocapnia should be prevented throughout the entire period of heating, as we did previously (24).

Acute restoration of eucap- nia after a period of hypocapnia during heating does not enable the relationships among hypocapnia, ventilation, and core temperature to be precisely characterized.


As mentioned, hyperventilation-induced hypocapnia con- tributes to the cerebral hypoperfusion during hyperthermia.

It is still debatable, however, whether this hypocapnia entirely explains the cerebral hypoperfusion during hyperthermia at rest.

When PET CO 2 is acutely restored to normothermic levels through CO 2 inhalation during hyperthermia, the decrease in middle cerebral artery blood velocity (MCAV; an index of anterior cerebral blood flow) is partially (9, 16) or largely (3, 31) reversed.

However, it is well recognized that restoration of PET CO 2 to eucapnia after sustained hypocapnia causes an over- shoot in MCAV beyond the original baseline value (37).

Thus restoration of PET CO 2 after sustained hypocapnia during heating could lead to overestimation of the contribution of hypocapnia to the reduction in MCAV.

To avoid that issue, it is necessary to clamp PET CO 2 at a eucapnic level throughout the passive heating, as was previously done during exercise in the heat (24).


Accordingly, in the present study we examined the effect of passive heating with or without clamping PET CO 2 at the eucapnic level on 1) the ventilatory sensitivity to rising core tempera- ture, and 2) the cerebral blood flow response.

We hypothesized that hypocapnia developed secondary to hyperthermic hyper- ventilation diminishes ventilatory sensitivity and accounts for only a fraction of the reduction in cerebral blood flow.


Fourteen healthy males (age: 24 Ϯ 3 yr; height: 173 Ϯ 4 cm; and body mass: 67 Ϯ 8 kg) volunteered to participate in this study.

The participants were nonsmokers, and none were taking prescription medication.

Written informed consent was obtained from all partici- pants.

The procedure used in this study was approved by the Human Subjects Committee of the University of Tsukuba and conformed to the provisions of the Declaration of Helsinki.


Participants performed two trials wherein they breathed room air (Control trial) or CO 2-enriched air (CO2 trial) that prevented a reduction in PET CO 2 during passive heating.

The two experimental trials were conducted in a counterbalanced manner and were separated by at least 7 days.

Participants were asked to abstain from strenuous exercise, alcohol, and caffeine for 24 h before the experiment.

The participants drank 500 ml of water on the night before the experiment and then consumed a light meal and 500 ml of water 2 h before the experiment.


Each participant arrived at the laboratory at 9:00 AM and rested in an environmental chamber regulated to an ambient temperature of 25°C and a relative humidity of 50%.

During this time, a thermocou- ple for measuring T es was inserted through the nasal passage to a distance equivalent to one-fourth of the participant's height.

The location of the probe in the esophagus was estimated to be posterior to the lower border of the left atrium (47).

The participant then voided his bladder, body mass was recorded, and an ultrasound Doppler probe for measuring MCAV was attached to the left side of his head.

Thereafter, the participant was instrumented with a heart rate (HR) monitor, thermocouples for measuring skin temperatures, and a cuff and electrodes for measuring arterial blood pressure.

Once instru- mented, the participant put on a water-perfused garment that covered his upper body with the exception of the head, left forearm, and hands, which were sheathed in cotton gloves.

On the left ventral forearm, a capsule for measuring sweat rate was attached using collodion and a probe for measuring skin blood flow was attached near the capsule.

The participant then sat in a chair in a bath (water tempera- ture ϭ 35°C) immersed up to his iliac crest and rested in a sitting posture while the water from the bath was circulated through the water-perfused garment.

After the participant put on a breathing mask, a sensor for measuring inspired and expired gases and a three-way valve customized to add CO 2 to the inspired air were connected to the mask.

Baseline measurements were then collected for 5 min while the subjects remained sitting.

The temperature of the water in the bath, and thus the garment covering the participant's upper body, was then increased to 41°C and held constant for the remainder of the experi- ment.

The water temperature was controlled using a heater and monitored using a thermocouple placed in the bath.


In the Control trial, participants breathed room air throughout the experiment.

In the CO 2 trial, they breathed CO2-enriched air (a mixture of room air and 100% CO2), beginning at the point when PET CO 2 fell below baseline due to hyperthermic hyperventilation.

We monitored PET CO 2 on a real-time basis and manually added 100% CO 2 to the inhaled air.

The fraction of inspired CO2 was Ͻ3.5%.

The flow rate of the added CO2 differed depending on the subject's breathing pattern as well as the levels of V E and the metabolic rate.

In a preliminary experiment, four participants voluntarily hyperventilated such that V E increased tõ 20, 30, and 40 l/min [breathing frequency (fB) of~20, 25, and 30 breaths/min, respectively] with CO2 supplementation to prevent hypocapnia under normothermic conditions.

This experiment en- abled us to roughly estimate how much CO 2 we need to add to the inspired air to prevent hypocapnia during passive heating during the CO2 trial.

When Tes reached 39.0°C or the subject could no longer tolerate the heating, the experiment was terminated.


Skin temperatures and Tes were measured using copper-constantan thermocouples, recorded on a computer (ThinkPad A21p; IBM,) at 1-s intervals via a data logger system (WE7000; Yokogawa), and aver- aged over 30-s intervals.

Mean skin temperature (T sk) was calculated from seven skin temperatures weighted to the following regional proportions: 7% forehead, 14% forearm, 5% hand, 7% foot, 13% lower leg, 19% thigh, and 35% chest (23).

HR was recorded every 5 s using a heart rate monitor (Vantage NV, Polar, Finland) and averaged over 30-s intervals.

Blood pressure was measured from the upper right arm every 1 min using an automated sphygmomanometer (STBP-780; Nippon Colin).

Mean arterial pressure (MAP) was cal- culated as the diastolic pressure plus one-third of the pulse pressure.


Inspired and expired gases were analyzed using a metabolic cart (AE310s; Minato Medical Science), and V E, tidal volume (VT) ad- justed to BTPS, fB, oxygen uptake (V O2), carbon dioxide output (V CO2), PET CO 2 and the respiratory exchange ratio were recorded breath-by-breath.

The flow sensor was calibrated using a calibration syringe able to blow a fixed volume (2 liters) of air.

The O 2 and CO2 .216.240.244) on May 18, 2024.

sensors were calibrated with room air and reference gas of known concentration (15.0% O 2, 5.0% CO2, N2 balance).


To determine MCAV, we used the transcranial Doppler ultrasound technique (WAKI 1-TC; Atys Medical).

A 2-MHz Doppler probe was secured with a customized headband to the left temporal region, and the signal was collected at a depth of 45- 60 mm.

Before the two experiments, the position and probe angle that provided the optimal Doppler signal were determined and photographed.

The pictures enabled us to obtain similar MCAV values at baseline in the two trials.

Cerebral vascular conductance (CVC) was calculated as MCAV divided by MAP.

Sweat rate was measured using the ventilated-capsule method.

Dry nitrogen was supplied to the capsule (3.46 cm 2 ) at a rate of 0.8 l/min.

The humidity of the nitrogen gas flowing out of the capsule was measured using a capacitance hygrometer (HMP 45ASPF; Vaisala).


Cutaneous blood flow was measured using a laser-Doppler flowmeter (ALF21; Advance).

Cutaneous vascular conductance was calculated as the cutaneous blood flow divided by the MAP and was normalized to the baseline value.

Breathing difficulty and thermal sensation were mea- sured every 5 min on a scale of 10 (e.g., difficulty of breathing: 1, very easy; 3, moderate; 5, hard; 7, very hard; and thermal sensation: 5, neutral; 7, warm; 9, very hot).


To evaluate the relationship between core temperature and venti- latory responses (V E, VT, fB, and PET CO 2 ), the ventilatory variables were plotted as functions of Tes, and linear regression analyses were conducted using 30-s averaged data.

For these analyses, we used a computer algorithm to determine the thresholds for the increase or decrease in the respective ventilatory variables, as evaluated from the inflection point where two calculated regression lines crossed (17, 44).

We chose the two best-fit regression lines based on the smallest residual sums of squares.

The slopes of the second regression lines relating V E, VT, fB, and PET CO 2 to Tes were considered as sensitivities.

The Tes strongly correlated with V E (r ϭ 0.74 Ϯ 0.13), VT (r ϭ 0.53 Ϯ 0.24), fB (r ϭ 0.66 Ϯ 0.2), and PET CO 2 (r ϭ Ϫ0.77 Ϯ 0.20).

Furthermore, we previously confirmed high day-to-day reproducibil- ity for the Tes threshold (r ϭ 0.83, P Ͻ 0.05) and the ventilatory sensitivity (r ϭ 0.97, P Ͻ 0.05) of hyperthermia-induced hyperven- tilation (17).


Two-factor repeated-measures ANOVA was used to analyze the time-dependent data using trial (Control and CO 2) and heating dur- ation (pre, 5, 10, 15, 20, 25 min, and end of heating) as factors.

Two-factor repeated-measures ANOVA was also employed to ana- lyze the core temperature-dependent data using trial (Control and CO 2) and Tes level (36.8, 37.2, 37.6, 38.0, 38.4, and 38.8°C) as factors.

When a main effect was detected, post hoc multiple compar- isons were performed to identify pairwise differences using paired t-tests with the Bonferroni correction.

Paired t-tests were also used for between-trial comparisons of the thresholds and sensitivities of ven- tilatory parameters to rising T es.

When the test of normality (Shapiro- Wilk test) failed, the Wilcoxon signed-rank test was used to assess differences between trials.

All data are reported as means Ϯ SD.

P Ͻ 0.05 was considered significant.

All statistical analyses were per- formed using a commercially available statistics package (version 21.0; SPSS).


Heating duration did not differ between the trials (Control vs. CO 2 , 32.9 Ϯ 6.4 vs. 33.8 Ϯ 8.2 min, P ϭ 0.58).

Body mass losses over the course of the experiment were nearly identical between the Control and CO 2 trials (1.5 Ϯ 0.5 vs. 1.4 Ϯ 0.5%, P ϭ 0.20).


Time- dependent changes in body temperature and thermoregulatory variables are shown in Table 1 and Fig. 1A.

There were main effects of trial on T es (P Ͻ 0.01) and of heating duration on T es and T sk (both P Ͻ 0.01).

Over the course of the hot water immersion, T es gradually increased, and the temperatures at the end of heating were similar in the Control and CO 2 trials (38.9 Ϯ 0.3 vs. 39.0 Ϯ 0.3°C, P ϭ 0.12).

Although T es at 5-25 min was higher in the CO 2 than Control trial, the difference was minimal (Ͻ0.19°C, Fig. 1A).

The rate of increase in T es did not differ between the Control and CO 2 trials (4.5 Ϯ 0.6 vs. 4.5 Ϯ 0.9°C/h, P ϭ 0.81).

Likewise, increases in T sk were equivalent during heating in the two trials.


Because we were unable to measure sweat rate and cutaneous blood flow on the forearms of three subjects due to technical difficulties, the data analyzed were from the remaining 11 subjects.

Forearm sweat rate and cutaneous vascular conductance did not differ between trials (P ϭ 0.52 and 0.68 for a main effect of trial, respec- tively), and thermal sensation increased similarly in the two trials (P ϭ 0.29 for a main effect of trial).


Similar changes in HR and MAP were observed in the two trials (P ϭ 0.25 and 0.99 for a main effect of trial, respectively, Table 1).

There was a significant interaction between trial and heating duration affecting MCAV and CVC (both P Ͻ 0.05).

In both trials, MCAV was decreased at and after 10 min of heating in comparison to the preheating value.

In addition, MCAV was higher in the CO 2 than Control trial at the end of heating (P ϭ 0.005, Fig. 1D) and explains 36.6% of the total decrease in MCAV occurring in the Control trial.

The changes in CVC were similar to those observed with MCAV, and CVC was higher in the CO 2 than Control trial at the end of heating (P ϭ 0.014, Table 1).


Time-dependent changes in respira- tory variables are shown in Table 2 and Fig. 1, B and C. A significant main effect of heating duration was detected on all the respiratory variables evaluated (all P Ͻ 0.01).

Similar increases in V E were observed in the two trials (P ϭ 0.52 for a main effect of trial).


From preheating to 15 min of heating, V E increased slightly and then increased as heating continued (at the end of heating, 23.8 Ϯ 8.3 and 22.9 Ϯ 7.3 l/min in the Control and CO 2 trial, respectively, Fig. 1B).

Both V T and f B did not differ between trials (P ϭ 0.31 and 0.37 for a main effect of trial, respectively).


At the end of heating in the Control trial, PET CO 2 was significantly lower than before heating.

As designed, PET CO 2 in the CO 2 trial remained unchanged over the period of heating and was higher than in the Control trial after 15 min of heating (Fig. 1C).


T es -dependent changes in ventilatory and cerebrovascular responses.

Figure 2 shows V E plotted against T es .

The thresh- olds and sensitivities of the relationships between ventilatory variables and T es are presented in Tables 3 and 4.

Twelve of the 14 subjects showed T es thresholds for increases in V E in both trials.

One subject showed no threshold in either trial and another subject exhibited no threshold in the CO 2 trial only.

There was no significant between-trial difference in the thresh- old (P ϭ 0.15).

We also observed T es thresholds for increases in V T in 8 subjects, for increases in f B in 11 subjects in both trials, and for decreases in PET CO 2 in 13 subjects in the Control trial.

There were no significant between-trial differences in the thresholds for V T or f B (P ϭ 0.06 and 0.56, respectively).


Individual data for the sensitivities of ventilatory responses to rising T es are shown in Table 4.

In 4 of 12 subjects, the sensitivity of V E to rising T es was higher in the CO 2 than the Control trial.

The sensitivities of V T and f B to rising T es were higher in the CO 2 than the Control trial in 3 of 8 subjects and in 3 out of 11 subjects, respectively.

There were no between- trial differences in the sensitivities of V E, V T , and f B to rising T es (P ϭ 0.31, 0.11, and 0.86, respectively).


Although there Ventilatory and cerebrovascular responses were compared between trials at various T es (Figs. 3 and 4).

There were main effects of T es on V E, V T , PET CO 2 , and MCAV (all P Ͻ 0.01).

The increase in V E and the respiratory pattern, as evaluated based on V T and f B , were similar in the Control and CO 2 trials (P ϭ 0.28, 0.74 and 0.18 for a main effect of trial, respectively, Fig. 3).

In the Control trial, PET CO 2 was lower at T es ϭ 38.8°C than 36.8°C, but it remained unchanged in the CO 2 trial.

A main effect of trial on PET CO 2 was detected.

At T es ϭ 37.6 -38.8°C, PET CO 2 was higher in the CO 2 than Control trial (all P Ͻ 0.01, Fig. 4A).

There was no main effect of trial on MCAV (P ϭ 0.30).

In both trials, MCAV was lower at T es ϭ 37.2-38.8°C than 36.8°C (Fig. 4B).

In the CO 2 trial, restoration of the PET CO 2 restored 37.6, 39.8, and 39.3% of the heating-induced decrease in MCAV in the Control trial at T es ϭ 38.0, 38.4, and 38.8°C, respectively.


Our main findings are that during passive heating at rest 1) the sensitivity of V E to rising T es did not differ between the Control and CO 2 trials; 2) the decrease in MCAV occurred early during heating, when there was no clear hyperventilation or resultant hypocapnia; and 3) at the end of heating (T es ϭ 39.0), MCAV was significantly higher in the CO 2 than Control trial, but hypocapnia explained only 36.6% of the observed reduction in MCAV.

These results suggest that during passive heating at rest ventilatory sensitivity to rising core temperature is not suppressed by the resultant hypocapnia and that hyper- thermia decreases cerebral blood flow largely independently of the concomitant hypocapnia.


Contrary to our hypothesis, the sensitivity of V E to rising T es was not greater in the CO 2 trial than the Control trial (Fig. 2 and Table 4).

Only 4 of 12 subjects showed a higher sensitivity in the CO 2 trial; the others showed somewhat lower sensitivity in the CO 2 than Control trial.

Similarly, V E at similar T es levels did not differ between the trials (Fig. 3A).


These results suggest that for subjects in a hyperthermic resting state, hypocapnia does not suppress ventilatory sensitivity to rising core temper- ature.

By contrast, we previously reported that, during pro- longed moderate-intensity exercise in the heat, restoration of Pa CO 2 to the eucapnic level led to a doubling of ventilatory sensitivity as compared with the sensitivity when Pa CO 2 was decreased secondary to hyperthermic hyperventilation (19.8 vs. 8.9 l·min Ϫ1 ·°C Ϫ1 ) (24).

We therefore suggest that hypocapnia diminishes ventilatory sensitivity during exercise but not at rest.


Furthermore, the fact that hypocapnia attenuates hyper- thermic hyperventilation only during exercise likely contrib- utes to the approximately three-times smaller ventilatory sen- sitivity to rising core temperature observed during exercise in the heat as compared with resting (17, 44).


The lack of change in ventilatory sensitivity upon Pa CO 2 restoration may be due to the location of the Pa CO 2 threshold for increases in V E. The threshold exists at a normal Pa CO 2 level at rest (4, 14), and V E remains unchanged when Pa CO 2 de- creases below the threshold.

This Pa CO 2 threshold is also reportedly unaffected by a 1.5°C elevation in core temperature at rest (4, 6).

Because hyperthermic hyperventilation during passive heating decreases Pa CO 2 below normal level, it is plausible that these changes in Pa CO 2 are below the threshold for increases in V E and so did not affect hyperthermic hyper- ventilation at rest.

On the other hand, although the Pa CO 2 threshold for increases in V E is reportedly similar during rest and exercise (11, 13), as mentioned, we previously found that V E increased with restoration of Pa CO 2 to the eucapnic level during exercise in the heat (24).

This implies that exercise in combination with hyperthermia shifts the Pa CO 2 threshold to lower Pa CO 2 levels than that at rest.


We previously observed that acute restoration of PET CO 2 to eucapnia does not influence V E at T es ϭ 37.6 or 38.0°C (16).

However, similar restoration increased V E at a higher T es (i.e., 38.7°C) (3).

In addition, restoration of normocapnia reportedly decreased V E at a rectal temperature of 39.1°C (39), although the level of hyperthermia should be regarded with caution, as rectal temperature responds more slowly than T es (42).

The reason for these disparate results remains unclear.

Because we detected no effect of hypocapnia on V E, irre- spective of core temperature in the present study (Fig. 4A), different core temperatures would not explain the lack of consistency between studies.

Other differences such as pos- ture [upright (16, 39) vs. supine (3)] might be involved but that remains to be determined.


With rising T es in the present study, MCAV in the Control trial decreased to levels below the preheating baseline, which is in line with previous studies (9, 16, 29, 31).

Early during the heating, MCAV decreased compared with baseline in both trials, despite hyperthermic hyperventilation, and thus resultant hypocapnia, being minimal (Fig. 1, C and D).

This implies that at rest hyperthermia can decrease cerebral blood flow indepen- dently of hypocapnia.


When the decrease in PET CO 2 was pre- vented in the CO 2 trial, MCAV was higher than in the Control trial during the latter part of heating (Fig. 1C); however, the restoration accounts for only 36.6% of total decrease in MCAV (Fig. 1D).

Hence, although hypocapnia appears to contribute to the decrease in cerebral blood flow during passive heating at rest, a larger portion of the response is explained by one or more hypocapnia-independent factors.


Furthermore, MCAV was similarly restored by 40% across the range of T es ϭ 38.0 - 38.8°C (Fig. 4B), indicating that the relative contribution of hypocapnia to the change in cerebral blood flow during passive heating is similar within this temperature range.


Previous studies reported that restoration of PET CO 2 through acute CO 2 inhalation during passive heating increased MCAV by~28 and~38% at core temperatures of 37.6 and 38.0°C, respectively (16) and increased MCAV by~50% at a core temperature of 38.5°C (9).

By contrast, the restoration of PET CO 2 reversed reductions in MCAV and posterior cerebral artery blood flow velocity by~67 and~84%, respectively, at a core temperature of 38.8°C (31), and it increased MCAV by~75% at a core temperature of 38.7°C (3).

It is however noteworthy that the all the results shown above could be influenced by prolonged exposure of hypocapnia, during which MCAV grad- ually increased despite the sustained hypocapnic state as a result of reductions in brain extracellular bicarbonate and recovery of extracellular pH (8).

Consequently, restoration of The mechanism underlying hypocapnia-independent de- crease in MCAV during heat stress is unclear.

Because there was no decrease in MAP throughout the heating in the present study, the possibility of decreased cerebral perfusion pressure is excluded.

On the other hand, blood distribution to the internal carotid artery, from which the MCA branches, reportedly decreases, possibly due to an increase in blood distribution to the external carotid artery, which supplies blood to the cutaneous vessels of the head for thermoregu- lation (35).

Furthermore, blood flow in the internal carotid artery gradually decreases with rising T es (35).

Thus modi- fication of the blood distribution in the brain during passive heating may explain the hypocapnia-independent reduction in MCAV.

In addition, cerebral vessels are innervated by nerve fibers that originate from peripheral (extrinsic) nerve ganglia and intrinsic nerves (22).

In humans, stellate gan- glion blockade increases cerebral blood flow (26, 46), sug- gesting that regulation of cerebral blood flow is associated with sympathetic nerve activity.

Given that hyperthermia increases skin and muscle sympathetic nerve activities (28), cerebral vasoconstriction during hyperthermia may be due in part to sympathetic vasoconstriction, as previously spec- ulated by Brothers et al. ( 9).


In our earlier study, which employed the same heating protocol (19), we showed that greater hyperthermic hyperven- tilation correlates with lower MAP.

This implies that hyper- thermic hyperventilation may lower MAP during passive heat- Table 1), indicating that hypocapnia does not influence MAP during passive heating at rest.

Hence, hyperventilation itself might decrease MAP during passive heating at rest.

This notion is indirectly supported by the earlier finding that lung inflation decreases MAP (36).


There were no between-trial differences in forearm sweat rate or cutaneous vascular conductance (Table 1), which sug- gests hypocapnia does not influence sweating or cutaneous blood flow during passive heating at rest.

This is in contrast to the previous observations made by Albert (1), Robinson and King (38) and Fujii et al. ( 15).

The discrepancy may reflect differences in core temperature.

For example, Fujii et al. ( 15) reported that hypocapnia reduced forearm cutaneous blood flow when T es increased by 0.6°C but not 1.0°C.

Since hyper- thermic hyperventilation and resultant hypocapnia during pas- sive heating occurred with increases in T es of~1.0Ϫ1.5°C, as shown in that study, it is plausible that the greater elevation in core temperature masked the hypocapnia-induced suppression of cutaneous blood flow.

It remains to be investigated whether the same explanation applies to the sweat response.


Coverdale et al. ( 12) recently reported that MCA diameter measured using magnetic resonance imaging decreased by 4% when PET CO 2 decreased from 36 to 23 mmHg, which is in contrast to previous reports that MCA diameter is unaffected by decreases in PET CO 2 to~25 mmHg (21, 41).

It may thus be possible that the degree to which MCAV was reduced in the Control trial underestimated the true degree to which MCA blood flow was reduced, as blood flow was actually decreased farther than was reflected by the change in velocity.


Reduced cerebral blood flow during hyperthermia reportedly leads to increased brain temperature (34) and central fatigue (39) without modulating cerebral metabolism (32, 43).

The present study demonstrated that hypocapnia induced through hyperventilation does not modulate ventilatory responses to rising core temperature but does contribute to the decrease in cerebral blood flow seen during passive heating at rest.

Based on this observation, we speculate that restoration of normocap- nia by inhaling CO 2 gas is not sufficient to fully restore cerebral perfusion during passive heating at rest, and a hyper- capnic state (i.e., increase in arterial CO 2 pressure above normocapnic level) or other countermeasures, such as whole body skin cooling (30), may be required to prevent cerebral hypoperfusion.

Furthermore, we found that inhaling CO 2 gas to prevent hypocapnia during passive hyperthermia does not in- duce additional hyperventilation or dyspnea, as evaluated based on difficulty of breathing.

This is in stark contrast to what was observed during exercise in the heat, wherein pre- venting hypocapnia by inhaling CO 2 gas increased ventilation (24).

Therefore, although inhaling CO 2 gas can reverse cere- bral hypoperfusion, irrespective of whether the subjects are resting or exercising, this maneuver can distress the respiratory system during exercise.

This is important information to know when employing CO 2 gas inhalation as a strategy to mitigate cerebral hypoperfusion during heat stress.


In summary, our results suggest that the ventilatory sensi- tivity to rising core temperature is not suppressed by hypocap- nia and that hyperthermia decrease cerebral blood flow largely independently of the hypocapnia during passive heating at rest.


Tsuji 2017 Jappl Physiol (doi 10.1152/japplphysiol.00232.2017) state on page 3 "we are the first to demonstrate that hyperthermia-induced hyperventilation is not suppressed by the resultant hypocapnia." This is incorrect.


Many previous studies in humans and other species have shown hyperthermia sustaining breathing and furthermore at PCO2 levels lower than Tsuji achieved.


Thus, Cunningham & O'Riordan [1] showed that hyperthermia in Man by 2.9 oC stimulated breathing at PCO2 levels of 25 mmHg, Iampietro et al. [ 2] showed hyperthermia in Man by 3oF stimulated breathing with a PCO2 fall of 25 mmHg, Rowell et al. [ 3] showed hyperthermia in Man by 1.5oC stimulated breathing with a PCO2 fall to 27 mmHg, Saxton [4] showed hyperthermia in Man by 2.5oF stimulated breathing with a PCO2 fall to 22 mmHg and Boden et al. [ 5] showed hyperthermia at 39.5 oC restarting breathing during hypocapnic apnea at 11 mmHg.


TO THE EDITOR: We thank Dr. Parkes [1] for the comment on our recent paper [2].

Dr Parkes points out that our statement “we are the first to demonstrate that hyperthermia-induced hyperventilation is not suppressed by the resultant hypocapnia” is incorrect as many previous studies have already reported that hyperthermia stimulates ventilation in the presence of concomitant hypocapnia.

It is true that several previous studies, including our work [3], have demonstrated that hyperventilation occurs during hyperthermia with concomitant hypocapnia.

However, given that arterial CO2 pressure is tightly and positively associated with ventilatory drive, it is possible that the resultant hypocapnia partly suppresses hyperthermia-induced hyperventilation.


That is, the reported hyperventilation during hyperthermia previously observed may be the result of combined influences of increased respiratory drive elicited by hyperthermia and reduced ventilatory drive due to hypocapnia.

To isolate the influence of hypocapnia on ventilatory responses during hyperthermia, we must compare ventilatory responses between hypocapnia and normocapnia, and the latter can be achieved by having individuals inhaling hypercapnic air during heating.

In contrast with our recent study, none of the previous studies cited by Dr. Parkes directly restored the reduced arterial CO2 pressure to normocapnic levels throughout heating procedure.


We are therefore the first to elucidate the effect of concomitant hypocapnia on ventilatory response during passive hyperthermia in resting humans, having used hypercapnic air breathing that kept arterial CO2 pressure at normocapnic level throughout our heating protocol.

Our results clearly indicate that the concomitant hypocapnia does not influence hyperthermic hyperventilation as assessed by core temperature threshold for increases in ventilation, as well as ventilatory sensitivity to rising core temperature.

With this direct evidence, we can conclude that hyperthermic hyperventilation is not suppressed by the resultant hypocapnia.

Further evidence is warranted to evaluate the role of more severe hypocapnia (e.g., arterial CO2 pressure of ≤30 mmHg) on hyperthermic hyperventilation, as observed in the previous studies cited by Dr. Parkes.


Over the past 40 years, the prevalence of diagnosed autism spectrum disorder (ASD) in the U.S. has dramatically risen [1], from fewer than 0.3 per 1000 children diagnosed with autism before 1980 [2] to 27.6 per 1000 children diagnosed with ASD in 2020 [3].

Changes in diagnostic definitions and guidelines and increased testing availability and Nutrients 2023, 15, 3772.

https://doi.org/10.3390/nu15173772 https://www.mdpi.com/journal/nutrients funding have made major contributions to this increase in diagnosed cases; under the added impacts of changes in dietary, environmental, and other exposures affecting the intrauterine environment, ASD prevalence has reached unprecedented proportions.

Males have been disproportionately affected: autism prevalence among boys is almost quadruple that among girls, and a recent study estimated that approximately 1 in 23 U.S. boys aged 8 years or older in 2020 had been diagnosed with ASD [3].

The degree to which ASD diagnoses have risen during this time highlights the potential role of non-genetic influences, including early prenatal exposures to heavy metals, organophosphate pesticides, and other environmental toxins, in offspring autism risk [4].


Maternal diet during pregnancy and breastfeeding represents an important additional non-genetic influence on offspring autism risk and has been studied increasingly closely over the past 15 years.

Zhong et al. [ 5], in their systematic review, reported evidence of the protective role of maternal intake of prenatal vitamins, folic acid, and vitamin D in reducing offspring autism risk.

Peretti et al. [ 6] also reported decreased offspring autism risk associated with a higher maternal intake of folic acid, omega 6 fatty acids, and vitamin D during pregnancy and an increased autism risk among the offspring of mothers deficient in omega 3 fatty acids and polyunsaturated fatty acids.

Although the results from different individual studies varied, the available data suggest that these selected nutrients may indeed exert neuroprotective effects during development.


In contrast, the maternal dietary intake of methanol during pregnancy was identified as a potential risk-increasing exposure for offspring.

Peretti et al. [ 6] called attention to the work of Walton and Monte [7], who found that the dietary intake of methanol during pregnancy was approximately twice as high among biological mothers of children with ASD as among mothers of neurotypically developing children.

Aspartame, a leading non-nutritive sweetener (NNS) in the U.S., is a ubiquitous source of dietary methanol in this country, and aspartame-sweetened products constituted the major sources of dietary methanol included in Walton and Monte's calculations of maternal methanol intake.

If their preliminary findings are correct, the possibility that pregnant women might unknowingly be exposing their unborn children to an increased autism risk through their consumption of aspartame-sweetened diet products during pregnancy is of particular concern.


The safety of aspartame, a leading sweetener used in diet sodas, other diet beverages, and over 6000 dietary, pharmacologic, and other products over the past 40 years [8], has been the subject of intense debate since before it entered the marketplace.

In 1981, the U.S. Food and Drug Administration (FDA) approved the use of aspartame as a tabletop sweetener and, in 1983, as an ingredient in diet sodas (DS) and other products [8].

Aspar- tame rapidly became the leading DS sweetener used in the U.S; DS consumption doubled, and aspartame consumption increased 17-fold, over the next 8 years [9].


By the end of 1983, however, the FDA had received numerous complaints of adverse reactions among aspartame consumers.

Prominent complaints included headache, anxiety, and depres- sion [10,11].

These and other neurological problems-including irritability, mood disorders, cognitive problems, and seizures-were subsequently reported to be increased among users of diet sodas/beverages (DSB) and other aspartame-sweetened products [10-22], which are leading vehicles of aspartame intake in the U.S. [23].


Impaired cognitive performance and increased anxiety-related behaviors also developed in aspartame-fed animals [24-30].

Such reports contributed to the prolonged controversy over the safety of aspartame [29,31-37].

While a review of this debate is beyond the scope of this report, we highlighted results with potential neurophysiologic relevance to our investigation.


Each of the three first-phase metabolites of aspartame has been studied with regard to its impacts on neurologic function.

Aspartame is metabolized in the intestine into aspartic acid, an excitatory neurotransmitter; phenylalanine, which is involved in neurotransmitter regulation; and methanol, the metabolites of which include formaldehyde, formate, and other toxins [38].

These three first-phase metabolites represent approximately 40%, 50%, and 10% of aspartame by weight, respectively [38].

The adverse neurological impacts following the consumption of aspartic acid, phenylalanine, and/or aspartame include changes in neurotransmitter levels [25,38,39] and excitotoxicity, with adverse impacts on neuron function/survival [40-42].

Primates, including humans, are uniquely vulnerable to methanol [43,44], the blood levels of which rise following aspartame consumption [44-46].

Exposures to methanol and formaldehyde resulted in increased neuronal apoptosis, neu- rodegeneration, and cognitive problems [43,47,48].


The potential mechanisms that might underlie the associations between aspartame in- take and long-term health problems in humans and their offspring were reported in animal studies.

Prominent among these is a recurring finding of the markedly decreased availabil- ity of the reduced form of glutathione-glutamine sulfhydryl (GSH)-in aspartame-fed animals [25,44-46,49-52].

This finding is crucial because GSH protects the developing brain by providing antioxidant defense against oxidative stress [53,54], by scavenging toxins [53,55], and by supporting methylation processes [56]Adverse impacts on the gut microbiota (GM) were also detected in animals exposed to aspartame [29,61,65-73] and two of its metabolites: formaldehyde [67] and pheny- lalanine [66].

Among other changes, aspartame-fed rats exhibited a doubling of serum propionate [65].

Propionic acid, a short-chain fatty acid produced by gut bacteria, was hypothesized to increase ASD risk through multiple pathways, including increased gut and BBB permeability [74,75]; decreased GSH and neurotransmitter levels; increased oxida- tive stress, excitotoxicity, and neuroinflammation [76,77]; and altered mitochondrial and immunological function [78].


Taken together, as summarized in Figure 1, such physiological impacts could create a perfect storm: increasing the access of toxins to the developing brain, decreasing the developing brain's antioxidant and detoxification capacities, and adversely impacting the microbiome-gut-brain axis, which is critical in neurodevelopment [79,80].

Further discussion of the processes summarized in Figure 1 is included in Section 4.


In order to address the question of whether maternal aspartame consumption in- creases autism risk in offspring exposed to aspartame in early life-in utero and during breastfeeding-we aimed to retrospectively assess the maternal intake during pregnancy and breastfeeding of diet soda and of aspartame from multiple sources, among biological mothers of offspring with either autism specifically or any ASD (cases) and among mothers of neurotypically developing offspring (controls), to evaluate the associations between daily early-life exposures to these products and current autism and ASD status in the case-control Autism Tooth Fairy Study (ATFS).


The primary aim of this study was to examine whether the odds of daily exposure during pregnancy/breastfeeding to ≥1 DS or an equivalent amount of aspartame, through the maternal diet, were significantly higher among offspring diagnosed with autism (autism cases), compared with neurotypically developing offspring (controls).

The secondary aim of the study was to assess whether these exposure odds were higher among offspring with any ASD (ASD cases), compared with controls.

Since males have increased developmental vulnerability to adverse early-life exposures and since the risk of autism is four times higher in boys than in girls, we performed both sex-specific and pooled analyses for both diagnostic categories.


times higher in boys than in girls, we performed both sex-specific and pooled analyses for both diagnostic categories.


Recruitment and data collection for the Autism Tooth Fairy Study were performed through the University of Texas Health Science Center at San Antonio (UTHSCSA) from May 2011 to June 2014.

The largest recruitment source (n = 255) was the Interactive Autism Network (IAN), an Internet-based U.S. registry [81] that included > 21,600 individuals with ASD and >22,000 parents of individuals with ASD [82].

Most families recruited through IAN had ≥1 child diagnosed with ASD (cases; n = 178).

Neurotypically developing children (controls) identified through IAN recruitment included siblings of cases (n = 61), and offspring of friends/associates of IAN parents (n = 16) [83].

Additional recruitment (n = 101) was conducted through media outreach in San Antonio and South Texas (SA/STX), through which 57 cases, including 7 referred from local autism services, and 44 controls were enrolled.

The ATFS, thus, included a total of 356 offspring: 235 ASD cases and 121 controls.

In addition to completing questionnaires regarding their children's early-life exposures, ATFS parents provided one or more of their children's shed deciduous teeth for compositional analyses, results of which are reported elsewhere [84].


The study was conducted according to the guidelines of the Declaration of Helsinki and was approved by the Institutional Review Board (IRB) of the University of Texas Health Science Center at San Antonio (UTHSCSA) (UTHSCSA IRB protocol number 11-313, approved on 12 May 2011).

In early data collection, all parents provided written in- formed consent before completing self-administered hard-copy questionnaires.

During later Internet-based enrollment and data collection, all parents reviewed an IRB-approved online study information sheet prior to completing online questionnaires through secure websites, in accordance with the protocol approved by the IRB of the UTHSCSA.


Parents provided demographic data about themselves and their households; the birth year and sex of each child; and whether each child had been diagnosed with either autism or autistic disorder, Asperger's disorder, pervasive developmental disorder-not otherwise specified, or childhood disintegrative disorder.

Children with any of these diagnoses were included as ASD cases.

Parents were also asked whether each child had been diagnosed with any additional behavioral, developmental, and/or learning disabilities/disorders; non-cases with none of these diagnoses were included as controls.


Parents were asked, "Was there ever a time when your child used at least three words you could understand (besides mama or dada) on a daily basis for at least a month, and then seemed to stop talking for a while (at least a month) where he/she used no words that you could understand?" Cases where parents responded "no" were considered unlikely to have experienced a regressive form of autism or ASD and were, thus, categorized as "non-regressive" cases.


Biological mothers completed retrospective questionnaires on their intake of diet sodas (DS) and other diet drinks (DD other ) during pregnancy/breastfeeding.

For each child they were asked, "While you were pregnant or breastfeeding your child, how often did you drink diet drinks containing artificial sweeteners?

Please count diet sodas first, such as Diet Coke, Diet Dr. Pepper, and Diet Sprite, and then other diet drinks, such as Crystal Light, sugar-free Kool-Aid, Slim-Fast, and other 'lite' drinks".

Within each of these two beverage subcategories, mothers recorded the number of cans or bottles of DS and the number of glasses, cans, or bottles of DD other that they had consumed per time unit; whether this time unit was daily, weekly, monthly, annually, or never; and the specific brand(s) consumed.

Mean intake of DS/day and DD other /day was calculated and summed to estimate total maternal intake of diet drinks/day (DD total /day) during pregnancy/breastfeeding.


Each mother was also asked, "While you were pregnant or breastfeeding your child, how many little packets of low-calorie sweeteners (such as Sweet 'N Low, Equal or Splenda) did you use in your coffee, tea, or other foods and beverages?

In answering the question, please keep in mind the number of drinks you had each day, and how many packets of sweetener you added to each drink.

Also, please include the number of packets you used in cereal or other food".

Intake of the following three leading NNS packets was specifically requested: "Equal/Nutrasweet (blue)", "Splenda (yellow)", and "Sweet'N Low (pink)"; space was left for recording intake of other NNS packet brands.

Maternal estimates of intake of these products during pregnancy and breastfeeding were part of an extensive questionnaire that gathered data on maternal and child exposures to a number of household products and other environmental exposures throughout pregnancy and the early life of the child.


Two case definitions were specified for analyses.

The primary case definition of interest was autism disorder (autism); the secondary case definition was any ASD.

In addition, subgroup analyses focused on offspring with non-regressive versions of each case definition, among whom the relative etiologic influence of intrauterine/neonatal exposures was expected to be maximized.

In all analyses, controls were neurotypically developing offspring and-for sex-specific analyses-of the same sex as cases.

Both sex-specific and pooled analyses were performed for each of the case definitions and exposures.


Maternal intake of ≥12 ounces (1 can) of DS/day during pregnancy/breastfeeding defined the primary, dichotomized exposure variable: daily early-life exposure to DS (DS early ).

The secondary, dichotomized exposure variable-daily early-life exposure to as- partame (ASP early )-was defined as maternal intake of ≥177 mg/day of aspartame during pregnancy/breastfeeding, from the sum of aspartame dosages in DS + DD other + packets consumed by the mother.

This cut-point was selected because it equals the minimum aspar- tame content per can of the leading diet colas sweetened with aspartame alone.

Estimates of total maternal aspartame intake from these three sources were possible because mothers were asked to report the specific brands they had consumed.

Dosages of aspartame and other NNSs used in each of the leading brands of DS were previously published [85]; for comparable products by other brands, NNS dosages were imputed according to the NNS types specified on product labels.

Leading DS brands exclusively sweetened with aspartame contained ≥ 177 mg of aspartame/can [85]; this dosage was, therefore, chosen as the cut-point for ASP early to identify offspring with daily early-life aspartame exposure comparable to that in 1 can of the leading diet colas [85].


For DD other , ingredient-labeling-specified types of NNS used, but not dosages, which are proprietary.

Based on limited available data, a 12-ounce serving of DD other exclusively sweetened with aspartame was estimated to contain approximately 100 mg of aspartame.

A tabletop packet contains 37 mg of aspartame [86].


Offspring born in 1984 or thereafter, with complete early-life DS exposure data and neurodevelopmental diagnostic data, were included in analyses.

The proportions of off- spring with DS early and ASP early exposures were separately calculated, by sex, for the following diagnostic groups: controls, all ASD cases, ASD cases without autism, autism cases, and the non-regressive version of each case definition.

Differences in the proportions of exposed controls and cases, in each category, were calculated using the Pearson χ 2 statistic in Stata/IC 14.2 for Windows.


To assess associations among diagnostic status and the primary and secondary ex- posures under study-DS early and ASP early -multilevel mixed-effects generalized linear models (MEGLMs) were used to consider potential influences at two levels: the individual child (level one) and shared intrauterine and familial influences, through the mother (level two).

Recruitment source was treated as a level-one predictor.

A total of 356 children, including children from 79 sibships, were included in analyses.

Models were fit using the meglm command in Stata/IC 14.2 for Windows.